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NM-SESES Tutorial - Numerical Modelling GmbH

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1. 1 40E 08 1 20E 08 N e 1 00E 08 4 vertical decay x e O horizontal decay pa 8 00E 07 theoretical decay a 6 00E 07 ne 4 00E 07 5 8 2 00E 07 0 00E 00 t 0 0 5 1 1 5 2 2 5 penetration depth mm Figure 2 53 Skin effect for 40 kHz AC current in the copper rod is plotted as a function of the penetration depth The computed data is compared with the analytical expression of the skin effect 2 12 The total and reduced formulation of magnetostatic In electrical machineries as well as in many sensors and micro devices one is often interested in computing magnetostatic forces caused by the flowing of time constant electrical currents J which by the Biot Savart s law generate an induction field B and by Lorentz s law forces on moving charged particles The starting point for comput ing magnetostatic forces is therefore the knowledge of the magnetic induction B For efficiency reasons this magnetostatic problem is generally solved numerically by com puting either a total or a reduced magnetostatic potential determining the B field by differentiation The most generic formulation is based on computing the reduced potential q however it may be that the problem to be solved is numerically un stable and strongly affected by numerical cancellation so that one has to resort to a formulation combining together both the reduced and total magnetic potential It is exactly this comb
2. X coord e OPDtherm e OPDexpan e OPDtot e n k xrOPD nrOPD res0 integrate Lattice OPD_ k 7 3e 6 Temp Tair resl integrate Lattice OPD_ k Refracx sqrt Strain YY Strain YY resOtresl Dump Temp Disp Stress If i gt 1 Append The two fields Temp and Disp are assumed bo be uncoupled and therefore they can be computed separately For each field the corresponding BlockStruct and the au tomatic mesh refinement is set Afterwards the field is calculated with the Solve Stationary statement The corresponding pump power is set in the first Solve statement by the expression ForSimPar P For each computed solution we then in tegrate the local OPD on the defined lattices by computing the contributions OPDin OPDena and their sum OPDiot OPDin OPDena Finally the mesh the computed fields Temp and Disp and the derived field St ress are written to a data file for graph ical visualization Numerical Results We have performed a parameter study for pump powers up to 20W per pumped side of the crystal in ten steps with a pump beam waist radius of w 250 um and a M factor of 55 The pump beam wavelength amounts to 809nm the absorption coefficient of the YAG crystal is 3 5cm and 40 of the pump power is assumed to be converted to heat The corresponding heat distribution for a pump power of 20 W SESES Tutorial September 2012 117 Maximum Temperature pasted 140 a 40402 396102 we aan 120 38 02 D 37E 02 100
3. V is given by solving the 2D Poisson equation V V 0 and the signals are transmitted with light velocity c We now want to compute the inductance L and capacity C for our transmission from the solution of 2 15 in order to obtain the impedance 2 13 From the definition of C and L we have Q CV and OV 0z LOJ Ot resulting in V cLJ with Q the total charge J the total current and V the potential on the surface of one wire when the second one is grounded The total charge Q and current J are given by the surface integrals Q f o with n the outward normal and t the unit tangent to the boundary OQwire But since the magnetic field is a rotation of the electric field times c we actually have J cQ The product of inductance and capacity is therefore a constant LC 1 independent from the wire s shape and the impedance is given by RC iw e au fee Following 3 for two parallel cylindrical wires the exact value of the capacity is given by E nds J a B tds 2 16 Orire Qwire TLE 2 17 log 2 17 with o 8 85x1071 C Nm the length of wire d the distance between the wires and r the wire s radius Results and Conclusion In the initial section of the input file we define the two wires of radius r 0 1m separated by a distance of d 2m together with the surrounding air To solve for the electro static problem 2 15 we enable the equation Elect roStaticand set the value of t
4. 2udev e 2 119 with the log strain tensor e ein n For small strains C x Id the log strain are e log log 24y 1 v with v A 1 2 the eigenvalues of the Green Lagrange strain E C Id 2 which is coaxial with C Since A 1 we have S IdAtr E 24E which is the standard St Venant Kirchhoff isotropic material law at infinitesimal strains If a correct behavior is given at small strains for extremely large ones the choice of 2 117 is questionable however we have to consider that the appearance of yield will limit the elastic range Because the stress 2 119 has the same form of the infinitesimal case previous infinitesimal plastic models can be reused here without changes by just considering the principal log strains instead of the infinites imal ones In order to quickly comes to an end we reuse here the von Mises J2 flow model of the previous example just accounting for linear isotropic hardening f t 4 DEV 7 2 y 4 a KE with oy K two material parameters If for the trial stress Turia T trial COM puted by 2 118 we have ftri f Ttrial g lt 0 then we return T 41 Ttrial and OTm 1 O0 trial K ONE 2u DEV Otherwise we return Tn 1 Ttrial 2uAAn A 2u Oi y ONE 2u DEV 24 DEV nnt ag OEtrial T 2u 5K with AA fwria 2u 2K T DEV tria and n DEV tria 7 Clearly the relation 07 41 0 tria may also be
5. 0 the main curvatures with respect to the normalized tangent vectors are given by k 02s n ti 02g 1 0 g and kz 03s n t2 Org r 1 Opg Numerical model The numerical example of a blister forming process by deep drawing of a elasto plastic laminate can be found at example Laminate s3d Since the geometry of SESES Tutorial September 2012 155 Figure 2 103 Initial and intermediary deformations with final state showing the spring back effect Just half of the structure is shown the model is rotational symmetric and the material laws used in this example are isotropic a 2D rotational symmetric solution may do the job However we present a 3D problem formulation since in practice anisotropic plastic laws are needed thus requiring 3D computations even for an initial symmetry of revolution In addition realistic computations will also require contact with frictions instead of friction less ones as done in this example An initial rectangular laminate clamped at its border is all what is needed as initial geometry if the closest point projection is computed by the user However for a vi sual aid in the graphical representation it is also useful to define the geometry of the rigid body stamps by dummy MEs where no numerical equation is ever solved see Fig 2 103 An additional advantage is that one can right use the geometry of these dummy MEs to compute the closest point projection numerically Thi
6. 0 0615 T XZXZ 0 0615 aDir ax ay az bDir bx by bz The laser is pumped by choosing a homogeneous pumped cylinder distribution with radius wo Parameter Heat conv P w0 w0 PI d_Disk sqrt x x tyx y lt w0 1 0 W m 3 As boundary condition we assume the back surface of the copper heat sink to be in contact with the cooling water and fixed to mechanical elongations all other surfaces are in contact with air and are free to expand As final step of the initial sectin we define a routine to integrate eq 2 100 The routine s input is a polarization state RePol ImPPo1 and the output is this state after the wave has traveled from the points p0 to p What we are actually doing is the computation of the exponential function E z dz exp Fe nold dz E z with the help of a spectral decomposion of B In the first part of the command section we define the lattices on which the OPD and the birefringence has to be determined The lattices are equally spaced in a square with the lower left corner at ropa fopa and the upper right corner at ropa fopa Every lattice starts on the back surface of the disk at the z coordinate doopper dindium and ends on the front surface of the disk with its direction parallel to the optical axis The number of integration points for all lattices is given by nopa The used parameters nrOPD rOPD and nOPD are defined in the initial section SESES Tutorial September 2012 123 For i From
7. Because the SESES input is very basic actually one has to specify the stress and its derivatives with respect to the infinitesimal strain there is quite some work involved before ending up modeling a plastic flow which however shows the flexibility of such an approach General Theory Plastic models are a particular class of hypoelastic models which are an extension of elastic models Within an infinitesimal theory of small displacements an elas tic model consists in defining the stress s as a function of the linearized strain Vu Vu 2 with u the mechanical displacement The extension to a hypoelas tic model consists in specifying a stress flow s instead of s which introduces a time dependency and the need for a time integration algorithm to be performed locally at each point The time however it is a pseudo time and intended is a slow mechanical deformation where all inertial forces can be neglected In this contribution we will not discuss details related to a particular choice of the time integration algorithm nor of an automatic step selection but we will use the implicit backward Euler algorithm with a fixed time step to be defined by the user This algorithm is first order accurate absolute stable and has been shown to be superior with respect to second order ones for several plasticity models The basic assumptions used by phenomenological models of plasticity are that the strain e e can be decomposed in an
8. E combined together Here you may first need to mirror the duplication along the zx or y axis before joining which is done by affine task and the affine panel f 1 6 Algebraic Mesh Definition If the computational domain is non constant and for example if its shape should be optimized then working with a mesh constructed by a preprocessor is not well suited since such meshes are hardly modified afterwards For an automatic search of the op timum better suited is to have a functional dependency of the domain s shape from some free parameters For this algebraic method the computational domain is con structed functionally with the help of the QME I QMEJ and QMEK statements defining an initial rectangular mesh and followed by a sequence of Coord statements defining geometrical maps for the node coordinates The maps are defined by a functional ex pression of the latest node coordinates represented by the built in variables x y z and the subdomain where the geometrical map is applied Although this algebraic method is very flexible one first needs to know the actual node coordinates and secondly the search of the proper geometrical map is not always a simple task Here several rect angular blocks of macro elements can be defined individually transformed and later joined together with the statement JoinME to form a complex geometry This is often more convenient than defining just one huge rectangular block and by deleting macro elem
9. SESES Tutorial September 2012 39 v x BC fix2 movement in x direction is not permitted BC fix no foveMent in y direction BC fix0 no movement in x direction v J Figure 2 2 The x displacement The values Figure 2 1 Three boundary conditions for are negative everywhere and the top left cor thermal expansion ner shows the largest value Disp 87605 1 5605 14605 30605 52605 75605 2 7605 12608 14608 18608 19608 2 1604 2350 25501 28601 30608 2608 34601 S701 3860 4 16 08 treo Figure 2 3 The y displacement Most val ues are negative which corresponds to an Figure 2 4 The displacement visualized in upward displacement The top right corner sito using an enlargement factor of 200 shows the largest value Dispor r timeo Alternatively one can select other boundary conditions As an example one can fix the cube so that it can freely expand and no stress is induced In conclusion starting the simulation with very simple preliminary models helps to get an intuitive under standing of the physics behind the problem considered 2 2 Designing a Hall Sensor This example presents the design and optimization of a Hall sensor device The Hall sensor consists of a rectangular silicon Hall sensor element as displayed in Fig 2 5 The sensor is manufactured with a 3 micron CMOS process Two voltage contacts are placed on two opposite sides and an external voltage is applied to
10. Tcoia 2 the solution process will only be successful if the Ra number is significantly lower In fact the limit is around Ra 10 The con vergence problems for higher Ra numbers can be overcome by starting the simulation from a previous already converged solution In this previous solution the temper ature and velocity distributions were obtained on the same computational domain but for a smaller Ra number There are two strategies to reduce the Ra number The first is to start with a similar geometry which is scaled down so that the factor h in the expression for Ra is reduced Alternatively one starts with a smaller temperature difference AT between the hot and cold plates which also reduces the value for Ra As mentioned convergence may be achieved starting the simulation from air at rest for a Ra number of Ra lt 10 After finishing the first simulation with a low value the Ra number has to be increased to its final value by enlarging the geometry or the temperature difference step by step Comparison of SESES with CFX TASCflow simulations and experimental verification Several results obtained from running the above described SESES file are shown in following graphs In Fig 2 120 the velocity distribution as obtained from the SESES simulation is compared with CFX TASCflow 4 simulation results that were obtained under identical conditions There flow field predictions of the two simulation tools agree very well with each othe
11. is not uniform and is represented by an ellipsoid with the principle axes o and e In general these principle axes do not coincide with the zx and y axis of the laboratory system The electric field vector E can be represented by its components E and Ee in directions of the principle axes of the ellipsoid The refractive indices no and ne for these two components are different and one component of the field vector E is retarded with respect to the other when passing through this element Therefore de pending on no and ne and the thickness of this element the polarization of this wave is changed When not a special thickness is chosen 2 or 4 4 plate the wave is el liptically polarized after this element see Fig 2 80 The field vector of this exit wave decomposes into a component in the primary direction Epo and into the perpendicu lar component Eaepoi The latter gives the amount of the wave which is depolarized by the element Anisotropic laser crystals as e g Nd YLF show a natural birefringence in contrast to isotropic crystals as Nd YAG This situation changes when the crystal is heated The thermal expansion leads to stresses and a local deformation of the crys tals lattice which leads to spatially varying stress induced birefringence called photo elastic effect 6 The optical anisotropic activity of a material is generally described by the dielectric impermeability tensor B which is the inverse of the permittivity tensor ce If
12. v the material derivative Steel at high temperatures is a visco plastic material with an elastic and an inelastic response determined by a yield function f s where the elastic region is characterized by values f s lt 0 Differently from plastic materials where we must have f s lt 0 and values f s gt 0 are forbidden for visco plastic material values f s gt 0 are permitted and the material will relax to the state SESES Tutorial September 2012 159 f s 0 ina characteristic time A common visco plastic law is the following J2 flow model s ktr e Id 2y dev e p f dev s 2 Boy n dev s dev s 2 123 0 if f lt 0 Diep V v ep m fale 750 with ep the plastic strain k the bulk modulus u the shear modulus 7 a relaxation time tr 0 dev Ep tr e the trace of a second order tensor and dev e e Idtr 3 the deviator operator In the following we assume a constant yield stress oy and so we do not consider any kinds of hardening Evaluating the material derivative D together with the assumption V v 0 and the steady condition O e Ot 0 yields the following system of PDEs pVv v V s f Ve v Ty 2 124 The first system determines the stress s and since the casting velocity is generally small the term pV v v can be neglected The second system determines the convected plastic strain it is a system of first order hyperbolic PDEs coupled just by the ri
13. 1 N the problem index and driving potentials V 6 where 4 is the Kronecker s symbol By post evaluating the wire currents Gij the driving potentials V to be applied for an overall consistent current in each wire of I are given by solving the linear system ee GijVj I By assuming the electric conductivity o to be independent from the temperature once we have solved the eddy current problem we can compute the temperature by defin ing the heat source as the Joule s dissipated heat For a time harmonic analysis this heat source is given by time averaging the real parts of current density jye and electric field Eye over a harmonic cycle T 2mw By considering that T A cos wt sin wt dt 0 and T71 la cos wt dt T71 JE sin wt dt 1 2 we obtain 1 T yy piwt iwt q T fo R jpe R Epe dt fo Rie cos wt Sje sin wt REg cos wt SEy sin wt dt Rjg REg Sjo SEy 2 jo Eo where as last we have used the definition a b Rakb Sab 2 As one would expect the total dissipated heat in the wires and in the steel is the nothing else than the supplied AC power W Josp qdV 2r foqpdA I V with V 0 V In fact the solution of the eddy problem V x p V x A j with Coulomb gauge V A 0 and rotational symmetry implies solving the scalar equation V2 p tu V2pA j with V2 p z so that for the induced electric field Fipa we have W 2m fo ig EPS pda 2nw
14. 2 30 Let us assume we want to compute the skin effect on a straight circular wire of ra dius ro a problem with a known analytical solution see 1 We start by fixing the frequency w choose a large enough 2D domain Q to represent the whole plane and define in the middle the cross section of the wire with a constant conductivity of On the boundary 02 we apply BCs that best represent an infinite domain and on the wire we apply a driving harmonic electrical field Eye constant over the cross sec tion Without loss of generality we may assume Fy to be real For a wire length of the applied voltage at the wire ends will be V Epe t which is also constant over the cross section The impedance of the wire is given by Z V I with J the total wire current computed by the integrals fy J dA paie u VA dn 2 31 wire Owire Owire By splitting the real and imaginary parts of Z w R w iwL w we obtain the effective resistance R w and inductance L w both depending on the frequency w L EoR Ie _ L EoS Ie p ee Hig 2 32 For a single wire one knows in advance that the inductance per unit of length is infi nite and a computation just yields the value of R whereas L grows with the logarithm 68 SESES Tutorial September 2012 of the computational domain size In general and for 2D computations just for a sys tem of straight wires with a total zero current is the inductance finite Therefore for a
15. 300 T21exp x T31exp X 4a T41 ex 290 Pa T10 sim 2D 5 T21 sim 2D g T31 sim 2D 2 200 T41 sim 2D 2 w 150 4 KKK AK I 2 X S e A 10 X EEEL AM y soh ie TECER GEF Ei gA Oss x y0 i 264 0 50 100 150 200 time min Figure 2 17 Time dependent temperature at and simulations with adjusted heating pa rameter h W m The prediction by the 1D analytical model is shown for comparison different positions on the aluminum stick Let us now compare the simulation results with measured data sets of temperature profiles and transients For this purpose four points along the aluminum stick were defined with Lattice statements whose temperature is written to a file during the calculation Moreover a line was defined along the stick along which the temperature profile was recorded In Fig 2 16 two measurements with distinct heating intensities are shown that can be reproduced successfully by our 2D simulation by adjusting the heating parameter h denoted by heaterflux For the simulation shown in this fig ure the heat transfer coefficient t coef f was set to 100 W m K This value seems quite large but includes a correction factor due to the reduced surface area in the 2D model The reduced surface area results from the 2D model dimensions that were de termined by the requirement of identical volume as in the real geometry thus leading to a surface area smaller by a factor of 11 with respect t
16. A good working example is h 5 a with a gt 1 and b 2 In order not to further slow down the non linear solution process we need to correctly specify the traction s derivatives 0 T with respect to the displacements u in order to assemble a global exact Jacobi matrix and to guarantee a quadratic convergence rate of the Newton iteration close to the solution For this task we need a second order approximation of the surface at xo given by s p1 p2 Xo tapa nkap 2 O p1 pol with free parameters pa and an implied sum over the indices a The closest point projection of any point x close to x9 is approximately given by Xo p1 p2 s p1 p2 with pa x xo ta 1 6kq O x xo By expressing the surface normal as Op x pS ti pi kin x t2 p2k n Op 8 X Op S Op 8 X pS t X to pokoty X n p k n x t pipekikon xn n Pakata Op 8 x Op S y1 pe ke we have pa Blpa 0 kata and by the chain rule 0 Op 1 Oxpa ka 1 ka ta 8ta Ina similar way we obtain xXo pa X0 OxPa kata ta It follows N pi p2 152 SESES Tutorial September 2012 that n amp Ox Xo n 0 and therefore OT nh on h On h _ A X x n w a m a ak a a Oh Ke an hota Ota 2 121 zr en Care a t This short presentation of the penalty method uses a continuum approach so that the tractions T actually define Neumann BCs for the elasticity equations The underl
17. BlockStruct Block Temp Block Disp To fix the cooled left end of the aluminum stick we set the z component of the dis placement field to zero in the boundary condition SESES Tutorial September 2012 51 Steady state Temperature Profile Temperature Transients at 4 Positions 400 r r r r r r r r r experiment 1 300 T10exo O J 350 experiment 2 x oe T21exp x simulation 2D h 130 a FFF T31exp xX simulation 2D h 160 x 250 l et on T41 ex 300 r simulation 3D h 60 ff fo 4 4 T10 sim 3l simulation 3D h 75 T21 sim 3D 250 1 Zao T31 sim 3D D D T41 sim 3D z 200 F kes z 3 g 50t o yx x A 5 150 7 8 4 De 100 X J 100 L 00 f ger os mon N n 50 50 PA wort OO OOO EE k y o i yo oc as 0 20 40 60 80 100 120 0 50 100 150 200 distance cm Figure 2 22 Comparison of the 3D simula tion results with the previously shown tem time min Figure 2 23 Comparison of the 3D simula tion results with the previously shown tran perature profile data sient data Steady state Displacement Profile Diso X ant SRLS ARR GR RRURRERREM SERRRRRR RRR RR ER displacement mm a 5 1E 04 Biot 0 5 4 6 7E05 eee Sten 0 on en L 1 1 0 20 40 60 80 100 120 Figure 2 24 Calculation of the displacement due to thermal expansion in the aluminum stick distance cm Figure 2 25 Calculation of the displacement pro
18. DiagDiff2 diffCoeff mmolA xmmolB A B SIunit Model TransConvDefined Enable Parameter Rate D_A D_B D_AGrad D_BGrad CONVECTION2 Rate A B Velocity density Material m_Fluid 1 to define and map the material m_F luid On its domain the Navier Stokes equations are set up together with the species mass balances However as will be discussed later when the velocity profile and the average pressure are given only the species mass balances need to be solved As last step the inlet concentrations are defined as Dirichlet boundary conditions BC b_inletl 0 0 JType ny 2 1 Dirichlet A 1 0 Dirichlet B 0 0 BC b_inlet2 0 ny 2 JType ny 2 Dirichlet A 0 0 Dirichlet B 1 0 Now that the initial section is complete parameters that control the actual computa tion need to be given within the command section First we specify initial values for the relevant fields A B Pressure Velocity Y and Velocity X Solve Init A 0 5 B 0 5 Velocity 1 velocity 6 velocity y ly y ly 2 0 Note that for Velocity X we allow for two different velocity distributions either the homogeneous plug flow or the parabolic Hagen Poiseuille flow Next we define the fields to be solved for using the BlockSt ruct statement set convergence criteria and execute the actual calculation with Solve Stationary ConvergGlobal AbsIncr A lt 1 E 12 amp amp AbsIncr B lt 1 E 12 BlockStruct Block A B Solve Stationary Finally post processing statements are add
19. EAS R fo oak a Bg C0OD0D0D 04t x Ry ct 0 2 L iy 0 2 L x XY FRR 0 0 HE PERRO DEX 0 0 0 0 0 2 0 4 0 6 0 8 1 0 0 0 y m y m Figure 2 116 Concentration profiles over different cross sections along the straight channel shown in Fig 2 113 The parameter values are left K 0 1 and right K 0 01 respectively Comparison of the analytical results based on 2 148 shown as the lines with the SESES predictions shown as the symbols where Vaver represents the average velocity 7 As expected there is better mixing in the Hagen Poiseuille compared to the plug flow case However for K 0 1 the differences are rather small but become larger as K decreases References 1 B HE B J BURKE X ZHANG R ZHANG F E REGNIER A picoliter volume mixer for microfluidic analytical systems Analytical Chemistry Vol 73 No 9 pp 1942 1947 2001 2 A E KAMHOLZ B H WEIGL B A FINLAYSON P YAGER Quantitative Anal ysis of Molecular Interaction in a Microfluidic Channel The T Sensor Analytical Chemistry Vol 71 No 23 pp 5340 5347 1999 3 F ISMAGILOV D ROSMARIN P J A KENIS D T CHIU W ZHANG H A STONE G M WHITESIDES Pressure Driven Laminar Flow in Tangential Microchannels an Elastomeric Microfluidic Switch Rustem Analytical Chemistry Vol 73 No 19 pp 4682 4687 2001 4 T J JOHNSON D Ross L E LOCASCIO Rapid Microfluidic Mixing Analytical Chemistry Vol 74 No 1 pp 45
20. For a practitioner is therefore of interest to know the range of applicability and properties of the various finite elements topics discussed together with the SESES file found at example Conicalshell s3d modeling a conical shell The shell is an axis symmetric shell of revolution with isotropic linear material laws and with loads ap plied symmetrically We use the geometric non linear model allowing large displace ments to be computed The shell is shallow in the topological K direction so that the three type of finite elements are easily selected by specifying either the Elasticity Elasticity2orElasticShellSingleK equation model Although the problem formulation is fully symmetric with respect to the axis of rota tion non linear solutions may not be necessarily symmetric as for warp deformations breaking the symmetry so that a reduced 2D axis symmetric approach only comput ing symmetric solutions is not followed here However to reduce a little bit the nu merical work we just compute a quarter of the 3D shell by applying symmetric BCs on the virtual cuts In doing so we cannot compute solutions with half symmetry The bottom of the shell is fixed but can freely rotate on the bottom edge and on the top we apply an increasing vertical downward traction All three types of finite elements just use Lagrange displacement degree of freedoms therefore the BCs are independent of the element type The conical shell starts to deform inward a
21. How ever we have the residual values R associated with the degree of freedoms u After a solution has been computed and thus after applying the contact constraints the residual values are always zero R 0 Therefore before amending the discretized equations for the glide constraints the residual values represent the sum of all forces acting on the boundary point without considering the contacts These residual values are concentrated force resulting from integrating over the domain the governing equa tions with various a priori unknown weighting factors The absolute value is therefore of no meaning to us not so its direction specifying the direction of the contact force acted by the rigid body Closest Point Projection A general approach to rigid body contacts consist in approximating the contact sur face by low order multivariate polynomials which are then used to evaluate the closest point projection If the normal should be a continuous function the global approxi mation must be C and for continuous curvatures C smoothness is required This general approach is the simplest to apply from the user point of view but a good ap 154 SESES Tutorial September 2012 proximation is memory intensive and requires well designed search algorithms In SESES a C interpolation on a 2D tensor grid of 3D sampling points is available but even with a very large number of points the curvature values may not be very close to the exact values of
22. Internal resistance and inductance values of a straight circular wire compared with analytical values of the left side On the symmetry axis the proper BCs are homogeneous Dirichlet In a series of N linear problems we set up the complex conductance matrix Gj in order to compute the driving voltages Voj In order to easily apply the driving voltages we have defined for each wire a corresponding domain of definition Wire lt i gt with i 0 N 1 which are also used to compute the integral 2 31 for the wire current For j To nWind 1 MaterialSpec Coil Parameter Current0 Z domain Wire j sigma 2 PI x Solve Stationary A mx 2 Define For i To nWind 1 VAL integrate Domain domain Wire i Current Z iCurrent Z MAT 2 nWinds 2 i 0 2 j 0 VAL 0 se A mie Hane eee rae a on MAT 2 nWinds 2 i 1 2 j 0 VAL 1 MAT 2 nWind 2 i 1 2 j 1 VAL 0 From the library NmBlas d11 we use the routine InvMat DotVec to solve the linear system 2 35 with the conductance matrix stored in the global variable MAT and to obtain the driving voltages stored in SOL This library is supplied together with the SESES binaries and its source code is available for reference in the include directory Since SESES just works with real numbers the conductance matrix MAT has actually a dimension of 2N x 2N and solution SOL and right hand side vector RHS a dimension of 2N The complex matrix inversion is simply ob
23. SESES Tutorial September 2012 119 Laser drystal Output coupler Figure 2 77 Optical scheme of a simple thin disk laser with quasi longitudinal pumping of the crys Figure 2 78 Representation of the elec tal tric field E in direction of the principal axes of the refractive index ellipsoid E and E There a crystal disk with a thickness smaller than the diameter of the disk is mounted with one of its faces on a heat think see Fig 2 77 which is high reflectivity HR coated for both the laser and the pump wavelength Therefore the excess heat is removed via this face If we assume a large heat transfer coefficient over the hole area a tempera ture field will be established in the crystal with the isotherms essentially normal to the optical axis resulting in significantly reduced temperature gradients in this direction The disk can efficiently be pumped by laser diodes in a quasi longitudinal scheme To increase the absorbtion of the pump radiation for a given thickness multiple passes of pump radiation through the disk must be used This disk fits into the resonator as an end mirror or as folding mirror The design is suitable for lasers in the power range of several watts to kilowatts The power can be scaled by increasing the pumped diam eter of the disk at constant pump power density and or by using more than one disk By employing the appropriate resonators and efficient cooling technologies it is pos sible to achiev
24. The maximum temperature amounts about 92 C Temperature Raise o 0 25 05 0 15 1 1 25 15 1 75 2 Figure 2 85 Temperature raise on radial straight lines starting in the center of the disk The lines are located on the front and back surface and in the middle of the disk SESES Tutorial September 2012 Figure 2 84 Heat flow in the central part of the disk The cones denote the direction of the heat flow Optical Path Difference Thermal Je End Effect ee Pee Cees e8ecee eee he eooo ki tes ttteseeeseesseeteesseoeoe Figure 2 86 Optical path difference OPD in duced by the thermal dispersion blue and the end effect red SESES Tutorial September 2012 125 The corresponding focal lengths of the averaged thermal lens are fi 29 1m and fena 11 7m which is definitively extremely weak Even when we deduce the focal length of the averaged thermal lens over a radius of 1 25 mm the values rest weak with fa 12 6m and feng 5 2m Lets last have a look onto the stress induced birefringence by checking the amount of local depolarization of a light ray following the corresponding lattice As the lat tices are equally spaced in a square we get from this depolarized part the information about the local depolarization of a homogenous beam traveling through the disk see Fig 2 78 The result with the well known Malte
25. WEBER Laser resonator with balanced thermal lens Optics Communication Vol 190 pp 327 331 2001 2 E Wyss M ROTH TH GRAF H P WEBER Thermo Optical Compensation Meth ods for High Power Lasers IEEE J Q E Vol 38 No 12 pp 1620 1628 2002 2 20 Shells or Thin mechanical structures A shell is a curved thin walled mechanical structure capable of supporting large loads This property stems from its curved nature and the ability to distribute internal trans verse loads towards in plane loads at the edges of the shell Shells are pervasive in nature and as artefacts and include skulls aortic valves blood vessels domes silos tubes cans car bodies balloons etc Because of their overall presence many me chanical simulations will then actually deal with shell structures A priori one may think this task being as simple as performing a mechanical simulation based upon the discretization of the 3D governing equations of continuum mechanics by some stan dard finite elements However this approach does not always work well when the mechanical structure is shallow in one of the three directions since low order finite elements are numerically unstable here and solutions are affected by large errors an effect known as numerical locking When numerical locking appears the displace ment are underestimated and the stress is overestimated i e the mechanical structure is by far more stiff than in reality The main reason of this failu
26. in the plot of Fig 2 131 The plots of the dynamic responses show that the constant heater temperature mode allows for faster response at the cost of power regulation In summary we reported a modeling method for the time dependent thermal analysis of flow rate dependent heat transport in a calorimetric flow sensor with a straight microfluidic channel Some general measurement principles have been discussed with the help of a simple 2D SESES model 186 SESES Tutorial September 2012 1 0e 03 0 017 0 016 0 015 0 014 0 013 0 012 0 011 0 01 1 0e 03 fawscaent 8 0e 04 J 8 0e 04 seno 6 0e 04 6 0e 04 piire 4 0e 04 4 0e 04 Velocity m s Power W Velocity m s poceeneeel 2 0e 04 J 2 0e 04 ATy K ho ONO O J 0 0e 00 0 009 300 400 500 600 0 100 200 300 400 500 60 Time s Time s 0 0e 00 0 100 200 10 Figure 2 130 Dynamic response of the heater Figure 2 131 Dynamic response of the heater temperature rise AT blue to a series of ve power blue to a series of velocity vp steps locity vo steps blue in constant power mode blue in constant temperature mode ATy p 10 mW 10K References 1 G GERLACH W D TZEL Grundlagen der Mikrosystemtechnik Hanser Verlag M nchen 1997 2 30 Flow Around a Circular Cylinder The flow around a circular cylinder has been the subject of intense research in the past mostly by experiments but also by numerical simulations
27. kg m 3 x calculated mass fluxes x mfluxFluidl 2 997e 06 kg s calculated effective permeability permEff 1 111e 09 m2 Based on 2 174 the permeability is calculated as permEff le3x Viscosity_AIR T_ref mfluxFluidl xlengthZ areaXY densiFluidlxdpress with mfluxFluid1 the computed total mass flux The small value k 1 111lex 10709 m means that the insulation assembly exhibits a small flow resistance A similar in formative output is generated when running in mode condYN 1 to compute the av erage thermal conductivity xz Based on 2 175 the conductivity follows as kappaEff le3xlengthZ T_Hot T_Cold qFluxWalll areaxy with gFluxWall1 the computed total heat flux As expected the value of kz 0 1448 W K m lies in between those for air and the insulation material References 1 M Roos E BATAWI U HARNISCH T HOCKER Ffficient simulation of fuel cell stacks with the volume averaging method Journal of Power Sources Vol 118 pp 86 96 2003 2 St WHITAKER The Method of Volume Averaging Kluver 1999 3 H BRENNER D A EDWARD Macro Transport Processes Butterworth Heinemann 1993 4 R B BIRD W E STEWART E N LIGHTFOOT Transport Phenomena Wiley 1960 204 SESES Tutorial September 2012 Figure 2 149 View of the heat exchanger Visible are two crossed channels which carry a hot and a cold air flow arrows 2 34 Heat Exchange between Air Flows Ai
28. q and insertion into V B yields a Laplace equation V LV Orea HH 0 2 41 to be solves for O q on a domain 2 The term reduced formulation originates from the fact that the magnetic potential O q only determines the irrotational part H VOreq of the H field and a total formulation is present whenever the magnetic po tential fully determine the magnetic field H VOt t which automatically implies Jo 0 within In this latter case the excitation of external currents must be included within the BCs on 02 when solving for Otot A total formulation can always be turned into a reduced one by defining Ho as the Biot Savart field of the current Jo external to the domain 2 thus eventually simplifying the setting of BCs Let us get acquainted a little bit with both the reduced and total formulations and for a case with Jo 0 let present an analytical solution The problem we are going to investigate consists of a ball of radius r and permeability u HoHre immersed in a constant magnetic field of Hp e x 1 A m generated by external currents The solution for either the total 04 4 or reduced ed potential is given by 7 Hra 1 zrgjx 3 for x gt r 7 7 Opa S and O11 rea z 2 42 z for x lt re Hrel 2 Since V Vz 0 and V Vz x 73 0 we have V VOt0t 0 and Otot is indeed a mag netostatic solution if the normal component of B and the tangential component of H are continuous on the sphere x r Using spheri
29. the given limit Store AtStep 1 CurrentAver Continuous 1 FreeOnRef Current Remesh AtStep 1 MaxNoElmt 20000 MinFractionElmt 0 05 Refine ErrPsi epsilon Global Solve Stationary The figure of merit for the Hall sensor is the Hall voltage which is written to the output together with some model parameters Write Magnetic field e T n bfield Misalignment Se um n misalign Hall ideal Se V n mobilxbfield vapplied width length Hall voltage se V n Halll Psi Shift Hall2 Psi Shift Here are some results obtained by running SESES for different parameter values The discrepancy of the Hall voltage with the ideal case is of ca 25 which can be reduced by augmenting the length width ratio With a misalignment of just 0 2 um we have an offset Hall voltage almost as large as for no misalignment and a magnetic field of LT agnetic field 1 000000e 00 T isalignment 0 000000e 00 um Hall ideal 3 162500e 02 V Hall voltage 2 455195e 02 V agnetic field 0 000000e 00 T isalignment 2 000000e 01 um Hall ideal 0 000000e 00 V Hall voltage 1 880438e 02 V agnetic field 1 000000e 00 T isalignment 2 000000e 01 um Hall ideal 3 162500e 02 V Hall voltage 4 333158e 02 V The electrostatic potential field Psi corresponding to the calculation with the third parameter set is plotted in Fig 2 8 SESES Tutorial September 2012 43 Temp K 3 5E 02 35E 02 35E 02 3 5E 02 34E 02 34E 02 34E 02 33E 02 33E 02 33E 02 33E4 02 32E4 0
30. 0 387 Ra 167 Nu P di ARES 1 255 J990 0 236 0 6 2 10 where both the Prandtl number Pr and the Rayleigh number Ra depend on material parameters of the surrounding air For our geometry the convective heat transfer co efficient is calculated as 10 1 W m K We note that this value is in good agreement with the value of 100 W m K used in the 2D model if the surface reduction factor is considered We shall now keep this value fixed and in addition consider heat transfer by radiation The Stephan Boltzmann radiation law states dQ LS eo A T Tree 7 2 11 50 SESES Tutorial September 2012 Alu olin eatin Figure 2 20 3D SESES Model of the heated stick shown from below with cooling on the Figure 2 21 Example of calculated temper right and heating on the left ature distribution in 3D assuming both con vective and radiative heat transfer where we have introduced the Stephan Boltzmann constant o and the emissivity e The emissivity is a unitless material parameter between 0 and 1 and for a given ma terial varies depending on the surface quality such as the roughness For aluminum a typical value above room temperature is 0 25 The new heat transfer boundary condi tion considering both convective and radiative loss thus reads BC Transfer OnChange 1 Neumann Temp D_Temp tcoeff refTmp Temp sigmaxeps refTmp 4 Tempx 4 tcoefftsigmaxeps 4 Temp x 3 W m 2 W m 2 K As for the geometr
31. 0 To nrOPD For j From 0 To nrOPD Lattice OPD_ i _ j Index m 0 nOPD rOPD ix 2 nrOPD 1 rOPDx j 2 nrOPD 1 d_copper d_indium 0 0 d_Disk m nOPD The next block defines the solution procedure where we first solve for the temperature Temp and then for the displacement Disp Finally we write out the temperature the heat flux the displacement and the stress field Convergence 1 BlockStruct Block Temp Block Disp Solve Stationary Dump Temp TempDiff Disp Stress In the last part of the command section we compute for all lattices the optical path difference OPD and the stress induced depolarization Write Text OPD X coord Y coord OPDtherm OPDend OPDtot Text Re Ex Re Ey Im Ex Im Ey n For i From 0 To nrOPD For j From 0 To nrOPD Text Pol 1 0 0 0 f 0 p0 zero Lattice OPD_ i _ j Pol ApplyPol p0 coord Pol Pol p0 coord f 1 Text 0f 0f i Q j Text 9 2e 9 2e 9 2e 9 2e 9 2e rOPD i 2 nrOPD 1 rOPD j 2 nrOPD 1 integrate Lattice OPD_ i _ j DRefrac Refracx sqrt Strain ZZ Strain ZZ DRefractRefracxsqrt Strain ZZ Strain ZZ Text 9 2e 9 2e 9 2e 9 2e n Pol Numerical results For this simulation we have chosen an absorbed pump power of 100 W and a pump spot radius of 1 mm The temperature distribution in the thin disk is shown in Fig 2 83 The maximum temperature localized in the center of the disks front surface amounts ab
32. 04 1 4 00E 04 6 00E 04 0 5 8 00E 04 1 00E 03 i 600 650 700 750 800 850 900 0 10 20 30 40 50 x mm Re Figure 2 134 x velocity at the center line be hind teeylinder Figure 2 135 Bubble length as a function of Reynolds number the size of these recirculation regions we plot arrows indicating the direction of the velocity After zooming in we notice two separation regions as shown in Fig 2 133 To determine their size we use the velocity distribution on the straight line running through the center of the cylinder and pointing in the down stream direction Behind the cylinder we expect a zero velocity on the cylinder s surface and a negative velocity along the line to the point where the recirculation area ends Further down stream the velocity will steadily increase towards the value at the inlet We therefore need the values of the velocity behind the cylinder on a straight line running through the center of the cylinder To this end we enter the statement x along channel center behind cylinder gt Lattice lat1 Index i 0 dataPoints 0 600 0 350 0 150 0 0 i dataPoints to define the location where the data is sampled The straight line starts at 600 350 mm i e on the surface of the cylinder and stretches 150mm in the z direction The parameter dataPoints specified earlier defines the number of equally spaced data points on the straight line To obtain the numerical data on the lattice points for plot ti
33. 1 100 605 703 1 1807 T00 6705 Table 2 2 Number of solver iterations to reduce the residual by a factor of 1078 as function of the frequency w slenderness factor a and number of pre and post smoothing cycles The system matrix of size 72318 is for second order nodal elements and it is preconditioned by the auxiliary preconditioner based on first order nodal elements The auxiliary system of size 19074 is solved either with an exact LU or an ILU 0 preconditioner value in parenthesis By considering the fact ITI Id we then have PeftSona Id The theory is actually a little bit more involved and in this form the method does not yet work Before or after applying Pier one has to smooth the high frequencies of the numerical error in the so lution not seen by the element of first order and here the numerical approach is exact the same as for multigrid methods The classical smoother is the Gauss Seidel method an iterative solver with less performance than the conjugate gradient but with excel lent smoothing properties 1 In general the smoother should also converge if used as an iterative solver but for this application where the system matrix is not positive definite nor an M matrix the Gauss Seidel solver promptly diverges As smoother cannot be used either and the fix is to apply the Gauss Seidel smoother to the original complex symmetric system This can also be done on the real system by applying a 2 x 2 block Gauss Sei
34. 1 0 0 009 0 0 0 0 0002 0 0004 0 0006 0 0008 0 001 1 1 L 0 004 0 006 0 008 0 01 Power W 1 0 0 002 Velocity m s Figure 2 127 Sensor calibration curve relat ing the sensor temperature difference AT gt _ 1 red and the heater power to the velocity vo in constant temperature mode heater tem perature rise above ambient AT 10 K Figure 2 126 Plot of the sensor temperature difference AT gt _ T T red and AT y blue versus the power applied to the heater velocity vp 2 1074 m s Constant Sensor Temperature Mode As indicated in Table 2 4 an alternative measurement mode adjusts the heater power density at constant heater temperature For this mode one needs to measure the lo cal heater temperature and then regulate the heating power to maintain a constant heater temperature The sensor temperature difference AT gt _ and the heater power is plotted in Fig 2 127 versus the flow velocity vp The temperature difference AT gt _ assumes a local maximum and decays at large velocities as in Fig 2 125 By contrast the heater power density increases monotonically with increasing flow velocity Dynamic Response For sensor applications the dynamic response behaviour to a stepwise increase of the flow rate is often critical In order to simulate the time dependent evolution of the temperature field in the flow sensor the thermal problem is solved dynamically while the velocity field is kept constant This me
35. 37602 ee 3602 ar 35E 02 E 35502 gt Poa 60 34E102 346302 Pak 396402 40 3 26402 a 3 26402 20 3416 02 e C SOE o 5 10 15 20 Pump Power W Figure 272 Temperature distribution in the Figure 2 73 Maximum temperature in C for laser rod at a pump power of 20 W per side the pump power range per side from 2 to The temperature scale is given in K 20 W Maximum Principal Stress 4 4E 07 25507 w a 5 5E 06 146307 P 396 07 a0 53E07 7207 wee oaeo EN 1 16108 136 08 1 5E 08 40 1 7E08 1 95408 ee 2 16 08 23E 08 2 5E 08 2 7E08 2 95408 o 5 10 15 20 D Pump Power W Figure 2 74 Distribution of the highest prin Figure 2 75 Maximum tensile stress in de cipal stress component pendence of the pump power per side per side is shown in Fig 2 72 The maximum temperature is located inside the crystal and amounts to 150 C Fig 2 73 shows this maximum temperature for the whole pump power range from 2 to 20 W Despite the non linear material parameter 2 97 the raise of the maximum temperature goes linear with the pump power As mentioned at the beginning of this example a too high stress may mechanically crack the crystal The stress forms a second rank tensor With suitable coordinate transformations the tensor can be diagonalized i e the stress components are reduced to the principal stress components Positive and negativ
36. 9 one applies recursively the constructed mesh As first the mesh is duplicated by copy amp paste Within the insert task or the affine task and move option fh se lect the mesh to be copied and either click the copy 4 and paste buttons or press the ctrl c and ctrl b key combinations The paste operation places the buffered object more or less at the actual pointer position The copied mesh is automatically selected and you can move it at the correct position by picking and dragging one of its edges or nodes As second a macro element is deleted in the copied mesh and the original mesh is inserted in the created hole or indentation Macro elements are deleted within the insert task t the affine task amp or whenever no task is enabled by selecting the macro element followed by a click of the cut button or by pressing the ctrl x key combination To insert the original mesh in the created hole or indentation with the join task and join fit option IH select the boundary to be joined then pick one of its edge and drag it over the corresponding edge of the hole or indentation until a snap takes place An automatic fitting of the inserted mesh is performed here Again be sure that that just edges on the boundary are drawn with a thick red line are other wise click on the internal red edge to join it with its neighbor To create a mesh with a circular shape one has first to define a NURBS curve with the curve task Z This butt
37. B along this line In order to compute the magnetic force we need to consider the magnetic energy as function of the air gap d The volume of the ferromagnetic material is constant and since B is constant as well the magnetic energy stored in the ferromagnetic material does not depend on d The only contribution stems from the magnetic energy stored in the air gap which is given by Wsap d A B 2u0 and derivative with respect to d as of 2 22 yields the force Fmag AB 2u0 This force grows quadratically in B but just for small B values since afterwards volume and saturation effects will limit the growth as schematically shown in Fig 2 40 Numerical model We present here some numerical results for the micro reed relay and the SESES input file can be found at example MicroRelay s3d Most of the Initial section is concerned with the construction of the macro element mesh for the relay and vacuum region For the ferromagnetic material we use the material law H hasim init Basim BsatHinit B H H H Minit Posen Beat 2 28 with Bsat the magnetic field value where saturation starts to act and init Hasim the slopes of the B H curve for zero and oo values _ OB H _ _ OB H _ Pay en aa e For this model shown in Fig 2 42 for Bsat 0 88T Minit 12000 and pasim 1 we can easily obtain the inverse relation H H B and the energy density integral 2 23 in analytical form In SESES this material law is defined through
38. Convergence AbsIncr Disp lt 1 E 12 1 nIter gt 5 isnan AbsIncr Disp failure 0 Solve Stationary At 6 Step 0 5 step 100 Failure step lt 1lE 3 0 step 2 Within the convergence criterion we set the failure criterion for Newton s algorithm as a maximal of 5 iterations and at the same time we catch NaN values showing up in the displacement norm With the Solve Stationary statement and the Failure option we set the response directive whenever a failure condition is meet inside New ton s algorithm If the case the incremental step of the pressure load is halved and if less than 107 we stop By conveniently changing the simulation parameter and predicting the solution it may be possible to pass the limit point and reaches the snap through point However it is not easy to follow the path in the unstable region since by reducing the simulation parameter one generally turns back on the solution path In order to easily obtain the displacement load curve one has to let making the changes of the simulation param eter to load control algorithms Here the simulation parameter becomes a dependent and thus controlled variable and a new independent variable must be introduced to gether with an additional equation connecting the solution the load and the new in dependent controlling variable The additional equation determines the type of load control applied and a common choice is spherical or cylindrical load control The 34 SESES
39. Current Z iCurrent Z c2 c 0 c 0 c 1 c 1 return Omega 2 PI c sqrt c2 c 0 c2 c 1 c2 Omega In a second example example Proxi s2d we compute an air coil with N windings at the single frequency w 27 x 100kHz The finite element mesh is constructed automatically starting from generic parameters like the number of wires the wire di ameter and the free space between the wires Because the wire diameter is small with respect to the skin depth we do not need to construct a finer mesh around the inner wire boundary The construction of the mesh is the most complex part of this example and the other definitions are similar to the previous example except that here we have to define the rotational symmetry Around the coil we have a cladding of free space which ideally should extent to infinity and on its boundary we have to set BCs for the dof fields Az and iAz Here we use the built in function Infinity as replacement for homogeneous Neumann or Dirichlet BCs which better models an infinite domain For this built in to work properly we have defined the coordinate origin in the middle 70 SESES Tutorial September 2012 2 4e 03 5 0e 08 2 2e 03 Y ie m 4 4 5e 08 2 0e03 A a 1 8e 03 L L analytic x 060a T Q 3 5e 08 5 1 6e 03 2 g 3 0e 08 amp 1 4 03 8 4 1 2e 03 2 5e 08 3 1 0e 03 2 0e 08 8 0e 04 4 1 5e 08 6 0e 04 1 0e 08 0 5000 10000 15000 20000 25000 Frequency Hz Figure 2 45
40. Flux To compare the values of the an and A constants we store the displacement uo and strain e uo fields under the name of Disp0 and Strain0O0 that otherwise would be lost by computing subsequent solutions Store Disp0 Disp Store Strain00 Strain Because the electric solution does not participate in norming the solution it is impor tant to start the eigen solver with a zero electric solution and therefore we initialize everything to zero after computing the inhomogeneous solution BC apply Dirichlet Phi 0 YV Solve Init Afterwards one computes a representative number of eigenpairs Solve EigenProblem nPair npair 48 Precision 1E 30 and for each computed eigenpair we write to the output the value of the mechanical frequency and the a values computed with three different integral methods Note that in computing the domain integral we have to consider the rotational symmetry and that this postprocessing related to the mechanical modes must be defined before computing the eigenpairs and it will be applied to each computed eigenpair Write AtStep 1 Text 2 0f frequency 10 3e Hz alpha 10 3e 10 3e 10 3e n eigennum l1 sqrt eigenvalue 2 PI apply Phi Flux 2 PIx xintegrate MechEne Strain00 Stress x 2 PIxintegrate ScalP Disp0 Disp x eigenvalue The results are obtained in the form 1 frequency 4 956e 04 Hz alpha 2 282e 01 2 282e 01 2 282e 01 2 frequency 1 281le 05 Hz alpha 2 076e
41. In the command section of the input file we define the fields to be solved set a con vergence criteria on the velocity and call Solve Stationary to compute a steady state solution BlockStruct Block Pressure Velocity Convergence MaxIncr Velocity lt 1 E 12 1 nIter gt 100 failure 0 Increment Standard 3 ReuseFactoriz 30 Solve Stationary Dump Pressure Velocity MassFlow The Dump Pressure Velocity MassF1ux statement defines the field variables to be displayed in the SESES GUI For the analysis of the simulation results we need to add post processing commands to write the desired information to external files This will be explained in the next section Numerical Results In order to check the accuracy of the results we compare them to experimental data available from 2 In particular we are interested in the force exerted on the cylinder by the fluid flow For a Reynolds number between 4 and 48 we expect two separation bubbles behind the cylinder with small velocities In order to get an impression of SESES Tutorial September 2012 189 1 00E 03 r Re 43 CFX TASCflow 8 00E 04 Re 23 CFX TASCflow 0 Exp Data d 6 00mm Re 45 7 SESES 25 A CFX TASCTlow 6 00E 04 Re 23 SESES SESES a a x_Exp Data d 2 95mm 4 00E 04 r 2 00E 04 T d E 0 00E 00 ais E 2 00E
42. Kappa_AIR 1 double T Heat conductivity of air in w m k temperature range 300k 1500k return 0 006292410832933179 Tx 0 00006858567183567118 3 7488608369754205e 9 1 2382966512126441e 12x T T As we already noted the problem at hand with reference to geometry and fluid flow is symmetric with respect to the pipe axis We can make use of this property by including the statement GlobalSpec Model AxiSymmetric Enable together with the specification of the physical properties Numerical Results We like to see how the hydrodynamic and the temperature boundary layer develop For this purpose we write the velocity and temperature profiles at various locations along the length of the pipe The statement Write File section txt Lattice lat 8 4f 8 4f 13 6f 13 6f n x y Velocity Y Temp writes the x and y coordinate values the y component of the velocity and the tem perature to the file section txt and the previous statement Lattice defines the points where the data is sampled The profiles consist of eleven equally distributed points on a straight radial line These data can be plotted with a suitable program The velocity profiles at different locations along the pipe length are shown in Fig 2 140 194 SESES Tutorial September 2012 The profiles show a constant velocity at the entrance of the pipe and a boundary layer which has extended to about half the radius at x 0 1m At x 0 2m the boun
43. N 89N Id The chain rule gives S OS 0E E 0S 0E 6 C 2 so that by expressing the variations S C with respect to the base N Nj we obtain 0S 0E with respect to the base N N N Ni Let e be an orthonormal constant basis and write N Q e with Q orthogonal then Q QT Id or 5Q Q Q 5Q 0 and 6Q Q 5Q Q As first we develop ON 6 Q e 6Q e Q 5Q Q e 0 N Dy N N Q N ae Qj Nj with Q Q 6Q and Q i N Q N and note that Q is anti symmetric since QT 5Q QT Q 5Q T Q Q T Q 5Q 2 As second for some values ai we develop aid Ni Ni 90 ai ONi Ni Ni dNi j iOji Nj Ni Ni Nj ig GLN Nj aij Ni Nj Vj jaiz Qj aj 4 Ni Nj We therefore have C 55 AFN Ni DD 2AA Ni Ni D0 5 M45 AF DN 2 N and S J SNIN DO jO Si NN IOD 45 55 S N N By noting that N N C 2A j j aie fori 4 j we have N SN N QN C 2055 AZ a we see that as E TD a NEON ONION 5 oN N N N N N ATA ij resulting in 2 111 by considering S 7 A SESES Tutorial September 2012 149 Figure 2 100 Blisters created by deep drawing of a laminate References 1 T BELYTSCHKO W K LIU B MORAN Nonlinear Finite Elements for Continua and Structures John Wiley amp Sons 2000 2 R A HORN C R JOHNSON Topics in Matrix Analysis Cambridge University Press 1991 3 J C SIM
44. Poisson ratio v Young modulus E and the coefficient of thermal expansion a In this example a cube of side length 1m with a 17x10 v 0 3 E 200x109 N m will be subjected to a temperature change of dT 14K By restricting the free expansion on portion of the boundaries a stress s will be produced in the material For the unidirectional case this behavior can be characterized by the following equations es Sear s ee i In our example we constrain the displacement on three side faces of the cube as shown in Fig 2 1 The material of the left edge is not allowed to move up or down ie the y movement is clamped the bottom edge is not allowed to move in the x direction and the right edge cannot move in the y direction These boundary conditions have the following consequences The top left corner is free to move in the x direction and the top right corner can freely move in the y direction The lower left corner is very limited in its movement in both directions In this corner we expect the largest total stress Fig 2 2 2 3 show the z and y displacement respectively whereas Fig 2 4 shows the absolute displacement An enlargement factor in our case 200 needs to be set to visualize the small displacements When displaying the stress field one observes the largest stress in the lower left corner as predicted The stress field matches the strain field in the following sense small strain large restriction large stress 38
45. a 0e 06 Slenderness 100 50 1 0e 04 gt 1 Vero E equency here we always have p oo For not too many elements the condition number can be computed in reasonable time so that for this example its computation is valuable in determining how strongly badly shaped elements and the frequency can impact on the convergence rate Fig 2 64 shows the condition number as function of these two pa rameters one sees that badly shaped elements strongly worsen the condition number as well as the frequency for good shaped elements Table 2 1 lists in the third column the number of iterations required for the conjugate gradient to converge when work ing on the unamended system matrix of dimension 19074 and by requiring a reduction of the initial residual by a factor of 1078 One can see a close correlation between the condition number and the number of required iterations to converge For large linear systems a number of iteration larger than a few percent of the matrix dimension can be considered as a slow convergence The problem of the slow convergence for large condition numbers can be overcome by preconditioning the original linear system Sx b which is multiplied left and right with the preconditioning matrices Peft Prigat yielding the new system S Pamt V with S Pns Rigu and Y Harb 2 72 The idea is for the preconditioning matrices to build a good approximation of the in verse i e to have S Id We then apply the ite
46. a dev a nn 1 n4 1 a Deng nn la 2 Heee r 2 108 Deny Sntilal ktr a id 24 1 pee dev a 24 feria A a EAGER finpe anal ett inga a Defining the model Let us compute the plane strain perfect plastic example presented in 3 The material parameters are given by Young s module Emoa 7x10 Pa Poisson s ratio v 0 2 yield stress Gy 2 30y 2 3 x 0 24x10 Pa and zero hardening H K 0 so that we also have q 0 The bulk and shear modulus are defined by k Emoa 3 1 2v 3 888x 10 Pa and u Eqog 2 1 v 2 916x10 Pa The SESES input file for this problem can be found at example J2Flow s2d Ina first step we define an element field St rainP to store the plastic strain e at each integration point and the global parameters BULK_MOD LAME_MUE to define the bulk and shear modulus k u ElmtFieldDef StrainP DofField Disp T22 GlobalDef Const BULK_MOD Pa GlobalDef Const SHEAR_MOD Pa GlobalDef Const YIELD Pa We then define a routine PlasticStress to specify the material parameter Stress Law defining both the local stress and the derivative with respect to the strain For this perfect plasticity example and ftria gt 0 the stress is given by Sn41 ktr n41 Id Oy Nn and from 2 108 we have the derivative dev a nn 1 nn41 a Deny 8n 1 a k tr a Id 2uay ea ria wn At the end of this return mapping algorithm we have to update the value of the plas
47. a con sequence the computation of the Taylor coefficients always involves real and smaller linear problems to be solved with respect to a complex coupled solution of 2 53 For our formulation the most convenient approach is to compute the admittance matrix Y w and so we are looking for its Taylor s approximation P w2 u Ero t Yw 2 57 p gt 0 with Y bY 0 and which is based on computing the Taylor s coefficients of the conductor current 2 55 L T 0 I J 0 dn I s LE 0 dn 2 58 ena Cr m Up to the second order the impedance Z w Y w is related to 2 57 by 2 Zw p o Yo lw PY Mi Yo t w YY 2 2 59 The governing equations determining the derivatives 0 E 0 in 2 58 are obtained by taking the w derivatives in 2 52 V x OPE 0 ipd A 0 0 Vx uw V x OP A 0 cd E 0 2 60 These equations build up a recursion of linear problems with the property that the two linear matrices involved are always the same we just have different right hand sides The recursion starts by computing E 0 and solves the electrostatic problem V x E 0 0 V cE 0 0 or VoVW 0 0 2 61 for the potential U 0 Since the domains Qn n 1 N are disjoint we actually have to solve N linear problems for the scalar potential V 0 in each domain 2 From 2 58 the Y 0 matrix of 2 57 is a diagonal matrix given by Fas f _oB 0 dn 2 62 The next step in the re
48. a scalar field O q is compensated by the fact that one has more flexibility and one can directly specify the current Jo This requires an element mesh fine enough to have a good approximation of the current Jo within Q but the previous Biot Savart approach by defining Jo 0 and Hp Hgs is still available This vector formulation is also the one required when computing approximations of the time dependent eddy current model since it requires the A field on supp Jo Mixing the scalar and vector formulation By choosing a domain Qyec with supp Jo C Qvec and assuming Qyec CC Q then on OQfree Q Qvec where the current is vanishing Jo 0 we do not necessarily have to compute the Biot Savart field Hgs In fact the reduced formulation can be replaced by computing the total scalar potential O with H V Otot and by solving the Laplace equation V LV Otot 0 2 47 Both the reduced formulation in Qyec and total formulation in free can be coupled and solved together by defining proper interface BCs on OQye Here however our major intent is to get rid of the Biot Savart field and therefore on Qy we do not solve for the reduced potential O q but for the vector potential A In doing this several questions arise e We need to define interface BCs on OQyec connecting the scalar formulation on Qfree and the vector formulation on OQyec e Since the domain Qfree is generally not simply connected the potential Otot is gen erally not
49. air cannot be neglected and that by rotating the disk this large error is also not constant Formula 2 20 is therefore a crude approximation and it is also expected that a 3D simulation will further improve the value of L considerably References 1 P KRITSCHKER T GAST Inductive Cross Anchor Detector with High Resolution Technisches Messen 49 No 2 pp 43 49 1982 2 J P BENTLEY Principles of Measurements Systems New York Longman Inc 1983 2 8 Modeling of a micro reed switch Micro reed relays with a magnetostatic operating principle were further miniaturized during the last years These electrical switches provide the galvanic separation of the control and load circuit so they are of special interest in micro system engineering SESES Tutorial September 2012 61 Figure 2 39 Schematic drawing of a micro reed re lay 3 Figure 2 38 Detail view of a micro reed re lay Components manufactured according to this process are 2 x 1 4 x 0 75 mm in size applications 1 Magnetostatic sensors are particularly valuable for portable applica tions because they require little energy or space they are sealed and can withstand high mechanical stresses The reed switches currently found on the market are made of two ferromagnetic overlapping metal blades placed in a glass tube 2 One of the blades is fixed to the substrate and the other one is free to bend down and to touch the fixed blade in order to close t
50. and is considered with this example Here the theoretical framework may get quite involved and the reader is referenced to 1 3 4 for additional back ground informations When modeling elasto plasticity at finite strain basically two different approaches are available The first one is based on a priori additive decompo sition of the strain rate into a plastic and elastic part combined with a hypoelastic law whereas the second one is based on a multiplicative decomposition of the deformation gradient into a plastic and elastic part combined with a hyperelastic law In the case of isotropy both formulations can be shown to be equivalent Historically the additive decomposition was the first to be developed and it is still widely used in commercial codes whereas the multiplicative decomposition is considered more precise and bet ter supported by physical arguments In the following we present the multiplicative approach for quasistationary elasto plasticity in the form proposed by 5 where in finitesimal plastic models can be reused without changes and just some wrapper code interfacing the numerical solver is different However the important assumption of isotropy has to be made here both for the elastic and plastic laws In addition we have a restriction on the form of the elastic law which however does not hold if one is ready to rewrite the infinitesimal plastic models Isotropic plasticity at finite strain Let Q C R be the domai
51. and lower shell surfaces are not easily modeled numerically The engineering community is therefore pursing another approach to shell modeling than the applied mathematical community which uses shell kinematic models The approach is based on the discretization of the native 3D non linear continuum formu lation and by trying to fix first or low order finite elements to avoid numerical locking This trend started with the reduced integration approach 11 and the mixed MITC plate elements 1 and has been further developed into what has become a solid shell approach These finite elements just use displacement degree of freedoms at the top and bottom of the shell from the external point of view they are the same as standard H Q solid elements but they are inherently anisotropic in the transverse direction which is assumed to be shallow The shallowness hypothesis is used to fix the various locking numerical behaviors The main advantage of this approach is that we can use the same 3D material laws and the same boundary conditions as for solid elements SESES Tutorial September 2012 183 _ a Y YPY wy Figure 2 95 Shell displacements for snap through and warp solutions Numerical example For the solution of mechanical problems SESES offers the choice between first and 2nd order solid elements and the solid shell elements of 9 all ones either for the ge ometric linear and non linear formulation and with generic 3D material laws
52. and only during the start up phase we have unsteady conditions This initial phase may be modeled as well but it is not done here As second we note that we have to consider a coupled thermo mechanical problem however due to the high tempera tures and the large amount of convected thermal energy the heat production due to mechanical inelastic deformations of the strand has little impact on the temperature and can be neglected Therefore the coupling is one directional one has first to de termine the temperature by solving the governing equation of energy transport and then with known temperature profiles one solves the governing equation of elasticity to compute displacements and stresses A correct computation of the temperature is important for the subsequent mechanical problem The most challenging part here are correct values for the thermal conductivity and capacity but otherwise the numerical solution of the convected thermal transport is robust and standard In the following we will not perform this step and instead we assume an unrealistic constant tempera ture of the strand Let us start with a Eulerian or spatial view of the casting problem as customary when dealing with steady flow conditions Momemtum conservation in spatial coordinates x Q with Q the domain representing the strand yields the governing equations pDiv Vstf 2 122 with p the mass density v the velocity s the stress f the body force and D e e V e
53. are generally solved iteratively based on the solutions of single lin ear steps Although for very large linear systems iterative methods may as well be used to find their solutions in the sequent we will consider the linear step as a black box and we will focus on the non linear iteration We start the discussion with the example example NonLin s2d and the simple linear 1D thermal problem for the temperature T x V KVT 1 with x 0 1 and T 0 T 1 0 The analytical solution for a heat conductivity of x 1 is simple T 27 2 2 2 A solution to this problem is computed with the statement Solve Stationary Before computing the solution we have added a print statement to improve the understand ing of the textual output Write Solving for the first time Solve Stationary The output for this solution step will be something similar to Solving for the first time AbsResid Temp 9 64e 02 After printing our label during the solution step the program writes the L norm of the residuals or out of balance values for the temperature equation we are solving for Since SESES identifies our thermal equation as linear a single linear step is performed By repeating the solution step a second time one obtains Solving for the second time AbsResid Temp 1 71le 16 Solving for the third time AbsResid Temp 2 10e 16 The values of the residual norm depends on internal scaling factors of the governing equations and so a pr
54. be found at example VolAverage s3d To run the cal culations in different modes some control parameters are introduced first The the flag permYN is set the effective permeability of the considered structure is calculated ap plying an external pressure drop of dpress 12 0 Similarly when the flag condYN is set the effective thermal conductivity of the considered insulation structure is cal culated applying the external temperature gradient that results from the difference between T Hot 600 and T Cold 350 202 SESES Tutorial September 2012 Define permYN 1 x permeability for isothermal system x condYN 0 thermal conductivity for stagnant system dpress 12 0 x given pressure drop in Pa T_Cold 350 x cold temperature in K T_Hot 600 x hot temperature in K T_ref T_Cold T_Hot T_Cold 2 x reference temperature in K x As next we construct a three dimensional cube as the computational domain repre senting the smallest repetitive part of the structure shown in Fig 2 147 To generate the tubular air channels we apply the homotopic function cy1zS which transforms a cube into a cylinder Two materials representing air AIR and the insulation material ISO are then defined and assigned to the computational domain see Fig 2 148 We next specify the physical parameters First a number of SESES routines are defined which provide the thermal conductivity of the insulation walls as well as the required fluid properties
55. calculate the hydrodynamic and the thermodynamic developing flow of the gas in the pipe As shown in Fig 2 138 we consider a pipe with a circular cross section and a diameter of d 0 0513 m and a length of L 0 70m The mass flow rate for a single pipe is m 0 73x10 kg s The properties of the flue gas are assumed to be those of air In this case with an absolute pressure of 1 bar at the pipe inlet the density of the air is p 0 2737 kg m which leads to an inlet velocity of v 1 29m s The fluid is entering the pipe with a top hat velocity profile The velocity at the pipe surface is zero and a boundary layer is developing as the fluid is moving through the pipe As the fluid passes through the pipe the boundary layer increases with increasing travel length After a certain length the boundary layer fills the whole pipe cross section The entering gas has a uniform temperature of Tg 1000 C The wall temperature is constant Tw 30 C A thermal boundary layer starts to develop and convective heat transfer occurs As the fluid flows through the pipe the physical properties of the fluid are changing according to the local temperature Thus the density increases downstream of the entry cross section of the pipe and as a consequence the mean velocity decreases With the gas properties at the inlet we calculate a Reynolds Re dv v 92 4 indicating a laminar flow regime The hydrodynamic entry length may be obtained from an expression of the
56. cations the Koiter model is used to a less extent since it requires finite elements for the Sobolov space H Q x H Q Here the main drawback is that finite elements with continuous derivatives as required by H Q are not easily constructed Instead the Nagadi model requires finite element approximations in the standard Sobolov space H Q Here similarly as for the discretization of the native 3D mechanical formu lation standard finite elements for H Q are subject to numerical locking However stable and locking free finite elements have been discovered for the particular cases of plates 7 3 and shells in the bending dominated state 4 This stability is asserted by proving the convergence rate to be independent from the thickness of the shell Even assuming that stable finite element based on the Nagadi shell model will be sometimes discovered thus paving the way for robust numerical simulations of shells their usage for practical applications is somewhat limited As first just the geometric linear case is generally considered so that for large displacements these linear shell elements need to be amended e g by a corotational formulation not free of disadvan tages see 8 Secondly the kinematic behind the Nagadi model assumes a vanishing transverse normal stress and no change in the shell thickness during the deformation As a result generic 3D material laws need to be amended for shells and mechanical contacts on the upper
57. constant T s the local crystal temperature on the surface 5 n the local normal to the surface and h SESES Tutorial September 2012 111 Thermally Loaded Crystal Thermally Loaded Crystal Plane Phase Front Phase Front etbbii OPD Integration Pathes Figure 2 67 OPDtn integrated along differ ent paths parallel to the laser beam axes as a function of the distance r to this axes Plane Phase Front Figure 2 66 Lens like influence of a thermally loaded laser crystal onto a traveling plane phase front the heat transfer coefficient 4 This boundary condition includes the cases of constant boundary temperature h oo insulated boundaries h 0 and convective heat transfer finite h Based on the computed temperature distribution T x the local displacement field u x together with the corresponding stress tensor is calculated As the local deformations are expected to be small the two fields T x and u x can be treated as uncoupled Optical Path Difference OPD The refractive index of a material depends on its temperature This effect is known as thermal dispersion and the change of the refractive index An as a function of the local temperature can be written as T On or To dT 2 90 n no An with To the reference temperature where the refractive index no of the material is de fined and On OT the thermal dispersion coefficient Because the speed of light in a material directly sca
58. deals with the actual computation The fields to be solved for are defined within a single Block and convergence criteria are set BlockStruct Block Pressure Velocity Temp Convergence AbsIncr Temp lt 1l E 8 amp amp AbsIncr Pressure lt 1 E 5 amp amp AbsIncr Velocity lt 1 E 10 1 nIter gt 15 failure too many iterations 0 As mentioned earlier the calculations can be performed in two different modes one where the temperature of the hot plate is raised in steps of dT with TVAR 1 and the other one where the system size is raised in steps of dlsys with 1VAR 0 For each mode separate instructions to control the calculations are defined to compute a stationary solution Solve Init Temp T0O Solve Stationary If TVAR ForSimPar Thot Step dT Until TO dT nMax Else ForSimPar lsys Step dlsys Until I1sys0 nMaxxdlsys For the analysis of the simulation results we add a number of post processing state ments to write the desired information to external files We first calculate the heat fluxes per area over the left and right boundaries the resulting heat transfer coeffi cient and Nusselt number as well as the Prandlt and Grashof numbers lm lsys x system dimension in m lt x A Im le 3 cross section in m 2 x qfluxC coldPlate Temp Flux A qfluxH hotPlate Temp Flux A alpha qfluxH Thot T0 Nu alphaxlm Kappa_AIR TO Pr Visco_AIR T0 Cp_AIR T0 Kappa_AIR T0 Prandtl number Gr gA
59. elastic e and plastic e part there is an energy function e depending on the elastic strain e and some additional inter nal strain variables specifying the stress s e and the internal stress variables q W together with a yield function f s q lt 0 constraining the stress variables s q and where the flow of the plastic strain variables is defined as follow If f s q lt 0 holds we are in the elastic region and there is no plastic flow 0 otherwise the plastic flow must maximize the plastic dissipation s q This maximum principle results in a yield function f being necessarily convex Although quite abstract this problem formulation is mathematically well sounded and exis tence uniqueness properties of solutions can be proven for some models 5 The SESES Tutorial September 2012 137 above formulation is equivalent to the following Kuhn Tucker equations where the independent variables are e NO f Og f with fA 0 A lt 0 2 103 This is clear since if f lt 0 we must have 0 and so there is no plastic flow otherwise if f 0 the above equations are just the classical Lagrange equations for the maximum of a constrained functional Let us apply the backward Euler algorithm to the eqs 2 103 by performing a single step The starting point is therefore a time t with a know configuration of the in dependent variables en h En and a time
60. following partial derivatives O c lt Hp are required SESES will then use in ternally these derivatives to solve the residual equations RY Re 0 associated with the degrees of freedoms using a Newton Raphson algorithm with exact derivatives 0 R R amp A i En In order to keep the same notation we give here directional derivatives D e a with respect to a vector a As first we note that since dev ktr Id 0 we have 0 fn 0 fn and then by considering the property n dev a n a after some algebra we obtain Desla 2pdev a Des a ktr a Id 2udev a D fla De fla 2un dev a 2un a Da nla Denfa 2 22 ea al nDefla fDenlal Numerical model The numerical example of a 2D the steel casting machine can be found at example Casting s2d The mold gives the strand an initial curvature and rolls are placed so that the strand makes a turn of 90 degrees then straighten and proceeds horizon tally see Fig 2 106 This shape defines the computational domain Q1 Without me SESES Tutorial September 2012 161 Stress Trace Pa 4 36E 08 3 94E 08 3 52E 08 3 11E 08 2 69E 08 2 27E 08 1 86E 08 1 44E 08 1 02E 08 6 08E 07 1 91E 07 2 26E 07 6 42E 07 1 06E 08 1 48E 08 1 89E 08 Mold Support 2 31E 08 Figure 2 106 The computational domain with mechanical contacts Figure 2 107 The stress free referential do main obtained by turning off all mechan
61. for the stationary rather than unstationary solution to be computed Since we solve for the single dof field Phi and the problem is linear no other statements are required to compute a solution Lastly we can export the computed dof field Phi and the electrical field Ef ie1d obtained from the relation E V to the graphics data file Data with the statement Dump Phi Efield In summary we have generated the following input container file and once read by the Front End program it will display the picture in Fig 1 6 cat gt Seses lt lt EOF aterialSpec Air Equation ElectroStatic Parameter EpsIso 1 QMEI 2 1E 6 QMEJ 2 1E 6 inimumRL 3 BC Tip 1 0 JType 1 Dirichlet Phi 1 V BC Substrate 0 2 IType 2 Dirichlet Phi 0 V Finish Solve Stationary Dump Phi Efield Finish EOF k2d SESES Tutorial September 2012 15 M EfieldPhi V m Nom Phi 1 0E 00 26E 06 95E 01 25E 06 9 0E 01 24E 06 85E 01 22E 06 8 0E 01 21E 06 75E 01 2 0E 06 7 0E 01 1 9E 06 65E 01 1 7E 08 6 0E 01 1 6E 06 55E 01 1 5E 06 5 0E 01 1 3E 06 45E 01 1 2E 06 4 0E 01 1 1406 3 5E 01 94E 05 3 0E 01 8 1E 05 25E 01 69E 05 2 0E 01 56E 05 1 5E01 43E 05 10E01 3 0E 05 5 0E 02 1 7E 05 3 0E 18 3 7E 04 time 0 time O Figure 1 7 Calculated electrostatic potential and electric field upon application of a bias of 1 V between tip and substrate The electric field is displayed with absolute values and arrows indicating the direction A simulatio
62. form see 1 len iam 0 05 Re 5 2 164 This leads in our case to a hydrodynamic entry length of xe 0 237 m The thermal 192 SESES Tutorial September 2012 1200 24000 T SESES T VDI W meatlas q SESES 1000 q VDI Warmeatlas 20000 800 16000 Tw 30 C E a Sy Air 3 600 1200 5 t 1000 C gt A T v 1 29 m s gt T T 400 8000 LL Wa L 0 7m 200 4000 Figure 2 138 Sketch of the pipe 0 o o 0 1 0 2 03 0 4 05 06 07 y m Figure 2 139 Temperature distribution and heat flux along pipe entry length may be expressed as see 2 le Sam 0 05 Re Pr 2 165 where Pr denotes the Prandtl number Here the entry length is 0 171m The Prandtl number is defined as ve here v cp and denote the dynamic viscosity the heat capacity and the thermal con ductivity respectively The decrease of the mean temperature due convective heat transfer to the wall can be calculate using the equations given in the VDI Warmeatlas 3 The convection heat transfer coefficient can be computed using the Nusselt num ber Pr 2 166 d Nu c 2 167 where a and d denote the convection heat transfer coefficient the thermal conduc tivity and the pipe diameter respectively According to 3 the Nusselt number is computed as Nu Nuj 0 73 Nuz 0 78 8 Nu 2 168 with d 2 d
63. formulation only decreases from left to right and so the computational cost but the whole domain as well as the finite mesh and Jo are identical in all five cases The first case uses a full vector formulation Q Qyec without scalar magnetic potential The second and third cases use a cladding layer around supp Jo The second and the fourth cases enlarge the domain Qe to be simply connected The third and fifth cases do not have a simply connected domain Qfree and require a jump definition for the magnetostatic potential The value is available as BC characteristics from the solution of the electrostatic prob lem and is stored in the global variable bcCurrent i specifying the jump BC value by the statement For i From 1 To NWIRE Write Current 0f e A n i bcCurrent i bcOne i Psi Flux Fig 2 56 shows the magnetic induction around Nye and on the wire s interior Since we are in empty space the magnetic induction is continuous and so for a more precise representation we have used a smoothing procedure averaging the contribution from neighbor elements All five cases yield very similar results Fig 2 57 shows the z component of H along the symmetry axis through the wire s center Here one sees that the most precise formulation is the full vector formulation which is also the most expensive to compute when using a direct solver We end these numerical experiments by noting that among all five models just the first one resu
64. in a reason able amount of time is fundamental to the creation of fully integrated on chip micro electromechanical fluid processing systems Effective mixing requires that the fluids be manipulated or directed so that the contact area between the fluids is increased and the distance over which diffusion must act is decreased to the point that complete mixing is achieved in an acceptable amount of time In macroscopic devices this is generally done by using turbulence three dimensional flow structures or mechani cal actuators As MEMS devices are fabricated in a planar lithographic environment design constraints mitigate against mechanical actuators or three dimensional flow structures Instead innovative static mixing concepts are pursued to increase the mix ing efficiency The mixing efficiency of the classical T junction mixer 2 3 for exam ple can be increased using the multiple splitting of streams 1 or through the use of slanted wells 4 Figure 2 113 Mixing of a fluid of species A with a fluid of species B in a straight channel This tutorial discusses the ability of SESES to accurately predict mixing phenomena in microfluidic devices by considering the simplest possible design of a static mixer i e a straight channel with two inlets as shown in Fig 2 113 where a fluid of species A is mixed with a fluid of species B Note that the transport of matter in the flow direction is mainly by convection whereas diffusion is the tran
65. in the crystal This may lead on the one hand to mechanical cracking of the crystal and on the other hand to stress induced birefringence All these effects have a strong influence onto the behavior of a laser device by reducing the output power and the beam quality Methods have to be found to reduce these ef fects as e g adequate resonator design improved cooling techniques or compensation by external elements Therefore a laser engineer must have the possibility to quantify the influence of different technical improvements onto these effects for estimating the laser performance during the development process Unfortunately analytical solutions only exist for some special cases as uniform pump ing 1 For real devices we have to resort to numerical methods With SESES its pos sible to deduce both the distribution of temperature stress strain and crystal defor mation and to quantify its influence by calculating the Optical Path Difference OPD and the birefringence To start a numerical simulation we have first to know the pump power distribution p x which acts at the same time as heat source in the crystal To get the temperature distribution the steady state heat transfer equation has to be solved by considering the cooling boundary conditions at the crystals surface Here we are concerned with Newton s law of heat transfer OT k hWo T s 2 89 On g with the heat conductivity Tc the coolant temperature assumed to be
66. induce a cur rent flow The silicon device is properly doped to act as a resistor On their way through the device the charge carriers are deflected in an external magnetic field by 40 SESES Tutorial September 2012 DC current Jo Silicon Vott Votz Halit Hal2 Figure 2 6 Geometry and contacts of the O Highly doped thin region modeled Hall sensor O Active sensing region Figure 2 5 Hall sensor layout with Jo sensor cur rent B outer magnetic field in z direction and Unan measured Hall voltage the Lorentz force and accumulate on the lateral sides By placing two contacts on the lateral sides a potential difference can be measured called Hall voltage The lateral Hall contacts are usually realized with highly doped thin adjacent regions to achieve optimum Ohmic contact behavior The modeling of our Hall device starts by posing the problem in mathematical terms Physical models of varying complexity and de gree of sophistication are available Here we use a simple single carrier model able to correctly characterize the device electrically The semiconductor drift diffusion model or even more complex models may be used but do not necessarily improve the results for this Hall device The single carrier model is based on the solution of the current conservation law for the current density J V J 0for xe 2 1 The material law for charge transport is given by _ WWhtetMe 1 pe B with qo the elementary charge the
67. is unused we can choose asym Lp 0 in order to get sym C Lp C Lp and the property of C Lp 0g f being three coaxial matrices The push forward of 2 112 gives lp Of r and if the yield surface is just a func tion of the deviatoric stress dev 7 with dev e e Idtr e 3 the deviator operator then we have f r dev f dev r Odev r and so tr Dp Atr Of d7 0 For a matrix A we have the relation Odet A 0A det A ATT and so the rate of plastic volume is given by J d dt det Fp Jtr EpF5 JPtr Lp J tr Fe Lp Fe J tr lp J tr Dp 0 showing that yield surfaces of the form f dev 7 have an isochoric plastic flow which is typical for metal plasticity Before proceeding further we have to decide which internal variables to use and as first we note that since the implementation is working with the displacement u x X as unknown the total deformation F Fe Fp 0x 0X is given as well as C FT F In 2 112 the time derivative is directly associated with the internal variables so that it is natural to store and update instead of q For the plastic flow the natural choice is to use Fp so that together with Lp Fp F we have of F os P By considering the pull back 0f OS F Of r Fe and the relation F F d dt F Fp we also have Fp A0 2 113 d dt Fp F 5 am F Fz Another possible choice as internal variables for the plastic flow
68. of air Note that for material AIR the mass momentum and energy balance need to be solved whereas for ISO only the energy balance is required average heat capacity of air for conduction Define Cp_AIR_ref CpITemp_AIR T_Hot CpITemp_AIR T_Cold T_Hot T_Cold global specifications GlobalSpec Parameter AmbientTemp T_O 293 15 K x air channels gt MaterialSpec AIR Equation CompressibleFlow ThermalEnergy Enable Parameter FlowStab Zero Parameter Density IdealGas AmbientPress le5 Pa STIunit Parameter Mmol M_AIR kg mol Parameter Viscosity Viscosity_AIR Temp Paxs Parameter PressPenalty le 5 s Parameter KappaIso D_Temp Kappa_AIR Temp KappaDTemp_AIR Temp W Kxm W m K 2 Parameter ThermStab zero Parameter ThermConv Velocity Cp_AIR_ref Density Val W Kxmx 2 insulation material x MaterialSpec ISO Equation ThermalEnergy Enable Parameter Density Val 1 STunit Parameter KappaIso D_Temp Kappa_CER_DURATEC Temp KappaDTemp_CER_DURATEC Temp W K m W m Kx 2 Before we can start our simulation a complete set of boundary conditions must be defined Depending on the mode in which we run the simulation as specified through the flags permYN and condyYN different sets of boundary conditions are used The boundary conditions BC_fluidi and BC_fluid2 specify the inlets and outlet of the air flow respectively In addition when thermal simulations are per formed condYN
69. of natural convection can be found in living rooms where the air cools down at cold surfaces mainly at the windows Due to the buoyancy forces the cold air which is heavier than the warmer air in the room falls off to the floor On the other hand the cold air is heated up again at the warm inner walls and at the hot surfaces of the heating radiators The interaction of heating in the center and cooling at the windows produces an air circulation in the room Human beings perceive this air movement as a draught Such a circulation can be found in any simple cavity where one side is heated and the other side is cooled The flow pattern which is established can be seen from Fig 2 118 The aim of this tutorial example is to develop a sim ple FE model which reproduces natural convection flow patterns similar to the one shown in Fig 2 118 To set up the model consider the two dimensional cavity shown in Fig 2 119 The cavity is heated from the left side with a temperature of Thot 310 K and cooled from the right side with Teo 300K The width and the height of the cavity are chosen to be the same i e h s 50x107 m To analyze convective heat SESES Tutorial September 2012 175 insulation insulation Figure 2 119 Geometry and boundary con ditions for the air filled cavity on which the presented SESES model is based width s Figure 2 118 Experimental investigation of 59 10 3m height h 50x10 3m and Ra the
70. of the refractive index An as a function of the temperature fol lowing 2 90 Here we assume the thermal dispersion 0n 0T to be a constant so that 2 90 can be rewritten as An T To On OT Next we define the Nd YAG material as inheriting all properties from YAG and by additionally defining the heat source The routine Pump is a user routine defined following 2 94 2 95 and 2 96 The first call is for the pumping from the left side and the second call for the pump beam from the right side of the rod We then build the macro element mesh and map two materials onto the macro element mesh The thermal and mechanical boundary conditions are then next The front and the back surface of the rod are in contact with air defined by the boundary condition BC Air The cylindrical surface of the rod is assumed to be in direct contact with the cool ing water define by the boundary condition BC Water As we have assumed a radial symmetric situation the longitudinal axis of the crystal is automatically fixed for de formations in the radial direction Further we assume the rod to be free to expand Therefore we have only to fix one point of the rod for displacements in the longitu dinal direction Therefore we fix the point in the center of the rod by the boundary condition BC Fixed 0 nytot 2 JType 0 Dirichlet Disp Y 0 m The command section of the input file starts with some settings concerning the used solver and desired information on the outpu
71. or stress power in the spatial configuration is known to be P s 1 s sym l By just considering the plastic part with respect to the volume of 2 we have PP Js lp Jtr Fe S FT Fe Lp F3 8 C Lp C 5 Lp By the assumption of isotropy from 2 109 we see that the tensors C and S are coaxial therefore they commutes and the matrix C S C C 8 is symmetric So for isotropy the asymmetric part asym Lp Lp L3 2 does not dissipate and does not need to be specified At this point we proceed as for the infinitesimal case by ex pressing the plasticity law with respect to the stress free configuration 1 We assume there are some internal variables q and related by a phenomenological law q q and that together with S they uniquely determine the yield condition f S q 0 with f a phenomenological convex yield surface By considering the model of associative plasticity based on the principle of maximal plastic dissipation which is a generally accepted model for hardening materials without softening the evolution of the plastic strain sym C Lp and internal variables is normal to the yield surface and we have the Kuhn Tucker equations of sym C L p Ase aa AZO f a lt 0 Afa 0 2 112 with the plastic multiplier to be determined by the additional consistency condition F S q 0 Since for isotropy C and S commutes similarly we can obtain sym C L dgf and since a
72. our example is activated after three iterations with full factorization The implementation of the model Coupling1D is optimal in the sense that for linear systems just a single linear step is required for the numerical solution The price to pay is that the dof fields on both the cathode and anode domains become matrix connected and the cost for a linear solution step increases accordingly This may be a necessity for strongly coupled point to point domains to assure acceptable convergence rates but in our case the coupling is weak and we may neglect some matrix connections between the dof fields on both domains with the options NoRow NoCol of the Coupling1D model Note that the coupling function is left unchanged we are just discarding cross coupling derivatives so that the convergence rate of New ton s algorithm will further slow down but each linear step will be cheaper A numer ical investigation has shown that we can freely neglect all cross coupling derivatives except in some circumstances for the electric potential This is indeed the case if we use a floating BC for the potential on a domain in order to set the total current flow since by neglecting the potential cross coupling derivatives the stiffness matrix be comes singular However we also see that since the nopples are quite close together and the electric conductivity is large enough the potential value at the cathode and anode GDLs is almost constant and its computation is n
73. predictor corrector technique When a family of solution is computed with respect to a parameter and the dependency is smooth one can use a Taylor expansion to approximate a solution for a new value of the parame ter This predicted solution is then corrected by starting the non linear solution algo rithm Since we are closer to the solution less iterations are required and a speed up is obtained Actually Taylor expansions are not practical to use since they require the derivatives of the solution with respect to the parameter Therefore they are com monly replaced by polynomial or rational extrapolation close related to Taylor ex pansions and continued fractions By repeating the previous solution and enabling extrapolation we effectively perform less iterations Write Computing predicted solutions n Solve Init Extrapolation Rat_2_2 Solve Stationary ForSimPar load At 0 Step 1 2 10 SESES Tutorial September 2012 33 In this example we have used quite a high degree of rational extrapolation in order to reduce the computational cost at the expense of increased memory requirements needed to store the computed solutions However there are many examples where high degree extrapolation is not better than low degree and can even be detrimental and this is especially true for polynomial extrapolation The pressure load of load 5 is not large enough to reach the snap through state therefore we may decide to increase further the load an
74. resulting velocities from this computation are written on a regular grid to files which are then imported into SESES and used in combination with the diffusion reaction and heat transfer calculations performed in SESES A one way coupling is considered here i e no in formation of the SESES computations are fed back to CFX 5 Use is made of the axial symmetry to set up the geometry of the reactor The Reynolds number of the free flow in the open channel is small hence the flow regime is laminar Due to the no slip con dition at the interface of open channel and porous region the velocity at the interface 196 SESES Tutorial September 2012 Air t 300 C vO0ms a p bar gt nM XZ A LE ML Burning Gas T 300 C x H2 0 02 x H20 0 98 Figure 2 142 Sketch of the single channel of a catalyst converter 8 00E 02 0 00E 00 2 00E 03 4 00E 03 u Velocity m s a F 8 w Velocity m s 3 00E 02 x 0 0m 6 00E 03 X 0 005m x 0 010m V x 0 015m e x 0 025m x 0 050m s x 0 10m e x 0 15m 7 2 00E 02 1 00E 02 0 00E 00 8 00E 03 0 00E 00 5 00E 03 1 00E 02 1 50E 02 2 00E 02 0 00E 00 4 00E 03 8 00E 03 1 20E 02 1 60E 02 2 00E 02 Radius m Radius m Figure 2 143 Velocity profiles in the left main and right radial flow directions at locations along the ope
75. rr This solution is obtained by first computing the linear mechanical solution without plastic rates and then with a coupled algorithm by suddenly turning on the plastic rates The plastic strain is always a continuous field due to the artificial diffusion introduced when solving the convected transport equations with H Q con formal finite elements Fig 2 109 shows the trace values of the stress At the begin ning of the straight zone we have singular stress values which are then convected and smoothed down wind For a more realistic model we have implemented the plastic rate as presented by 1 This model is solved by setting the variable MODEL S IMPLE to zero and it is apparent that the solution is now much more hard to compute The plastic rates are slowly turned on by the homotopic parameter HOMOT and the solu tion is found at HOMOT oo The step length of HOMOT is chosen adaptively but due to the large and increasing system condition the homotopic process stops long be fore This is an example showing that singular stress values generate very stiff and ill conditioned problems SESES Tutorial September 2012 163 Inlet L Figure 2 110 A rectangular flow channel References 1 S Koric B G THOMAS Efficient thermo mechanical model for solidification pro cesses Int J Numer Meth Engng Vol 66 pp 1955 1989 2006 2 25 Hagen Poiseuille Model of Viscous Laminar Flow To introduce the basics of fluid flow we presen
76. s algorithm for the solution of the residual equations Rg 0 for all species converges quadratically dropping the computation of one species and solving just for Rg Za ZN 0 with a 6 No and by applying the off block dof field correction for Zn generally leads to slow and or non convergence of Newton s iteration The reason is that if we do not solve for y the native derivatives aR OX are not ex act derivatives anymore since we are missing terms stemming from the dependency ty 1 Yy Zaw in R Za Zy For user defined derivatives within material laws the user may amend these values to obtain exact derivatives however this is not pos sible for the internal derivatives computed by SESES Although an ideal quadratic convergence of Newton s algorithm will be unavailable the situation is not so dra matic if we can still get good convergence rates In order for this to happen we have to reduce to a minimum the dependency of the residual equations R Za Zya from the x mole fractions So the first step consist in the choice of the most inert species N as redundant species which has the minimal impact on the reaction rates The second contrivance is a little more technical and has to do with the Stefan Maxwell diffusion law used for the transport process Within this law there is cross diffusion among all species and the diffusion matrix Dag is in general a full matrix Therefore the transport equation for each species is st
77. since it grows linearly with the number of elements and therefore for large problems the linear solver time will always dominate the overall solution time References 1 K J BATHE Finite Element Procedures Prentice Hall International 1996 2 M BERNADOU Finite Element Methods for Thin Shell Problems Wiley 1996 3 D CHAPELLE R STENBERG An optimal low order locking free finite element method for Reissner Mindlin plates Mathematical Models and Methods in Applied Sci ence Vol 8 No 3 pp 407 430 1998 4 D CHAPELLE R STENBERG Stabilized finite element formulations for shells in a bending dominated state SIAM J Numer Anal Vol 36 No 1 pp 32 73 1998 5 D CHAPELLE K J BATHE The Finite Element Analysis of Shells Fundamentals Springer 2003 6 P G CIARLET T LI EDS Differential geometry theory and applications Series in contemporary applied mathematics CAM 9 World Scientific 2008 7 R DURAN A GHIOLDI N WOLANSKI A finite element method for the Mindlin Reissner plate model SLAM J Numer Anal Vol 28 No 4 pp 1004 1014 1991 8 C A FELIPPA B HAUGEN A unified formulation of small strain corotational finite elements I Theory Comput Methods Appl Mech Engrg Vol 194 pp 2285 2335 2005 9 S KLINKEL F GRUTTMANN W WAGNER A robust non linear solid shell element based on a mixed variational formulation Comput Methods Appl Mech Engrg 195 pp 179 2
78. small constant in the diagonal before performing the incomplete factorization This amendment should be done on the diagonally scaled matrix i e after diagonal preconditioning and the value of 0 05 seems to be an optimum How ever the overall performance and robustness is poor but get better with increasing frequency The really bad performance at low frequencies may have its origin on a light inconsistent right hand side which is generally the case for w 0 and a given stiff current This failure should be further investigated and a consistent right hand side should be enforced by finding a field Hp with V x Ho Jo and using the term V x Ho in the dicretization instead of the stiff current Jo 4 The good numerical results obtained for first order nodal elements and the ILU 0 preconditioner can be again used to precondition the system matrix obtained for edge elements with the auxiliary preconditioning method exposed above The only dif ference is that we do not change the order of the finite element approximation but the type i e from edge elements to nodal elements The ingredients are all the same just the prolongation matrix is different and here II represents the interpolation be tween nodal and edge elements This auxiliary preconditioner suffers as the previous auxiliary one whenever we have badly shaped elements and has the same poor per formance at low frequencies as the above ILU 0 preconditioner with diagonal shift We have us
79. solving the complex linear system of equations with the conductance matrix G N Oyu hiara iN 2 35 j l and effective resistance and inductance are computed as N N R RV wl I gt 3a 2 36 i 1 4 1 Numerical Results In a first example example Skin s2d we compute the internal resistance and in ductance of a straight circular copper wire for a frequency range of 0 25kHz The skin effect will limit the current approximately to a layer of thickness 6 which need to be resolved by the computational mesh For copper we have o 56x 10 s m result ing in 0 4mm at 25kHz Having defined the geometry and a fine enough mesh near to the boundary we enable the equation Eddy Harmonic for eddy harmonic anal ysis The driving voltage is applied by defining the external current Current0 Zas Jo oo within the wire At the wire boundary the current should be Jo at any frequency resulting in homogeneous Dirichlet BCs for the dof fields Az and iAz We define the frequency as a simulation parameter frequency and compute several so lutions in our frequency range with the statement Solve Stationary ForSimPar Omega Skip Step 1 Omega 0 95 Until 25000 2 PI At each solution step we compute the total current 2 31 resistance and inductance after 2 32 which are plotted in Fig 2 45 together with the analytical values 2 33 Write AtStep 1 f e Hz I1 9 2e 9 2e 39 2e A R 39 2e Ohm L 9 2e H n double c 2 integrate
80. step At where we use the subscript n to specify a time function evaluated at tn With the backward Euler algorithm the solu tion for f 11 n 1 at tn 1 tn At is computed by performing a single time step of length At and using the time derivative evaluated at t 1 From 2 103 we therefore obtain the system of equations j Eh AdOs fn 1 En 1 Ent Ad0q fini Sny Ogee W Ensi Eh p1 n41 Qn4 1 Y Enq1 B Cnt En 1 2 104 fansi lt 0 fnti Ar 0 AX gt 0 with fn 1 f Sn 1 qn 1 and AA AT The value of the strain En11 at the new time t 41 is considered know and determined by incremented solution u 1 The common procedure used to solve the system 2 104 is first to assume that the plastic flow does not change i e AA 0 and from 2 104 we obtain the trial solution ae a k En 1 En Sn 1 Oce U En 1 Eh n gt GQn 1 Qn 2 105 The important point here is that this solution satisfies fn 1 lt ftria which is a property following from the convexity of the yield function f see 3 4 Therefore since fr41 lt 0 if ftri lt 0 we must necessarily have AA 0 and our trial solution 2 105 is actu ally the right solution at t 41 otherwise we know that AA gt 0 and the unilateral con straint of 2 104 turns into simple a constraint and we have to find e 419 n41 Sni Gn 1 AA so that e Eh Ad s fn 1 En 1 Ta En Aq fn 1 Sn 1 Oee V En 1 gt eb i E
81. the analytical description is summarized by the following equations U x t Uls da t RJ a t de L222 4 H R418 2 12 ee ae Gu c with U J the voltage and current along the wire and R L C the resistance induc tance conductance and capacity pro unit of length of the wire Combining 2 12 to gether we arrive at the following equation of telegraphy for the voltage U oU oU oU U RC LG LC Sox RGU RC LG LO The solutions are damped harmonic waves propagating along the cable with the damping constant 7 R iwL G iwC and impedance Z w U w J w y R iwL G wl 2 13 In this tutorial example example CrossTalk s2d we are going to compute the capacity C and the inductance L for two ideal cylindrical parallel wires and compare the values with analytical results In doing this we also briefly present the theory of transversal modes in transmission lines The conductive cross talk is not considered since we assume perfect electrical insulation between the wires i e G 0 The ideal case is characterized by ideal conductors with R 0 where the current is flowing over the surface of the wires Therefore the current flow in the wires does not need to be considered and the damping caused by the physical non zero value of R does not enter the computation of L and C For this ideal case we are therefore left to solve the Maxwell s eqs in the free space surrounding the wires wher
82. the homogeneous BCs we obtain Un 4 sin Anz Pns 7 2 sin n z sin An L with n n 1 o the positive solutions of the equation E 1 An L tan L 0 The constants A are choosen so that the functions are orthonormal lt un um gt Onm and so A 1 lt sin A z sin Anz gt The inhomogeneous solution with BCs u 0 0 s L 0 0 0 and L 1 is given by T as so that by considering zero damping the solution 2 79 is given by oo 2 u _ ug W lt U0 Un gt Un iwt tao wa CS _ ai and the impendance by Vget t L g co aye 1 which is the same as 2 86 although this is not easily seen except for the case w 0 Appendix We want to prove here the equivalency of a w2A We first note that if we solve for V F 0 in Q using a Dirichlet BC on 0 and natural BC otherwise then we have JgVf FdV 0 for any function f zero on 0p since fao fF dn 0 From 2 78 we are actually solving V C e uo g Eo 0 and V e Eo g e uo 0 and therefore for f u we have f e u C e uo g Eo dV 0 and for f bn we obtain fo En Eo g uo dV 0 Because o 1 on OM and V e E g un 0 on Q we can write a Jan E E g e u 9 dn fo V e En 8 Un d0 dV fo En g e un EgdV fo En g e uo e un C e uo dV fo s un e up dV w2 lt un Up gt w2 An References 1 N Guo P CAWL
83. the material param eter Mue requiring the relative permeability p 9 H B H and its logarithmic derivative H Op 0H The relay is embedded in a homogeneous external magnetic field with only a z component and so on the bottom face of the rectangular domain we set the magnetic potential to zero and on the top face we prescribe the magnetic field B which may be done with either a Neumann or a Floating BC To compute the force on the relay with the surface integral 2 27 we define the boundary of the bendable lamella with the statements BC SurfaceReed Restrict material Vacuum OnChange material ReedL BC SurfaceCheck Restrict material Vacuum OnChange material Vacuum It is important to use the Restrict option to only select the boundary part external to the lamella since it is in the vacuum region that we want to perform the numer ical integration The second boundary SurfaceCheck selects the whole vacuum s boundary and it is used to check the accuracy of the surface integral We end up the Initial section by defining postprocessing routines for computing the Maxwell s stress tensor and magnetic energy density In the Command section we define the solution algorithm and post processing tasks The ferromagnetic material law is non linear and so we set up a non default conver gence criterion to stop Newton s algorithm and to speed up the computation we reuse SESES Tutorial September 2012 65 the factorized linear system f
84. the medium is isotropic we then have B No Td with no the refractive index Under a deformation given by the strain e the dielectric impermeability is changed due to the photoelastic effect according to 6 B B P e 2 99 with P the photoelastic tensor To investigate the depolarization effects of strain in duced birefringence we follow the change of a plane wave Eo z exp i wt kz traveling in the z direction which to a first order is given by OEo z 2ri as eee ngId 2 100 zZ This equation is then integrated along the ray path and this is done by the post processing routine ApplyPo1 for a small increment of dz The observation of the depolarized part of a linear polarized plane wave traveling through a pumped rod is one possibility to analyze stress induced birefringence The SESES Tutorial September 2012 121 Top view Side view Figure 2 81 Top and side view of the crystal lattice axis Figure 2 82 Macro element mesh a b and c in case of a 1 1 1 cut cubic crystal after application of the cylz routine set up is schematically shown in Fig 2 79 A horizontal polarized plane wave enters the pumped rod Inside the rod the polarization is locally disturbed After passing the rod a vertical polarizer transmits only the depolarized part of the wave which is observed on the screen With SESES the photo elastic effect can be calculated and an additional post processing leads to the pattern of
85. the solution algorithm and the operating principle we choose a simple 2D model of the sensor geometry and solve the heat transport equation only I e rather than solving a fully coupled thermo fluidic model we shall solve the heat transport equation only and prescribe the fluid flow independently as a parabolic velocity profile Ay r 2 157 v v 1 where vo is the maximum speed at the center y 0 of the channel and d the height of the channel Equation 2 157 is the Hagen Poiseuille law for viscous laminar flow see also the example on page 163 This simplification is justified whenever the tempera ture rise is relatively small and thus not changing the fluid properties themselves Let us discuss now the key parts of the input fileexample FlowSens s2d provided for this example The geometry of our sensor is comprised of a horizontal fluid chan nel a medium above pyrex and a medium below plexy glass see Fig 2 123 The temperature sensors and the signal analysis electronics are both located on the upper side of the thin pyrex plate For the fluid flow we define the material Fluid solving for the convective and diffusive heat transport MaterialSpec Fluid Equation ThermalEnergy ElmtFieldDef veloc DofField Temp ForMaterial Fluid 2 MaterialSpec Fluid Parameter Kappalso 0 585 W Kxm Parameter Density Val rhoFl kg mx 3 Parameter Enthalpy cpFl Temp cpF1 0 J kg J Kxkg J Kx 2 kg Parameter ThermConv Enthal
86. trated in Fig 2 145 where the reactor temperatures are shown for for three differ ent thermal conductivities of the porous material The lowest temperature of 300 K is at the inlet shown in dark blue As expected the small thermal conductivity of k 0 1 W mK leads to a hot spot within the porous region with a peak temper ature of about 490 K shown as the light yellow region As the thermal conductiv ity increases the temperature distribution within the reactor becomes more uniform The concentration distributions of O2 N2 H2 and H20 along the reactor are shown Figure 2 145 Temperature distribution along the reactor for three different thermal conduc tivities of the porous material a x 0 1 b x 1 0 and c x 5 0 W m K in Fig 2 146 Note that the distributions of oxygen and nitrogen look very similar This can be explained by the very small hydrogen concentration i e for hydrogen burning only small amount of oxygen are required Due to this excess in oxygen the distributions of O2 and N gt are primarily influenced by the flow conditions and the gas diffusivities However the distribution of H is different in that the Hy conversion which happens mainly at the boundary between the free and porous regions creates a strong H flux in the radial direction Figure 2 146 Concentration distributions along the reactor for a O2 b N c H2 and d H20 References 1 CFX 5 CFD Tool ANSY
87. uV Orea UV 90 V UVOtot O for XE tot V uVOrea Ho O for x Orea and since the Hp field is generally the Biot Savart field of 2 40 the potential Oo is also a well known function of the problem Now let us drop the subscripts red and tot on the magnetic potential O then we have to solve the equations VuVO 0 for x Mtoe V uVO pHo 0 for x rea which are equivalent to the previous ones if at the interface boundary OQinter Qrea N Piot we assume the magnetic potential is a double valued function and undergoes a jump of Oo expressing the difference between a total and reduced formulation The point here is that from a numerical point of view is quite simple to apply a well de fined jump on an internal boundary when computing a scalar potential and in doing so we can easily combine the total and reduced magnetostatic formulation For our analytical example we have Op z and this value can be used as a jump value on the sphere x r if one want to solve for the total formulation within the ball and a reduced one outside When defining a jump BC we can decide on which side of the jump the dof values are defined so that on the other side their values are obtained by adding the jump value Since within the sphere the magnetic potential tends to be flat it is better to define here the dofs which in our example is done by BC jump Restrict material Sphere OnChangeOf 3 4x material Sphere tmaterial Air In a ge
88. vector normal to the cylinder and e the unit vector in x direction which coincides with the main stream direction In order to compare the results from the computations and the experiments the drag coefficient per unit length of the cylinder 2F 2 2 163 Cd pu d SESES Tutorial September 2012 191 is calculated and the results for both Reynolds numbers Re 23 and Re 45 are plotted in Fig 2 137 together with results from CFX TASCflow The coefficients of both simulation are somewhat bigger than the experimental data from literature This is again due to the channeling effect of the lateral BCs as explained above References 1 M M ZDRAVKOVICH Flow around circular Cylinders Vol 1 Fundamentals Oxford University Press 1997 2 S TANEDA Experimental Investigation of the Wakes behind Cylinders and Plates at Low Reynolds Numbers Journal of the Physical Society of Japan Vol 11 No 3 1956 2 31 Developing pipe flow with heat transfer In this example we consider the heat transfer in a pipe as it occurs in a heat ex changer or a boiler In the case of a household boiler hot flue gas from a gas or fuel oil burner passes through a number of vertical pipes The water to be heated moves freely around the pipes The entry temperature of the flue gas is around 1000 C The pressure of the gas is assumed to be equal to the ambient pressure We assume the wa ter temperature to be 30 C The goal of this tutorial is to
89. we are going to use the same model of the previous tutorial example with some adaption towards non linearity The 1D species transport within the GDLs as well as the oxygen ions O transport in the electrolyte are left unchanged For the potential jump at the anode electrolyte interface we now use a non linear model 2Reasl AW AUNernst ta S asinh jF 2a 2 181 qNavo with j the positive current flow q the elementary charge Navo the Avogadro s con stant and where the Nernst potential UNernst Yq and the function F xa are now de pendent from the species concentrations a at the interface The asinh dependency SESES Tutorial September 2012 219 of the overpotential from the current j stems from the Butler Volmer electrochemical reaction model with an equal symmetry factor for anode and cathode and it is a gener ally accepted model We also use the same model at the cathode electrolyte interface to model the potential jump for the oxygen s reduction For this simplified 1D sys tem where we use constant diffusion coefficients without cross coupling diffusion of the species it is not hard to write down the scalar governing equation r j p 0 for the current j as function of the input parameters p x4 Y Yt representing the species concentration and potential values at the borders of the anode and cathode GDLs The solution to this equation j j p as function of pis generally not avail able in analytical f
90. with Jo a given external current If we assume that no electrical current flows in the lamellas then Jo 0 since no current flows in vacuum The relation V x H 0 62 SESES Tutorial September 2012 x10 0 6 x 10 Abstand m B Feld T Figure 2 40 Magnetic force and mechanical force as function of the contact distance and the magnetic field B expresses the fact that the H field can now be given as the gradient of a magnetic po tential H V O and by using the constitutive relation B uH a Poission equation for the magnetic potential is left to be solved V Ho Hr VO 0 2 21 with u uoutr H po the vacuum permeability and uy ur H the relative perme ability in general a non linear function of H H For this switch we are mainly interested in the magnetostatic forces acting on the lamella responsible for turning on off the switch and two methods are available here The first method is based on computing the magnetic energy inside in the domain Q given by W f E B dV 2 22 Q with E B the magnetic energy density 4 B E B f H B dB 2 23 0 Let d represents the distance between the lamellas and view this value as an indepen dent parameter The energy becomes a function of this parameter W W d and for small changes d d 6d we can write to a first order W d W d OW 0d d The change in energy OW Qd d can now be interpreted as the work done by the
91. with n Fe A the orthonormal eigenvectors of T For a material description used here by the numerical algorithms the Kirchhoff stress T or the intermediate P2K stress S is pulled back on 2 from 9 or 2 resulting in 5 vA Ti S aN QN 2 110 with vectors N F7 n F5 1 not generally orthonormal anymore In addition we need a material description of the elasticity tensor S E with respect to the material Green Lagrange strain E As shown in the appendix for the intermediate configuration Q we have cB MOF Oei Ti 2Ti ij iJ J rA _ _ _ 7 N o Nje N Nj N N 2 111 By considering that the plastic strain F is frozen during the solution of a single time step and does not depend upon E the relation OS 0E follows from the above one by a simple pull back i e by replacing the vectors N with pulled back ones N F5 N We now need to identify a quantity representing the plastic flow and to formulate a rate equation in the stress free configuration Q By writing the spatial velocity gradient in the form o V l e F P F Fe Fp Fpl F3 le lp with le F and lp Fe Fp F F31 we may now identify the pull back of lp given by Lp Fp F 1 as the plastic velocity gradient in Q where we have L F 1 Fe Fo Fe Fp F5 Le Ip SESES Tutorial September 2012 143 with Le Fzt F the elastic velocity gradient in Q The rate of mechanical work
92. x A ndA faa VHi x A ndA Stoke since the condition Qyec CC Q implies DOQyec Therefore these two linear contri butions are given by ae Oj f H x VH ndA 2 49 OOvec and LAs f VH x H ndA 2 50 OOvec and are seen to be anti symmetric This fact may be used to symmetrize the linear system however if iterative solvers are used the convergence rate may be slower than solving non symmetric matrices which however per default have positive semi definite diagonal blocks The answer to the second question is two folds The most simple one is to avoid at the beginning a non simply connected domain by choosing an enlarged domain Qyec and thus getting closer to a full vector formulation with Quy Q The second one is given by selecting surface cuts Cj i 1 N in Qfree so that the domain simple Qfree Ui Ci is simply connected and therefore Otot is a simple function on OQgimpie On each cut Cj one then defines a possible constant jump A for the potential Otot determined as follow For a point on Cj let be y a path in Osimpie joining this point By applying Stoke s theorem on any surface S with boundary 0S yi we have f mwa Veido H di f Vaw d A Si Si Ji i and so the constant A is given by the total current crossing the surface S N Qyec Since V Jo 0 in Q and V x H 0 in Ofc we see that the jump value A is actually invariant with respect to the choice of Ci yi and S Further although th
93. 0 1 exp z log 2 for x gt nx 2 10 1 exp nx x log 2 forz lt nx 2 with the statements QMEI nx 20 1 x nx must be even x QMEJ ny 1 20 Coord 10x 1 exp x log 2 y block 0 nx 2 1 0 ny Coord 10 1 exp nx x log 2 y block nx 2 1 nx 0 ny The resulting mesh is shown in Fig 1 11 This type of mesh dimensioning with large variations in the element size is used when one knows in advance that a high density of elements is required somewhere to accurately compute solutions Although an ex perienced user may easily locate these critical regions as for example around corners tips and boundary layers this information is not always available or simply the task to devise suitable element meshes may be clumsy Therefore one generally resorts to automatic procedures to create computational meshes as will be illustrated in some examples of this tutorial SESES Tutorial September 2012 21 5 5 eee 6 Aa ln x 0 9 9 11 12 16 5 21 Figure 1 12 Example of the 2D geometrical maps with bump and sphere Predefined geometrical maps The flexibility of algebraic approach defining geometrical maps to change the initial rectangular shape is counterbalanced by the fact that one has to find the proper map to obtain the shape of choice To simplify this task and for commonly used geometries the standard distribution provides the include file Homotopic sfc defining several useful geometrical maps For instance thi
94. 01 2006 10 G M RKUS Theorie und Berechnung rotationssymmetrischer Bauwerke Werner Verlag 1976 11 O C ZIENKIEWICZ R L TAYLOR J M TOO Reduced integration technique in general analysis of plates and shells Int J Numer Meth Engng Vol 3 pp 275 290 1971 136 SESES Tutorial September 2012 2 21 Plasticity Models Plasticity theory plays an important role in metal forming and geophysics where sim ple linear elastic laws cannot correctly reproduce mechanical deformations Even for infinitesimal strains plastic models are non linear they depend on the history of the deformations and may be on the overall quite involved for an overview see e g 1 2 Since SESES accepts general non linear elastic laws with the possibility to use user defined element fields to implement the history dependency a possible approach to plasticity is to define models of interest within template files and use them on demand Although the generic user does not need to understand the model set up and the de tails of plasticity theory this approach has the advantage that one clearly see what the models are doing and how they are written so that small adaptations and extensions are readily implemented In this example we will discuss such an approach and com ment the steps required to define commonly used plasticity models for infinitesimal strains In particular we will compute the 2D example of J2 flow with plane strain presented in 3
95. 01 2 076e 01 2 076e 01 3 frequency 2 016e 05 Hz alpha 2 069e 01 2 069e 01 2 069e 01 46 frequency 1 184e 06 Hz alpha 9 316e 00 9 316e 00 9 315e 00 47 frequency 1 21le 06 Hz alpha 3 597e 00 3 597e 00 3 596e 00 48 frequency 1 222e 06 Hz alpha 1 326e 00 1 326e 00 1 325e 00 108 SESES Tutorial September 2012 bins 80 60 40 20 peepee Impedance dB 20 i ji i i 0 200 400 600 800 1000 1200 1400 Frequency kHz Figure 2 65 The impedance Z w for the PZT5A disk of 1 without damping The first 10 frequencies well agree with the ones published in 1 with the higher ones being close within a few percent The cause is to be found in a too coarse mesh used by 1 and our results well agree with the precise computations done by 2 The first two versions of the an values are the same but just if a small enough precision Precision 1E 30 is selected when computing the eigenpairs This value is gen erally smaller than the one required to achieve the same precision within the eigen frequencies and with respect to this precision the more stable value is the second one The small discrepancy of the third version for high eigenvalues has origin in a low order integration scheme not suitable for quadratic elements and all three values are the same by increasing the accuracy of the integral s evaluation Simultaneously to the generation of the above output we also create the plot file Plot with the analyt
96. 1 the temperature boundary conditions BC_wal1l1 and BC_wal12 are set at the outer surfaces of the insulation material No slip boundary conditions have to be applied to all fluid solid interfaces These surfaces are elegantly deter mined using the OnChange material statement which specifies all surfaces of a given material in this case ISO BC BC_noF low OnChange material ISO x no flow at channel walls x Dirichlet Velocity 0 0 0 m s Finally we give symmetry conditions with respect to the flow field in those air chan nels where only 1 4 of total the cross section is considered With the specification of the boundary conditions the initial section of the input file is now complete SESES Tutorial September 2012 203 In the command section of the input file we first perform a manual refinement of the air channels initialize the temperature field and run the solution procedure depending on the setting of the flags condYN and permyYN At the end we add post processing commands to write the desired information allowing us to compute averaged values of permeability and conductivity based on 2 174 and 2 175 Numerical Results The content shown below is an excerpt of the output when running the example in mode permYN 1 x input parameters x T_O 293 15 K p_O 1 000e 05 Pa dpress 1 200e 01 Pa properties of air x viscosity 293 15 K 0 0185 cp x calculated densities densiFluidl 0 7306
97. 2 32E4 02 32E 02 V pica Aluminum 32E 02 3 1E 02 3 1E 02 3 1E 02 3 1E 02 3 0E 02 3 0E 02 Figure 2 9 A double layer cantilever heated by a current flow time 0 Figure 2 10 Computed temperature with the hottest spot at the cantilever s tip 2 3 Electrothermally Driven Cantilever Microactuator This example presents the modeling of a cantilever heated by a resistor as displayed in Fig 2 9 By applying a voltage between the contacts Vo and Vapplieq a current will flow in the aluminum layer and Joule s heat dissipation will heat up the cantilever Due to mismatch in the thermal expansion coefficients of the aluminum and silicon layer the thermal induced strain will bend the cantilever This example is not a proper real ex ample but serves the purposes of explaining the modeling of a multiple field coupled problem Is it kept as simple a possible so that we just perform a 2D simulation and use the bare minimum of details The SESES input file for this cantilever problem can be found at example E1Therm Mech s2d In a first step we define the material properties for bulk silicon For this material we are going to compute the temperature and the mechanical displace ment but since bulk silicon is an insulator no current flow is computed The isotropic thermal induced strain is defined with the parameter AlphalIso of the the built in LinElastIso defining a linear isotropic elastic law The thermal strain is made a function
98. 2 1 30e 03 1 00e 08 3 a a ts 3 1 206 03 gt J 9 80e 09 5 00e 06 1 076 09 7 1 10e 03 1 00e 03 1 9 60 09 0 00e 00 1 1 06e 09 0 20 40 60 80 100 o 20 40 60 80 100 Frequency kHz Frequency kHz Figure 2 60 Resistance and inductance matrix coefficients up to w 27x 10 Hz 1 156 03 r 1 00e 03 2 00e 05 r r r 1 206 04 Ro f BON S g 1100 03 F ROO st 9 000 04 gz g 0 00e 00 PF 1000 04 eo ia La er eee ee X00 1 7 8 00004 EF S XOT tte 1 05e 03 X00 approx J 7 00e 04 O 2 00e 05 XOt approx a00e 05 S s S T 4 6 00e 04 S 5 D l 5 1 006 03 Q 4 00e 05 pae g 7 4 5 00e04 F 4 6 00e 05 2 9 50e 04 F 4 4 00e 04 2 2 6 00e 05 K 2 5 Tooo a S 4 4 00e 05 S B 9 00e 04 H _ 7 8 00e amp B 8 00e 05 ie by 3 NJ 2 00e 04 V weer k o oe 04 H s c 7 7 eet 4 2 00e 05 8500 04 F Y 4000 04 1 006 04 ee 8 00e 04 b L L i 1 0 00e 00 1 200 04 L L L 1 0 00e 00 0 2 4 6 8 10 0 2 4 6 8 10 Frequency kHz Frequency kHz Figure 2 61 Resistance and reactance matrix coefficients compared with a second order low frequency approximation potential Y from the solution process and the disappointing observation is that e g the three presented methods to compute the inductance matrix at w 0 will then give different results with up to 20 discrepancies Up to machine precision the computed resistance R and inductance L matr
99. 20 6 206e 07 2 106e 05 352 499 0 025 9 697e 07 3 290e 05 382 030 and may now be used for plotting with third party software Another popular file format for data is the comma separated format with the suffix csv for the MS Excel application To generate such a format one just needs to use a comma instead of a tabulator inside the format string 36 SESES Tutorial September 2012 Cantilever Microactuator 4e 05 Cantilever Microactuator x displacement 3 5e 05 y displacement i a 8 107 T 3e 05 z z 2 5e 05 T 5 205p E S 4 107 1 5e 05 cf a 2 e 1e 05 er kar 5e 06 a ee a j i l 0 005 0 01 0 015 0 02 0 025 0 005 0 01 0 015 0 02 0 025 0 03 current A current A Figure 1 22 Plot of the microactuator tip dis Figure 1 21 Plot of the microactuator tip dis placement generated by Mathematica placement data generated by Gnuplot Plotting with Gnuplot Gnuplot is a freely distributed scientific plotting software available for Linux and Win dows The following commands can be entered interactively in the Gnuplot mode or be pasted in the settings file script gnu set terminal postscript color 20 set output curveplot ps set title Cantilever Microactuator set xlabel current A set ylabel displacement m set xzeroaxis set yrange 0 4e 5 set grid plot curve using 1 2 title x displacement with lines c
100. 24 Continuous casting of steel The discover and the first production of steel started long time ago in Anatolia around the 15th century BC and todays we are approaching a worldwide production of a bil lion of tons p a Around 1930 Siegfired Junghans developed a continuous casting process for melted brass Its application in the steel industry was advocated and pio neered by Irving Rossi who in 1954 founded in Z rich the Concast company now part of the SMS group and a major supplier of steel casting machines Since then the con tinuous casting of steel in forms of billets blooms and slabs has become the standard production process From a hot furnace the melt is transferred into a tundish reservoir allowing to continuously feed the casting machine The melt is then drained into a water cooled copper mold where the hot metal directly in contact with the mold so lidifies into a thin shell Here the mold may oscillates or may be lubricated in order to prevent sticking The thin solidified shell acts as a containment for the melt inside the 158 SESES Tutorial September 2012 now called strand After exiting the mold the strand is further cooled by water sprays and its movement is redirected and supported by rolls The cooling and solidification process continues until the whole strand section solidifies and the strand can be cut into fixed lengths The steel is still very ductile and can be further formed into its final shape by milling Not onl
101. 51 2002 174 SESES Tutorial September 2012 Conc A Figure 2 117 Concentration profiles at different cross sections for K 0 1 Comparison of the already shown SESES results based on a plug flow shown as the lines with those based on a Hagen Poiseuille flow shown as the symbols 5 I MULLER Grundziige der Thermodynamik 3rd Ed Springer 2001 6 M N ZI IK Boundary Value Problems of Heat Conduction Dover Reprint 1989 7 R B BIRD W E STEWART E N LIGHTFOOT Transport Phenomena Wiley 1960 2 28 Heat Transfer and Natural Convection in a Closed Cavity Natural convection can be found nearly everywhere in nature For example the cir culation of the earth atmosphere and the global wind systems are mainly driven by natural convection In the presence of a gravity field natural convection is initiated by density differences in liquids or gases These differences may be due to the presence of different chemical species or due to temperature differences within the field of a pure fluid Natural convection also is important in numerous technical applications In en gineering natural convection plays e g an important role in the heating ventilation and air conditioning of buildings as well as in the field of solar energy A well known phenomena
102. 7 SESES Tutorial September 2012 23 Gradual mapping It is sometimes useful to gradually apply a geometric map in order to obtain smooth results and for this purpose simple shape functions are required We have defined the ramp function Routine double ramp double val double type as the function R v with R oo 0 0 R 1 00 1 and otherwise continuous between on 0 1 with the degree of smoothness determined by the parameter t ype With this function we can define the bump function Routine double bump double val double v 4 double type as B v R v v i vo 1 R v v2 v3 v2 with values B oo vo U V3 00 0 B vi v2 1 and the previous ramp behavior between vo v and v v2 This bump function is typically used to apply a full map within the interval V1 v2 but only gradually in the intervals vo v v2 v3 and not at all outside vo v3 A last useful shape function is the following double triangle function zero outside 1 1 antisymmetric with respect to the origin and with the maximum of 1 at vmax Routine double triangle double val double vmax double aval abs val return val lt 0 1 1 aval lt vmax aval vmax aval lt 1 l aval l vmax 0 which is used to define the function Routine double dilat double val double c double sc double sm double d return val d sc triangle val c sm sc sm The dilat function leaves the point c
103. CoordNonConst Coord x yt y y3 l_cp 1 block 0 nxtot nny3 nny4 1 Coord x ytl_cp 1 block 0 nxtot nny4 nytot The Coord statements are introduced due to the varying thickness lep of the com pensating disk which enforces a redefinition of the mesh for each solution step As thermal boundary condition we assume the uncooled part as well as the front and back surface of the rod to be in contact with air The cooled part is of course in con tact with water For the mechanical fixation we assume the uncooled part where the mount is located to be fixed for radial elongations and due to symmetry we fix lon gitudinal elongations in the center of the rod see Fig 2 89 As last part of the initial section we define the rotational symmetry with the statement GlobalSpec Model AxiSymmetric Enable We start the command section with the definition of the information in the output stream and the used solver Info Refine HideBC LinearSolver NonSymmetric and define the lattices on which the OPD has to be calculated For k From 0 To nr_OPD Lattice OPD_ k Index m 0 d_OPD k r_OPD nr_OPD 0 0 1_OPD m d_OPD We then initialize the temperature field with the air temperature Solve Init Temp Tair We ingrain the solution procedure into a loop to perform the parameter study For each solution step the automatic mesh refinement algorithm is used for the tempera ture and the displacement field At the first Solve Stationary statement th
104. ESES Tutorial September 2012 with Rgas the gas constant k a pre exponential factor and Ea a typical activation energy of the process For the overall reaction rate we use ad hoc the expression THTCO y Ah TAS 2 180 TCOTH20 Real r Ttwa l with Ah the reaction enthalpy and AS the reaction entropy While the expression 2 179 is correct far from equilibrium in the forward direction it is no longer correct near thermodynamic equilibrium and 2 180 represents an amendment in order to attain equilibrium Even for compositions favoring moderately the reverse reaction the ex pression seems to be acceptable since it assumes small negative values forcing the gas composition back to equilibrium Far form equilibrium with reverse reaction condi tions the applicability is questionable since the reaction kinetics will almost certainly fail to be correct However this condition will not be attained during sensible SOFC simulations From the reaction rate 2 180 in unit of mole s we obtain the species production rates as Ia Mavar with Ma the species molar mass and va the stoichio metric reaction coefficients and since mass is not created nor destroyed we satisfy the necessary condition 5 Ha 0 Both gas and air channels are interleaved with nopples used to electrically contact the anode and cathode surfaces No convective transport is possible within the nopples so that here we just need to model the 2D conductive transport
105. EY D HITCHINGS The finite element analysis of the vibration characteristics of piezoelectric discs Journal of Sound and Vibration Vol 159 pp 115 138 1992 110 SESES Tutorial September 2012 2 J KOCBACH P LUNDE M VESTRHEIM Resonance frequency spectra with conver gence tests for piezoceramic disks using the Finite Element Method Acustica Vol 87 pp 271 285 2001 2 17 Longitudinally Diode Pumped Composite Laser Rod The following examples deal with solid state laser devices A good introduction into this field including set ups and pumping techniques is e g given in 1 2 3 When a solid state laser crystal is optically pumped by flash lamps or diode lasers only a part of the pump light is converted to laser or other radiation The other parts acts as a veritable space resolved heat source leading to a spatially varying temperature distri bution in the crystal This temperature distribution will influence the laser behavior The three main effects are e Thermal Dispersion Due to the dependency of the refractive index of a material on the temperature the laser beam may be disturbed by its travel through the crystal e Deformation Due to the temperature raise the crystal will expand locally leading to deformation of the crystal surface The deformed surfaces will have a lens like influence onto the laser beam traveling through the crystal e Stress and strain Based on the local deformation stress and strain will occur
106. For our example however we just solve linear problems and so it is safe to remove all zero or numerical close to zero coefficients from the matrix which are detected when building the sparse matrix This non default behavior is obtained with the statement LinearSolver CoeffCutOff 0 which results in some speed up However the most important speed up is obtained by keeping the factorized matrix when solving a sequence of linear problems with an invariant matrix The admittance matrix is computed by solving N problems and by applying separately in each wire a driving voltage For the same frequency w a closer look shows that the linear matrix obtained by discretizing 2 53 is unchanged and we just have different right hand side vectors Therefore the most expensive nu merical operation of factorizing the system matrix must be done only once and the N problems are then solved by performing N backward forward substitutions which are by far less expensive see 5 This non default behavior is enabled with the state ment Increment ReuseFactoriz which tries whenever meaningful to reuse a previously factorized matrix As first we compute the low frequency approximation by solving an alternate se quence of electrostatic 2 66 and magnetostatic 2 63 problems with the help of the following command procedures Procedure double SolveElectroStatic MaterialSpec Wire Equation EddyFreeNodal Disable OhmicCurrent Enable BlockStruct B
107. Here u Cp and a represent the dynamic viscosity the heat capacity and the thermal diffusivity of the considered fluid respectively Note that for air we have Pr 0 7 For natural convection problems the heat transfer coefficient then can be expressed as Nu Nu Gr Pr 2 153 176 SESES Tutorial September 2012 It turns out that the type of flow field observed depends on another dimensionless number the so called Rayleigh number Ra defined as the Grashof times the Prandtl number gh AT va T For Ra lt 108 the fluid flow is always laminar Note that at the present time flow simulations using SESES are restricted to laminar flow only For setting up the prob lem we therefore have to ensure that the applied temperature difference AT as well as the characteristic length h are chosen such that the condition Ra lt 10 is fulfilled For the situation of free convection at vertical walls results from a large number of experiments were combined to derive the following empirical relationship see VDI Warmeatlas 2 Ra Gr Pr 2 154 1 6 Nu 0 825 0 387 Ra f 2 155 1 9 167 16 9 i pee To evaluate the accuracy of the obtained SESES simulation results they will be com pared with the above correlation For this to apply 2 155 to the cavity shown in Fig 2 119 AT as it appears in 2 152 is defined as AT Thot Teola 2 fi 2 156 Setting up the problem To boot up the calculations in two
108. NM SESES Multiphysics Software for Computer Aided Engineering Tutorial Version September 2012 oy NM Numerical Modelling GmbH J Borth BGhnirainstrasse 12 T Graf CH 8800 Thalwil T Hocker Switzerland E Lang http www nmtec ch B Neuenschwander mail info nmtec ch B Ruhstaller G Sartoris H Schwarzenbach Copyright 1996 by NM Numerical Modelling GmbH This work is subject to copyright All rights are reserved whether the whole or part of the material is concerned specifically the rights of reprinting reuse of illustrations translation broadcasting reproduction or storage in data banks Duplication of this publication or parts thereof is permitted in connection with reviews personal or schol arly usage Permission for use must be obtained in writing from NM NM reserves the right to make changes in specifications at any time without notice The information furnished by NM in this publication is believed to be accurate how ever no responsibility is assumed for its use nor for any infringement of patents or other rights of third parties resulting from its use No license is granted under any patents trademarks or other rights of NM The author NM Numerical Modelling GmbH makes no representations express or implied with respect to this documentation or the software it describes including without limitations any implied warranties of merchantability or fitness for a partic ular purpose all of which
109. Newton iteration At the end of the inward movement the stamp is retracted to its original position in order to observe the spring back effect We note that for this particular geometry and depending on the yield value for the plastic flow the structure may buckle and our simple solution method may not be able to pass this singular point For the penalty method we have quadratic convergence both with a closest point projection computed analytically or by interpolation In the latter case a slight degra SESES Tutorial September 2012 157 Tundish Figure 2 104 Steel casting machine made by SMS Concast Z rich Figure 2 105 Schematic view of steel cast ing dation of the convergence is noticed The algebraic contact approach does not work for this example The problem lies in the sudden change of the constraint equations at each iteration so that the global residual equations are not continuous The New ton iteration gets stuck in a dead lock cycle of opening and closing constraints and the convergence rate stales The simple algorithm described above for enabling and disabling the constraints which generally works well for a simple half space rigid body needs to be amended However it is not clear if a robust method valid for any geometric form can be found References 1 T A LAURSEN Computational Contact and impact Mechanics Springer 2002 2 T A WRIGGERS Computational Contact Mechanics Springer 2007 2
110. Nom Strain Trace 5 3E 06 5 0E 06 4 8E 06 1 6E 03 4 5E 06 1 5E 03 4 2E 06 1 4E 03 4 0E 06 1 3E 03 3 7E 06 1 2603 3 4E 06 11603 3 2E 06 9 5E 04 2 9E 06 84E 04 2 6E 06 7 3E 04 2 4E 06 6 3E 04 2 1E 06 5 2E 04 1 8E 06 41E 04 1 8E 03 1 7E 03 1 6E 06 31E 04 1 3E 06 2 0E 04 11 06 9 1E 05 7 9E 07 5 3E 07 2 6E07 2 3E 29 1 6E 05 1 2 04 2 3E 04 34E 04 time O time 0 Figure 2 11 Computed mechanical displacement left and strain right in the cantilever electric field computed in the first step to evaluate the Joule s heat Electric potential and temperature are then used in a third step to compute the mechanical displace ment In this problem however we just use the temperature to evaluate the thermal strain the electric potential is only used in the second step To compute a solution with the least amount of time we therefore define three blocks solving for the poten tial temperature and displacement fields The system automatically detects that all three blocks define linear equations and so no block convergence criteria must be de fined and a single block iteration will be done When multiple blocks are defined the default solution behavior is to solve each block once and then exit which will correctly solve our unidirectional coupling problem BlockStruct Block Psi Block Temp Block Disp Solve Stationary When solving for the mechanical displacements SESES will additionally solves for the mechanical rot
111. Nu 3 Nu 1 615 Pr 1 6 u1 3 66 Nuz 615 4 Re ro Nus FP Re Prz 2 169 Based on the Nusselt number calculation and the computation of the convection heat transfer coefficient the mean fluid temperature distribution from the inlet to a certain point downstream can be determined The temperature distribution along the pipe axis is shown for our case in Fig 2 139 SESES Tutorial September 2012 193 Setting up the problem The SESES file for this example can be found at example HeatPipe s2d In the first part of the input file the geometry is defined where the mesh is denser close to the wall and at the inlet as the velocity and the temperature gradients are large due to the developing boundary layer QMEI nx radius nx QMEJ ny length ny Routine double shift double val double v 2 double fac double f val v 0 v 1 v 0 return val facx v 1 v 0 fx f f Coord shift x 0 radius 0 9 y 1 Coord x shift y 0 length 0 9 1 The fluid considered is air with the temperature dependent physical properties We implement the temperature dependency by defining SESES routines which are taken from a library database Routine double Viscosity_AIR 1 double T Viscosity of air in pa s temperature range 250k 2273k x return 3 4746851634130747e 7 Tx 7 254574139642276e 8 Tx 4 474301400558843e 11 1 5696157920120257e 14 1 2112463749591405e 18x T T Routine double
112. O T J R HUGHES Computational Inelasticity Springer Verlag 1998 4 J C SIMO Numerical Analysis and Simulation of Plasticity Handbook of Numerical Analysis by P G Ciarlet and J L Lions Vol VI North Holland 1998 5 J C SIMO Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory Comput Methods in Appl Mech Engrg Vol 99 pp 61 112 1992 2 23 Stamping and mechanical contacts There are several methods to shape an industrial mass product one of them is by stamping a laminate made of a material showing a plastic behavior The method is used to give cars their body shape for the production of cooking pans beer cans blis ters enclosing pharmaceutical pills as shown in Fig 2 100 etc Modeling the stamp ing or drawing process is an important task allowing precise statements to be made about the final product as the shape thickness stresses points of rupture or delami nation and so on Important aspects of the modeling are the correct definition of the plastic material laws and of the mechanical contacts between the stamp and the un derlying material Since plastic laws are already discussed in other contributions we focus here on the modeling of the mechanical contacts and start with a clamped thin laminate modeled by non linear finite strain solid shell elements together with a sim ple isotropic plastic law To this initial configurat
113. OA BC jump Neumann MagnPot BC fix Dirichlet MagnPot Solve Init Solve Stationary Store jumppot MagnPot BC jump Jump MagnPot jumppot A All these four approaches i e total reduced total reduced with Biot Savart potential as jump and total reduced with jump computation by a Laplace problem are com puted by the given SESES example The magnetic potential solution is displayed in Fig 2 54 and the example reports the maximal numerical error of the solution The magnetic potential is computed accurately by all formulations however for values of u gt 10 the reduced formulation wrongly computes the H and B field within the ball Rel Perm 1 000000e 25 Total MagnPot error 5 581309e 02 Bfield error 2 47493le 01 3 233611le 00 Reduced MagnPot error 8 527405e 01 Bfield error 1 081236e 11 9 095459e 10 Tot Red 0 MagnPot error 5 581309e 02 Bfield error 2 47493le 01 3 233611e 00 Tot Red 1 MagnPot error 6 60182le 01 Bfield error 2 805619e 00 5 698386e 00 One can argue that so large permeability values never show up in practice and indeed this total reduced formulation is generally not required in the case of a large but con stant permeability However this may be a necessity whenever we have a non linear material law u u H SESES Tutorial September 2012 81 For large problems even when the permeability u does not cause numerical instabili ties for practical reasons one also resorts to this reduced total formulation The rea
114. Parameter iCurrent0 Z For i To nWind 1 SOL 2 i 1 domain Wire i 0 sigma 2 PI x A m 2 Solve Stationary Although we solve several linear problems the frequency is constant and so the linear system matrix The linear problems just differ in a different right hand side vector so that if we use a direct solver the linear matrix need to be factorized just once at the beginning Keeping the factorized matrix in memory is done by defining the following statement at the beginning Increment ReuseFactoriz As result the time expenditure required to solve the N linear problems is almost the same as to solve a single problem Clearly if the number N of windings is large then a large amount of finite elements are required to spatially resolve each wire which therefore is the major limiting factor References 1 S RAMo J R WHINNERY T VAN DUZER Fields and waves in communication electronics Wiley 1994 2 10 Steel Hardening Steel can be thermally treated to change and give it important properties as hard ness softness or to remove internal stresses generated by fabrication processes Many treatments are available which include normalizing anealing hardening and temper ing Normalizing and anealing involve heating the steel above its critical temperature value followed by holding this temperature for some time and a final cooling This is done to remove internal stresses or to soften the steel By anealing the steel i
115. S Inc Canonsburg PA USA http www waterloo ansys com cfx 200 SESES Tutorial September 2012 20 mm 10 mm 20 mm p4 gt lt gt 2mm 18mm air gap Tesia Thot Figure 2 148 Computational domain with air channels AIR light blue and insulator ISO yellow representing the smallest repetitive Figure 2 147 Section of the thermal insu structure of our thermal insulation structure lation structure consisting of two insulation walls with air channels and an air gap be tween the walls Not shown is an offset of 18 mm in the depth location of the air chan nels between the two walls 2 33 Effective Transport Parameters from Volume Averaging In many complex technical devices such as fuel cells a multitude of coupled physical and chemical processes take place within the assembly fluid flow diffusion charge and heat transport as well as electro chemical reactions For design and optimiza tion purposes direct numerical simulation of the full 3D structure using CFD tools is often not feasible due to the large range of length scales that are associated with the various physical and chemical phenomena However since many fuel cell compo nents such as gas ducts current collectors or thermal insulation assemblies are made of repetitive structures volume averaging techniques can be employed to replace de tails of the original structure by their averaged counterparts 1 2 3 An efficient approach is based on the f
116. The numerical and the Blasius solutions for the x velocity component on the line at x 0 5 are compared in Fig 2 112 for the kinematic viscosities v 0 01 1074 10 For v 0 01 the numerical solution has a clear overshoot caused by the mass conser vation and the setting of the BCs however the thickness of the boundary layer is for both solutions the same The mesh is chosen so that for v 10 it can still resolve the boundary layer not so for v 107 which is then given by the thickness of the first element The drag on the plate of length is given by integrating the shear stress component Try Drag of u yu y odz buU f of ynde 2bpU f 0 V Uml with b the thickness of the plate and f 0 0 33205733 The numerical values agree well with the analytical ones for thin boundary layers resolved by the computational mesh The incompressible Navier Stokes defined by the statement Equation Incompres sibleFlow Enable uses residual based stabilization to circumvent the various in stabilities associated with a Galerkin discretization of the equations 2 127 In prac tice to the residuals equations obtained by the classical unstable Galerkin discretiza tion one adds the integrals of the strong residuals weighted by special test functions Since the strong residuals tends to zero this stabilization method is consistent how ever the strong residuals can be freely scaled and therefore we always have free pa ram
117. Tutorial September 2012 change of solution and load together with a proper norm definition must be the same as the change of the new controlling variable Load control is not only used to com pute and pass limit points but also in general as replacement for simple continuation methods Load control has the property to stabilize the continuation methods and so larger steps can be taken Load control methods require the derivative of the resid ual equations with respect to the simulation parameter begin controlled In SESES this derivative is by default computed by numerical difference it can be computed exactly but then must be defined explicitely For this example the default difference algorithm works nicely but there may be times where an exact definition is necessary In order to use load control a second simulation parameter playing the role of the new independent variable must be defined and used similarly and in replacement of the simulation parameter without load control In our example the new simulation parameter is control This parameter will control the length of each new step by considering displacement and load together Write Computing with load control n Solve Init Dump AtStep 1 Disp Write AtStep 1 File plot Text f load Lattice point Sf n Disp Y Convergence double ok AbsIncr Disp lt 1 E 12 write AbsIncr Disp 2e n AbsIncr Disp if ok write load f control f n load control return ok I
118. Vis Oy For convenience we introduce the dimensionless length variable 7 y L in the y direction together with the abbreviation DAB K AA 2 139 so that 2 138 can be written as OcA O7c 2 140 Ox Og This equation subjected to the following BCs x 2 OcA cA calx 0 9 A0 2A 0 2 141 A c4 0 aha 2B loos represents a homogeneous boundary value problem and can be solved by the method of separation of variables 6 Inserting the Ansatz c4 z y O x U g into 2 140 leads to 10 Vig 2 Z _ 2 142 KO wv ee with m a constant and a particular solution for c4 x y is given by elt eK mz C1 sin m gy C2 cos m 2 143 It follows from the BC 2 141b that Ci e mx 0 which leads to C4 0 Evaluation of 2 141c gives e mE Oy sin m 0 which implies that m 0 7 27 im with i 0 00 The general solution for c4 x then follows as the superposition of the particular solutions calx 9 Coot gt eK i TC cos im 2 144 i 1 The coefficients C2 with i 0 00 yet have to be adjusted to the BC 2 141a i e they have to be determined from ch C20 X Coi cosir 9g 2 145 i l SESES Tutorial September 2012 171 ooopo0o0000 NMWARUOONOO 6 Y axis Figure 2 114 Predictions of 2 148 for the concentration field along the straight channel shown in Fig 2 113 for K 0 1m7 Suppose c9 is given by
119. Yx x integrate Bound b_pos i a TempDiff Y PI x integrate Bound b_pos i b TempDiff Y PI x integrate Bound b_pos i c TempDiff X 2 PI x E aY Moo Lattice pos i a node nx i dy Lattice pos i b node nx i dy 1 Write Lattice pos i a y 6 3f Temp 5 2f DFlux 6 2f n Lattice pos i b y 6 3f Temp 5 2f DFlux 6 2f Text 6 2f n F_c a F_b y TL y T Fig 2 139 compares the computed and estimated temperature distribution and one can see that these errors are very small The graph also shows that the computed average temperature along the pipe is smaller than the estimated one This means that the heat transfer to the wall is larger in the simulation Several effects may cause these discrepancies The Nusselt number calculation of the VDI W rmeatlas is based on a large number of experiments However sensitivities studies has shown that the inlet conditions influence the temperature distribution as well as the fluid properties In our estimation we used fluid properties at the mean temperature of the inlet and outlet of the pipe We suspect that these effects then lead to the different temperature distributions In Fig 2 139 the convection heat transfer coefficient is also shown As expected the coefficient is very high close to the inlet and drops to a value of a 3 W m downstream SESES Tutorial September 2012 195 1000 900 800 4 y 0 0m y 0 1
120. a function but a multi valued function e What about uniqueness and accuracy of the coupled solution The answer to the first question is obtained from the Maxwell s equations requiring the normal component of B and the tangential component of H to be continuous Let n be the normal to the boundary OQyec between the current domain Qye with B V x A and H u V x A and the free space domain Qfree with B uH and H Ho VOo The equality of the normal component B n of B gives u Ho VOtot m V x A n and the one of the tangential component H x n of H gives Ho VO tot x n wt V x A x nor ut V x A Ho x n VO tot x n Following the SESES manual these conditions are satisfied by setting the following Neumann BCs V x A n for Otot 3 V Otot xn forA 2 48 In order to solve our linear problem with a single linear step also the correct deriva tives should be specified for these interface BCs It is interesting to note that both Neumann BCs together may result in a global sym metric contribution Let H x and H x be the shape functions associated with the SESES Tutorial September 2012 83 dof fields and A i e we have 5 H O and A 5 H A with O and A the unknowns of the problem to be solved The residual contributions of both Neumann BCs 2 48 are given by RA fia Hi VOxn dA a He x VO ndA and Re Sona ay x A ndA ho V x H A VH x A ndA e H A ds foo VHi
121. a longitudinally diode pumped laser rod with undoped end caps used in novel laser devices The SESES input file can be found at example LongRod s2d Fig 2 70 shows a X folded resonator as it is used to realize diode pumping from both sides The pumping radiation is focused through the two SESES Tutorial September 2012 113 Deduction of the Averaged Thermal Lens 0 0045 0 0040 e Simulation Pee Fit i Pee tegy 0 0035 0 0030 0 0025 OPD mm 0 0020 0 0015 0 0010 0 0005 0 0000 0 00 0 10 o2 o3 0 40 0 50 0 60 0 70 0 80 r r mm Figure 2 68 Parabolic fit to the deduced OP Diot over the extent rft Figure 2 69 Nd YAG laser rod with two un doped end caps folding mirrors directly into both sides of the laser rod as shown in Fig 2 71 The fold ing mirrors have to be dichroic coated e a highly reflective for the laser wavelength and highly transmittive for the pump wavelength The laser rod consists of a 1 1 Nd doped YAG center part and two undoped end caps as shown in Fig 2 69 With out end caps longitudinal pumping leads to high temperatures and stresses on the pumped surface going with a significant bending of it 5 Undoped end caps signif icantly reduce temperatures and stresses on the pumped surface and prevent surface bending The heat source q r z directly scales with the normalized but arbitrary shaped pump distribution p r z In the presen
122. a pseudo time and since the integration pro cess is quite stable small or large steps yield similar results The necking breakdown is approached with a trial amp error algorithm where the failure criterion is given by a slow convergence rate If failure occurs the step is repeated with a lower increment until the minimal value of 1x107 Convergence if finite AbsIncr Disp nIter gt 5 failure 148 SESES Tutorial September 2012 volume s rate log10 ORM ON FD OO Figure 2 98 The necking of a circular bar un 0 01 02 03 04 05 06 der large elongation shown is one half of the pseude time last d solution ep PONVE TBE SOSUN Figure 2 99 The rate of volume s change in logarithmic scale else if AbsIncr Disp lt 1E 15 return 1 else return 0 write Solve Stationary Step 0 001 0 256 maxs 0 005 step x1l 2 gt maxs maxs stepx 1 2 Until 1 Failure return step lt 1lE 10 0 step 2 The necking breakdown appears suddenly and without some sort of load control nu merical instabilities prevent any further computations The instability is due to the rapid change in volume as shown by Fig 2 99 where the displacement for the last converged solution is shown in Fig 2 98 Appendix To avoid an overloaded bar notation let consider Fp Id and so N In order to compute 0S J0E we first note that by coaxiality of Sand C Id 2E we have S X SiN 9N and C X 2N N with S T A and Xa
123. ad by defining the total current to be a linear function of the floating potential In summary the materials are defined with the statements MaterialSpec Electrolyte Equation OhmicCurrent Enable Parameter SigmaIso 5 S m MaterialSpec Cathode From Electrolyte Equation TransportO2 Enable Parameter Sigmalso 1 2E4 S m Parameter Diff 02_02 1 0e 5 kg s m MaterialSpec Anode From Electrolyte Equation TransportH2 Enable Parameter SigmalIso 3 0E4 S m Parameter Diff H2_H2 1 0e 5 kg s m and the boundary conditions with the statements BC Cathode 0 ny IType nx Dirichlet Psi OV Dirichlet 02 x02 BC Anode 0 0 IType nx Floating Psi current mA Dirichlet H2 H2 Define NAVO 6 02202E23 Ginv 1 1 0E5 O2flux 1 4 mmol02 NAVO Q0 H2flux 2 4xmmolH2 NAVO Q0 Macro CUR Current X Normal X Current Y Normal Y BC OxyRedox Restrict material Cathode 0 nyOtnyl IType nx SESES Tutorial September 2012 209 Neumann 02 D_Current O2flux CUR Normal 0 O2flux kg s m 2 kg Axs 3 BC HydOxid Restrict material Anode 0 ny0 IType nx Jump Psi D_Current 1 10 CUR Ginv Normal 0 Ginv V V A mx 2 3 Neumann H2 D_Current H2flux CUR Normal 0 H2flux kg s m 2 kg Axs 3 Since for this simple model the potential jump is just a function of the current the potential is independent from the chemistry and should be solved in a first step and the mass flow equations for O2 Hz in a second step These two equ
124. al Vector material VectorJ OnChangeOf 2 3 material Scalar material Vector tmaterial VectorJ Neumann MagnPot D_Bfield SetAlways return A_DOT_N__DA Bfield 1 STunit Symmetric Linear HideFlux The value of V x A is not available as a model parameter to specify the BC value but this is nothing else as B which is therefore used for the specification This interface BC is a linear function of the computed primary field A and therefore it is important to correctly specify the field derivative in order to get the solution in a single linear step So for the numerical BC we specify the value B n together with the B field derivatives given by n The additional minus sign with respect to 2 48 has to do with a change of sign of the normal n and the following technicality For a boundary surface defined inside the computational domain Q we have the choice on which side we want to evaluate the BC value which is done by defining a one sided BC boundary Since the vector potential A is not defined within the domain Qfree of the scalar formulation we are forced to define and evaluate the BC on the other side inside Qyec However here the magnetostatic potential is undefined and the default behavior will not set this BC value expect if using the additional option SetAlways Generic Neumann BCs with derivatives are considered as generic non linear functions of the primary fields yielding non symmetric matrices to be solved The Linear option ju
125. all that difficult since the default view shows the whole domain at once However for a 3D domain this is not generally the case and one typically wishes to adjust the viewing angle and look from behind look inside a device or limit the view for very complex structures This will be explained in the following and the user is advised to open some 3D examples for practice and follow the instructions to quickly get acquainted with the tools For a 3D visualization one of the most important tools is the capability to choose another viewing position i e to rotate or to move the displayed object For this task one has to select the toggle button gf or t By moving the mouse while keeping the 16 SESES Tutorial September 2012 Figure 1 8 Illustration of some graphical visualization tools available in the Front End pro gram Rotating and reducing the mesh domain to be displayed left cutting through the mesh domain along an arbitrary plane middle and zooming right first button pressed you change either the viewing angle or the viewing position and you can quickly toggle between both choices by pressing the second button In order to look inside a 3D domain there is the possibility to make a planar 3D cut of the domain This is done by selecting the toggle button and pressing the first mouse button to define the origin of the cutting plane While keeping the mouse button pressed you change the normal of the plane by moving the mouse a
126. alysis and the pressure load as a boundary condition To compute a family of solutions depending on the pressure load we define the global variables load and cont rol but for the moment we will just consider the first one in order to specify a pressure load with the help of the Pressure BC The definition of a global variable let us choose its value before computing a solution Therefore we start computing some solutions for values of load 1 2 3 4 5 Write Computing solutions for different loads n Convergence AbsIncr Disp lt 1 E 12 1 nIter gt 15 failure Too many iterations 0 Solve Stationary ForSimPar load At 0 Step 1 5 Since the mechanical displacement is mainly in the vertical direction and the problem is non linear we have chosen the convergence criterion solely based on the conver gence of the vertical displacement and a maximal number of 15 iterations In the Solve Stationary statement we select the simulation parameter load and define 5 steps with an increment of one Six different solutions will be computed starting with a null value of the pressure load For this non linear problem the derivative of the residual equations with respect to the unknowns representing the displacement field are computed exactly so that the Newton iteration converges quadratically We may apply the techniques presented in the previous section using fast increments to speed up the computation but here we present another method using a
127. an interpolated analytical function This is due to the interpolating character which must be C and exact at the sampling points If the rigid body surface is simple then an analytical projection together with some well designed if else state ments can be a valid alternative providing exact values of the projection and surface characterization In this contribution we will present both approaches for a surface of revolution consisting of two piecewise linear segments and two parabolae with global C continuity as displayed in Figs 2 101 2 102 The most difficult part of the analytical projection algorithm consists in computing the orthogonal projection to a generic 2D parabola F x ax with x R a gt 0 and a generic point P P Py By symmetry we can assume P gt 0 and since for P 0 we always have the projection 0 0 we can further assume P gt 0 The or thogonal projection is stated here as an orthogonality condition between the segment P F and the tangent d dz F as P F d dx F 0 and resulting in the cubic equation f x 2a x3 1 2aP x Px 0 Since f 0 lt 0 and f 00 00 we always have a solution for x gt 0 For 1 2aP gt 0 we have d dx f x gt 0 so that f is monotone and just a single solution exists For 1 2aP lt 0 we have d dx f 0 lt 0 so that the other two solutions if they exist they are necessarily nega tives Therefore the projection for P gt 0 is
128. and important findings have been made The flow situation is of relevance for many industrial applications e g tubes in heat exchangers chimneys towers antennae masts cables and offshore risers The main interest is focused on the wake behind the cylinder and the forces on the cylinder as they may cause the cylinder to oscillate with a certain frequency If this frequency equals the eigen frequency of the cylinder the movement of the cylinder will increase and a material failure due large tensions may occur Depending on the Reynolds number the flow around the cylinder can roughly be divided into five regimes For very small Reynolds numbers up to 5 the flow is domi nated by viscous forces and the flow field is the same in front and behind the cylinder In the Reynolds number range from 5 to 48 the flow starts to separate behind the cylinder forming two separation regions with recirculating flow The length of the bubbles increases with increasing Reynolds number From Reynolds number 40 to 300 the vortices detach periodically from the cylinder and travel down stream This phenomenon is called the Karman vortex street For higher Reynolds numbers the shedding of vortices becomes irregular and the flow becomes turbulent At very high Reynolds numbers the vortices are shed periodically again A detailed description of the phenomenon of the flow around a circular cylinder can be found in 1 In this tutorial we will concentrate our interes
129. and region outside the interval c sm c sm invariant it maps this interval onto itself and the points c sc are moved to c d and it is typically used to construct pyramidal geometries Shifting and stretching with bump and first example of Fig 1 14 Coord coord 1 5 1 3 xbump x 1 4 6 9 1 bump y 1 4 6 9 1 91 bump z 1 0 9 10 1 block 1 0 10 Te oe Shifting and stretching with di lat and second example of Fig 1 14 Coord x dilat y 5 5 4 x 2 0 5 dilat z 5 5 4 x 2 0 5 1 Shifting and stretching with dilat and third example of Fig 1 14 Macro ZD z 1 8 3 2 Coord dilat x 0 5 4 9 ZD dilat y 0 5 4 9 ZD z block 0 4 04 1 9 Coord dilat x 0 5 4 9 ZD dilat y 10 5 4 9 ZD z block 0 4 6 10 1 9 Coord dilat x 10 5 4 9 ZD dilat y 0 5 4 9 ZD z block 6 10 0 4 1 9 Coord dilat x 10 5 4 9 ZD dilat y 10 5 4 9 ZD z block 6 10 6 10 1 9 DeleteME block 4 6 0 10 1 9 block 0 10 4 6 1 9 24 SESES Tutorial September 2012 Figure 1 14 Gradual shifting and stretching Spherical geometries The basic function used to define spherical geometries is given by Routine double sphere T1l double val T1 double cen T1 double mp which gradually maps any cube surface centered at cen into its in sphere depending on the morphing parameter mp If mp 0 the cube surface is left unchanged and for mp 1 we obtain the in sphere To obtain a ball centered at cen just apply this function to a who
130. and skips the first line of the data file that contains comments i e the columns labels With the ListPlot command the plot shown in Fig 1 22 is generated Controlling with Matlab Matlab is a mathematical software tool by MathWorks Inc addressed to scientists and engineers for both mathematical calculations and communication to measurement in strumentation Here we shall discuss how to create an external control system with Matlab that uses simulation results obtained by launching the SESES kernel in batch mode The most straight forward approach is to write to and read from intermedi ate SESES files Within Matlab we can generate a file named MatLab_2_SESES to be loaded by SESES with an Include statement afterwards we run the SESES calcu lation and let SESES write a result file SESES_2_Mat Lab to be read by Matlab The Matlab code for our simple interaction example reads as follows clear all close all Transferring some data to SESES par 0 5 fid fopen MatLab_2_SESES w fprintf fid Define par 20 12e n par fclose fid Invoke SESES dos k2d example s2d Read the SESES results fidl fopen SESES_2_MatLab r res fscanf fidl s fclose fid1 eval res Comment lines in Matlab start with To invoke a command line statement Matlab provides the command dos At the end of the above example the simulation results are read from th
131. ansport models accounting for multicomponent fluid flow and possibly heat transport The layer in between consisting of the gas diffusion layer GDL for the cathode and the anode together with the electrolyte is modeled locally by a 1D model connecting both the 2D cathode and anode domains point to point i e each point within the cathode is connected with a point at the anode through a point to point transversal cross coupling possibly accounting for porous flow multicomponent dif fusion heat transport charge transport and electrochemical reactions The definition of the 1D model is completely left to the user and it is part of the input Simple analyt ical or semi analytical models are generally directly specified within the SESES input syntax however there is also the possibility to solve sophisticated 1D problems by finite difference or finite element methods with the help of numerical routines embed ded in dynamic libraries to be loaded at run time The question is of course if such 2D 1D models make any sense It depends but for SOFC and proton exchange mem brane PEM fuel cells where the lateral dimensions are order of magnitude larger than the electrolyte and GDL layers together they surely do Because of the large ratio be tween thickness and lateral dimensions you may assume that lateral transport within the thin vertical layer may be neglected with obvious advantages The modeling of the 1D vertical transport is completely decouple
132. ap pear in blue and comments in violet to the graphics mode as displayed in Fig 1 3 by pressing the graphics button J Here the geometry of the 2D problem is displayed The boundary of the dam with the lake water is on the left hand side whereas the basis of the dam is at the bottom of the figure The content of the graphics window is updated manually by pressing the right mouse button or the graphics button A or automatically by enabling automatic updates with the button In the graphics mode we can now inspect the computed fields by map ping the field values to the problem geometry This visualization is launched with the field control lad opening up a panel showing a list of computed fields available for display see Fig 1 5 Select the toggle and either the mechanical displace ment or the stress in the choice menu After selecting the field the graphics window will look like Fig 1 5 The cross section of the dam is now colored according to the displacement values with the color legend given on the right side of the figure Optionally one can select the option causing the calculated dis placement itself to be applied to the geometry However the mechanical displacement is negligible small relative to the dimensions of the dam so that to visualize the defor mation one has to choose a multiplicative factor of ca 1000 with the dialog If the field is selected for visualization then the graph reveals that the highest stress occurs
133. arameter Depth 0s 7T5E 3 m to be the same of the channel thickness of 0 75 mm so that the computed mass flows within the channels as displayed by the BC characteristics are effectively the real ones This however will yield wrong values for the electric current through the nopples since this current do not need to be multiplied by the virtual thickness of the device In order to use the point to point domain cross coupling the cathode is defined as the master domain and therefore for the cathode material we enable the built in model Model Coupling1D BlockOffset 1 NoRow H2 H20 O02 CO N2 CO2 Pressure Psi Psi H2 H20 02 CO N2 CO2 Pressure The BlockOf fset model s parameter is used to locate the slave domain i e the anode domain within the ME mesh In this example the cathode is the whole ME block 0 and the anode is the ME block 1 so that we have a ME block offset of one The parameter NoRow will be discussed shortly As last model s parameter we list the dof fields being cross coupled together between the cathode and anode domains which in our example are all defined dof fields This list is followed by the numerical values of the right hand sides associated with the governing equations solved both on the cathode and anode domains together with all non zero cross coupling derivatives As discussed previously these numerical values are obtained by solving locally for a 1D transport model using Newton s algorithm to first obtain the electric
134. arameter that effectively changes its value during animation is left to the user For example it may be the time or some other freely defined parameter Animation is typically used to quickly visualize the changes in the numerical solution due to different parameter values In the example example ParamStudy s2d a SESES Tutorial September 2012 17 JAIE Alba A J P gA DAAN J AN Figure 1 9 Construction process for meshes with local refinement and circular shapes m 2 sequence of calculations is performed using different values of the pressure from 0 to PMax For each computed solution a write operation is performed to write the values of temperature displacement and stress fields to the graphics data file Data The relevant input lines are the following Define Press 1 pressure MPa Dump AtStep 1 Temp Disp Stress Solve Stationary ForSimPar Press At 0 Step PMax 10 9 Note that the sequence of these statements does matter In particular the statement Dump AtStep 1 to write the data for each computed solution must be defined prior computing solutions with the Solve statement When viewing the results of our example in the draw window an animation can be selected by choosing the option when one of the Temp Disp or St ress field has been selected for visualization A scrollbar is also available t
135. are expressly disclaimed The author NM its licensees distributors and dealers shall in no event be liable for any indirect incidental or con sequential damages NM wishes to record its gratitude to the numerous individuals that contributed ei ther through discussions proposals support or testing to the development of this project In particular we wish to thank Prof Dr Edoardo Anderheggen who has over many years supported the design and development of SESES The following institu tions supported the authors during the initial development of SESES Swiss Federal Institute of Technology in Ziirich the Commission for Technology and Innovation of the Swiss Federal Government and Landis amp Gyr Zug Contents Introduction 1 Getting Started 1 1 The SESES Environment o s n pe Re wa a ee ee Be Sle FR s 12 B ilding Your Fist Model s es ghee He SG ERE ERA LER EYES 1 3 Graphical Visualization 4 444625 ebb ew CEA REDE RE RE OS 14 AMimation 2 2 kee ee ee RR eR Ee ee Ee 1 5 Preprocese Mesh Definition os s cos sacd a sacre eanes wR RR 1 6 Algebraic Mesh Definition 4 2 sa caa eR EPR Oe a 1 7 Non Linear Algorithms ananasa aaaea aaa 18 Contingation Methods spe rosta oira ee ee be E a e 1 9 External Post Processing and Controlling 0 2 Application Examples 21 UNCTINGIGHIAID 24 52 2 eo 4 eee Oe Saw SERRE ES PH Ow eH 2 2 Designing a Hall Sensor gt o coss socs eA DRO PR pipron 2 3 Electrothermally Driv
136. are the symmetric tensors Cp F Fp or C5 which are independent from any rotation associated 144 SESES Tutorial September 2012 with Fp By considering the symmetry of and Of 08 together with p Cp d dt C5 Cp from 2 113 we have of S where the last equality just a if we have sym C7 0f C7 df O8 or equiv alently C 0f 0S Of OS C This is indeed the case for isotropy where f S can only depend on the ae hi Tif A of the spectral representation 2 109 of S The relation 0f 0S f m N 2 N shows that S and 04 08 are coaxial and from 2 109 this is also the case for C and S therefore C and 0f 0S are simultaneously diagonalizable and they commute oF d dt C5 2 F5 sym G7 Fo SSP ti SRC eii P P Several numerical algorithms are available to integrate 2 114 and it is important to note that they must all be formulated with respect to the fixed stress free configuration Q By applying the first order implicit solution 241 exp n41 2n of 2 t a t z t ont tn tr41 to 2 114 we obtain CF aa Fp 0 2A 0p farsa Pavan Ce 2 115 Since det exp A exp tr A holds for any matrix A see 2 if the plastic flow is pressure insensitive f T f dev r then 0 f dev 0 f and tr dev 0 f 0 so that there is exact numerical conservation of the plastic volume det Fp det C This condition is generally required on the onset for metal plasticity and it is the rea
137. as ily lead to divergence In this example by taking just one third of their values with setincr Incr 3 divergence in the following steps is avoided After the first step we specify for 100 times the computation of fast increments with ReuseFactoriz and again we are forced to divide their values by 4 in order to avoid divergence Al though some tricks were necessary to avoid divergence the algorithm converges with a slow rate but each single step is fast since the linear system matrix is factorized just in the first step Write Solving with fast increments n Convergence write Step 2 0f AbsIncr Temp e Rate e n niIter AbsIncr Temp quot AbsIncr Temp return AbsResid Temp lt 1 E 15 Increment setincr Incr 3 ReuseFactoriz setincr Incr 4 100 Solve Stationary For this small 1D example running very fast the speed up when using fast increments is not clearly identifiable but for 2D and 3D large problems it will Solving with fast increments Step 1 AbsIncr Temp 1 855581le 00 Rate inf Step 2 AbsIncr Temp 2 489984e 01 Rate 1 341889e 01 Step 3 AbsIncr Temp 5 581262e 02 Rate 2 241485e 01 Step 25 AbsIncr Temp 4 052745e 15 Rate 3 768345e 01 Step 26 AbsIncr Temp 2 515550e 15 Rate 6 207028e 01 Step 27 AbsIncr Temp 1 509828e 15 Rate 6 001977e 01 From the output we see that the derivative computed in the first step is good enough to reach convergence although with a slow rate and by cutting the increments If the
138. as before but with possibly inhomogeneous values In par ticular we assume the driving voltage to have a value of 1 V A solution proposal satisfying the same BCs of the inhomogeneous solution but now with a driving volt age of Vo exp iwt can be given in the form Fi Jve anle ra 2 79 with the time dependent modal participation coefficients x to be determined After inserting the proposal 2 79 into 2 75 taking the scalar product with um assuming the cross coupling coefficients Jo YUn UmdV to be zero for m n we arrive at the equations for the x n nin wrn Vet Anw iynu 2 80 with u2 An lt Up Un gt ug s u dV and yy T Yun UndV 2 81 n JQ Q The time dependent solutions for the coefficients x are then given by wy w wn Enlt Vo Anw iynw yn w e with yn w CE ER 2 82 106 SESES Tutorial September 2012 The piezo crystal is generally characterized by its impedance Z w V w I w V w Q w with V the voltage J the current and Q the charge at the contact which is given by the surface integral Q D dn f e E g e u dn 2 83 No No with n the outward normal to the surface If we denote by ag and a the charge of the inhomogeneous stationary solution and of the eigen solutions then the contact s charge is given by CO CO Q Voaoe X antn t Voaoe Vo X an Anw tynw yn w e n l n 1 In the appendix we
139. ast example we have presented the thin disk laser design which is well suited to overcome this draw back But this design is limited to quasi longitudinal pumping requiring high quality diode pump modules and beam shaping optics The pump radiation has to pass the disk several times to guarantee a high absorbed part of the pump power in the thin disk In practice this is realized by a special and costly pump optics Furthermore the maximum pump power per disk is limited and for high power lasers two ore more disks have to be used in one and the same resonator The scaling of this design to higher output powers requires therefore an additional effort In contrast the scalabil ity of transversal diode pumped rods is much easier In principal we only have to increase the length of the laser rod and the pump modules Also the requirements to the beam quality of the pump modules and the beam shaping optics are much lower Therefore transversal diode pumping represents still attractive design for high power solid state lasers In the present example we will introduce a new and very promising transversally diode pumped laser design with an internal compensation of the thermally induced lens The design bases on the idea to take advantage of the thermal effects themselves and to use a heated optical element as a compensating element 1 2 The thermal lens in a rod is evoked by the temperature dependent refractive index of the laser gain medium which mainly s
140. at the lower left corner of the the dam which faces the water You may wish to rerun this simulation with different parameters For instance find out which parameter values can be modified to give a mechanical displacement that is no longer negligible but leads to a significant distortion of the dam geometry due to the water pressure of the lake 1 2 Building Your First Model In the example of the previous section the reader was introduced to navigating in the Front End program and running a simulation In the next introductory example SESES Tutorial September 2012 13 applied Substrate Air Tip Substrate Figure 1 6 Illustration of the electrostatics problem for which a SESES simulation is being built and its representation in SESES found at example E1StatTip s2d we will walk through the process of building a complete model i e of writing an input file We will document on how to implement the approach discussed in the section about the modeling strategy For this purpose let us consider a 2D dielectric domain with two electrical contacts where an electrical bias is applied to Our model shall correspond to the cross section of a tip near a substrate surface at ambient air conditions both contacted electrically as in the tunneling mode of a scanning probe microscopes see Fig 1 6 We wish to calculate the electrical potential and the corresponding electric field denoted by the SESES keywords Phi and Efield respecti
141. ate is so good that the problems to be solved are generally bounded by the large memory requirements and not by the computational time This is also true in view of the fact that the solution time can be further reduced by parallelization and multigrid methods technologies not discussed here For nodal elements of second order and the ILU 0 preconditioner the dependency on the convergence rate from the frequency is again low but much more pronounced for the slenderness factor with an impressive slow down here an example for w 1 a 1 10 20 30 40 50 60 70 Iter 17 29 34 46 77 263 643 1838 As next we present a speed up method similar to a two cycle multigrid method based on the robustness of the ILU 0 preconditioner for first order nodal elements If we have some additional memory at our disposal we can apply a technique known as auxiliary preconditioning where we use the solution obtained with first order ele ments to precondition the system of second order elements On the same mesh and with the same BCs one can show that the system matrices Sgrs Sona obtained with element of first and second order are related by Sang ISarst I with I a prolon gation matrix which is easily computed numerically The left preconditioner is then defined as Peft ISL IT and its application yields Paa M a i 2 73 102 SESES Tutorial September 2012 Freq w Slend a Pre 1 Post 0 Pre 0 Post 1 Pre 1 Post 1 Pre 2 Post 2 s T00 6703
142. ations are also uncoupled to each other they just depend on the normal current flow at the anode electrolyte interface and so we can solve each equation in turn to get a solution of this SOFC system However this 1D model is also very small that we may decide to solve all equations together and since the full system is linear just a single step is required Convergence 1 BlockStruct Block Psi H2 02 Solve Stationary Numerical Results For this example Fig 2 152 shows the potential distribution along the vertical line The potential jump 2 178 is given by 1 01 V and we have quite a large potential drop within the electrolyte due to the small electric conductivity Because of the small vertical dimensions the concentrations of Oz Hz are almost constant and equal to the ones set with the Dirichlet BCs The current is overall constant and given by the prescribed value at the floating BC for the potential For this example the only figure of merit is the knowledge of the potential at the anode which gives us the electric power supplied by the cell This value is given by the computed potential shift at the floating BC A summary of these working conditions is given with the following statement Write Text Potential 3f V Current 3f mA Power 3f mW n Anode Psi Shift current Anode Psi Shift current As stated before for this simple example the potential is completely independent from the chemistry so that the figure of merit may
143. ations within the shell elements defined for the thin cantilever whereas for the bulky part we use standard displacement volume elements At the inter face between these two types of elements additional kinematical constraint equations must be satisfied among the various mechanical dofs but they are set automatically so that the mechanical rotations are somewhat hidden to the user except for the speci fication of Dirichlet BCs For illustration of the results the temperature field Temp can be displayed together with the mechanical displacement as shown in Fig 2 10 The hottest spot is at the outermost region of the cantilever while the greatest heat flux is located at the anchor of the cantilever The displacement and strain fields are shown mFig 211 2 4 Heat Conduction in a Cylindrical Stick Heat issues are of common interest since they occur in many distinct application fields and in general are the result of coupled effects Thus it is worthwhile to consider a basic heat problem here in order to get familiar with the relevant simulation concepts Let us consider a cylindrical aluminum stick of total length 125 cm and a diameter of 4cm being cooled at one end and heated by a gas burner at the other end For an overview of the experimental setup see Fig 2 12 Essentially the elongated geometry 46 SESES Tutorial September 2012 water cooling measurement slots x atu ick Y AUUUTANUNLUAEVORRAGVOURAENUNUTROUAAAGLT Alu Heating Cooling Trans
144. be obtained by just solving for the poten tial However we have computed the O2 H2 concentrations so that if one defines the Nernst potential as a function of these concentrations one can right away study a full coupled system As a next step in refining the modeling of a SOFC one should con sider the temperature dependency of the heterogeneous reactions and the fuel supply which may decrease the reaction rates whenever a shortage occurs References 1 J LARMINIE A DICKS Fuel Cell Systems Explained Wiley 2000 2 A A KULIKOVSKY Analytical Modelling of Fuel Cells Elsevier 2010 210 SESES Tutorial September 2012 2D Cathode Flow Field Local Interaction 1D aca cathode vVVVVVVY 2D Anode Flow Field electrolyte anode interconnector Figure 2 154 The 2D 1D modeling approach with 2D cathode and anode Figure 2 153 View of a single layer in a SOFC stack domains and a 1D point to point cou system pling interconnector gas inlet 2 36 A planar 2D 1D SOFC model In this example we further develop the previous tutorial example and go a step fur ther in the modeling of a solid oxide fuel cell SOFC system like the one shown in Fig 2 153 In particular we are going to use a 2D 1D modeling approach which al lows us to avoid computationally intensive 3D calculations without impairing too much on the model s quality see Fig 2 154 The anode and cathode are modeled sepa rately by 2D tr
145. been suggested to prevent the leak of the liquid metal at the end of the cylinders Experimental studies indicate SESES Tutorial September 2012 75 Figure 2 50 Two dimensional model of the experimental situation Figure 2 51 Finite element mesh for the circular rod and surrounding large repulsion forces between a liquid metal bath and a copper rod subject to alter nating current Fig 2 49 demonstrates the electromagnetic repulsion between liquid bismuth tin and a copper rod carrying 40 kHz AC current The ditch in the liquid is about 0 5 cm deep For a half cylinder shaped ditch in the liquid with the dimensions r 0 5cm and 1m and the density p 8700 kg m we calculate after Archimedes principle a buoyancy force of 1 m Fhioyny p 37 78 9 81 z7 34N 2 37 To expand the experimental data to different excitation frequencies and currents the following model calculations have been performed The result of 2 37 will serve for validation of the computations The repulsion principle and its SESES model could then be applied to design the turns of a coil which prevents the leak of the liquid metal at the end of the rotating cylinders in Fig 2 48 Implementation of the Model A two dimensional model is used to describe the experimental situation of Fig 2 50 The input file example Repulsion s2dis divided in four parts The first part com prises the definition of parameters and dimensions as well as the material propert
146. c view of a SOFC andin Figure 2 152 Potential distribution volved chemical reactions along the vertical efficiency d Teold out Teold in _ 292K 280K 0 48 Thot in Toold in 305K 280K According to our definition at the beginning of this example this value corresponds to a good heat exchange efficiency 2 35 A first model of a SOFC fuel cell In this example we are going to model a simple solid oxide fuel cell SOFCs are solid state devices working at high temperature of 800 1100 C not requiring complex catalytic reactions at the electrodes to run the electrochemical fuel oxidation which may be hydrogen or carbon monoxide In their simple form they consist of a cathode and an anode made of porous materials and in between a solid porous electrolyte see 1 2 Fuel cell modeling is in general a complex task involving chemical reac tions current flow mass flow and energy conservation In this example we consider the combustion of hydrogen and focus on the chemistry of a 1D system at constant temperature leaving out for other examples topics like convective flow fuel delivery fuel recombination temperature distribution and fuel cell optimization Our fuel cell system is depicted in Fig 2 151 the hydrogen fuel is delivered at the anode and oxygen is delivered at the cathode by an external pipe system not show in the figure Both gases diffuse towards the electrolyte and at its boundary we have two heterogeneous rea
147. c2 requiring the definition of the material parameter Mue Val defining the relative permittivity j1 j19 and by setting the above Dirichlet BCs for the dof field MagnPot By splitting the evaluation of 2 19 in two parts with the contribution from the core as before but now with a numerical contribution from our computational domain Q 60 SESES Tutorial September 2012 we have I yie H as f Ve ds Blcore 2 A OANQDE OANQ H core with l re the length of the core outside the computational domain The value B core can by approximated by the average Blcore ee f B dn Score A N with F A the BC flux directly available as SESES output Due to the linearity of the problem we have F A F 1 A so that NI Soph ucctelSore 1 1 A and evaluation of the second integral in 2 18 now yields d dA lr F 1 l dI N B dn NF 1 N F 1 2 e dt D e dt m a 1 dt i Score resulting in the inductance L N Sore ze Hcore F 1 It is apparent that the relative error between both approximations of L tends to zero with core gt and so we limit ourselves by comparing the different terms within the denominator Score leore 2a loe d h d n Se De F 1 core Hdisk Ho where a value of R 1 implies equivalency From the textual output generated by running the example it is apparent that with a value of R 0 5 the scattering of the magnetic field in the
148. cal coordinates with x p and z pcos for a generic function g f p cos we have V g 0 f p cos ep f p p sin e9 With H VOtot we see that the tangential component H x n H eg is continuous if O 4 is continuous and the normal component B n B ep is continuous if LO tot is continuous which is indeed the case Since within the ball x lt r both the B and H fields are constant even for a non linear permeabil ity rel Mret H is Oot a solution provided H is a solution of equation H ret H 1 Hrei JH 2 This solution is rotational symmetric and the accompagning SESES example can be found at example TotRedFormul s2d In order to compare the numerical solution with the exact analytical one we use exact BCs For the total formulation we just set a Dirichlet value of on OQ and for the reduced formulation we set the Dirichlet value eq and define the Ho field as e x 1 A m by specifying the material parameter Hfieldo For moderately values of ure gt 1 both formulations yields the same results and convergence with respect to mesh refinement is according to the theory i e for first and 2nd order when using H Q conformal elements selected with the equation statement MagnetoStaticorMagnetoStatic2 Even for very large values of Hrej the total formulation is very stable however not the reduced one The problem lies in the cancellation of the terms uV Orea WHo within SESES T
149. cally unstable For these reasons we do not try to obtain the derivative from the function SESES Tutorial September 2012 29 itself but the user should specify the derivative in order to get the correct convergence behavior We thus provide the derivative and try to solve again the non linear thermal problem MaterialSpec Silicon Parameter KappaIso D_Temp 1 Temp 1 W m K W m KxK Write Solving with exact derivative n Solve Stationary This time we obtain a quadratic convergence behavior Solving with exact derivative AbsResid Temp 9 64e 02 AbsResid Temp 8 73e 01 AbsResid Temp 2 06e 01 AbsResid Temp 3 42e 02 AbsResid Temp 1 61le 03 AbsResid Temp 3 74e 06 AbsResid Temp 1 56e 11 The default convergence criterion is a value of 10 for the residual norm and when ever the norm value is less than this tolerance the Newton s iteration is successfully ended This convergence criterion however can be specified as a function of any com putable norm and the actual iteration number nIter Since for our thermal problem an accuracy of 10 K for the temperature may be considered as acceptable we solve again by setting the new convergence criterion We also visualize the conver gence rate by using the built in function quot to obtain the quotient of the function value in the previous iteration Write Solving with reduced accuracy n Convergence write Step 2 0f AbsIncr Temp e Rate e n nit
150. ccel pow 1m 3 Thot TO TO pow kinVisco 2 Together with the material properties of air already defined in the initial section for each run these results are then written to external properties files Solution strategies As explained in the introduction the dimensionless Rayleigh number Ra character izes the flow regime that follows from exposing a fluid in a gravity field to an ex ternal temperature gradient In our example where h s 50x10 m and AT Thot Teora 10K this yields a Rayleigh number of Ra 1 08x10 lt 108 ie a laminar flow field is achieved If the height and width of the square cavity is enlarged to a value of h s 200x10 3m the Rayleigh number increase to Ra 6 9x 10 i e it is now much closer to the Ra number of Ra 108 where the transition be tween laminar and turbulent flow occurs In an experiment this would mean that any disturbances which might occur in the flow need some time to be dampened by viscous forces eventually these disturbances completely disappear The numerical simulation shows a similar behavior However a simulation under these conditions is likely to fail Convergence might not be achieved due to numerical errors which SESES Tutorial September 2012 179 will not be dampened out during the iteration process If the simulation is initiated with the initial condition that the air is at rest uini Vini 0 and with a uniform temperature field of Tini Thot
151. ce when one has to apply voltages and measuring currents or doing the reverse the wires are not closed in 2 since they need to be connected to external measuring equipment From a numerical point of view however we may assume ideal contacts where the measuring equipment has no influence on the system In order to define ideal contacts for each conductor Nm m 1 N we select a generic cross section Cm with C Cz being the two distinct oriented surfaces of Cm We then allow the SESES Tutorial September 2012 89 potential Y to have a constant jump in its value when crossing this surface The jump value is given by the path integral Vn U C C2 f Vd I E amp A dl 2 54 with Ym any directional path in Qn joining the surfaces Ct gt C at the same point The invariance of V from the choice of Cm is given by the fact that a constant jump of WV does not change the gradient E V Y Since there is no current flowing through the conductor boundary Nm and by considering V J 0 together with Gauss s the orem we see that the computation of the conductor current Im is also invariant from the choice of Cm However we have to consider the correct side in order to get the correct sign and by the above choice of Vm we have Im J dn 2 55 Cr It is customary to study the dynamic of a linear system by using the complex nota tion for all involved fields with the real part representing the physical value and by assu
152. ce of the crystal This influence is summarized in the end effect optical path difference OPDeng and it is given by OPD ait fo 1 yn ndz 2 92 0 with e the mechanical strain tensor and n the normal vector tangent to the integration path In the case where the crystal surface acts at the same time as resonator mirror the factor no 1 in 2 92 has to be replaced by no The sum OPDtot OPDin OPDena can be compared to the OPDiens produced by an ideal thin lens of focal length flens given by r2 2 flens l The thermal lensing effect including both the thermal dispersion and end effect OPD can in first order be described by the focal length of an averaged thermal lens The value of the focal length is obtained from a parabolic least square fit using 2 93 to the calculated OPDio r OPDtens 7 OPDo 2 93 Depending on the shape of the pump distribution almost arbitrary OPD profiles are possible Especially for longitudinal pumping the thermal load is very inhomoge neous and the radial dependence of the OPD is far from being parabolic The focal length of the fitted lens is therefore not the same whether the fit is performed over the whole rod radius or just over a part of it For the reminder of this example the fit was extended over one pump spot radius as the laser beam is usually adjusted to meet its size Fig 2 68 shows this situation Laser set up This example presents the simulation of
153. ch can be found with the help of the Ansatz Ty Ce Back substitution yields the characteristic polynomial and the general form k a 0 gt AL 2 gt Ty 2 Ce O 2 7 with the two coefficients C and C2 determined by the boundary conditions 2 6 At x 0 we obtain C2 C C and since dT dx h k at x L we find C h KA2 cosh AL with a k The solution to 2 5 2 6 is therefore T x Tye Ta 2 The sinh Ax 2 8 h 2KA cosh AL The larger the heat transfer coefficient a the stronger the temperature profile devi ates from the linear solution obtained for a 0 In other words a determines the curvature of the temperature profile SESES Tutorial September 2012 47 Building a 2D model for the heated stick We shall start with a 2D model of the heated stick which will be used to calculate the time dependent temperature at selected points as well as complete temperature pro files The input file can be found at example HeatStick s2d These calculations will be compared to measurement data acquired at the slots indicated in Fig 2 12 The mesh of the 2D model is shown in Fig 2 13 In order to get accurate simulation results one needs to consider heat loss by radiation and convective transport apart from the cooling and heating boundary conditions mentioned above in 2 6 In heat problems it is common to introduce a phenomenological heat transfer coefficient a as the proportionality
154. computational expenditure may be of concern Therefore we are not going to present in full details the modeling of a SOFC system ready for production but limit ourself to the main features In partic ular we make the isothermal assumption and assume a constant working temper ature of T 1223k Extensions to include the temperature dependency are quite straightforward once the temperature dependency of the the material laws is known One just has to provide these dependencies and for fast convergence their deriva tives with respect to the temperature We also are not going to consider to many species since they just increase the computational cost without necessarily improv ing the insight understanding We thus consider the species O2 No at the cathode and H2 H20 N2 CO CO at the anode The 2D cathode and anode domains are ac tually two distinct flow channels the cathode is an in air flow suppling oxygen and at the cathode we have a gas flow suppling hydrogen for a proper operation of the the electrochemical reaction in the fuel cell We assume both flows to be laminar and irrotational i e a potential flow which allow us to use a simple linear model to com pute the velocity v and pressure p with the model equation PorousFlow Within both channels additionally to the overall mass transport there is mixing of species by diffusion convection and possibly chemical reactions Therefore for the cathode we solve additionally for the equation
155. convergence rate is too slow or if divergence shows up then one has to update again the derivative It is possible to do it on a static base by specifying the computation of for example once standard increment followed by three times fast increments and by repeating this cycle However it is also possible to do it dynamically with the help of increment control Write Solving with increment control n Increment Standard setincr Incr 3 ReuseFactoriz Control quot AbsResid Temp gt 0 1 1 0 Standard Control 1 Solve Stationary SESES Tutorial September 2012 31 Cylindrical roof under pressure Pressure load 0 18 1 r r y 0 16 0 14 0 12 F 0 1 F 0 08 F 0 06 F 0 04 0 02 0 L L j L L 1 Displacement 0 2 4 6 8 10 12 14 416 Pressure Load Tip Displacement Figure 1 19 A cylindrical roof under pres sure Figure 1 20 Load displacement curve for the cylindrical roof under pressure load In the first step of this example we compute standard increments and divide them by 3 to reduce initial overshooting We then define fast increments and if the conver gence rate falls below 0 1 we update again the derivative by performing a full linear step This is achieved with the Control statement allowing to jump back and forth between the different increment directives The first increment directive Standard setincr Incr 3 is executed just once at the beginning then the second directive ReuseFa
156. ct with the flowing cooling water and are transversally pumped pumped through this cooling water For simplicity we will assume that the compensating disk shows no absorption at the pump wavelength Model Specification With the present example example Tosca s2d we will perform a parameter study to obtain the optimal thickness of the thin compensating disk for a set of given geo metrical and pump parameters where for simplicity we will perform just a rotational symmetric simulation Fig 2 90 shows the simulated device consisting of two rods and the compensating thin disk The outer rod parts of the length luc are in contact with the mount see Fig 2 89 and are therefore not directly cooled by water Adjacent we have two parts of length cup which are directly cooled by water but not pumped by diode radiation The disk with thickness lep is located between the two pumped rod parts of length These geometrical parameters together with the rod radius r are defined at the beginning of the input file Define luc 1 0E 3 nl_uc 1 x uncooled length m l_cup 2 0E 3 nl_cup 1 x cooled and unpumped length m x l_p 20 0E 3 nl_p 10 pumped length m r 2 0E 3 nr 5 rod radius m x nl_cp 4 128 SESES Tutorial September 2012 l_cpmin 0 6E 3 x minimum length of compensator m l_cpmax 1 3E 3 x maximum length of compensator m gt n_l_cp 3 x steps in length of compensator For the parameter stu
157. ction calls and graphical output First example of Fig 1 16 Coord cylx coord 5 5 1 block 2 7 0 10 0 10 Second example of Fig 1 16 Coord cyly coord 5 5 0 5 block 1l1 9 3 101 9 Third example of Fig 1 16 Coord cylz coord 5 5 1 block 0 10 5 10 0 10 DeleteME block 2 8 5 8 0 10 First example of Fig 1 17 Coord cylxmix coord 5 5 3 5 2 1 Second example of Fig 1 17 Coord cylymix coord 5 5 3 5 2 1 Third example of Fig 1 17 26 SESES Tutorial September 2012 y Figure 1 17 Cylindrical geometries with cylxmix cylymix cylzmix Figure 1 18 Cylindrical geometries with cylxmix cylymix cylzmix Coord cylzmix coord 5 5 1 2 2 1 Coord cylzmix coord 5 5 5 2 2 block 0 2 0 10 0 10 Coord cylzmix coord 5 5 5 2 2 block 8 10 0 10 0 10 Coord cylzmix coord 5 5 5 2 2 block 3 7 02 0 10 Coord cylzmix coord 5 5 5 2 2 block 3 7 8 10 0 10 First example of Fig 1 18 Coord cylxmix coord 5 5 5 2 1 1 Second example of Fig 1 18 Coord cylymix coord 5 5 5 2 1 block 0 10 4 10 0 10 Third example of Fig 1 18 Coord cylzmix coord 5 5 5 2 1 block 0 10 0 10 0 7 1 7 Non Linear Algorithms This section illustrates some of the available features when solving non linear prob lems A linear problem is solved in one single step by finding the solution to the linear system of equations obtained when discretizing the governing equations Non linear SESES Tutorial September 2012 27 problems instead
158. ctions taking place At the interface between cathode and electrolyte we have the reduction of oxygen where electrons from the external circuitry combine with the oxygen The oxygen ions diffuse then towards the anode electrolyte inter face where the oxidation of hydrogen takes place delivering electrons to the external circuitry The stoichiometric coefficients of both reactions are given by Oxidation of Hy 2H 20 gt 2H O 4e Reduction of O Oz 4e gt 20 2 176 By closing the external circuitry a current starts flowing because at the anode electro lyte interface the oxidation of hydrogen pumps carriers uphill resulting in a discon tinuous potential of ca 1 1 2 V This discontinuity called the Nernst potential is an SESES Tutorial September 2012 207 expression of the difference in the free Gibbs energy between educts and products of the heterogeneous reaction It is an optimization task to minimize the electrical lost within the fuel cell and have most of the Nernst potential available to drive the ex ternal load The rates of the reduction and oxidation reactions 2 176 are the rates of production and consumption of electrons at both interfaces and therefore proportional to the normal current flow at the interface and given for the stoichiometric coefficients of 2 176 and a unit of moles per surface and time by j n 2 177 4 qNavo rate with j n the electrical current normal to the interface anod
159. ctoriz takes place and the control function is evaluated to eventually jump to another directive Here if the convergence rate is below 0 1 we jump to the next directive Standard which performs a full linear step and jumps back to the second directive Solving with increment control Step AbsIncr Temp 1 855581e 00 Rate inf tep AbsIncr Temp 2 489984e 01 Rate 1 341889e 01 tep AbsIncr Temp 1 545840e 01 Rate 6 208232e 01 AbsIncr Temp 1 635748e 02 Rate 1 05816le 01 AbsIncr Temp 5 149250e 03 Rate 3 147948e 01 AbsIncr Temp 3 002182e 05 Rate 5 830330e 03 AbsIncr Temp 3 765816e 07 Rate 1 254360e 02 tep AbsIncr Temp 4 923727e 09 Rate 1 307479e 02 tep AbsIncr Temp 6 560768e 11 Rate 1 332480e 02 tep 10 AbsIncr Temp 8 815400e 13 Rate 1 343654e 02 tep 11 AbsIncr Temp 1 189537e 14 Rate 1 349385e 02 tep 12 AbsIncr Temp 5 359217e 16 Rate 4 505298e 02 a oO ue tep tep ODAHDUBWNHE Oe Oe eenee pep ee envep oO oO From the output we can see the speed up in the convergence rate as soon as we per form a full linear step updating the derivative and its inverse 1 8 Continuation Methods This section illustrates some of features available in SESES when computing families of solutions depending on user parameter We will go through the discussion by pre senting the example example ContMethod s2d of a cylindrical roof under a pres sure load from above as displayed in Fig 1 19 We will compute different solutions for different val
160. current and then by setting all required input values according to stoichiometry of the electro chemical reaction For convenience we have used Maple to generate once the rather lengthly input text which is not shown here At the bottom of the cathode we prescribe an air flux of 0 0001kg sm with a floating BC for the pressure and fix the mole fraction s value with Dirichlet BCs by respecting the constraint a 1 At the top of the cathode we do not prescribe any BCs so that convective flow will be free to leave the cathode however conductive flow will be blocked In a similar fashion we set the BCs on the anode with a gas flux of 0 001kg sm Larger values of air or gas fluxes will impair or prevent the convergence of the solution algorithm due to the coarse element mesh around the nopples where the velocity field is forced to change direction The dof fields are initialized in order to satisfying from the beginning the constraint ta 1 and they are set to the values of the Dirichlet BCs As noted previously since the mole fractions must sum to one a species is completely redundant and for speed up it would be nice to remove this redundancy As first we have to choose the species to become the dependent species and as will be clear in mo ment one should take the most inert one in our case N2 One then proceeds exactly as when computing solutions with N2 and the problem specification is virtually un changed up to compu
161. cursion computes A 0 and solves the magnetostatic problem V xp Vv x A 0 J 0 V A 0 0 2 63 SESES Tutorial September 2012 91 with the current J 0 gt gt Jn 0 gt 7En 0 computed by 2 61 By linearity we have A 0 gt 0 An 0 with A 0 the solution of 2 63 with source term J 0 instead of J 0 The major difference with 2 61 is that each singular problem for A 0 has to be solved in Q and not just in Qn Without giving a proof this magnetostatic solution can be used to compute the induc tance matrix L w of 2 56 at w 0 which is actually the magnetostatic inductance L determining the energy W By assuming B 0 on 0Q we have w 8 B 3 H vVxa 5 YVx A 5 JJA 2 Jo 2 Jo 2 Jo 2 Jo 1 1 2 64 m An R Lrinlmlns 2D fn A 3 with i Imtn Qm The next term in the recursion 2 60 computes 0 E 0 and solves V x 0 E 0 iA 0 0 V cd E 0 0 with the vector potential A 0 5 An 0 computed in 2 63 The source term iA 0 is pure imaginary and so the solution 0 E 0 as well Therefore we define the real scalar field with 70 E 0 V A 0 determined by solving in each conductor Qn the real equation V cV cA 0 0 2 66 By linearity we have 0 E 0 5 0 En 0 with 8 En 0 given by solving 2 66 with source term a An 0 instead of o A 0 The solution of this problem is very close to the electrostatic problem 2 61 and in fact we have the same linear matric
162. d compute additional steps Write Computing further solutions n Extrapolation NoExtrapolation Solve Stationary Skip Step 1 2 10 Since at the actual value of the simulation parameter 1oad 5 the solution has already been computed in the last solution step we use the Skip option to skip computing again this solution At the pressure load of load 7 Newton s algorihtm slows down and needs much more initial iterations to reach the basin of quadratic convergence At the load of 1oad 8 it does not converge anymore and the solution process is stopped As discussed previously no solutions exist for this pressure load close to the actual computed solution Since many mechanical structures due to their limited elasticity do not survive a dynamical snap through process knowing the maximal value of the pressure is equivalent to determine the ultimate collapse point We may combine here the impossibility to find solutions with a try amp error technique to determine the maximal load This method is expensive and not optimal from a computational point of view but it is easy and simple to use We approach the maximal load by below with an adaptive incremental procedure As soon as a solution is successfully computed we increment the load otherwise we restart the computation with a smaller incremental step The step length will converges toward zero and when small enough we stop the computation Write Computing limit point by try amp error n
163. d defining the input file Optionally a similar section defining the input SesesSet Up file may follow containing settings for the Front End program The input container file ends with a section only used when the input file is used as a shell script Here the Kernel k2d is started to run the simulation Running Seses k2da Starting a Calculation and Viewing the Results Rather than walking through all the details of this example we proceed by comment ing how to compute and display numerical results Regardless of the active graphics or text mode pressing the run button will start the calculation We may now return 12 SESES Tutorial September 2012 SESES 2D home seses develop example tutorial Dam s2d rie ojsjuje sjaje sjef afe b Disp Norm m M A 1 71E 02 Li 1s 1 536 02 M Dray 1 44E 02 File amp Slot Dat 135 02 o 4266 02 m 1 17E 02 Fi p 109 02 Disp Stress 3 89E 03 8 99E 03 8 09E 03 7 198 03 om 62 03 __fidd Disp Dis ac s 4or o3 DiS Fact Style aes e 3 606 03 V Legen 2 708 03 1 80E 03 ae Lj ble Bao 0 006400 MoO O Figure 1 5 Graphical visualization within the Front End pogram showing the computed mechanical displacement 3 a ele am s top width m s bot tom width Figure 1 4 The text editor of the Front End program The input file is displayed with syntax highlighting causing keywords to
164. d from the 2D transport and so nu merical difficulties related to the large geometric ratios are avoided Further no matter how complex the 1D model may be the overall cost is just given by its evaluation growing linearly with the number of elements used by the 2D transport model So the limiting factor is always represented by the solution of the 2D transport problem which is mainly determined by the number of dof fields and the type of cross coupling between the cathode and anode domains For this 2D 1D approach there is a special built in model Coupling1D allowing to couple point to point two non overlapping domains through the right hand side of the modeling equation associated with some dof fields The list of dof fields which can be coupled is free and the implementation is optimal in the sense that if the whole system to be modeled is linear then the solution is obtained with a single linear solution step However the computational work grows SESES Tutorial September 2012 211 with the number of dof fields involved in the cross coupling so that if the coupling is weak one may be faster in computing solutions by using a Gauss Seidel solution strategy and setting to zero some or all cross coupling derivatives Mathematical model Even for the simplified 2D 1D approach there are many possible choices available for the model s complexity as e g the number of species involved thermal depen dency dynamic behavior so that even here the
165. d proximity effect in wires and coils and from finite el ement computations derive effective values for resistance and inductance For a single straight wire a 2D computation is well suited but for coils an exact analysis would require a 3D computation This is still a very expensive operation but if one is ready to accept a small analysis error then valid 2D formulations are available and will be presented General Theory Starting from the Maxwell s equations one arrives at the eddy current equations by as suming zero displacement currents D 0 which for good conductors are typically small and therefore can be neglected For an harmonic analysis at a fixed frequency we may use the complex notation with a time dependency of the form t e which save us from directly integrating the governing equation with respect to time Within the conducting wire we have the material law J cE and by considering a homogeneous conductivity a from the property V J 0 we obtain V E 0 With these assumptions and for a 2D domain one then derives the following governing equations Vu IVA iwo A Joz 0 2 29 with A 0 0 A the vector potential to be computed and Jo 0 0 Joz the driving external current both complex quantities The resulting magnetic field is given by B V x A and the electric field by E 0 A iwA The effective current has only a non zero z component J 0 0 J given by J Joz iwo Az
166. dary layer has almost penetrated the whole pipe cross section This is in accordance with the estimation of the hydrodynamic entry length The maximum velocity at the pipe center line is decreasing with travel length of the fluid This is due to the temperature reduction which in turn leads to a density increase and in the end to smaller veloci ties In Fig 2 141 temperature profiles at various locations along the pipe length are shown The development of the thermal boundary layer is also clearly shown The de velopment of the temperature profiles proof that the thermal entry length is correctly estimated If we like to compare our result of the numerical simulation with the calculation ac cording to VDI Warmeatlas we must compute average temperatures Here we calcu late a mass weighted average of the temperature as Jf pvTdaA which is formulated in the following way T_a integrate Bound b_pos0a Temp Velocity Y x integrate Bound b_pos0a Velocity Y x We consider a slice and calculate the energy balance for this control volume The diffusive heat transfer through the two cross sections and to the wall is computed To do this at different locations along the pipe we define a loop as shown below For i From 1 To noData 1 Define Ta integrate Bound b_pos i a Temp Velocity Y x integrate Bound b_pos i a Velocity Y x _b integrate Bound b_pos i b Temp xVelocity Y x integrate Bound b_pos i b Velocity
167. de induces a large inhomogeneous thermal strain caus ing the cantilever to bend if the temperature is not 300 K Since the aluminum layer is a thin layer we use here the same finite elements used for the thin silicon layer In a real structure we may not use aluminum as resistor otherwise large dissipation will be found at the connecting wires We may still use aluminum for its large expansion coefficient but electrically insulated from a doped silicon channel acting as resistor MaterialSpec AluShell From Silicon Equation ThermalEnergy OhmicCurrent ElasticShellJ Enable Parameter Heat JouleHeat W m 3 Parameter KappalIso 235 0 W m K Parameter SigmaIso 2 1e5 S cm Parameter StressOrtho LinElastIso Emodule 69 GPa PoissonR 0 333 AlphalIso 2 3E 5 Temp 300 STIunit The device is defined as a stack of two thin layers made of silicon and aluminum attached to a bulk of silicon material Define cur 0 01 for one um depth nx0 4 nxl 4 nx nx0 nxl ny0 4 nyl 4 ny ny0 2 nyl QMEI nx0 50 nx0 nxl 300 nx1 QMEJ ny0 50 nyO nyl 10 ny1 ny0 50 ny0 Coord coordx1E 6 1 DeleteME block nx0 nx 0 ny0 DeleteME block nx0 nx ny ny0O ny Material AluShell block nx0 nx ny0 ny0 1 Material SiliconShell block nx0 nx ny0 1 ny ny0 The BCs fix mechanically the bulk silicon material and embed it in a thermal bad at 300 K Two electrical contacts are defined at the extremes of the aluminum laye
168. del smoother mimicking the complex algebra The SESES approach to auxiliary preconditioning for this example is as follows One first assemble and backstore the system matrix for first order elements Sfrst under the name firstorder as well as the prolongation matrix II with name proj MaterialSpec Wire Equation EddyFreeHarmonicNodal Enable MaterialSpec AirEddy Equation EddyFreeHarmonicNodal Enable BlockStruct Block EddyFreeHarmonicNodal ReuseMatrix nodall LinearSolver None Reordering ComplexCluster Solve Stationary Misc Matrix Afield FirstTo2ndOrder proj In a second step one assembles and solves the system for the second order elements by enabling the auxiliary preconditioner composed by some pre and post complex Gauss Seidel smoothing cycles the auxiliary system matrix firstorder prolonga tion matrix proj and the choice of a solver for the auxiliary system MaterialSpec Wire Equation EddyFreeHarmonicNodal2 Enable MaterialSpec AirEddy Equation EddyFreeHarmonicNodal2 Enable BlockStruct Block EddyFreeHarmonicNodal LinearSolver CG InfoIter 0 DiagPrecond Disable Reordering ComplexCluster ConvergSolver SITER nSolveriIter 0 NormR lt 1 0E 8 NormRO MaxIter 500 Preconditioner Auxiliary PreSmooth 2 PostSmooth 2 ComplexSmoother Projector Afield proj iAfield proj Matrix nodall Precond ILU ILU LU Solve Init Solve Stationary The auxiliary problem S can be solved either exactly inexactly w
169. depolarized light as observed on the screen in Fig 2 79 To quantify this effect its useful to calculate the relative amount of depolarized light in a circle of radius r whose center is on the axis Model Specification The file example ThinDisk s3d starts by defining some geometry parameters of the thin disk device with Yb YAG as active material We then define the cooling pa rameters represented by the heat transfer coefficients to air and water hair and hwat as well as the absolute temperatures Tair and Twat As we have chosen Yb YAG a quasi three level system as active material only 15 of the pump power is assumed to be converted to heat Furthermore since we use a thin disk we take benefit of the approximation that the pump power density is constant along the direction of the op tical axis As we assume a radial symmetric pump distribution here we only have to specify the total absorbed pump power and the pump spot radius w0 1 0E 3 pump beam focus radius m x P 100 x Pump power W conv 0 15 x fraction converted to heat x For isotropic material behavior and for the photo elastic effect we have to specify the orientation of the crystal lattice with vectors a b and c relative to the lab system Yb YAG is a cubic crystal and is normally 1 1 1 cut With z being the optical axis we will obtain the situation as shown in Fig 2 81 in top and side view In SESES we only have to specify the x y and z components of t
170. different modes the input fileexample FreeConv starts by defining the control parameter TVAR Define TVAR 1 x 1 variable temperature 0 variable mesh x lsys0 50 le 3 x minimum system dimension in m gt nMax 3 x number of solutions aT 1 x temperature increments dlsys 15 le 3 x length increments in m gt lsys lsys0 x system dimension in m x Thot TVAR 301 305 x hot plate temperature in K x Using the mode TVAR 1 the temperature Thot of the hot plate is raised in incre ments of dT Similarly for 1VAR 0 the system dimension 1sys is raised in incre ments of dlsys This allows one to increase the Ra number which depends on both the applied external temperature gradient as well as on the system dimension by small increments see Section 2 28 for details The temperature on the hot side starts with the value of 301 K in case of TVAR 1 The temperature is increased in 10 steps nMax 10 by temperature increments of 1 K to its final value of 311 K The temper ature of the cold side remains unchanged and is specified through the parameter T0 which is defined in the section physical parameters The procedure is similar in the case when the system dimensions are enlarged There the initial system length of lsys 50 1e 3mis increased in increments of dlsys 15 1e 3m to its final value of 200 le 3m The next paragraph deals with the system geometry We consider a square domain with a minimum siz
171. drop in accuracy is due to the fact that finite elements used to dis cretize the Navier Stokes equations cannot reproduce the quadratic solution 2 125 the relation 2 126 does not hold exactly and the superconvergence behavior is lost 2 26 Blasius Plate Flow At Zero Incidence Due to the non linearity of the stationary incompressible Navier Stokes equations Vv 0 plv V v pV v Vp Ff 2 127 with v the velocity p the pressure f the body forces p u the constant density and viscosity very few analytical solutions are known see 1 For small viscosities or large Reynolds numbers it is well known that solutions are characterized by thin lay ers around boundaries These layers can be studied by solving the simplified Prandtl boundary layer equations obtained by removing small terms after a dimensional anal ysis at large Reynolds numbers and solutions to these equations can be found e g in 166 SESES Tutorial September 2012 pse onasan 0 8 L 0 8 gt H 5 0 6 i ra 7 0 6 8 x P v 1E 2 numeric sol S 04 Pi v 1E 4 numeric sol J 0 4 v 1E 6 numeric sol sss v 1E 2 Blasius sol 1E 4 Blasius sol J 0 2 1E 6 Blasius sol 0 2 a ya 0 2 i p 0 2 0 0 1 0 2 0 3 0 4 0 5 y coord Figure 2 111 Velocity distribution for v 0 01 and computational mesh lo Figure 2 112 Comparison of numerical and Blasius so cally adapted to resolve the boundary lutions at x 0 5 for the kinematic vi
172. duction however this value is small ca 0 1 V and it is not considered here at this point At the interface electrons are replaced by negative oxygen ions drifting and diffusing towards the anode As noted before in the electrolyte the ions diffusion can be neglected so that we just solve for the drift current like within the cathode and anode just with another electric conductivity since now we have ions instead of electrons When the ions arrive at the interface with the anode the potential undergoes another negative jumps due to the hydrogen oxidation which is modeled following 2 178 by a jump BC depending on the normal electrical current Although the Nernst potential AWNernst is a function of the educt and product concentrations we neglect this small non linear dependency and use a constant value of AWNernst 1 1 V so that we can also include here the neglected jump at the cathode electrolyte interface At this interface the ions are replaced again by electrons drifting towards the anode contact and here we have two choices either defining the contact s voltage with a Dirichlet BC for the electric potential or the total current through the device with a floating BC We prefer the latter and set a value of 3mA mm7 since a zero current corresponds to a turned off system whereas for a full fledged non linear model it is not always clear which voltage corresponds to this state It may also be useful to connect both contacts with a resistive lo
173. dy we have also to define the minimum and maximum thick ness of the compensating disk _min and fcp_max as Well as the number of parameter steps necp This section is followed by the definition of the element dimensions co ordinates node numbers of the mesh in the direction of the optical y axis As default value for the thickness of the compensating thin disk we set 1 mm This value will be automatically changed during the parameter study After the definition of the cooling refinement and OPD parameters we start with the specification of the pump distribu tion In the case of transversal pumping we usually specify the total absorbed pump power per length dP dy from which the fraction conv 40 in case of Nd YAG is con verted to heat Inside the rod we assume a parabolic shaped pump distributionsee of the type p 1 ar with pe the pump power density on the optical axis and a a decreasing factor see Fig 2 91 If at the distance R along the rod s radius the power density drops to n pe we use here 7 0 5 then we have a 1 7 R The integrated pump power density in a cross section of the rod equals the pump power per length dP dy so that we have 27 R dP dP drd dP dy Pez dr Pe n JA P Ty eT pR 0 0 The corresponding laser pump values are then defined as parameters dP_dy 10 pump power per length W mm x conv 0 4 x fraction of power converted to heat eta 0 5 power drop a
174. e a high beam quality and a high efficiency simultaneously This design is particularly well suited for quasi three level systems such as Yb YAG because a low mean temperature and a high pump power density are necessary for efficient opera tion Therefore Yb YAG was chosen as the preferred active medium but the thin disk concept is also suitable for a variety of other laser materials as e g Nd YAG Nd YVO4 and Tm YAG Recently the thin disk lasers of very high beam quality band high out put powers became industrial available More literature about this concept can be found in 1 2 3 In the present example we will explore the benefits of the thin disk laser concept and will analyze the stress induced birefringence called photo elastic effect Birefringence Birefringence is an involved matter and therefore we abstain from a detailed descrip tion an introduction into this subject can be found in 4 5 Here we focus mainly on the effect of birefringence induced depolarization Fig 2 78 shows the situation of a linear polarized plane wave entering a depolarizing element The refractive index 120 SESES Tutorial September 2012 Entrance Exit Screen Epot Ein Vertical Polarizer Eaepot Eout Figure 2 80 Depolarization by birefringence orizontal polarized Wave The entrance wave is linear polarized whereas the exit wave shows elliptic polarization Figure 2 79 Set up to analyze depolarization
175. e all currents and charges are zero Non zero currents and charges will just show up just at the surface of the wires In signal theory and for linear systems it is customary to perform a harmonic analysis where the participating fields have complex values and the time dependency is taken to be of the form e t e exp iwt For this setup the Maxwell s Eqs read VB 0 VE 0 Vx E iwB 0 Vx B iwpoegE 0 2 14 56 SESES Tutorial September 2012 The particularity of transmission lines with two or more wires compared to waveg uides with one single wire is that for the former a signal can be transmitted at any fre quency whereas for the latter there is a cutoff frequency below the one no signal can pass through The modes with zero cutoff frequency are transversal modes so called TEM modes where the longitudinal components of E and B are zero i e E B 0 and just for these modes we seek here solutions of 2 14 Because of the z invariance we may as well consider the z dependency to be of the form z exp tkz and construct general z dependent solutions by superposition of these spatial modes With these assumptions and the property poco 1 c the Maxwell s Eqs 2 14 yield the dispersion law k w c and result in the equations 1 2 OB Ey OE OE OE egy OO Ey de ee 2 15 Ox Oy Ox Oy C2y y a i e the magnetic field B is given by the rotating clockwise the electric field E by 90 the electric field E
176. e center of the domain width The fluid is assumed to be air of den sity p 1 156684kg m and viscosity u 1 849830x10 kg ms We compute two flow fields with Reynolds numbers 23 and 45 where the Reynolds number is defined as Re T 2 160 with d the diameter of the cylinder us the velocity of the undisturbed flow and v u p the kinematic viscosity Model Specification We start the problem specification of the input file example FlowAroundCyl s2d by defining some user parameters to be used within algebraic expressions for pre and post processing purposes DensityAIR 1 156684E 00 x air density in kg m3 x ViscosAIR 1 849830E 05 x air viscosity in kg m s rCyl 0 050 x cylinder radius in m CharLength 2 rCyl x characteristic length scale in m gt NueAIR ViscosAIR DensityAIR x kinematic viscosity in m 2 s x EpsP le 2 x regularization parameter in s TauHydro CharLength 2 NueAIR hydr time scale in s to est regul par x Reynolds 23 Reynolds number gt Veloc ReynoldsxNueAIR CharLength x mean flow velocity in m s x The parameter epsP is a regularization parameter used to attenuate numerical noise see the SESES manual for further details As second step we define the problem s 188 SESES Tutorial September 2012 geometry and an initial rectangular mesh is reshaped into a domain modeling the circle and its exterior with the help of homotopic functions loaded with the Include
177. e cor responding thickness of the compensating disk is set by the ForSimPar statement Afterwards the fields Temp and Disp are calculated using the adaptive mesh refine ment algorithm and written out together with the Stress field Finally the OPD is calculated and written out to the file OPD_i dat while 1 i Store StressAver Continuous 1 FreeOnRef Stress TempDiffAver Continuous 1 FreeOnRef TempDiff Remesh Refine ErrTemp TErr Global Refine ErrDisp MErr Global Solve Stationary ForSimPar l_cp At l_cpmin ix 1_cpmax 1l_cpmin n_l_cp if REMESH break Dump Temp Disp Stress If i gt 0 Append Write File OPD_ i dat Text r t OPDtherm t OPDend t OPDtot n For k From 0 To nr_OPD S12 8e t 12 8e t 12 8e t 12 8e n k r_OPD nr_OPD integrate Lattice OPD_ k DRefrac Refrac 1l sqrt Strain YY Strain YY DRefract Refrac 1l sqrt Strain YY Strain YY 130 SESES Tutorial September 2012 d 0 6 mm dP dV d 1 3mm gt r R time 0 Figure 2 91 Radial pump power density profile Figure 2 92 Temperature distribution in the center re gion of the device The temperatures decreases in the compensating disk especially in the outer part This decrease is more pronounced in the case of thicker disks Numerical results We perform the simulation for a Nd YAG rod of r 2mm radius a pumped length of lp 20mm a cooled and unpumped length of leup 2mm and an uncooled length of l
178. e electrolyte q the elemen tary charge Nayo the Avogadro s constant and the factor 4 stemming from the stoichio metric coefficients of e in 2 176 The Nernst potential is actually the open circuit voltage since as soon as a current flows irreversible processes reduce this voltage bar rier In general this voltage drop also called overpotential is a complex function of the normal current but for SOFC under normal working conditions a linear dependency is a good approximation and for the cell voltage we have j n AV AV Nernst Q 2 178 with G a proportionality constant called the effective surface conductance Setting up the problem In the previous section we have given all necessary details for the working of a SOFC and we are now ready to set up a SESES example example FuelCell s2d corre sponding to our description Although we are going to make a 2D simulation our model will be laterally invariant so that we are actually performing a 1D simulation embedded in a 2D model For this first example we are going to model just a small neighborhood of the electrolyte Because of the small dimensions we can consider the temperature to be constant and the mass flow of the species to be diffusive since there is no external pressure gradient driving a convective flow As a further simplification we are not going to model the diffusion of oxygen ions O in the electrolyte since here the electrical drift current is dominant In summa
179. e energy and entropy balances additionally come into play These balance equations are also known as the first and second laws of thermodynamics 5 6 SESES Tutorial September 2012 e Material Laws Balance equations by themselves are insufficient for a complete problem description i e they usually contain more unknowns than available equa tions Furthermore they often contain unknowns such a flux quantities that cannot be directly measured However it turns out that further relationships exist between these unknowns i e they are not independent of each other These relationships are called material laws or constitutive equations As the name suggests they do not hold universally but only for particular materials or material classes such as the class of ideal gases or the class of incompressible fluids e Boundary Conditions Are required to cut off a model i e to separate the model ing domain from its surroundings The number of boundary conditions required is determined by the degree highest derivative of the governing equations involved Often the solution strongly depends on the chosen boundary conditions and an improper choice may render the results useless irrespective of the numerical accu racy Using the wrong boundary conditions means that the model is incorrect or its connection to the exterior world is not correctly specified Combining these ingredients together we obtain a coupled system of governing equa tions bel
180. e file SESES_2_Mat Lab and evaluated within Matlab With the help of such an approach it is possible for instance by using optimization code available within Matlab to implement a search for input values producing an optimized result Such optimization algorithms may be helpful whenever systematic parameter sweeps are too time consuming and possibly not independent of each other This simple ap proach has the drawback that a new instance of SESES is stared each time If may not be so and with the help of e g named pipes it is possible to write some simple code implementing a master slave communication channel to be compiled into dynamical libraries and to be loaded both by Matlab and SESES Chapter 2 Application Examples In this chapter we present collected SESES examples ranging from single field linear problems to more complex multiphysics non linear coupled problems All examples discussed here are found in the examples directory of the distribution and can be run without any SESES license This allows the user to visualize all numerical results by its own and to verify or eventually criticize our assertions All examples are explicitely kept small in size in order to run on commonly used PC hardware 2 1 Thermal strain In this example example ThermStrain s2d we present a 2D thermal induced mechanical displacement by focusing on setting up the boundary conditions An isotropic material with induced thermal strain is characterized by the
181. e obtain zi core I pai f E ds N Scorer N Sore pa g A a ag ot core Hdisk Ho dt core with U the induced coil s voltage Since the inductance L is defined by the relation U L dI dt we have dla h d a L i L N Soore pa y 2 20 Hcore Hdisk Ho Numerical model The hand formula 2 20 for the inductance L has been derived by considering the in duction field B to be constant along the integration path A see Fig 2 36 Within the air gap see Fig 2 37 this may be a crude approximation and therefore for comparison purposes a numerical model is developed including 2D scattering effects within the air gap The input file for this example can be found at example AngSens s3d The shaded simply connected 2D computational domain Q Qeore U Qaisk U Qair Shown in Fig 2 36 is chosen such that there is no current j 0 and therefore V x H 0 holds in Q This condition implies the existence of a scalar magnetic potential in Q with H V Together with the Maxwell s equation V B 0 we obtain the Poisson equa tion V uV 0 to be solved with u Heore in core Qeore H fdisk in the disk Qaisk and u po in the air Qair On the two surfaces ONN OQeore ONG UON1 we set Dirichlet BCs with constant values of 0 and A a free parameter On 02 OQ9UON1 we use homogenous Neumann BCs with B n 0 This problem is solved with 2nd order elements by enabling the equation Magnet oStati
182. e of 1 sys0 and nx ny macro element subdivisions in x and y directions To be able to change the system size within a single SESES run to the size lsys we apply a scaling transformation s2d SESES Tutorial September 2012 177 QMEI nx 10 Ilsys0 nx QMEJ ny 10 lsys0 ny CoordNonConst Coord coord lsys lsysO 1 scale by a factor of lsys lsys0 We next specify the physical parameters of air First SESES routines are defined for the following properties of air the heat capacity cp the density p the molar mass M the shear viscosity u and the thermal conductivity as well as its derivative with respect to temperature As the reference state air at a pressure of p 10 Pa and a temper ature of T 300K is considered Since we only allow to vary with temperature all other material functions are evaluated at T The force vector in the momentum balance is identified with the gravity force f p 0 g 0 acting in the y direction GlobalSpec global parameters Parameter AmbientTemp TO K MaterialSpec fluid fluid parameters Equation CompressibleFlow ThermalEnergy Enable Parameter FlowStab zero Parameter Density IdealGas STunit Parameter mol M_AIR kg mol Parameter Viscosity Visco_AIR TO Paxs Parameter Kappalso Kappa_AIR TO W Kxm Parameter Force D_Temp D_Pressure x buoyance y force x Force X 0 Force X_DTemp 0 Force X_DPressure 0 Force Y Density Val xgAcc
183. e potential has a jump the physical field H VOtot is well defined on Ofrec Since this example is not meant to be a mathematical treatise on existence unique ness and approximation error of solutions the third question is answered by doing numerical experiments 84 SESES Tutorial September 2012 Preprocessing Basically we want to compute solutions without the need to evaluate the Biot Savart integral 2 44 and therefore on Qyec we solve for the vector formulation 2 46 and the current Jo needs to be specified fulfilling the solenoidal condition V Jo 0 For simple geometrical shapes one may look for analytical expressions but this approach is generally avoided for the following reasons As first the user needs to construct a FE mesh with good approximation properties with respect to Jo and at the end one does not really know which are the discretized current values Secondly nothing can be said about the discretized solenoidal property which generally does not hold any more Now since the mesh must be in anyway adapted to the geometrical shape of supp Jo on this mesh the preferred way is to compute the current numerically in stead of specifying it analytically The first reason is that this preprocessor step is generally much faster then the magnetostatic solution itself since one just needs to solve the current flow model on the domain supp Jo Secondly since the current is computed by a numerical model the meaning of the p
184. e solid structure through which fluid flow happens To characterize the internal flow resistance of the thermal insulation assembly shown in Fig 2 147 we solve 2 172 for the permeability __ bl kz Ge wz 2 173 Here we have assumed that the main flow goes along the z direction Consequently k can be obtained from a simple simulation where for a given external pressure drop Ap the resulting average fluid velocity wz is determined Using the integral version of the continuity equation for steady state flow m A p wz with rn the total mass flux eq 2 173 can be re written as LL mz kz 2 174 In this example we will then show how to compute k for the considered thermal insulation assembly With this information we are then able to approximate the flow characteristics of the original structure by performing a simple simulation of an un structured porous material Similarly to the porous flow the overall characteristics of an assembly with respect to heat conduction can also be described by effective thermal conductivities They are obtained from Fourier s law 4 VT 1 k F stating that the heat flux F is proportional to the external temperature gradient VT The proportionality constant is the inverse of the thermal conductivity and solving for x gives _ dz Lz 2 17 Kz As for the permeability we will then show how to compute z Model Specification This numerical example can
185. e solution of the above equations Let use the unbold notation 7 for the vectors 7 and c log i 1 3 From the first equation we see that be 1 is coaxial with be triq and by taking ihe log on both sides we are left with Entl AA0 fn 1 Etrial En 1 En A AX0g fn 1 fru F T En 1 G En 1 0 SESES Tutorial September 2012 145 which are formally equivalent to the equations of the infinitesimal case If we do not consider the trivial equation for 41 we have to solve four equations for the un knowns A and 7 1 generally solved by a Newton Raphson algorithm In order to compute the elasticity tensor in material coordinates 2 111 we need the derivatives sT Which in our case are actually 0 7 and are computed as follows Let 7 be a com ponent of cui by defining H 7 41 0 01 En 1 n41 Eo Etrial n Of 0 0 f and the matrix G by the relation 0 II GO the first two equa tions are given by AA f Xo and by taking the derivative we have GO 5 Z Of 1O AA 0 X0 with Z G 1 A 0 fG G 14 Ad0 f From fii 0 we obtain 0 fr41 OfGO x 0 and by contracting the previous value of Gd on the left with Of yields an equation for 0 A and the final result is Z Of Of Z Of Z af with the derivatives 0 741 found in the first row Since the relation q q is gen erally derived by a potential through differentiation the matrix O q is symme
186. e state at rest an analysis based on linear system theory is a good framework which is generally per formed in the time Fourier space i e the system is fully characterized by determining its response to an external harmonic excitation over the full spectral range In this ex ample we present the fundamentals of the impedance theory for piezoelectric crystals together with a computational example Impedance of a piezoelectric crystal Let us assume the crystal to be at rest and in a unstressed state By applying a pressure somewhere on the crystal surface a stress state is induced mechanical displacements are generated and charges are transported to the contacts Due to the small magnitude of the displacements a linear theory for the governing equations can be used If u is the mechanical displacement and the electric potential in the crystal then we have the following local relations between mechanical and electrical fields s C e u g E D g e u e E i with E V the electric field D the dielectric displacement field e u Vu Vu 2 the linearized strain tensor u the displacement s the stress tensor the dielec tric permittivity tensor C the elasticity tensor and g the piezoelectric tensor The gov erning equations of a dynamical mechanical system are in the linear theory of small displacements given by p yu V s f with p the mass density f the body force and y a phenomenological damping factor pro
187. e values denote tensile and compressive stress respectively The largest positive value of the three remaining diagonal elements therefore denotes the maximum tensile stress which is a valuable indicator for thermal crystal damage Typical fracture limits for Nd YAG are reported in the literature and confirmed by the author to range from 130 to 260 MPa Fig 2 74 shows the distribution of the highest principal stress component It is obvious the the inner part of the rod is under compression whereas the outer part is under tension The maximum tensile stress is located on the rod axes at the surfaces of the crystal Fig 2 75 shows this maximum tensile stress as a function of the pump power It raises with the pump power up to about 100 MPa at 20 W per side which is below the fracture limit Due to the non linear thermal expansion 2 98 the maximum tensile stress shows a small nonlinearity with respect to the pump power per side The focal length of the averaged thermal lens following 2 93 is deduced from the 118 SESES Tutorial September 2012 Thermal Lensing SESES Simulation Hyperbolic Fit Focal Length mm 0 5 10 15 20 Pump Power W Figure 2 76 Focal length of the averaged thermal lens as a function of the pump power per side Dots indicates the values obtained from the OPD data whereas the solid line represents the hyperbolic least quare fit to this data OPD data and plo
188. e we have dropped the compu tation of the scalar potential Y the stiff current Jo is generally used as the driving source for the magnetic field On a domain Q C R with conductors Qm m 1 N 98 SESES Tutorial September 2012 where o Q const and o Q UmQm 0 in a preprocessing step we apply voltages to the conductors resulting in quasi static stiff currents Jo For w gt 0 we then have additional eddy currents J Jo cE Jo iwo A induced by the time dependent magnetic field B V x A When solving the eddy current problems in SESES we have to take a decision either to use H Q or H curl Q conformal elements to approximate the vector potential A the former also known as nodal elements and the latter as edge elements Then from a weak formulation associated with the PDEs 2 70 and by plugging nodal or edge shape functions we obtain a complex symmetric linear system of equations to be solved whose solution give us A The pro and contra of both finite element types have already been discussed in the Example 2 14 and here we just remind that with nodal elements the system matrix is regular but singular for edge elements if somewhere in the domain 2 we have 0 Although it would be natural to use a complex lin ear solver to solve for the complex vector potential A in a multiphysics environment where several real and complex dof fields may be computed simultaneously in a cou pled manner the most general solver is a r
189. eal solver and complex dof fields are simply split in a real and imaginary part This approach can double the memory usage but from a numerical point of view is equivalent to a complex solution In particular since the linear system obtained is complex symmetric as explained in the Example 2 14 it is possible to obtain a symmetric real system of the form A B S Hi 2 71 with A B symmetric real matrices For symmetric matrices the iterative solver of choice is the conjugate gradient method In exact arithmetic and for positive definite matrices it converges in at most N step with N the size of the system 1 2 3 Our real symmetric matrix 2 71 is not positive definite but has pairs of real eigenvalues of opposite sign since if A x y is an eigenpair of S then also A y x Despite this fact for our application the conjugate gradient generally converges smoothly but stalling cannot be excluded and in general the chances of failure can be reduced with a good preconditioner We document here some numerical tests done with the model found at example EddyPrecond s3d a simple voltage driven rectangular coil as displayed in Fig 2 62 The electrical conductivity is 5 8x10 A Vm and the driving voltage is ap plied as stiff current Jo computed in a preprocessing step with the OhmicCurrent model The current is stored in a user field cur0 used to define the material parameter iCurrent 0 The eddy current problem is th
190. ealing for first quantitative esti mations However since in general the contact surface is unknown its determination is part of the problem solution which turns even a linear mechanical problem into a non linear one and generally it further slows down the convergence rate of non linear problems It is important to note that the usage of sticky contacts always results in a path dependent problem not however so for friction less contacts In the following we will just consider the more common case of friction less contacts which are also more complex to implement numerically At the base of any contact algorithms there is a fast algorithm checking the impenetra bility condition with the rigid body for every point x on the computational boundary where a mechanical contact has been enabled If this condition is violated the contact algorithm will then force the point x back on the rigid body surface To accomplish this task one first computes the closest point projection on the rigid body surface xo formally defined so that the vector x xo is parallel to the surface normal n at xo In general this projection is not unique however it will be for points close enough to the surface with an upper bound for the distance depending on the surface curvature During the solution process and at each linear step the friction less contact algorithm will then force any invalid movement x to stay on the plane having normal n and passing through the point
191. eched The second method is quite close to the energy methods but in general tends to give lower force values References 1 J SCHIMKAT Grundlagen und Modelle zur Entwicklung und Optimierung von 66 SESES Tutorial September 2012 MagnPot A 8 37E 01 8 53E 01 BField Norn T 8 B3E 01 8 70E 01 8 40E 01 AE 7 STE OL 9 02E 01 7 53E 01 9 19E 01 7 105 01 6 67E 01 9 35E 01 6 23E 01 9 51E 01 5 802 01 9 67E 01 5 376 01 9 84E 01 4936 01 1 00E 00 4 505 01 407 01 1 02E 00 3638 01 1 03E 00 3208 01 71 08 00 2 77E 01 1 07E 00 2 33E 01 1 08E 00 1 90E 01 1 47E 01 1 10E 00 1 035 01 1 11E 00 6 02E 02 tine 0 1 13E 00 1 15E 00 o tine Figure 2 44 Computed magnetic field B with arrows indicating the direction and Figure 2 43 Numerical solution for the magnetic 2 intensity around the lamellas scalar potential Silizium Mikrorelay Dissertation Technische Universitat Berlin FB Maschi nenbau und Produktionstechnik 1996 2 URL http www asulab ch de composants_de html 3 F GUEISSAZ D PIGUET The microreed an ultra small passive MEMS magnetic prox imity sensor designed for portable applications Proceedings of the 14th IEEE In ternational Conference on Micro Electro Mechanical Systems MEMS 2001 Interlaken Switzerland January 2001 4 J D JACKSON Classical Electrodynamics John Wiley amp Sons 1999 2 9 Skin and Proxi
192. ectrum we may as well perform a spectral analysis on e g rectangular shaped signals and follow their dispersive evolution when traveling along the line References 1 D J GRIFFITHS Introduction to electrodynamics Prentice Hall Inc 1989 2 J D JACKSON Classical Electrodynamics John Wiley amp Sons 1999 3 HORST KUCHLIN Taschenbuch der Physik 2 7 Variable Gap Sensor The physical principle of variable gap magnetic sensors is based on the variation of an air gap within a magnetic circuit These sensors may be realized by winding a coil on 58 SESES Tutorial September 2012 armature rotating disk Figure 2 34 Example of a variable gap sensor Figure 2 35 Angle sensor system a C shaped magnetic core as shown in Fig 2 34 A displacement change of a magnetic armature leads to a change in the inductance L of the coil and therefore the electric impedance can be used as a measure for the armature displacement Variable gap sensors show some advantages in the measurement of small displacements They are highly sensitive with a resolution of less than 1 nm 1 and the maximum non linearity typically is given as 0 5 2 They are used for non contact applications and can be used to measure angles in a form as shown in Fig 2 35 with a rotating disk of variable thickness In this example simple formulas for estimating the inductance of the angle sensor as a function of design parameters are derived followed by a compar
193. ed here a variant of the auxiliary preconditioner presented in 5 6 since this latter fails for the case 0 and although mathematically quite reliable on the overall these two auxiliary preconditioners are generally inferior to the simple ILU 0 preconditioner with diagonal shift References 1 Y SAAD Iterative Methods for Sparse Linear Systems SLAM 2003 2 W HACKBUSCH Iterative Solution of Large Sparse Systems of Equations Applied Mathematical Sciences 95 Springer Verlag 1994 104 SESES Tutorial September 2012 3 H A VAN DER VORST Iterative Krylov Methods for Large Linear Systems Cambridge University Press 2003 4 Z REN Influence of the R H S on the Convergence Behaviour of the Curl Curl Equation IEEE Trans Magn Vol 32 No 3 pp 655 658 1996 5 R HIPTMAIR J XU Nodal auxiliary space preconditioning in H curl and H div spaces SIAM J Numer Anal Vol 45 No 6 pp 2483 2509 2007 6 R HIPTMAIR J XU Auxiliary space preconditioning for edge elements IEEE Trans Magn Vol 44 No 6 pp 938 941 2008 2 16 Harmonic Analysis of a Piezoelectric Crystal Piezoelectric crystals find a broad application in technology from LC oscillators to sensors for pressure force acceleration as well as transducers and actuators for ultra sonic applications The piezoelectric effect is a small physical effect coupling together electric field and mechanical strain Due to its small perturbation of th
194. ed to write the concentrations at different cross sections to external files Numerical Results Fig 2 115 shows the concentration fields as computed by SESES for the cases K 0 1 and K 0 01 respectively One sees that the mixing efficiency decreases for smaller K values In Fig 2 116 the same numerical results are compared with the analytical solution In general there is an excellent agreement between the two methods and the agreement becomes better for smaller K values The reason could be that in the derivation of 2 148 the diffusion transport has been neglected in the x direction which is only valid when K lt 1 Fig 2 117 shows again the numerical results for the plug flow but now compared with the results obtained for the parabolic Hagen Poiseuille flow Note that for a slit the Hagen Poiseuille profile is given by 2 yY y Vz y 6 Vaver a 5 2 149 Ly Ly STunit SESES Tutorial September 2012 173 Figure 2 115 Concentration fields in the straight channel for left K 0 1 and right K 0 01 as predicted from the SESES model 1 0 gt 1 0 XXX K 0 1 k X 0 001 0 8 F he S Gs 0 8 0 6 2OQQQQ z 0 6 L lt 2 J lt fo
195. el Force Y_DTemp Density DTemp xgAccel Force Y_DPressure Density DPressure x gAccel STunit Parameter ThermStab zero Parameter ThermConv Cp_AIR TO Density Val Velocity W m 2 K Parameter ThermConvDVelocity Cp_AIR TO Density Val J m 3 K H Before we can start our simulation a complete set of boundary conditions must be defined Since our simulation domain is the 2D cross section through a closed box no slip boundary conditions are applied to all four boundaries BC zeroVelocity OnChange 1 Dirichlet Velocity 0 0 m s zero velocities at boundaries In addition the boundaries on the left right are kept at the constant temperature of Thot TO BC hotPlate 0 0 JType ny Dirichlet Temp hot K x temperature left plate Thot x BC coldPlate nx 0 JType ny Dirichlet Temp O K x temperature right plate TO x Note that SESES uses Natural boundary conditions when nothing is specified oth erwise This means that the normal component of the flux associated with a certain field variable is set to zero Since we do not specify explicit thermal conditions for the upper and lower boundaries this implies zero heat flux over these boundaries which is consistent with the boundary conditions shown in Fig 2 119 With the specification of the boundary conditions the initial section of the input file is now complete 178 SESES Tutorial September 2012 Computation and postprocessing The following section
196. electric potential B the outer magnetic field He the charge carrier mobility and ne the carrier density We assume here the carrier density to be equal to the doping concentration Vo pUeVG x B 2 2 For our Hall device the domain boundary X is partitioned according to the electrical contacts oN ODN U OOD U ON p U ONG U ONG with Ncc the two current contacts OQ 4 17 the two Hall contacts and Ny the re maining boundary On N y no electric current is flowing out of the device on OQ c the potential is given and on the Hall contact 0Q p m the potential is floating constant and no current is flowing This physical situation is represented mathematically by the following BCs I nlooy 0 dlane 9 lar Vapplied Pom J dn 0 and any n const 2 3 SESES Tutorial September 2012 41 From the theory of boundary value problems the governing equation 2 1 together with the material law 2 2 and the BCs 2 3 is a well posed problem with a unique solution for the electric potential For an ideal Hall plate of infinite length with a vanishing Hall contact width the Hall voltage is a function of the current density Jo and the magnetic field B according to JoB b He BzVapplieab qone a Uva 2 4 For a real device we have to consider the finite device length a and contact width w as given in Fig 2 6 A possible optimization goal would be the requirement that the discrepancy of the measured Hal
197. emat ical maps and the Coord statement Since we defined a simple and coarse mesh with only 4 elements but expect the electrostatic potential to vary strongly for instance near the tip we choose a mesh refinement level of 3 with the statement MinimumRL 3 Therefore each macro element is subdivided 3 times such that each original element is replaced by 8 x 8 elements In order to specify the electrical contacts we define the mathematical boundary conditions BC with the BC statement as follows BC Tip 1 0 JType 1 Dirichlet Phi 1 V BC Substrate 0 2 IType 2 Dirichlet Phi 0 V We name our boundary conditions Tip and Substrate To define the BC geometry we use the keywords JType and IType to state that a mesh segment between 2 mesh nodes in j and i direction is specified respectively For instance for the horizontal Substrate we specify the starting node 0 2 and a length of 2 in the i direction thus defining a boundary segment between the nodes 0 2 and 2 2 Among the different types of BCs we select here the Dirichlet BC allowing to prescribe the value of a dof field on the boundary For applying an electrical bias of 1 V between the contacts we therefore use Dirichlet Phi 1 VattheTipandDirichlet Phi 0 Vatthe Substrate The Finish statement tells the parser to stop reading the initial section of the input file We now proceed with the command section in which the solution method is specified The statement Solve Stationary asks
198. ement field and electric field as well as the activation of the piezoelectric model For this analysis quadratic elements are best suited In the definition of the material parameters from the literature one has to pay attention at the different direction of the polarization axis Since we are performing a rotational symmetric computation we have also to specify the material parameters for the azimuthal direction i e the ones normal to the computational domain SESES Tutorial September 2012 107 MaterialSpec PZTA5 Equation Elasticity2 ElectroStatic2 Enable Model PiezoElectric Enable In a second step we define a simple geometry fix the structure mechanically and de fine two electric contacts where the external voltage is applied Because the electric contacts are placed at the top and bottom of the disk the disk is symmetric with re spect to the middle plane and only such modes can be exited Therefore is enough to compute half of the disk The modeling starts by defining a coupled block structure both for the inhomogeneous and the eigen solutions BlockStruct Block Phi Disp Afterwards we compute the coupled inhomogeneous solution with the potential set to zero on one contact and to 1 V on the other one Solve Stationary Write alpha0 e n apply Phi Flux and compute the total charge ag at the contact This charge is nothing else than the SESES contact characteristic and can be accessed with the built in symbol apply Phi
199. ements QMEI Start width 2 nx 5 width nx QMEJ ny 4 length ny The physical properties as well as the equations to be solved are defined with the next statements Here we have to solve for the incompressible Navier Stokes equa tions and to define the viscosity as well as the mass density The mass density p does not really enter the incompressible Navier Stokes but it is used together with BCs and therefore must be defined The parameter PressStab is the penalty parameter used to solve the mass conservation law Since this equation is not an elliptic it has been stabilized with a second order dissipative term in the pressure proportional to PressStab and its value should not be chosen too small or too large MaterialSpec Air Equation CompressibleFlow Enable Parameter Density Val density kg m x 3 Parameter Viscosity visco Paxs Parameter FlowStab Zero Parameter PressPenalty le 6 s The defined material Ai r is then mapped on the whole domain with the statement Material Air 1 The next statements define the BCs as stated at the beginning Since the solution only depends on the pressure drop we define a value of 0 Pa on the outlet and the constant pressure drop dp at the inlet BC Inlet 00 IType nx Dirichlet Pressure dp Pa Dirichlet Velocity X 0 m s BC Outlet 0 ny IType nx Dirichlet Pressure 0 Pa Dirichlet Velocity X 0 m s BC NoSlip 0 0 JType ny nx 0 JType ny Dirichlet Velocity 0 0 m s The next statements are pa
200. en Cantilever Microactuator 2 4 Heat Conduction in a Cylindrical Stick oaa 2 5 Image Acquisition in Scanning Probe Microscopy 26 Cross lalkand Telegtaphy s necra d eK SE pe Be eh eR RR 27 Va ble Gap Sensor se ceea eia HSS ae TRIES e ege G 2 8 Modeling of a micro reed switch aoaaa 29 Skmand Proximity Effet lt x soi se ord e hk i k a Re Owe aE e ARR OH ZIO Seel Hardening oro na r ee R Eee a OS PRD EPS SEG 2 11 Eddy Current Re epulsi n lt c o sossar daga darada i 12 15 16 17 19 26 31 34 212 219 2 14 2 15 2 16 217 2 18 2 19 2 20 221 PAD 223 2 24 2 29 2 26 Lar 2 28 2 29 2 30 2 31 fs R39 2 34 2 39 2 36 SESES Tutorial September 2012 The total and reduced formulation of magnetostatic 77 The scalar and vector formulation of magnetostatic 81 Eddy currents in a linear system lt cuca e sn bee DODD e eee eS 88 Preconditioning eddy current systems 6 6 64 be se ee Be 97 Harmonic Analysis of a Piezoelectric Crystal 104 Longitudinally Diode Pumped Composite Laser Rod 110 Thin Disk Lager gt e cece ee Roe EER EME Lee RRS BRM 118 Thermo Optically Self Compensated Amplifier 126 Shells or Thin mechanical structures 2 454 4444 64 be be RS 131 PiastiGry Models s e ma dae ta maea eed SG SE SES ES PEREESS 136 Necking of achcular Dal 6 c4454 Psu yeas a RE Ee EES 141 Stamping and mechanical co
201. en solved using the equations EddyF ree HarmonicNodal or EddyFreeHarmonic for nodal and edge elements The typical streamlines for the computed magnetic field are given in Fig 2 63 In this example we want to study the impact of the frequency and the element shape on the performance of the iterative solver Therefore the example is parameterized so as to freely change the frequency w and the slenderness a length width of the rectangular elements forming the coil Iterative solvers generally work both for regular and singular systems if these latter have a solution For regular matrices the condition number p gives us a first impres sion how good iterative solvers will work not however for singular matrices since SESES Tutorial September 2012 99 Figure 2 63 Streamlines of the real part of the Figure 2 62 View of the single coil geometry magnetic field RB Frequency w Slenderness a Iter No P Iter Diag P Iter ILU 0 P 21540 4554 29591 5235 31597 5887 Table 2 1 Number of solver iterations to reduce the residual by a factor of 1078 as function of the frequency w slenderness factor a and for no diagonal or the ILU 0 preconditioner The system matrix of size 19074 is for first order nodal elements 100 SESES Tutorial September 2012 4500 4000 3500 3000 2500 A 2000 Figure 2 64 Condition number of 1500 the system matrix as function of ae the frequency w and slenderness 0 factor
202. ent as a function of 0 30 60 90 120 150 180 Reynolds number Dregees from forward statgnation point Figure 2 136 Pressure distribution on the surface of the cylinder As next we compare the pressure distribution on the surface of the cylinder As be fore we first define a lattice of sampling points around the cylinder The data points are located on a circle with the distance dr away from the cylinder surface The first point is placed at the forward stagnation point of the cylinder and the last at the rear stagnation point of the cylinder Plotted is the dimensionless pressure coefficient cp defined as 2 Cp oe 2 161 The computed pressure distributions based on SESES and CFX TASCflow are shown in Fig 2 136 They are nearly identical and in fair agreement with the experimental data The discrepancy is due the size of our computational domain in the lateral di rection In the experiment the cylinder diameter is very small compared the channel width whereas in the simulation the cylinder blocks a considerable part of the chan nel cross section This leads to increased velocities around the cylinder and in turn toa higher stagnation pressure at 0 and to a lower pressure at 90 away from the forward stagnation point The force exerted on the cylinder is given by evaluating on the cylinder s surface the integral Fe f Tan n pn ee dA 2 162 8 with p the pressure Twa Vv Vv the wall shear stress n the unit
203. ent values in the conductors by 2 55 yielding each time the m column of Y w In general the computation of eddy currents is delicate although the equations to be solved are linear In particular at high frequencies one has to pay attention to have a good mesh resolution in the conductors since by the skin effect the current tends to flow just beneath the conductor surface For a planar conductor the current skin depth is given by 6 2 jwwo and since in our example we are going to use a copper wire cross section of 1mm with o 5 8x10 A V m and just few elements we see that our numerical example will work up to wmax 10 kHz 90 SESES Tutorial September 2012 Low frequency approximation For our system of conductors the time averaged energy dissipation W lt RV RI gt IT RZ w I 2 V RY w V 2 is just determined by the real part of the impedance or admittance matrix Since by a Taylor expansion of Z w or Y w at w 0 the real linear term in w is missing at low frequencies the energy dissipation is a quadratic function of w If one is just interested in this quadratic term or in general in the first few Taylor coefficients of Z w or Y w these values may be computed in a less expensive way than using the classical approach of computing several eddy model solutions We first note that for w 0 the eddy model 2 52 decouples into the electrostatic and magnetostatic formulations of Maxwell s equation and as
204. ents with the Del et eME statement Some 2D examples In this section we present some simple examples of defining 2D meshes with the help of geometrical maps In the first example we start with a rectangular mesh of dimension 1x ly and of nx ny elements We then translate a subdomain in the left half of our domain along the x axis and rotate a subdomain of the right half about the 20 SESES Tutorial September 2012 rotation center 0 1 Ix 2 ix 20 Figure 1 11 Manual mesh di mensioning according to a ge ometrical sequence Figure 1 10 A mesh created with two Coord statements to perform a rotation and a translation and a DeleteME state ment to create the void space angle angle with two Coord statements A void space is created by deleting some elements with the DeleteME statement Our final mesh is shown in Fig 1 10 and it has been created with the statements Define angle PI 10 QMEI nx 8 1x 20 nx QMEJ ny 4 ly 10 ny Coord xtl y block 1l 2 1 2 Coord 1lx 2 cos angle x 1x 2 sin angle y ly 2 ly 2 sin angle x 1x 2 cos angle y ly 2 block 5 7 1 3 DeleteME block 5 7 1 3 Our next example illustrates a mesh defined using a function to adjust the width of neighboring elements according to a geometrical sequence We first start by defining a 1D mesh nx x 1 with elements of side length 1 For domain z coordinate x 0 nx we then change its value according to the function x peie 1
205. er AbsIncr Temp quot AbsIncr Temp return AbsIncr Temp lt 1 E 2 Solve Stationary For our reduced accuracy requirement only 5 solution steps are now required Solving with reduced accuracy Step 1 AbsIncr Temp 1 855581e 00 Rate inf Step 2 AbsIncr Temp 8 451505e 01 Rate 4 554640e 01 Step 3 AbsIncr Temp 3 025883e 01 Rate 3 580289e 01 Step 4 AbsIncr Temp 5 41918le 02 Rate 1 790942e 01 Step 5 AbsIncr Temp 1 99445le 03 Rate 3 680355e 02 The effort required to solve a non linear problem based on the full solution of lin earized problems is the number of iteration times the almost constant effort to solve a single linear problem Even for weak non linear problems the number of iterations re quired is generally in the order of 5 to 10 which makes the solution of non linear prob lems computationally much more expensive However by changing the approach of solving the underlying linear problems one can effectively solve at least weakly non linear problems with little more effort than a single linear one For optimized algorithms the total effort is then determined by the character of the non linearity Devising optimized algorithms principally differs whether we are using a direct or an iterative linear solver We present here a speed up method based on the utilization of a direct linear solver which is also the solver used per default However methods are also available for iterative linear solvers The general property of direct solver
206. ermost atoms of the metallic tip maintain the tunneling current and are thus responsible for the lateral resolution A schematic overview of the scan ning probe setup is shown in Fig 2 26 In most commercial SPM setups the tip holder is kept fixed while a piezo moves the sample relative to the tip in order to scan the sample surface A method for simulating the surface scan and the dependence of the tip geometry will be discussed next SESES Tutorial September 2012 53 4 0E 00 9 5E 01 9 0E 01 85E 01 8 0E 01 7 5E 01 7 0E 01 6 5E 01 6 0E 01 5 5E01 5 0E 01 4 5E 01 4 0E 01 3 5E 01 3 0E 01 Air Tip Substrate aS EE ee timeo Figure 2 28 Left 2D simulation domain of sinusoidal tip and surface Right Calculated electrostatic potential distribution between tip and sample Numerical model of a tip sample geometry In the introductory example of the first chapter Chapter 1 Getting Started a simpli fied SPM geometry with a vertical line segment as tip and a flat sample surface both defined as boundary condition of an electrostatics problem was presented As a first refinement to that two dimensional model the example example SpmSine s2dde fines the tip geometry as well as the sample surface with a sinusoidal function Since both the tip and the surface contour are assumed perfectly conductive it suffices to implement the medium in between them as the modeling domain In particular we may start with a single rectangular macro elemen
207. ersing material boundaries As expected the temperature field is moving downstream with increasing flow rate The sensor calibration curve is the curve relating the measurement signal and the quantity of interest i e the flow rate In our example we calculate the dependence of the sensor temperature difference AT gt _ T T on the mid channel velocity vo Fig 2 125 shows the dependence of AJ gt _ and the heater temperature rise above ambient temperature AT at a given power density applied to the heater We note that AT gt _ exhibits a maximum value while ATy decreases monotonically with in creasing velocity in the channel The slope of the AT gt _ curve decreases and thus the sensitivity of the sensor is reduced at higher velocities The heating power influences the flow rate range in which the sensor can be operated As an example Fig 2 126 shows such a dependence for the same geometry as above 184 SESES Tutorial September 2012 ATp 4 K ATy K fi 1 fi 2 6 0 0002 0 0004 0 0006 0 0008 0 001 Velocity m s Figure 2 125 Sensor calibration curve relating the sensor temperature difference AT gt _ red and the heater temperature rise AT y blue to the velocity vo in the center of the flow channel in constant power mode heating power P 107 W 0 017 1 2 Hy 0 016 A o 3 0 015 OB seen 7 g 0014 a 6 x 0 013 osh ee 5 wee 4 5 J 0 012 5 O4 eee 3 0 011 ae 2 0 01 0 2
208. es From 2 58 the 0 Y 0 matrix of 2 57 is pure imaginary and reads OY mn _o sEn 0 dn To compute the next coefficient 02Y 0 we need to solve for 0 A 0 a problem similar to the magnetostatic problem 2 63 and for 02 E 0 a problem similar to 2 66 but just with other source terms And so on for each additional Taylor s coefficient of Y w which are alternatively real and imaginary Model specification When solving the eddy current model 2 53 in SESES we have to take a decision ei ther to use H Q or H curl Q conformal elements the former also known as nodal elements and the latter as edge elements The edge elements are the natural frame work when working with the curl curl operator of 2 53 since they do not enforce the normal component of the vector field to be continuous and differently from nodal elements they are well suited to model sharp corners and sharp discontinuous ma terial laws see 2 4 However the eddy current formulation cannot be gauged with 92 SESES Tutorial September 2012 Figure 2 58 View of the two coils geometry Figure 2 59 The computational domain show ing the finite elements just on material bound aries edge elements and the linear system to be solved is singular If the equations are as sembled in a consistent manner this is not a drawback for iterative solvers However the convergence rate is strongly i
209. eters to fit A priori error analysis of the stabilized discretization schemes give us the order of magnitude of these parameters however a fine tuning is generally required for optimal numerical results Here the singularity at 0 0 of the Blasius flow may be used to calibrate the incompressible Navier Stokes solver By refining the mesh around this singularity the calibration is performed by defining the stabilization parameters through the material parameter FlowStab in order for a coarse mesh to have approximately the same solution and by minimizing over or under shooting for both the pressure and the velocity The dependency of the stabilization parameters from the velocity v and viscosity u is according to the theory 2 however one needs to calibrate the constant scaling factors The proposed values are also used by the default setting of FlowStab 168 SESES Tutorial September 2012 References 1 H SCHLICHTING K GERSTEN Boundary Layer Theory Springer Verlag gth Ed 2000 2 M BRAACK E BURMAN V JOHN G LUBE Stabilized finite element methods for the generalized Oseen problem Comput Methods Appl Mech Engrg No 196 pp 853 866 2007 2 27 Microfluidic Mixing in a Straight Channel Microfluidics a part of the microsystems domain is about flows of liquids and gases single or multiphase through microdevices fabricated by MEMS technology In microfluidics the ability to mix two or more fluids thoroughly and
210. ew constrained SESES Tutorial September 2012 153 Figure 2 101 Rotational symmetric shape of the _ i S upper and lower stamps Figure 2 102 Radial shape consisting of two lines and two parabolae and the four regions for point projection onto the basic shapes system This can be done by amending the native system with simple algebraic oper ations performed on the rows and columns of the assembled matrix After solving the amended system the linear constraint equations are fulfilled exactly The solution of this exact formulation is robust however it suffers from the fact that once applied we cannot say for sure if we have a contact point or not i e if by releasing the glide con straint the boundary point will violate or not the impenetrability condition A brute force method will combinatorially turn on and off all glide constraints upon reaching consistency with the impenetrability condition an unfeasible approach Much better is to check if the rigid body sets the contact point under tension or compression Just for compressive points we continue to apply the constraints whereas for tensile ones we release them At the end of the non linear solution all contact points must be com pressive points and all other boundary points must not violate the impenetrability condition The question arises how to know if a contact point is under compression or not since at the discretization level the numerical stress is not at our disposal
211. factor between the heat change and the temperature difference with respect to a reference point of temperature Tref T2 o A T Tet 2 9 with A being the surface area The transfer coefficients for convection can be derived tabulated figures of merit that depend on the specific geometry and surrounding ma terial In the section discussing a 3D model we will also introduce heat radiation and consider a realistic value for the heat transfer coefficient In our model heat transfer is defined with the following statements BC Transfer OnChange 1 OnBC Heating Disable OnBC Cooling Disable Neumann Temp D_Temp tcoeff refTmp Temp tcoeff W m x 2 W mx 2 K with the reference temperature ref Tmp We note that since the Neumann BC depends on the temperature itself one needs to supply the derivative of the temperature dof field with the D_Temp statement in order to enable a successful Newton s algorithm The cooling conditions enforced with water flow through the left end of the heated stick is implemented with a simple Dirichlet boundary condition BC Cooling 0 0 JType 1 Dirichlet Temp refTmp K The heat source located at the right end of the stick is defined as follows BC Heating nx 5 1 IType 1 Neumann Temp time lt toff heaterflux 0 W cm 2 x time dep heating wherea test true false statement was constructed to implement switching off the heat source after the time toff The duration of the time dependent simulation i
212. fer holder gas burner E mm lt Figure 2 13 2D model for the heated stick with cooling on the left and heating on the right Figure 2 12 Sketch of the horizontal alu minum stick heated on the right and water cooled on the left side of the stick suggests a simplification to a 1D problem and so we will first present an analytical solution to a simplified 1D problem As a next step heat radiation loss at the stick surface is considered and an analytical solution method is proposed Then a model for a 2D case and lastly a 3D case are calculated in order to refine the simulation results and make them comparable to experimental data of the stick in transient and steady state Analytical model in 1D In one dimension the heat equation reads qr da Cp P Raz q rea aa 4 with T the temperature the heat conductivity q the heat density rate cp the spe cific heat capacity and p the specific density As a first approach let us consider the stationary equation with a heat loss proportional to the local temperature PT Mizz a T Tief 2 5 with a a phenomenological heat loss coefficient As boundary conditions we choose T 0 T and F L h 2 6 with Tef the temperature of the cooling water at x 0 F VT the heat flux and h a measure of the heating power at x L The general solution is a particular solution to the inhomogeneous equation that we simply choose as Tr Tyrer plus the homogeneous solution whi
213. ffine Translating rotating and scaling These geometric maps are generally directly defined using the available vector algebra once the translation vectors the rotation matrices and the scaling factors are known and so the role of Homotopic sfc is limited to provide utility functions to easily define rotation matrices In particular the function Routine double rot T1 double val T1 double alpha double axis T1 defines a rotation of the vector val around the axis axis of angle alpha and returns the rotated vector The rotation axis does not need to be a normalized vector If the rotation is not with respect to the origin then one calls rot by first subtracting the rotation point which is again added after the call This is done by the function affine for the rotation center cen with an additional single scaling of the coordinates by fac as follows Routine double affine T1 double val T1 double cen T1 double fac T1l double alpha double axis T1 return centrot val 0 cen 0 fac 0 val 1 cen 1 fac 1 val 2 cen 2 fac 2 alpha axis Direct scaling and first example of Fig 1 13 Coord 10 0 x 10 0 0 7 8 5 y 8 5 1 8 1 5 z 1 5 1 2 block 8 10 8 10 0 10 Rotation with rot and second example of Fig 1 13 Coord rot coord 9 9 0 P1I 8 0 0 1 9 9 0 block 8 10 8 10 0 8 Affine map with affine and third example of Fig 1 13 Coord affine coord 5 5 0 5 0 4 0 8 1 PI 4 0 0 1 block 4 6 4 6 0
214. file in x direction due to thermal expan sion BC Cooling 0 0 0 JKType ny nz Dirichlet Temp refTmp K Dirichlet Disp X 0 m The resulting total elongation of the stick is the value of the displacement field zx component at the right end of the stick For our example with h 60 W m K and an expansion coefficient of AlphaIso 2 310 K t we get an elongation of 3mm The displacement profile is shown in Fig 2 25 and is essentially the integral of the temperature profile times the expansion coefficient From the solution of the simple 1D analytical model 2 8 one would thus predict a hyperbolic cosine function for this x displacement profile In conclusion we have demonstrated the use of SESES to model the temperature dis tribution and thermal expansion in a heated aluminum stick by considering both con vective and radiative heat loss The simulation results were compared successfully with experimental data in transient and steady state 52 SESES Tutorial September 2012 LASER LA IN Figure 2 27 Illustration of image acquisi tion artefact in scanning probe microscopy Cantilever Figure 2 26 Schematic overview of a scanning probe microscope SPM showing the detection principle for measuring the tip sample interac tion 2 5 Image Acquisition in Scanning Probe Microscopy In scanning probe microscopy SPM a tiny and sharp tip scans across a surface while the tip sample interaction is monitored The signal of this
215. flow pattern in a cavity where an exter 1 08x10 nal temperature gradient is maintained 1 transfer phenomena dimensionless numbers can be derived from similarity consid erations These numbers characterize the type of flow field as well as the observed heat transfer In the case of natural convection three numbers the Nusselt number the Grashof number and the Prandtl number are sufficient to describe any situation 3 The Nusselt number Nu is a dimensionless form of the heat transfer coefficient a defined as aL E Nu 2 150 It relates a to its limiting value in the case of pure heat conduction L where de notes the thermal conductivity and L is a characteristic length The Grashof number Gr represents the relation between buoyancy forces and viscous forces It can be inter preted as the Reynolds number in natural convection problems and is defined as gh AT Gr gt 2 151 The parameter h is the characteristic length of the considered flow problem e g the height of a heated vertical wall Furthermore g is the gravity constant v is the kine matic fluid viscosity T Tauia and AT Tyan Tania is the external temperature gradient applied Note that in the case of a cavity the applied temperature difference is AT Thot Teoja The third dimensionless number considered is the Prandtl num ber Pr which represents the ratio of the thickness of the hydrodynamic to the thermal boundary layer Peo 2 152 a K
216. fo V2 pu Van Ag iAg pdA mw fol Va p pV opRAg pSAg V2 pote a ee 2rw Jog ut V2pRAg SAg V2pSAg RAg d 2mw Jo PW VapRAg V2pSAg Wop BAe VapRAg dA 2rw fog u VapRAg SAg V2pSAg RAg dn 0 since the last boundary integral is zero by the symmetric choice of boundary condi tions Hence for the total dissipation we obtain we Jo o Jo Go ES ED 2mp d A _ a Hol E bonp aA Jy ane jo Vdd ye 1 Vi U V A steady thermal problem is solved to compute the temperature of the steel The tem perature of the surrounding air and wires is not considered instead at the interface between steel and air we use the linear mixed boundary condition F n a T Thir with F the thermal flux and n the normal to the boundary The coefficient a includes here thermal radiation and heat transfer to the surrounding air with an ambient tem perature of Tair This mixed BC is enough to stabilize the thermal problem thus making it unique solvable This example is found at example Hardening s2d The computational mesh has been constructed with the built in mesh generator and is shown in Fig 2 46 Due to the 74 SESES Tutorial September 2012 liquid metal solid metal sheet rotating cylinder Figure 2 48 Drawing of the metal sheet fab rication Figure 2 49 40kHz alternating current of 50 A in the four copper turns give rise to a ditch in the liquid metal The metal d
217. g a fixed amount of memory and floating point operations or an iterative solver requiring a fixed amount of memory but running an iteration loop up to a satisfactory convergence criterion is reached Here the convergence rate depends on the condition number p Amax Amin With Amin and Amax the extreme eigenvalues of the system matrix For p 1 we have the identity matrix and our linear system is easily solved but for increasing values p gt 1 the matrix departs from the identity matrix and the convergence rate slows down Iterative solvers requires the storage of the system ma trix and some additional working vectors to run whereas direct solvers requires much more memory since many zero coefficients of the initial sparse system matrix become non zero during the factorization For 3D applications the memory and computa tional requirements are generally too large for direct solvers to be used and so one has to resort to iterative solvers Quite some iterative solvers can be found in numer ical libraries and used out of the box however they do not generally converge at a satisfactory rate or may even diverge and therefore are not robust as direct solvers This problem can be partially overcome by preconditioning the linear system i e by solving a similar system much closer to the identity matrix in order to increase the convergence rate and thus the robustness of the solution process Clearly in order to be effective the associated similar sys
218. ght hand side Ilp The steady state condition implies that for II 0 we must have v 0 otherwise there is a contradiction Our model formulation for continuous casting is almost complete except that we still have a spatial formulation and we do not yet know where to solve the equa tions 2 124 In other words we do not know the exact shape of the spatial domain Q and therefore an Eulerian or spatial solution approach typical of fluid dynamics can not be used The matter is also a little bit more involved than for classical elasticity where with the help of the Piola identity the spatial formulation is pushed back to ob tain a Lagrangian or material one with respect to a referential domain 9 Due to the continuous flow a steady solution of a material formulation with respect to an initial stress free referential system with particles at rest does not exist since it would imply infinite deformations Therefore to be able computing steady solutions of 2 124 we need an Arbitrary Lagrange Euler or ALE approach There is nothing special about an ALE approach and everything boil down in the definition of a referential stress free system with particles not necessarily at rest This is the key point and the matter de pends on the problem at hand Therefore we assume a stress free strand at the exit of the mold and its free flow in space with all mechanical loads turned off will de termine our initial reference system Q9 Due to the continuo
219. he contact see Fig 2 38 2 39 The magnetic and mechan ical forces acting on the bendable lamella are shown in Fig 2 40 The mechanical force Fmech is represented by the flat surface it is a linear function of the contact distance and it does not depend on the magnetic field The magnetic force Fmag is a nonlinear function of the contact distance and the magnetic field Both forces are at the inter section of both surfaces in equilibrium If the magnetic force drops below the value at the point 3 then the relay opens and enters the new equilibrium state as indicated by point 4 If the external magnetic field is increased and exceeds the value at point 1 then the relay closes again The mobile lamella snatches down and the new equilib rium state is indicated by point 2 An optimal relay is characterized by a low value of the magnetic field necessary for closing the contact The switching hysteresis is repre sented by the path along the points 1 2 3 and 4 and for a reliable operation of the relay it is important for the hysteresis not to be too small in order to prevent a flutter of the contact at the time of switching Mathematical model For this micro reed relay the mathematical model is represented by the magnetostatic equations obtained from the Maxwell s equations with the assumptions of stationarity zero charges and zero electrical fields The equations to be solved for the magnetic field B and H field are given by VB 0 VxH Jo
220. he electric potential to 0 V and 1 V on both wires The value of the potential SESES Tutorial September 2012 57 EfiekdPhi Norm V m 18E 00 1 7E 00 1 6E 00 1 5E 00 14E 00 1 3E 00 1 3E 00 1 2E 00 1 4400 995 01 9 0E 01 8 1E 01 72E01 63 01 54E 01 45E01 36E 01 27E01 18E01 9 0E 02 0 0E 00 time 0 Figure 2 32 Electro scalar potential Phi be tween two charged cables Figure 2 33 Electric field absolute value E on the exterior boundary should be set to represent an infinite domain and a homo geneous Neumann BC is the preferred choice Fig 2 32 Fig 2 33 show the computed potential and electric field distribution The total charge 2 16 on the wire is computed with the statement Write Charge e e n Wirel Phi Flux integrate Bound Wirel Dfield X Normal X Dfield Y Normal yY whereas the first value being computed internally over residual assembling is gen erally more precise than the second one computed by direct integration From the charge we obtain a capacity of C 8 08 pF compared to the analytical value 2 17 of C 9 3pF It is possible to compute the inductance and capacitance of more complex structures following the procedures discussed above and so the impedance Z according to 2 13 Using the EddyHarmonic model it is also possible to include the effects of frequency dependency of material laws and thus to model non ideal transmission lines Once the dispersion relation is know for the full sp
221. he intersection of the first two do mains defines the wire s cut section where we apply the driving electrostatic voltage and measure the current the intersection of last two must defines the cut surface used to apply a possible magnetostatic potential jump Finally the macro bcGround must specify a grounding point for each wire If other current configurations are chosen then just this geometry section needs to be changed In the second part of the Initial section we define the three materials VectorJ Vector Scalar we declare the element field current 0 to store the current com puted in the preprocessing step and define the BCs The material Vect ord is used to compute the electrostatic current Jo and has a non zero conductivity of o 5 8x 107 A V m In the materials VectorJ Vector we will use the vector formulation 2 46 and in Scalar the scalar one 2 47 On the domain boundary 0Q we define both the scalar and vector potential to be zero With the macro bcGround we define all wire s grounding points and for each wire together with some logical ME opera tions we define a one sided boundary to apply a jump BC of 1 V for the electrostatic potential Y Again with logical ME operations we define the two one side boundaries of OQvec used to define the interface Neumann BCs of 2 48 For the magnetostatic po tential we have Macro A_DOT_N__DA A A O Normal 0 A 1 Normal 1 A 2 Normal 2 Normal 7 bcl Restrict materi
222. he lattice vectors a and b the vector c is computed internally From Fig 2 81 we see that the z component of a and b are equal and given by 1 e EE 2 101 a 2 101 The length of all axes in the top view of Fig 2 81 equals 2 3 and we obtain for the zx 122 SESES Tutorial September 2012 and y components 2 2 y 5 cost e 3 cos a 8 2 102 2 2 j 3 sin a bj 3 sin a b For simplicity we chose a 0 and define alpha 0xPI 180 x angle between Projection of a and xx beta 120xPI 180 x Angle between Projection of a and bx ax sqrt 2 0 3 0 cos alpha ay sqrt 2 0 3 0 sin alpha az sqrt 1 0 3 0 bx sqrt 2 0 3 0 cos alphatbeta by sqrt 2 0 3 0 sin alphatbeta bz sqrt 1 0 3 0 After the defintion of the relevant problem parameters we construct the mesh shown in Fig 2 82 and define the material properties For Yb YAG the photo elastic coeffi cients are not reported in the literature We therefore use the values of the well probed material Nd YAG because the coefficients will mainly be given by the host material YAG These values are defined with respect to the crystal lattice whose orientation is given by the vectors 2 101 and 2 102 and are to be transformed to the global system with the built in function TransformT4 Parameter Photo TransformT4 T XXXX 0 029 T vyvyyy 0 029 T 22224 0 029 T XXYY 0 0091 T YYZZ 0 0091 T XXZZ 0 0091 T XYXY 0 06157T YZYZ
223. hese paths are equally spaced perpendicular to the axis from it to the distance roPD Each path starts at the longitu dinal coordinate 10PDO has a length of 10PD and contains n10PD integration points for the OPD evaluation SESES Tutorial September 2012 115 Next the material parameters of Nd YAG NdYAG and undoped YAG Cap are speci fied MaterialSpec Cap Equation ThermalEnergy Elasticity Enable Parameter StressOrtho LinElastIso Emodule 307 GPa PoissonR 0 3 AlphalIso 1 78e 6 Temp Tair 1 65e 8 Temp xTemp Tair Tair STIunit Parameter Refrac 1 82 Parameter KappaIso D_Temp 1 9e8 pow log 5 33 Temp 7 14 331e2 Temp 331e2 Temp Temp 1 9e8 7 14 Temp pow log 5 33 Temp 8 14 W m K W m K 2 MaterialSpec NdYAG From Cap Parameter Heat conv Px Pump x y w0 M lambda alpha ref Pump x yt l w0 M lambda alpha ref W m 3 We start with YAG and since we model both for the temperature and the mechani cal displacement we have to enable their computation with the Equat ion statement The heat conductivity and the thermal expansion 5 o are functions of the temper ature T and following 6 we use the models 8 2 E a 7 a ean and Se 1 78x10 76 T To 1 65x1078 T T 2 98 with T K The parameters Emodule PoissonR and Refrac denote Young s modulus E the Poisson s ratio v and the refractive index no The parameter DRefrac denotes the change
224. hows a positive thermal dispersion In contrast some selected SESES Tutorial September 2012 127 Compensating thin disk luc Tcup lp lep Figure 2 90 Simulated device consisting of two laser rods and the compensating thin Figure 2 89 Scheme of a thermo optically disk self compensating amplifier with a com pensating thin disk placed between two transversally pumped laser rods glasses liquids and curing gels show a negative thermal dispersion whose modulus may be two magnitudes higher than for the laser gain medium Let us cut a transver sally pumped laser rod into two parts and place between them a thin disk of such a material then this disk will quite exactly adopt the radial temperature distribution of the rod The negative thermal dispersion in the disk leads to a negative thermal lens which will be much stronger than the ones in the two parts of the rod Therefore such a thin disk of suitable thickness should be able to mainly compensate the thermally induced lens of the pumped rod To guarantee a good contact of the thin disk and the two parts of the laser rod on the whole surfaces the most advantageous materials are liquids or curing gels Fig 2 89 shows a scheme of this set up called Thermo Optically Self Compensating Amplifier TOSCA The thin compensating disk is placed adjacent between two laser rods The main part of the rods and the compensating disk in case of a non hygroscopic material are in direct conta
225. i cal loads The solution shows some residual 2 72 08 3 14E 08 3 56E 08 stress StrainP XX Stress Trace Pa 75E 03 8 55E 07 7 83E 03 7 78E 07 6 90E 03 7 O1E 07 5 98E 03 6 24E 07 5 06E 03 5 47E 07 4 14E 03 4 70E 07 3 22E 03 3 93E 07 2 30E 03 3 16E 07 1 38 03 2 39E 07 4 61E 04 1 62E 07 4 59E 04 8 55E 06 1 38E 03 8 53E 05 2 30E 03 6 84E 06 3 22E 03 1 45E 07 4 14E 03 2 22E 07 5 06E 03 2 99E 07 5 98E 03 3 76E 07 6 90E 03 4 53E 07 7 82E 03 5 30E 07 8 74E 03 6 07E 07 Figure 2 108 Plastic strain component p 22 Figure 2 109 Trace of the stress tr s 162 SESES Tutorial September 2012 chanical loads the strand shape will be circular and to consider the bending forces straightening the strand on the horizontal part of the domain we define the strain Exx y yo R with R the radius of the strand and yo the middle vertical posi tion By just fixing the strand at the exit of the mold this initial strain with a non linear geometric formulation will result in a circular strand of constant curvature R see Fig 2 107 There are no predefined dof fields for pure convected transport by a velocity v therefore the dof fields St rainP representing the plastic strain tensor p must be declared with the statement ConvectDef StrainP T2Z Afterwards we define the material laws as discussed previously and BCs with a clamp ed strand at the exit of the mould and as crude approximation also
226. ic eigen problem Omii C C5 0 5 X with N C305 N The first eigen problem can also be replaced by a quadratically convergent Denman Beavers iteration directly computing C5 and running a little bit faster ll Gd 5 O55 ZO Eis Numerical model Up to now the model has been kept quite general the only assumptions have been made are that of a multiplicative decomposition of the deformation isotropic mate rial laws and the special choice of a first order time integration algorithm involving an exponential map The plastic models need now to be implemented in SESES as a 146 SESES Tutorial September 2012 material law for non linear elasticity where one has to define the P2K stress S as func tion of the Green Lagrange strain E together with the symmetric derivatives 0S J0E Since the relations 2 110 2 111 together with the spectral decomposition of be are model independent we see that each isotropic plastic model just needs to define the principal stresses 7 together with the derivatives 07 0e as function of the log strains For our example we use the following quadratic potential w OP ci uO e2 2 117 i with A u the Lame s constants of the infinitesimal theory For the stress we have T w e MS e 1 1 1 7 Que KONE 2 DEV e 2 118 i with x 2 3u the bulk modulus ONE 1 1 1 1 1 1 and DEV Id 1 3 ONE or in tensor notation T gt Tini Q n Idetr e
227. ical value of the impedance 2 84 for zero damping used to create Fig 2 65 Analytical solution In this section we present a 1D analytical solution and assume the piezo disc to have a thickness L along the z axis and use the material laws s E0 u g0 and D g zu 0 with E the Young s module g the piezoelectric coupling coefficient and e the dielectric constant The governing equations 2 77 reduce now to the simple form 0 s p and 0 D 0 and we look for a solution with a clamped mechanical BC u 0 0 at z 0 a stress free BC s L 0 at z L a grounded potential 0 0 at z 0 and a time harmonic driving potential L Voe at z L A straightforward computation shows that the solution is given by e sin Az u g sin v iwt 5 EERE Jre os with p a w a E g9 e v A cos L 1 E g9 and the electric impedance reads Voet LA 1 tan L iwD iwe 1 Z w 2 86 We may of course compute the same solution using the inhomogeneous and eigen so lutions decomposition as described previously The eigen solutions u z e for the SESES Tutorial September 2012 109 homogeneous BCs u 0 0 s L 0 and 0 L 0 are computed by solving the equation a0 uUn w2pun 0 The generic solution reads un ko cos n 2 ky sin An z with A2 p a w2 The potential n is found from the condition D const and reads g tun k2z k3 By applying
228. ical results Afterwards we will present a second example in more detail by explaining the meaning of the input statements used to describe the modeling prob lem as well as the statements required to compute and display numerical solutions We then continue to review the basics of SESES by presenting and explaining with simple examples the most common features that a user frequently applies when solv ing engineering problems As an example we review how to define algebraically a mesh of finite elements or how to solve non linear governing equations However for a complete review of all available features and possibilities the user manual should be consulted 1 1 The SESES Environment Fig 1 1 shows a schematic overview of the simulation environment The basic con cepts the Front End and the computational Kernel programs are thoroughly docu mented in the manual here they will be briefly discussed and illustrated with an ex ample The Seses and SesesSetUp files are input files and for convenience they may all be defined in a single container file The modeling problem is specified inside the initial section of the Seses file while numerical computations data extraction post processing and similar tasks are specified in the command section In the op tional SesesSetUp file settings of the Front End program are stored The Data files contain the simulation data used to visualize numerical results Please note that the use of an input conta
229. ices are the same as well as the three inductance matrices The eddy current problem is solved with the statements For n From 1 To NWIRE BC v n Jump iVint 1 Frequency V s Natural Vint Solve Stationary Define For m From 1 To NWIRE ADM 2 NWIRE x m 1 2 n 1 0 ADM 2 NWIREx m 1 2 n 1 1 x residual Bound v m Current iCurrent x v m iVint Flux v m Vint Flux BC v n Natural iVint For each wire we set the driving voltage of 1 V solve the linear system and compute the N wire currents yielding each time one column of the admittance matrix Since we solve for the time integrated electric potential Y W iw a jump BC value of 1 w is used for the dof field iVint It is a must to use the BC characteristics for the dof fields Vint iVint to compute the conductor currents 2 55 or a valid alternative is given by the built in function res idual however one should not directly evaluate boundary integrals This is the only available method if one removes the electric 96 SESES Tutorial September 2012 2 106 03 1 106 08 3 006 05 1 1 1 1 09e 09 sowas ROD er ieee R0 E 1 90e 03 1 08e 08 2 50e 05 Se L01 4 09e 09 _ lt I x Tia as 6 1 80e 03 1 06e 08 gt O a z 1 70e 03 P 8 5 2 00e 05 Pe 1 08e 09 3 03 E 04e et z 1 60e 03 g 150e 05 b Se 4 1 08e 09 8 2 1 500 03 H 1 02e 08 8 aas 5 amp 1 40e 03 F 1 00e 05 Ss 4 1 07e 09 8
230. ices of the impedance 2 56 are symmetric and Fig 2 60 shows the Roo Roi and Loo Loi co efficients for a frequency sweep up to Wmax 27x10 Hz Fig 2 61 shows a perfect agreement with the resistance Rog RZoo Ror RZoo and reactance Xoo SZoo X01 SZo1 coefficients compared with the values obtained from a second order ap proximation of the admittance Y w Z w References 1 H AMARI A BUFFA J C NEDELEC A justification of eddy currents model for the Maxwell equations SIAM J Appl Math Vol 60 No 5 pp 1805 1823 2000 2 O B R Edge element formulations of eddy current problems Comput Methods Appl Mech Engrg No 169 pp 391 405 1999 3 O B R A VALLI The Coulomb gauged vector potential formulation for the eddy current problem in general geometry Well posedness and numerical approximation Comput Methods Appl Mech Engrg No 196 pp 1890 1904 2007 4 P MONK Finite element methods for Maxwell s equations Oxford University Press 2003 5 G H GOLUB C F VAN LOAN Matrix computations 2nd edition The John Hop kins University Press 1989 SESES Tutorial September 2012 97 2 15 Preconditioning eddy current systems The modeling of multiphysics systems leads to the successive solution of generally large but sparsely filled linear systems of equations To solve these linear equations one has the choice between a direct solver based on a LU factorization and requir in
231. ics 160 T T T T 80 140 F m a 120 p 4 60 100 F 4 50 n S 80 J 40 5 9 60F J 30 3 t A Oo 40 H 20 20 10 0 1 1 1 1 0 0 4 0 5 0 6 0 7 0 8 0 9 1 m_C_Flow m_AFlow n_C_Seal m_A_Seal Voltage V Figure 2 155 Computational domains for Figure 2 156 Electrical characteristics of the cathode and anode with nopples SOFC system obtain V pv Io Ve ev V ja Ma o The left hand side of the species transport equation are actually the equations solved by SESES so that if Ig 0 we need an amendment for the right hand side With this amendment we see that the sum of the new productions rates is always zero thus assuring the validity of the constraint gt a 1 during the solution It is to be noted that it is well possible to compute mass and species transport problems with out requiring the mole fraction s sum to be one In fact it is up to the user to enforce this constraint by properly choosing correct BCs and production rates however if this is not the case we cannot use anymore the Ste fanMaxwe11Dif f built in model Clearly if the mole fraction s sum must be one and this constraint remains valid dur ing the solution one mole fraction is completely redundant and in fact for speed up purposes we are not going to solve the transport equation for one judiciously chosen species taken as N2 However some numerical amendments are then required which however run much faster then the so
232. ies In particular the magnitude of the electrical field in the rod along the z axis is spec ified with the copper properties The electrical field is given as the quotient of the maximum DC current that copper can stand and the DC conductivity of copper To vary the total current the value of the electrical field can be increased by a factor CurrCorr In order to resolve details of the conductor and its close surrounding the mesh has been additionally refined in the central region via the parameter nn Note the distinction between logic coordinates integer numbers and absolute coordinates real dimensions This distinction is important for the geometrical transformation in part two Part two defines the mesh and the geometry as well as the minimum refine ment level for calculation The homotopic transformation of the rectangular geometry 76 SESES Tutorial September 2012 Figure 2 52 Left skin effect at 40 kHz AC excitation Only the outer shell of the copper rod carries current Right magnetic field lines in the vicinity of the copper rod at 40 kHz alternating electrical field of the copper rod and its close surrounding to a spherical geometry is optional The effect of the statements Include Homotopic sfc Coord sphere coord ax2 ax3 2 ay2 ay3 2 1 block nxl nx4 nyl ny4 is shown in Fig 2 51 In part three the space is filled with material and the boundary conditions are defined The vector potential A is chose
233. ilicon Parameter KappaIso 1 Temp W m K Convergence restore default Write Solving the non linear problem n Solve Stationary After restoring the default convergence criterion this time the output will be some thing similar to Solving the non linear problem AbsResid Temp 8 73e 01 AbsResid Temp 5 52e 02 AbsResid Temp 1 26e 04 AbsResid Temp 6 58e 05 AbsResid Temp 3 41le 05 AbsResid Temp 1 77e 05 SESES SOFT ERROR Coupled Newton Raphson loop did not converge SESES recognizes the new problem to be non linear and so it automatically starts a Newton s iteration consisting of linear solution steps The convergence of the resid ual norm is low and since after 15 steps the convergence criterion is not fulfilled the Newton s algorithm gives up and returns The coupled Newton s algorithm how ever converges in the vicinity of a solution quadratically but only if the derivatives of the residual equations with respect to the solution are available For this problem we have specified the heat conductivity to be a function of the temperature but SESES is not designed to figure out the derivative Actually with some programming effort it would be possible to obtain the derivative either in analytical form or numerically by difference However the former method is rather complex and may result in long and complex analytical expressions to evaluate while the latter may be numeri
234. in the presence of the rolls which for simplicity are defined all along the straight zone Here a slight improvement of the J2 flow model II f dev s n is used by considering a rate expressed in the form II f dev s ep n This new form allows e g for isotropic hardening and the required derivatives Ds f Dz f readily follow from the two scalar ones D devs f and D lt f which are returned together with the rate f dev s by the routine F lowRule For the solution algorithm and the dof fields u and p we can choose between a fully coupled or uncoupled algorithm however a coupled algorithm is generally required due to the stiffness of the system This is mainly due to the fact that for a velocity v 0 and if Il 4 0 then Ve and the problem does not have a solution In practice smaller velocities will result in larger condition numbers of the linear system to be solved and for a fixed length floating point format there is a velocity lower bound below the one no solutions can be computed To improve the condition of the system one should check that the linear mechanical solution without plastic rates should be smooth without any singularities in the stress If the stress is singular the plastic rates Il are unbounded with strong gradients making the problem numerically ill conditioned and very hard or even unsolvable Fig 2 106 shows for some arbitrary material parameters the convected plastic strain component p
235. in the solid GDL We use here the same Stefan Maxwell diffusion laws as for free convection but we scale down all coefficients by a constant factor However as we will see later underneath the nopples and because of the 1D cross coupling the total mass production rate is not zero The governing equations for the species transport cannot take this into account and so in order to collect together and not to loose this mass and to have a correct ac count on the mass flow we still solve for the convective mass transport but by forcing a slow velocity Electrial current transport is also considered at the anode and cath ode by defining the model equation OhmicCurrent Since it is not possible to set BCs over an open domain just the nopple s boundary is defined as an electrical contact and within the nopples we use a large fictive electric conductivity to numerical enforce an equipotential condition over the nopple s area A factor of 10 100 larger than usual will do fine without impairing too much the numerical stability With the description of the 2D transport models for anode and cathode we are done and we are left with the 1D point to point model specification Several choices are possible here but it is also true that these 1D models are easily changed adapted to any particular situation and plugged in without the need to change the structure of the 2D transport model so that for our purposes any 1D model will actually do fine Essentially
236. ination of potentials which makes the problem a little bit involved and this contribution aims to clarify this situation We start with the formulation of the magnetostatic problem characterized by solving the subset of the Maxwell equations VB 0 VxH J 2 39 for the induction field B the magnetic field H and for a given external time constant current Jo having the property V Jo 0 Induction and magnetic fields are assumed to be related by the constitutive relationship B uH with u the permeability In order to have a numerically attractive formulation we would like to solve this problem by computing a single scalar potential which however requires an irrotational i e curl free governing equation which is not yet the case for 2 39 For the moment lets assume we are working in empty space Q R with u po 4r x 107 Vs m Then the solution for the H field in 2 39 is well known and given by the Biot Savart integral Hps x f a ayy 2 40 78 SESES Tutorial September 2012 The major idea behind the magnetostatic reduced formulation is to split the H field in two parts H Ho Hj one consisting of the known Biot Savart integral Hp Hgs having the property V x Ho Jo and the irrotational part H to be computed by a scalar formulation and taking into account material and bounded domains effects In fact for the Hj field we now have V x H 0 so that a reduced magnetic scalar po tential O q can be defined with H V
237. iner file has the advantage that it can be either processed by the Front End program or executed as a script file or passed as argument to the Kernel program in a command line fashion without invoking the Front End The Front End program is an interactive GUI program for the definition and visualiza tion of simulation domains finite element meshes and numerical results Within this SESES Tutorial September 2012 9 NM SESES GUI Front End Container File NM SESES Computational Figure 1 1 Schematic overview of the SESES simulation environment application it is possible to edit the input file to graphically construct a 2D mesh to start a simulation in the background and to visualize numerical results as shown in the Fig 1 3 1 4 1 5 The computational Kernel program is a batch oriented program reading the input file and generating results by sending information to the output stream or by writing data files Modeling Strategy We briefly discuss here the solution approach common to many design and optimiza tion problems The numerical modeling process can be subdivided into the following tasks e Geometry Specification The geometrical domain relevant to the problem must be identified Check if geometrical symmetries exist that allow a reduction of the computational effort e Physical Model Specification Identify the governing equations relevant to the problem Check if the governing equations and their variables are co
238. interaction allows the se quential construction of an image of the sample surface to be investigated Two dis tinct operating principles for scanning are used constant height and constant distan ce In the former principle the tip height is kept constant and the varying signal is recorded In contrast for the constant distance mode an electronic feedback loop adjusts the z position to ensure constant tip sample distance and the z position is recorded The most common measurement principles of the tip sample interaction are the atomic force employed in atomic force microscopy AFM and the electronic tunneling current used in scanning tunneling microscopy STM In this tutorial example the operation of SPMs shall be simulated in order to illus trate the different operating principles and the limitations for the lateral resolution In particular a virtual experiment with a sample surface shall be carried out in order to study the acquired surface image with regards to the tip geometry In mathematical terms the acquired image is the convolution of the tip geometry and the surface to pography With simplified models we shall calculate the electrostatic field between a metallic tip and a conducting sample surface and interprete this quantity as a measure of the resulting tunneling current In a real STM experiment the geometry of the tip is not that critical since the tunneling current depends exponentially on the tip sample distance Thus the out
239. ion we then add a moving stamp impressing a shape to the laminate We assume here the drawing process does not 150 SESES Tutorial September 2012 deform the stamp itself the mechanical contacts just show up between the laminate and the stamp and the process is so slow that inertial forces can be neglected These assumptions greatly simplify the solution of the problem since the stamp s rigid body geometry is a mere input parameter known at any time there are no self contacts to be considered and we need to solve a series of stationary solutions parameterized with respect to a pseudo time parameter determining the movement of the stamp The physics of mechanical contacts may be a complex matter and tribology is the sci ence describing the interactions at the contact surface However for the ideal cases of a friction less and sticky contact the underlying physics is completely blended out and replaced by pure kinematic conditions There are some practical interests in these two ideal situations since they represent extreme cases with real contacts lying in between For both cases we have first the impenetrability condition of the bodies involved Then if this condition occurs for the friction less case both bodies at the contact surface glide without friction for the sticky case they are as glued together No other physical laws are used here and the simple specification combined with ro bust solution methods makes these ideal contacts app
240. iori its absolute value may not say much However in this ex ample we see that when solving the for second and third time the value is almost 15 orders of magnitude smaller than the first time and it does not change much with the third step With the second and third step the temperature value is already the solution of the governing equation and so the residuals are numerically zero Due to the limited size of the number representation in computer memory a numerical zero is almost never an algebraic zero and must always be interpreted with respect to numerical values being computed The machine precision e is defined as the largest number such that 1 1 holds and for common applications we have e 107 which explains the drop of 15 orders of magnitude in the residual norm between the first and second solution step The convergence behavior can also be characterized with respect to increment norms i e the change in the numerical solution Increment norms have the advantages to be easier to interprete since their reflect the change of the independent variables solved for and not the dependent out of balance values of the governing equations We there fore repeat the previous example by printing the L increment norm Solve Init Convergence write AbsIncr Temp 2e n AbsIncr Temp return 1 Write Solving for the first time Solve Stationary Write Solving for the second time Solve Stationary Write So
241. ison with more precise finite element models considering scattering effects within the air gap Estimating the magnetic inductance For induction phenomena the displacement current term D within the Maxwell equations can be neglected hence to estimate the inductance L we can work with the Maxwell equations V x H jcona and V x E 6B 0 By considering an oriented surface F with boundary OF and normal n their integral form after application of Stoke s theorem reads H ds jena dn and E ds f Bean 2 18 aF F Ot Jr OF By considering the surface F A as shown in Fig 2 36 from the first integral in 2 18 we obtain NI H ds fp B ds 2 19 OA OA with N the number of coil windings and J the coil s current The normal component of the magnetic induction B is continuous across the boundary between air and magnetic material hence to a first order we can assume B to be constant along the closed path OA resulting in leore h d N T e Blea eee core Hdisk Ho with h d a as defined in Fig 2 36 and core the lenght of the magnetic core with large permittvity core By considering the surface F Score to be a section of the core and SESES Tutorial September 2012 59 Figure 2 36 Simply connected simulation do main Q without currents Figure 2 37 Magnetic field around disk that here the induction field B due to the high permittivity is almost constant from the second integral in 2 18 w
242. itch radius is 0 5cm deep to be compared with 2 37 rotational symmetry the left edge of the domain must be located at x 0 Further it is important to correctly resolve the boundary layer caused by the skin effect as shown in Fig 2 47 for the dissipated heat The problem is defined in a way that given the coil current I C we solve the consistent eddy problem to compute the coil voltage V C which is followed by the thermal computation On the output as show below CURRENT WIREO 3 000000e 02 1 439329e 11 CURRENT WIRE1 3 000000e 02 2 026934e 11 CURRENT WIRE2 3 000000e 02 2 239964e 11 CURRENT WIRE3 3 000000e 02 2 508244e 12 OT DISSP HEAT 0 7 8354381554594045e 02 W OT DISSP HEAT 1 7 8354381554478482e 02 W PPP Pp we check the consistency condition for the current J on each wire and that the com putation of the total dissipated heat can indeed be computed either by the relation W 27 fa qedA or W I V which are equivalent to machine accuracy 2 11 Eddy Current Repulsion The fabrication of thin metal sheets by solidification of the liquid metal between ro tating cylinders has not attained industrial production yet Fig 2 48 shows the idea of the sheet fabrication For a successful industrial production several technical prob lems have to be surmounted One such problem is the confinement of the liquid metal at both ends of the cylinders Eddy current forces have
243. ith a nested iter ative solver and its own preconditioner or inexactly by just applying once the auxil SESES Tutorial September 2012 103 iary preconditioner We point out that if a nested iterative solver is used the auxil iary solution is inexact and non constant and the CG algorithm may not work well anymore A generalized CG algorithm with additional orthogonalization steps for a non constant preconditioner matrix can be used to fix the problem but by our expe rience this is not generally required Smoothing plays an important role as given in Table 2 2 where we have solved the auxiliary problem both with an exact LU and inexact ILU 0 preconditioner and a variable number of pre and post smoothing cycles performed before and after solving the auxiliary problem By applying a suf ficient number of pre and post smoothing cycles the number of iterations needed to reach convergence is similar as for first order nodal elements However smoothing is quite an expensive numerical operation with each symmetric Guass Seidel cycle being equivalent to two matrix vector multiplications but not easily parallelized A single pre and post smoothing cycle looks to be an optimum For edge elements we have not yet found a robust preconditioner as for nodal ele ments of first order The major reason is that since the system matrix S is singular the incomplete factorization generally fails This unfavorable situation can be in part saved by adding a
244. itution into 2 104 yields the following equations Ehi Eh AANny Enti nt AA nn41 Va Snt1 ktr En41 Id 2u dev enqi eh41 gt Qn 1 4H 1 n41 Ko n41 fn 1 0 In order to obtain a solution to these equations we note that the normal vector n to the yield surface f does not depend on any values at t except for the strain En 1 which is a know quantity 2u dev En 1 Eh Gin _ dev Sirial ain P nnti a al aMMa S g 2u dev En 1 En inl dev Strial i inl with Strial k tr n41 Id 2u dev En41 2u the stress of the trial solution 2 105 This decoupling is a property of the von Mises yield function and does not hold for general models A final evaluation of fn 0 yields the value of AX AA dev Strial Gian 2oy 2 n Fog Sa DS ee a ee ee In summary the stress to be defined as input to SESES is given by Sn 1 Striai if Serial lt 0 otherwise by ftrinn S g i 2 YD S n l trial ire 2H 2K SESES Tutorial September 2012 139 To obtain an optimal convergence behavior we need further the derivative of the stress Sn 1 With respect to the strain 41 In order to keep the same notation we compute here directional derivatives D e a with respect to a vector a By noting the properties tr dev a 0 and n 41 dev a n 1 a after some algebra we obtain Doras ftriaila 2u Nn 1 a De Strial a ktr a Id 2u dev
245. ivergence However by using linear extrapolation of the solutions with respect to the pseudo time Extrapolation Linear the number of steps can be reduced to 10 Fig 2 97 shows the displacement u shear stress Syy and yield region where f s 0 holds for the solution at time t 0 07 corresponding to a vertical displacement of the upper edge of 0 07 m with the bottom one being clamped in the vertical direction An amplification factor of 20 has been used to visualize the displacement on the computational domain References 1 J LUBLINER Plasticity Theory Dover Publications 2008 2 A S KHAN S HUANG Continuum Theory of Plasticity John Wiley amp Sons 1995 3 J C SIMO T J R HUGHES Computational Inelasticity Springer Verlag 1998 4 J C SIMO Numerical Analysis and Simulation of Plasticity Handbook of Numerical Analysis by P G Ciarlet and J L Lions Vol VI North Holland 1998 5 W HAN B D REDDY Plasticity Mathematical Theory and Numerical Analysis Springer Verlag 1999 SESES Tutorial September 2012 141 2 22 Necking of a circular bar In this example we extend the previous infinitesimal elasto plastic model to the finite strain case by considering the necking of a rod subjected to a large elongation For de formations larger than 10 20 the modeling of anisotropic plasticity is still an active research field and open questions still remain whereas the isotropic case is much less controversial
246. k stored matrix lt name gt is loaded in memory and stored back at the end of the block so lution so that the ReuseFactoriz option will work again The matrix back storage can be released with the statement Misc ReleaseMatrix The sequence of linear problems starts with the electrostatic problem 2 61 computing in one single step all wire currents Jm 0 m 1 N All these initial currents are required later on to compute the inductance matrix L 0 by the energy method 2 65 and therefore they are stored in the element field current 0 Afterwards for each sin gle wire we solve the sequence of electro and magneto static problems We compute here the admittance matrix 2 57 up to the second order and from 2 59 we obtain the resistance R 0 and inductance L 0 matrices at w 0 After the first magne tostatic solution we can compute L 0 by the energy method 2 65 but the volume integral 2 64 over 2 may be used as well From the residual equations 2 67 together with A 0 on OQ it is possible to show that the same numerical values are obtained but just if we exactly have V A 0 which is not generally the case These resistance and inductance matrices can also be obtained by solving the eddy current model 2 53 at very low frequencies A frequency value of w 0 cannot be used but the problem is stable with respect to w 0 and so we use w 271x107 Hz The result is that up to machine precision the two computed resistance matr
247. l voltage be less than 10 of the ideal case by choos ing the device length a as small as possible Another important optimization step is the minimization of the offset Hall voltage relevant for the device precision This is the Hall voltage measured when the external magnetic field is zero B 0 and is determined by slightly misaligned Hall contacts introduced by the CMOS fabrication process Other studies requiring a 3D model may include the influence of the device thickness or a magnetic field B not orthogonal to the sensor plane In the following we will limit ourselves to a 2D model and obtain the device character istics as function of the geometric parameters as device length device width contact dimension and contact misalignment The SESES input file for this Hall sensor problem can be found at example Hall Sensor s2d The geometry of the device and the definition of the contacts is shown in Fig 2 6 The device is fully parameterized with the help of user variables Define width 5 um x length 20 0 x um x cont_w 2 0 x um x misalign 0 2 um x bfield 1 x Tesla x mobil 0 1265 m x 2 Vx s x vapplied 1 V x The governing equation 2 1 and the material law 2 2 are selected by defining mate rial Silicon as follows MaterialSpec Silicon Equation OhmicCurrent Enable Parameter SigmaUns HallRes SigmaIso Q0 1 0e24 mobil S m BfieldZ bfield T MobHall mobil m 2 V s STunit The BCs f
248. le cube centered at the same point with mp 1 The next function Routine double spheremix T1 double val T1 double cen T1 double rs double rq double type uses the function sphere with a non constant morphing parameter to gradually maps cube surfaces into in spheres For rs lt rq if the half side of the cube is larger than rq the cube surface is left invariant and for values less than rs you will get exact spheres so that you actually obtain a ball in a cube For rs gt rq the role is reversed and you obtain a cube in a ball The parameter t ype determines the ramp used to gradually apply the morphing parameter A cube is gradually transformed with respect to the x coordinate first example of Fig 1 15 Coord sphere coord 5 5 5 x 10 1 A ball in a cube second example of Fig 1 15 Coord spheremix coord 5 5 5 2 4 2 block 1 9 1 9 1 9 A cube in a ball third example of Fig 1 15 Coord spheremix coord 5 5 5 3 2 1 block 1 9 1 9 1 9 Cylindrical geometries The cylindrical maps are equivalent to the 2D spherical maps and for each sphere function there are three x y or z versions corresponding to the yz xz xy planes SESES Tutorial September 2012 25 y Figure 1 15 Sphere examples with sphere and spheremix y Figure 1 16 Cylindrical geometries with cy1x cyly cy1z where the 2D sphere functions are applied Since no new concepts are meet here we limit ourself in giving application examples with fun
249. les with the refractive index following c c n a plane wave en tering a thermally loaded crystal is disturbed during its travel through the crystal as shown in the upper part of Fig 2 66 This situation can be compared to the phase front deformation caused by an ideal thin lens shown in the lower part of Fig 2 66 A valuable means to quantify this phase front deformation is the OPD Let us assume a homogenous material of length with the refractive index no The time a light ray needs to travel through the crystal is given by E no At c Within the same time the light would travel a distance of c At no in free space We therefore call fo no the optical length of the material In our case the refractive 112 SESES Tutorial September 2012 index of the crystal depends on the local temperature and therefore varies along the beam path following 2 90 We call this the disturbed case and the corresponding optical length The difference between the disturbed and the undisturbed case 4 opt is the so called thermal OPD given by opt a n OPDin fot o Ane dz f f ar dT dz 2 91 0 0 To This integral is evaluated along different paths parallel to the laser beam axes in the thermally loaded crystal as shown in Fig 2 67 This leads to the OPD r as a func tion of the distance r to this axes Another lens like effect the end effect is introduced by the bending of the front and back surfa
250. let gt 5 22497695e 15 N lt Field VelocityY Flux ForceY TypeBC Dirichlet gt 0 0005 N lt BC_data gt File Data written The maximal error for VelocityX is 4 726968e 14 The maximal error for VelocityY is 1 95080le 13 From this output we can see that the mass flow of 0 5kg s at the inlet outlet differs from 2 126 by ca 4 the viscous flow exerts a total force of 0 0005 N on the channel and the accuracy of the velocity is close to machine accuracy The precision of the velocity field seems surprising if one considers the very limited number of used elements and the error affecting the total mass flow This type of superconvergence is occasional and a little modification of the problem will remove it For the Hagen Poiseuille problem and from the relation 2 126 we see that the total mass flow and the pressure drop in the channel are equivalent for the problem specification In practice however the pressure at the inlet outlet is unknown so that one prefer to specify the total mass flow We can do this by a little amendment at the inlet BC where instead of setting the pressure we define the total mass flow BC Inlet IType 0 nx 0 Pressure Floating massflow kg s VelocityX Dirichlet 0 m s Differently from the previous case the maximal error in the velocity is now about 4 and by changing the number of elements you will see a linear convergence behavior The dramatical
251. lock OhmicCurrent ReuseMatrix electromat Solve Stationary For m From 1 To NWIRE Define CUR m 1 v m Psi Flux Store current1 DofField Afield Psi IpFrom FIELD Current Procedure double SolveMagnetoStatic MaterialSpec Wire Equation EddyFreeNodal Enable OhmicCurrent Disable MaterialSpec AirEddy Equation EddyFreeNodal Enable BlockStruct Block EddyFree ReuseMatrix magnetomat Solve Stationary Store currentl DofField Afield Psi IpFrom FIELD Afield SigmalIso For both problems the source term is an initial current and is defined by the material parameters MaterialSpec Wire Parameter CurrentO currentl A m 2 Parameter Current00O currentl A mx 2 SESES Tutorial September 2012 95 We use here the element field current1 as link between both formulations At the end of the electrostatic problem we store in current 1 the computed current J and at the end of the magnetostatic problem we store the value of oA This is a necessity since when solving e g the electrostatic problem by enabling the material equation OhmicCurrent afterwards the value of A from the previous magnetostatic solu tion is lost Since we solve alternatively two different linear problems the option ReuseFactoriz discussed above will not work here since it requires the same block matrix However we can use the back storage of block matrices enabled with the option ReuseMatrix lt name gt Here at the beginning of a block solution the bac
252. low region m_void MaterialSpec m_void Equation CompressibleFlow ThermalEnergy TransportH2 TransportH20 TransportO2 TransportN2 Enable Parameter Density IdealGas AmbientPress p0 Pa SIunit Macro Renew BURNH2_RATE 1 0e 3 x mol s m 3 x Macro Renew BURNH2_DH 2 48e5 x J mol x Macro Renew BURNH2_DS 55 0 x J K xmol x Parameter RREA BURNH2 RREA Here the Equation statement activates the balance equation NavierStokes for the overall mass and momentum the mass balances H2 H20 O2 N2 for each species as well as the energy balance Temp Within the Parameter section we define the ideal gas density law and the producion rates for the hydrogen burning reaction by calling the macro BURNH2 By newly defining macros before the macro call of BURNH2 we can redefine default parameters used by the macro ifself The large negative enthalpy change indicates that the burning of hydrogen is strongly exothermic i e it produces a large amount of heat The other Parameter statements define the following prop erties the overall thermal conductivity Kappalso the viscosity Viscosity of the gas mixture the molecular weights Mmo1 lt A gt the heat capacities Cp lt A gt and the pair wise diffusion coefficients Di f lt A gt _ lt B gt of all species Note that the properties of the porous flow region m_reac are very similar to those of m_void To transfer the prop erties of m void to m_reac we use the statementm_reac From
253. lts in a regular linear system You may check this fact by setting the variable displaymatcond and thus running some Maple code on the linear system matrix written to the file tmp mat by the statement Info LinearMatrix Format 17e MatrixFile tmp mat HideBC 88 SESES Tutorial September 2012 When using both the scalar and vector formulation there are excess of dofs at the interface OQyec which are not eliminated internally In this case the direct solver is still capable of computing the correct solution but attention since this may not always be the case as for example when just computing an half quarter or eighth of this symmetric solution Also do not except the same symmetry for the vector potential A which however must be the case for B and H 2 14 Eddy currents in a linear system The numerical characterization of a system of conductors wires or coils with time varying currents is in the industrial developing cycle an important topic and may be used to characterize e g inductive proximity sensors For many problems of interest the geometry is truly 3D so that 2D models cannot be used and numerical expensive 3D computations have to be done However by assuming a system with a linear response the analysis performed within the context of linear theory simplifies the problem at hand Further if just the low frequency range is of interest we will show some valid approximations to additionally reduce the computational work Theref
254. lution of one transport equation If the amend ments of the right hand side of the species transport equations are strictly required the one on the right hand side of the total mass transport is not In fact one can argue that in our case Ip does not have a strong impact on the velocity v profile and there fore a value of II 0 may be considered in order to completely decouple the total mass flow from the other transport equations resulting in a computational speed up However since the coupling through a non zero IIo is weak there is no need to specify cross coupling derivatives and therefore there is almost no slow down in considering this type of coupling For a correct account of the mass flow we need to solve for the mass transport everywhere where IIo 4 0 and this is include the area underneath the nopples where virtually there is no convection at all Model specification The SESES input file for this model can be found at example SOFC2p1 s2d For the cathode and anode we define two disjoint domains with the same geometry and a total of four materials see Fig 2 155 Two materials are for the cathode and anode flow channels and other two materials model the nopple s area where convective flow is almost turned off Some details about the material s properties have already been SESES Tutorial September 2012 215 discussed and are not repeated here The thickness of the third dimension is set with the statement GlobalSpec P
255. lving for the third time Solve Stationary 28 SESES Tutorial September 2012 As first step we initialize again to zero the solution in order to destroy the previ ously computed one Then we substitute the default printing statement displaying the residual norm with a statement printing the incremental norm of the temperature This is done by defining a user convergence criterion and by embedding a call to the write function Since the problem is still linear the criterion always returns 1 mean ing unconditional convergence As last we solve as before three times and obtain the following output Solution newly initialized Solving for the first time AbsIncr Temp 1 86e 00 Solving for the second time AbsIncr Temp 6 60e 16 Solving for the third time AbsIncr Temp 1 77e 16 The numerical behavior is the same as before but now we can state that the tempera ture is computed up to a precision of 10 This is the precision of the numerical solu tion when solving the discretized governing equations which is by far much smaller than the precision with respect to the exact analytical solution of the governing equa tion Now that we have acquainted some feeling on norm values for numerical solutions of the linear thermal problem we change its character and turn it into a non linear problem Here we define the thermal conductivity to be a function of the temperature k K T 1 T and try to compute a solution MaterialSpec S
256. ly the sensor and heater pads are separated from the flow channel by a thin membrane If the medium is at rest a symmetric temperature distribution around the heater is expected and the two sensors one upstream and one downstream from the heater location measure identical temperatures For non zero flow rates however the temperature distribution is not symmetric any more The difference of the two sensor temperatures is then proportional to and therefore a measure of the flow rate The design of a specific sensor geometry will depend on the application of interest and the sensor electronics at hand The simulations documented here allow for efficient sensor characterization prior to fabrication In particular a qualitative and quantita tive understanding of sensor operation is enabled by the simulation We attempt to illustrate what kind of questions might arise when developing a calorimetric flow sen sor 1 Table 2 4 gives an overview of the operating principles commonly used with calorimetric flow sensors Each operation mode has its benefits and drawbacks For instance in constant power mode a regulation of the temperature of the heater is not necessary but the sensor signal will react slowly to an abrupt change in the flow rate The simulation results for different operating principles will be discussed below 182 SESES Tutorial September 2012 Model Specification Let us now build a SESES model of the flow sensor Since we focus on
257. ly couples the different transport phenomena The syntax requirements for the input files the numerical methods and the physical models for the different application fields are described in detail in the SESES user SESES Tutorial September 2012 7 manual While the user manual serves as reference for everyday use this tutorial rep resents a step by step introduction to solve design and optimization problems The basic concepts are explained with the help of typical application examples Several selected SESES features and postprocessing techniques allowing the information ob tained with the numerical simulation to be passed to other applications such as Mat lab or Excel will also be illustrated Thus the reader of this tutorial will learn how to use SESES to solve engineering problems by describing them in mathematical terms modeling executing the numerical calculations simulating analyzing the data ina useful format postprocessing and how to generate new designs optimizing Chapter 1 Getting Started In this chapter the reader will learn the basics of SESES We will start by reviewing the SESES environment and after presenting some basic strategies for multiphysics mod eling a first example is presented to check a correct running environment Although the example is not thoroughly explained it serves as an introduction on how to use the SESES Front End program how to run a numerical computation and how to visualize numer
258. m amp y 0 2m gt lt y 0 3m t y 0 5m 400 y 0 7m 700 600 500 Velocity m s Temperature C 400 300 200 0 0 005 oo 0015 Ge 0 025 0 03 0 0 005 0 01 0 015 0 02 0 025 0 03 Radius m Radius m Figure 2 140 Velocity profiles across pipe Figure 2 141 Temperature profiles across cross sections at various locations pipe cross sections at various locations References 1 H L LANGHAAR J Appl Mech Vol 64 A 55 1942 2 W M Kays M E CRAWFORD Convective Heat an Mass Transfer McGraw Hill 1980 3 WARMEATLAS VDI Diisseldorf 1980 2 32 Hot Spots in a Tubular Reactor In this example we show how SESES can be coupled with other simulation software As illustrated in Fig 2 142 we consider a developing flow in a chemical reactor The open channel of the reactor has a circular cross section Air at ambient temperature enters the channel with a homogeneous velocity of 0 04m s A porous shell is placed around the open channel with an annulus cross section A mixture of hydrogen and water vapor enters the shell from the face surface and diffuses through the porous me dia Oxygen from the air penetrates the porous material and reacts with the hydrogen according to 2H O2 gt 2 H20 2 171 The air flow is simulated based on the CFD tool CFX 5 1 The
259. m cfx 2 29 Calorimetric Flow Sensor In this tutorial example the operating principle of a calorimetric flow sensor and its implementation in SESES is discussed Rather than focusing on the geometrical de sign of realistic flow sensors we wish to illustrate a modeling method for this specific microfluidic device that relies on the solution of the heat conduction equation with forced convection Many different physical principles for measuring flow are employed in practice For conductive fluids an electromagnetic measuring principle based on the induced volt lsee for instance www flowmeterdirectory com SESES Tutorial September 2012 181 Sensor 1 Heater Sensor 2 Figure 2 122 Geometry of a typical calorimet ric flow sensor Figure 2 123 Simulation domain with the boundary conditions highlighted Operation Mode Characteristics Application Comments Constant AT gt _ f vo small fluid flows no temperature Power P gaseous media regulation necessary Ty Fe fluid flows Constant AT gt _1 f vo small fluid flows temperature regulation Temperature TH P f vo medium fluid flows fast response Table 2 4 Operating principles of the calorimetric flow sensor age in a magnetic field is employed For microfluidic systems the calorimetric flow sensor principle is widely used Such sensors contain a heater and two temperature sensors that are in good thermal contact with the flow channel volume see Fig 2 122 Typical
260. m_void In addition the material m_reac contains the parameter setting 198 SESES Tutorial September 2012 MaterialSpec m_reac From m_void Parameter Kappalso 0 10 W m K to adjust the thermal conductivity to its value within the porous material Finally the defined models are assigned to their computational domains using the Material statement and boundary conditions are set Here we give the compositions and tem peratures at the air and burning gas inlets With this the SESES model of the reactor is complete What remains is to control and start the actual simulation and to specify the kind of output generated Both is done within the command section of the input file We start with the initialization of the various field variables using the Init statement As already mentioned the velocity fields within the free flow region are imported from CFX 5 simulations We initialize those fields with the above defined functions w veloc x y and u_veloc x y The temperature and pressure are set to their reference values and the compositions on the regions of free and porous flow are set to their values at the inlets In SESES this is accomplished using the function material lt mat gt which is 1 on lt mat gt and 0 elsewhere We then use the Remesh statement to locally refine the mesh at the boundary between the regions of free and porous flow Once all initialization parameters are set Solve Init performs the actual initialization The nex
261. merical data is presented and then examples for postpro cessing the data with mathematical and plotting tools including Mathematica and Gnuplot are given We also discuss how to create an external control system with Mat SESES Tutorial September 2012 35 Lab that uses feedback from simulation results and launches SESES in batch mode in order e g to automate the search for an optimized design Let us first discuss how to store simulation data in a multicolumn ASCII format for subsequent import in other software tools We shall illustrate this with the cantilever microactuator example example E1lThermMech s2d Analogous to the previous example discussed in the context of animation we run a Solve Stationary state ment with the sweep specification for the parameter cur according to Solve Stationary ForSimPar cur At 0 005 Step 0 005 4 Here we compute five solutions starting with a current of 5mA and with increments of 5mA For plot post processing we want to create a column table with the current values in the first column the zx and y displacement of the cantilever tip in the second and third column and in the last column the average temperature of the device The first table row is reserved for a comment and the vertical size of the table is given by the number of solution steps i e the number of different current values These are the statements used to create our table and they must be defined before the Solve statement Lattice tip n
262. ming a harmonic time dependency of the form exp iwt which is always implied and for conciseness not written explicitely At each harmonic frequency w our lin ear system is fully characterized by the impedance matrix Z w with V Z w I and V I CN being the voltage and current values at the contacts From 2 54 we see that at low frequencies w 0 the impedance has a constant real part and an imaginary part linear in w showing the inductive character of the system Therefore the complex impedance is also written in the following form splitting the real and imaginary parts into a resistance matrix R and inductance matrix L Z w R w iwL w 2 56 Since the equations 2 52 are uniquely solvable the impedance matrix is regular and can be inverted Therefore for numerical computations we have the free choice in selecting either V or I as independent variables but even a mix would be possible Because we solve for the primary fields A and Y W iw here the most convenient way is to compute the admittance matrix Y w Z w by choosing the contact voltages as independent variables which are specified by setting Dirichlet and jump BCs for Y The choice of the contact currents as independent variables is not so conve nient in this formulation although possible We then solve N problems m 1 N by selectively setting the value Vm 1 on each conductor Qm and Vp 0 on all other ones Qn n 1 N m n and then compute the N curr
263. mity Effect Skin and proximity effect belong to the class of eddy current problems in electromag netism where time dependent magnetic fields induce electrical currents in conducting wires For an AC driven current in a wire the harmonic magnetic field generates eddy currents but with opposite direction of the driving one so that the current tends to flow within a thin layer at the surface of the wire This is the classical skin effect and it is dissipative in nature so that the wire effective resistance is larger than the one for a DC current and grows with the square root of the frequency Due to the reduced current flow the inductance of the wire will be smaller at higher frequencies When multiple wires are close together as for example in a coil on each wire acts the skin effect but it is also true that the magnetic field created by each wire will reduce the current flow in the other wires This disturbance is called proximity effect due to the fact that the wires are close together and they disturb each other The proximity effect is also dissipative in nature but differently from the skin effect the effective resistance of the wire grows quadratically with the frequency and it is dominant at low frequen cies As an example already at audio frequencies resonant circuits with high quality factors need to consider effective resistance values for coils In this example we are SESES Tutorial September 2012 67 going to investigate the skin an
264. mm pump beam focus radius m M 55 x pump beam quality factor lambda 0 000809 mm pump beam wavelength m ref 1 82 x crystal refractive index alpha 0 35 mm x crystal abs coefficient 1 m P_Max 20 maximum pump power W P_n 10 pump power steps conv 0 4 x fraction converted to heat It is mentioned here that the pump power P denotes the power per pumped side of the crystal The total pump power double side pumping therefore amounts 2P In the present example we will perform a parameter study with increasing pump power The front and the back side of the rod are in contact with air whereas the cylindrical surface is assumed to be directly cooled by water The heat flow through the surface S of the rod defines the boundary condition It is given by Newton s law of heat transfer 2 89 and the corresponding parameters h and To for water and air are defined as hair 0 0015 x heat tr coefficient to air x in W cm 2 K x hwat 1 0 heat tr coefficient to water x x in W cm 2 K x Tair 293 x temperature of the surrounding air Twat 290 x temperature of the cooling water x The section OPD calculation concerns the parameters for the OPD evaluation As shown in Fig 2 67 the OPD has to be evaluated along predefined paths by calculating the integral 2 91 and equation 2 92 The number of paths parallel to the optical axis of the crystal is defined with the parameter nroPD T
265. n 1 2 106 Qn 1 O W En 1 ois Ent fn 1 0 For general plasticity models this system of equations is solved with an iterative New ton s algorithm generally started with the trial solution 2 105 J Flow with isotropic kinematic linear hardening For particular models we can simplify the equations 2 106 and for linear models we can find a closed solution as done for the following J2 flow example with hardening 138 SESES Tutorial September 2012 which is valid for a 2D plane strain 2D rotational symmetric and 3D analysis not however for a 2D plane stress analysis The energy function Y e Um e Vp generally splits in a mechanical and plastic part For the mechanical part we assume the most simple linear isotropic strain stress law given by s Vm ktr e Id 2udev e 2 107 with k u the bulk and shear modulus tr e e the trace of a tensor and dev e e tr e Id 3 the deviator operator For the plasticity model we use the classical J2 flow model with the von Mises yield function and isotropic kinematic linear hardening f s a dev s ail 3 oy WE FH 4KG with 1 2 and q qj q2 dev qi 0 the strain stress plastic variables and oy H K three model s constants The von Mises yield function has the properties dev s qi fame ais Idevis ail af n V3 3f n with n dev s q l which are readily verified and subst
266. n channel is zero A boundary layer starts to develop The development of the boundary layer can be observed in Fig 2 143a where the velocity component in the main flow direc tion is shown The cross sectional area where the velocity is constant becomes smaller at each position in the downstream direction The transition from a plug flow to a parabolic velocity profile causes the fluid to move also towards the channel axis This secondary flow is shown in Fig 2 143b Model Specification Within the initial section of the input file example Reactor s2d we first specify the the four different species associated with the burning of hydrogen in air 2 171 The species are defined with the help a macro list which is useful later on for automatically generating input statements Macro SPEC_LIST M A B M H2 A B M H20 A B M O02 A B M N2 A B Macro DEF_SPEC X A B X Species SPEC_LIST DEF_SPEC Next all geometry parameters are defined and an initial mesh is specified with the QMEI and QMEJ statements To ensure a fine mesh between the regions of free and porous flow the homotopic function shift is used to decrease the mesh size at this boundary Within the next section the SESES routines SESES Tutorial September 2012 197 Routine double u_veloc double x double y FromData u_veloc txt Routine double w_veloc double x double y FromData w_veloc txt are defined to specify the
267. n is started by pressing the run button Runtime information will be displayed in the standard output and for this example the graphics data file Data will be generated We may now investigate the simulation results by switching to the graphics mode with the graphics button M and opening the field control panel with the button lad Selecting the toggle and the electrostatic field Phi in the choice menu generates the content of Fig 1 7 Obviously the electrostatic potential transforms gradually between the two BC segments In addition we may also want to inspect the electric field by selecting Efield The vector field direction can be visu alized by selecting the option in the dot control Note that the vector field direction is always perpendicular to the equipotential field lines and the highest electric field is located at the end of the tip facing the substrate surface However the value at the endpoint of the tip is strongly mesh dependent since for this simple model the electric field is singular at this point 1 3 Graphical Visualization Although the SESES manual describes the Front End GUI program in details for a beginner it is not always straightforward to find out the actions and doings required to visualize something We have already presented some examples on how to display numerical fields so that here we mainly discuss how to choose a view which is a task independent from the displayed content For a 2D domain this is not
268. n of a body at rest Under actions of some quasi static external forces the body deforms and so let Q gt N Q t be the unknown motion to be computed F 0x 0X the deformation gradient with X Q a material point and x o X t 2 a spatial point In the multiplicative decomposition one postulates F Fe Fp with Fe and F the elastic and plastic parts of the deformation gradient The idea behind this decomposition is that for any small neighborhood U of x X by removing all tractions acting on OU we have an elastic relaxation into a stress free configuration with a remaining irreversible plastic deformation Fp a physical effect known as spring back The stress free intermediate configuration is labeled however since in general Fp nor Fe are gradients we cannot define the stress free configuration as Q Q with o but just through the action of Fp or Fe Let for the moment assume there is no plastic deformation F 0 then the whole elasto plastic formulation should turn into an elastic formulation with 2 Q We use here a generic hyperelastic law with the Cauchy stress s given by the derivative of an elastic potential Although material and spatial points are connected by the diffeomorphism the material description is required to formulate anisotropic material laws which however is not the case in our example Here the second Piola Kirchhoff P2K stress S J F3 t s FZT with J det Fe is expres
269. n the quadratic one do not in general perform as well so that the only free parameter for the penalty method is the rigidity factor defined by the constant RIGIDITY For the algebraic method we use the built in BC routine Glide which applies the constraint equa tion 2 120 at any displacement dofs found at the BC surface At the present time the routine just implements constant constraints so that if the values of the normal n and n Xp depends upon x quadratic convergence of the Newton iteration cannot be guaranteed Actually the routine applies the constraint just if one passes a non zero value of the normal and this decision is taken by the routine AlgebraicContact which similarly to PenaltyContact takes as input the data structure CONTACT We enable the constraint whenever the condition 6 lt 1x10 AND n R gt 0 OR 6 lt 1x10714 is fulfilled with the distance of the boundary point x to the contact sur face and R the assembled residual forces at x If the boundary point was clamped on the contact surface on the previous iteration then we have 6 lt 1x10 4 and we release the lock just if at x we have a tensile residual force n R gt 0 Further we always enable the constraint if the boundary point violates the impenetrability con dition 6 lt 1x10 We use here the same isotropic plastic law of the example Necking of a circular bar which we refer for further explanations For the discretization we use n
270. n the stick denoted by T10 T21 T31 T41 The transient behavior is shown in Fig 2 17 during a 200 minute time window The heat source was switched off after 100 minutes Fig 2 17 exhibits good agreement between measurement and simulation during the heating cycle However after turn off the temperature drops significantly faster in the simulation In addition we can compare the time dependent temperature profiles during heating as well as cooling The for mer is shown in Fig 2 18 The data plotted in constant time increments demonstrates that the heat conduction is fast initially but slows down when approaching steady state Building a 3D Model for the heated stick In the previous 2D approach we have achieved satisfactory agreement with measure ment data by assuming convective heat transfer only and by adjusting the two param eters heating intensity heaterf1lux and the transfer coefficient tcoeff The former adjusts the resulting temperature scale and the latter the curvature of the temperature profile With the optimum value t coef f 100 the data is reproduced nicely How ever a closer look into the heat transfer literature reveals that typical values for a are around 5 to 20W m K In particular the convective heat transfer coefficient is de termined by the Nusselt number Nu through a Nu d where d is the characteristic length scale in our case the cylinder diameter For horizontal cylinders the Nusselt number is given by 2
271. n to be zero at the domain s boundary This implies that no magnetic field lines cross the boundary Finally part four is the command section and performs the FEM computations and prepares the graphical visualization of the results The total force on the rod as well as the total electrical current in the rod are computed as integral of output fields and displayed Discussion and Validation of the Data The left of Fig 2 52 shows the skin effect of 40kHz alternating current through the rod A total current of 50 A yields the magnetic field lines in the right of Fig 2 52 SESES calculates an induced electromagnetic force of 1 28 N on the rod This value is smaller than the estimated buoyancy force of 3 4 N given by 2 37 However nor the exact current through the rod nor the detailed shape and dimensions of the ditch in the liquid metal are well known The difference between the computed and estimated repulsion force is therefore not surprising Fig 2 53 plots the exponential current de cay as a function of the rod penetration depth The penetration length has been cal culated with 2 38 SESES also calculates an exponential current decay however with a superimposed oscillation see Fig 2 53 The origin of this oscillation is not yet clear For sure it is not influenced by mesh refinement 7 2 EN ESE EEE E ir 2 38 dw Y gow V 2566x100 5 7107 40x10 27 oe 220 SESES Tutorial September 2012 77
272. nalty parameter can be chosen very very large resulting in an exact Dirichlet BCs However when n has at least two non zero components by considering large values in the global system we have the singular linear system n n 05h u h n It cannot be solved singularly but has to be solved together with the global system In this case the problem is ill conditioned and the value of the penalty cannot be chosen to large otherwise the global solution is dominated by a singular system with infinitely many solutions Algebraic method We shortly present an algebraic method to contact mechanics directly working on the nodal displacement dofs u of the discretized equations When the nodal movement x X u violates the impenetrability condition the constraint equation 2 120 is added to the system and of the three degree of freedoms uj one picks a slave and two masters The slave is generally the one with the absolute largest coefficient of the nor mal n we assume here n gt nz n thus uiz is the slave degree of freedom and we have the additional equation u i nz n xX9 n X Nptjz NyUjy Clearly adding any constraint equation to an already uniquely solvable linear system make the sys tem unsolvable However by considering that we are actually looking for solutions which are saddle points of a functional adding constraint equations is perfectly legal and the method of the Lagrange multiplier tells us how to solve the n
273. ncrement Standard ReuseFactoriz 3 LoadControl Spherical FacIncr 1E3 sqrt 2 Predic step load Extrapolation Quadratic Define load 0 Solve Stationary ForSimPar control Step 5 25 By running this example we first start all over again and so we reinitialize the dis placement to zero with Solve Init and the pressure load with Solve Define load 0 We write data for graphical visualization and to display the displacement load curve of Fig 1 20 Under load control the user specifies the controlling variable cont rol whereas the changes of the controlled pressure load 1oad are made by the load control algorithm In order to follow the changes of the pressure load we print its value at the end of each solution step together with the Convergence statement The load control algorithm is specified with the Increment LoadControl state ment and the Spherical function corresponds to spherical load control This func tion requires the specification of the parameter load to be controlled a scaling factor FacIncr for weighting the contribution of displacement and load with respect to the step length of cont rol and a prediction function P red for the controlled parameter used to speed up the computation 1 9 External Post Processing and Controlling This section illustrates how SESES can be interfaced with other software tools to ana lyze or visualize simulation data or to control numerical computations First a general method for exporting the nu
274. nd after a while it approaches an unstable snap through region see first row of Fig 2 95 Ina dynamical simulation the shell will snap through and will start oscillating around the next stable solution For stationary computations however solutions around this turning point can only be computed with the help of a load control algorithm The particularity of this instability is that increasing displacements are characterized by decreasing loads By just changing the load we cannot pass the turning point since by decreasing the load we go back on the path of already computed solutions Load control algorithms are devised to pass these points since they use another solution s parameter than the load typically something related to the norm of the computed displacement A brief overview is given in the Section 1 7 Non Linear Algorithms By further increasing the displacement we arrive 134 SESES Tutorial September 2012 6e 08 4e 08 2e 08 Z ke 0 fo Ke 2e 08 Elasticity2 L ElasticShellSingleK RL Elasticity RL L 0 4e 08 Elasticity 6e 08 0 2 0 4 6 0 8 1 1 2 1 4 1 6 displacement m Figure 2 96 Applied load versus displacement L norm for various finite elements and re finement levels at a bifurcation point where warp deformations show up see second row of Fig 2 95 At the present time no robust algorithms are implemented to pass bifurcation points and so it is a matter of luck if we can pass these
275. nd you shift the plane along its normal with the mouse wheel To reset the 3D cut view just press the mouse button on a point not on the displayed object There are two possibilities to restrict the view The first and most common one is to use the mouse wheel to zoom in and out To reset the zoom press once the reset button A second possibility to restrict the view is given by selecting a subdomain of the macro element mesh This is done by clicking on the domain button f which opens a control panel to select combinations of user defined domains These tools for selecting a view are summarized in Fig 1 8 Once you have found a proper view you can save the settings by pressing the state button a which opens up a control panel and then press the button STORE STATE Each time you double click on the reset button 9 your default settings will be restored If you press the button your settings will be written in the SesesSetUp file and restored each time you newly start the Front End program 1 4 Animation The Front End program offers the possibility to display animated sequences of numer ical results A prerequisite for an animation is a graphics file with multiple data slots A slot is the data written in a single write operation and multiple slots are defined by appending data to a previously created file Any type of numerical result can be animated supposed the data has been written to a graphics file The choice of the in dependent p
276. neral case the value of the jump Oo to be prescribed at the interface boundary OQinter is not known analytically but must be evaluated numerically We can directly 80 SESES Tutorial September 2012 Figure 2 54 The magnetic potential for the analytic solution of 2 42 computed with the total reduced and total reduced formulation arrive at an expression for Op by inserting 2 40 into 2 43 and thus obtaining an integral expression for this Biot Savart potential as a function of the external current Jo This is indeed possible and the resulting expression is given in the SESES manual However its numerical evaluation is rather delicate and error prone and therefore its usage is discouraged A better approach is obtained by considering that the value of the jump Oo on AQinter is a potential field on Qt satisfying the Laplace equation V Ho V V Oo 0 and having the Neumann BCs Ho n on OQ inter Fixing the value of Oo on a single point of each simply connected component of Qo makes the solu tion of this problem unique Therefore in practice the solution of the magnetostatic problem with a reduced and total formulation is generally preceeded by the computa tion of the jump Oo by solving a linear Laplace problem The solution of Oo at OQinter is then stored in memory with the help of a boundary user field jumppot and used as a jump value when solving for the reduced total formulation which is done in our example by Normal Y MUEO mue T
277. nfluenced by the frequency the electric conductivity the shape of the elements and optimization techniques used to reduce the number of unknowns e g symmetrization and the removal of the electric potential in the con ductors with constant conductivity Therefore when using edge elements the success of the method is basically given by the convergence rate of the iterative solver and the availability of a good preconditioner On the other side nodal elements suffer from an inferior matrix conditioning but the formulation based on nodal elements is gauged and so the linear system to be solved is regular Here a robust direct solver can eventually be used for not too large problems with the advantage that optimizations reducing the number of unknowns but not the accuracy will directly result in a speed up and that the time expenditure does not change by varying the frequency w For this example we have chosen a numerically robust formulation with nodal elements and a direct solver so that one has to be careful when extending this problem with regions of high permeability u gt uo When solving the eddy model 2 53 for a constant conductivity we see that Y 0 is a valid solution enforcing the Coulomb gauge V A 0 and so removing the elec tric potential Y from the solution process will speed up the computation Although it is not possible anymore to specify the driving voltage at the wire contact an alter native is available In a preprocessor
278. ng we use a single Write statement Write For i From 1 To 6 File data i txt Lattice lat i 9E 9E 9E 9E 9E 9E n x y 2 Pressure Veloc 2 Density Val Velocity norm Velocity writing the x y coordinates of the sampling point followed by the normalized pres sure density or a velocity component In Fig 2 134 the results of the SESES calcu lations for Re 23 45 7 are compared with the results obtained from CFX TASCflow simulations Notice the close agreement between these predictions The length of the separation region is the distance between the end of the cylinder where the x velocity is zero and the point where the graph of the x velocity intersects the line of zero z velocity We compared our results with data from 2 where the length of the separa tion bubbles were determined in an experiment by observation Fig 2 135 shows the computed and the observed lengths of the bubbles the agreement is very good for all computations 190 SESES Tutorial September 2012 Re 43 CFX TASCTlow Re 22 CFX TASCflow Re 43 SESES Re 23 SESES 1 50 xX Re 45 Exp Data A _Re 36 Exp Data 100 SESES momentum eq ASESES forces on cylinder ACFX TASCTiow Drag Coeffcient Cp 04 1 E 01 1 E 00 1 E 01 1 E 02 1 E 03 1 E 04 1 E 05 1 E 06 1 E 07 Reynolds Number Figure 2 137 Drag coeffici
279. no depolarization losses The thin disk laser concept is therefore very promising for the future References 1 K CONTAG M KARSZEWSKI CHR STEVEN A GIESEN H HUGEL Theoreti cal modeling and experimental investigations of the diode pumped thin disk Yb YAG laser Quantum Electronics 29 pp 697 703 1999 2 A GIESEN H HUGEL A Voss K WITTIG U BRAUCH H OPOWER Scalable concept for diode pumped high power solid state lasers Appl Phys B 58 pp 365 372 1994 3 C STEWEN M LARIONOV A GIESEN K CONTAG Yb YAG thin disk laser with 1 kW output power 4 E HECHT Optics Addison Wesley publishing company pp 282 26 2nd Ed 1987 5 B E A SALEH Fundamentals of Photonics pp 210 234 John Wiley amp SonsInc 1991 126 SESES Tutorial September 2012 Depolarisation Figure 2 88 Depolarization loss of the disk for a homogenous beam of radius r passing the disk forth and back Figure 2 87 Depolarized part of a plane wave traveling through the disk The grey dotted circle indicates the pump spot 6 J F NYE Physical properties of crystals Clarendon Press Oxford 1993 2 19 Thermo Optically Self Compensated Amplifier The thermally induced lens in a laser rod is by far the most critical issue in the de velopment of high power solid state lasers and limits the output power of lasers with good beam quality IM 10 20 to a narrow power range In the l
280. ns of many air slabs or a complicated param eterization of the flow profile The numerical example can be found at example HeatExchanger s3d it defines a cold channel in the center and half a channel of hot air above and below the cold channel see the left of Fig 2 150 In the Geomet ry definition section an initial rectangular geometry is transformed to a rotated rhom bus with the following statements Include Homotopic sfc SESES Tutorial September 2012 205 TempLk 3 096 02 3 056402 3 026 02 2 996402 2 956 02 2 906 02 2 886402 2 856 02 2 826402 air aluninium isolation hot Inlet coldInlet _hotQutlet_ coldOut let Figure 2 150 Left Computational domain with two air channels Right temperature profile 2 786 02 Coord coord 0 x3 0 bump x ax2 ax3 ax4 ax5 1 xbump y ay0O ayl ay2 ay3 1 block nx0 nx7 ny0 ny4 nz0 nz5 Coord coord 0 x3 0 bump x ax2 ax3 ax4 ax5 1 xbump y ay3 ay4 ay5 ay6 1 block nx0 nx7 ny3 ny6 nz0 nz5 In the input file it is further important to define several blocks and domains in order to define the velocity profile later on Inthe Material definition section we define a user field velocity to store the velocity profile in the flow channels The velocity is just initialized later on and not directly computed Therefore only the temperature equation for conduction and con vection of heat need to be solved For this heat exchanger model the P clet mesh numbe
281. nships 2 110 2 111 To this routine we also pass the element field CpInvMOneEF used to compute the logarith mic strain trial log A before calling our plastic model The update of the plastic variables is done in the routine J2FlowPrincipal in the form of Ce Id Fria bent Fray Id X exp 2en41 N Q N Id and amp 41 amp AX The updates are first stored in additional buffer element fields with suffix u which are first copied into the permanent ones at the end of a successful solution step In addition we supply the routine J2FlowFiniteStrain which may be used as a replacement to the built in call of Elast LogSt rain It shows in details all numerical operations done by Elast LogSt rain and it is completely equivalent This routine computes the spectral decomposition of be trial performs a call of our plastic model J2F lowPrincipal and implements the relations 2 110 2 111 The numerical point of interest is the possible numerical cancellation in 2 111 when computing the term Tid5 7X2 7 with A Aj In this case we use a symmetrized H pital s rule given by Aibe Tj Aj Ti OziTj Tj FT 4d Aye 2 The geometrical setup of this example is quite simple we define an axis symmetric problem with a rectangular domain and set Dirichlet BCs on both ends of the rod to prescribe the elongation We then compute solutions incrementally by increasing the displacement at the BCs with respect to
282. ntacts 6 2 ka He be Seba eee 149 Continuous casting ofst el o o osoo ecs S SSR ETE MRE Te RS 157 Hagen Poiseuille Model of Viscous Laminar Flow 163 Blasius Plate Flow At Zero Incidence 02550006 165 Microfluidic Mixing in a Straight Channel 0 168 Heat Transfer and Natural Convection in a Closed Cavity 174 Calorimetric Flow Sensor 2 6 6 606 6 eee eee eH 180 Flow Around a Circular Cylinder 6 4 a ss cocto n rasana as 186 Developing pipe flow with heat transfer aooaa aaa 191 HotSpots ina Tubular Reactor 5 242542 24 anot 4 ere aaa 195 Effective Transport Parameters from Volume Averaging 200 Heat Exchange between Air FloWS i ea ie eae sen ae ee es 204 A first model ora SOFC fuel cell 5 6 4 w vee Be be Se 206 A planar 2D 1D SOFC model 2 1065 ee eee GERD SE oR ON 210 Introduction Computer Aided Engineering CAE focuses on the application of physical mathemati cal numerical models to solve real world multiphysics engineering problems The past decade has been a period of rapid progress for CAE The area of possible appli cations has been expanded and numerical methods have become increasingly sophis ticated and adapted to exploit the available computational power of modern micro processors In particular nowadays CAE methods play a key role in the industrial product development process and help to e Speed up the development cycle of products e Optimize the pr
283. o go through the individual calculated data slots 1 5 Preprocess Mesh Definition Mesh generation is an important task and sometimes also a time consuming task per formed either with a preprocessing static or algebraic dynamic approach For com plex geometries and if the domain s shape is given and not part of the modeling prob lem the preprocessing method is generally preferred Here with the help of the built in 2D mesh builder or another CAD and FE preprocessing tool one creates a static mesh which is imported afterwards For an external preprocessor one can use the con venience routine ReadMesh to import unstructured meshes defined by the common 18 SESES Tutorial September 2012 element node data format In this section we shortly present useful tips when work ing with the built in 2D mesh builder You can experiment with its functionality and basic geometrical shapes with the example found at example MeshBuilder s2d and shown in Fig 1 9 As a rule of thumb try to use a divide and conquer approach to construct the mesh it is faster As first example since meshes are quickly refined it is better to start working with coarse meshes Secondly since single mesh pieces are quickly joined together try to exploit symmetries and start with one quarter or one half geometries With cut amp paste mirroring and joining operations the complete geometry is then quickly obtained We want to apply these simple rules for constructing meshe
284. o that we are left to solve the Blasius equation f f 2f 0 with the boundary conditions f 0 f 0 0 and f co 1 The first two conditions represent the no slip condition u 0 on the plate and the last one the stream velocity u U 0 at infinity Due to its non linearity this boundary value problem needs to be solved numerically as done for example with the program Maple and the input below by considering 7 150 as infinity INFITY 150 ODE 2 diff f e e 3 e diff f e e 2 0 SESES Tutorial September 2012 167 SOL dsolve ODE f 0 0 D f 0 0 D f INFITY 1 type numeric maxmesh 1024 PEA IRA f e e plots odeplot SOL e diff f e e e 0 5 Computations of the Blasius flow are done on a unit square domain for various kine matic viscosities and the input file can be found at example BlasiusFlow s2d The boundary conditions for this incompressible solution are v 1 0 at x 0 and y 1 and v 0 0 at y 0 on the point 1 0 we fix the pressure p 0 to avoid floating values The boundary layer at x 0 must be resolved by the grid in order to obtain the correct solutions therefore as shown in Fig 2 111 we use a geometrically spaced mesh in the y direction By the setting of the boundary conditions the velocity is discontinuous at 0 0 and the pressure has a singularity there Therefore to re solve this singularity we also use a geometrically spaced mesh along the x direction
285. o the real cylinder surface The exact value of tcoeff will be discussed in more detail in the next section on a 3D model of the heated stick In addition the analytical solution of 2 8 is also plotted The factor of sinh representing the heating intensity was adjusted By contrast the prefactor in the argument a where a corresponds to our parameter t coeff was calculated as 0 021 cm using a 10 W m K and 235 W mK x Keeping the simulation parameters fixed we can now consider the time dependent SESES Tutorial September 2012 Heating Temperature Profiles vs Time wo a fo 49 Heating Temperature Profiles vs Time 350 10 min exp i 100 min exp 20 min exp x 110 min exp 300 30 min exp 300 120 min exp 40 min exp O R 130 min exp 90 min exp a 140 min exp _ 250 10 min sim 2D a _ 250 F 100 min sim 2D x 20 min sim 2D is x 110 min sim 2D 200 30 min sim 2D eo J 200 120 min sim 2D a 40 min sim 2D d 3 130 min sim 2D S 90 min sim 2D n f 140 min sim 2D amp 150 a M S 150 a a g R 2 100 a Pa 100 Bo a a 50 Nees ae ee 50 a pee eet Ok a ae ce See 0 1 1 1 1 1 0 1 1 0 20 40 60 80 100 120 0 20 40 60 80 100 120 distance cm Figure 2 18 Time dependent temperature profiles during heating distance cm Figure 2 19 Time dependent temperature profiles during cooling temperature at 4 selected points o
286. obtained by the general method of 2 116 but the algebra get a little bit involved By noting that G KONE 2u DEV and DEV Tn 1 is parallel to DEV tria1 with DEV m m41 7 2AA we have f DEV tn41 DEV m 1 nT and 8f DEV m4 41 7 Id nn DEV 7 SESES Tutorial September 2012 147 2uAX DEV nn giving A Ty Z GUA a G 1a ANDE See F 2uAXr SC u Ney nn T 2 ONE 24 DEV u AA iy nn where we have used a non trivial matrix identity to compute the inverse just holding for a vector n with the properties n 1 and DEVn n as here the case Together with Z 0 f 2un 0 fZ 0 f 2u and 0 fZ 0 f 2 3K the formula 2 116 yields the above result This elastic plastic law together with the computations of an axis symmetric rod sub jected to a large elongation and showing a typical necking behavior can be found at example Necking s2d The value of the plastic deformation F p is stored in the element field CpInvMOneEF in the form of C _ Td the shift with the unit ma trix is used to avoid numerical cancellation at infinitesimal strains The value of the equivalent plastic strain is stored in the element field XiEF The above rela tions T 41 OTn 1 0Etria together with the conditional update of the internal variables are coded in the input routine J2FlowPrincipal passed to the built in routine ElastLogStrain taking care of the interfacing relatio
287. ode nx ny 2 Write File curve Text current A tx displacement m ty displacement m temp K n Write AtStep 1 File curve Append Text 3f t cur Lattice tip 3e t 3e t Disp X Disp Y Text S 3f n integrate Temp integrate 1 Solve Stationary ForSimPar cur At 0 005 Step 0 005 4 In order to obtain the displacement values at the tip we have first to define a lattice where numerical fields can be evaluated We do this with the Lattice statement followed by the lattice name and the lattice points here a single point located at the cantilever tip To write the first comment line to our data file named curve we use the Write statement followed by a format string and the output file specification Within the string we use the tabulator t and the newline character n At each value of the swept current we append to the output file the value of the current of the displace ments Disp X Disp Y at the tip and the averaged temperature The average value is computed with the built in function integrate The format strings 3f t and 3e t following the Text and Lattice keywords specify floating point f and engineering e notation respectively with a precision of three digits As column separator we choose the tabulator t The generated file curve reads current A x displacement m y displacement m temp K 0 005 3 879e 08 1 316e 06 303 281 0 010 1 552e 07 5 265e 06 3h 35 125 0 015 3 491e 07 1 185e 05 3292531 0 0
288. oduct performance e Develop a thorough understanding of the underlying physical chemical processes e Visualize device functionality not accessible through experiments e Support the decision making process at different product development stages e Eliminate potential failures and identify pitfalls In general the design of a physical mathematical numerical model for the solution of an engineering multiphysics problem by CAE tools is a creative and often challenging task For practical applications simple recipes usually do not exist Rather it is the combination of physical insight a sound basis in mathematics and last but not least comprehensive modeling experience that leads to the correct model and strategies for model validation respectively Often a problem can be described by a network model on a high level of abstraction However when it comes to understand the physical behavior and properties of system components the modeling approach is often expressed by formulating governing equations consisting of the following parts e Conservation Laws Exist for quantities such as mass number of molecules elec trical charge momentum and energy Conservation laws are universal i e they are independent of the materials considered and they always need to be satisfied As an example the conservation laws in fluid mechanics are the mass and momentum balances of which former is also known as the continuity equation In thermody namics th
289. of the temperature we assume it vanishes at the environment temperature of 300 K and is a linear function of the temperature i e we use a first order approxima tion for the thermal strain MaterialSpec Silicon Equation ThermalEnergy Elasticity Enable Parameter KappaIso 150 0 W m K Parameter StressOrtho LinElastIso Emodule 190 GPa PoissonR 0 3 AlphalIso 2 3e 6x Temp 300 STIunit The cantilever is a thin structure and care must be taken when computing the mechan ical displacement since low order finite elements do not perform well Two options are available here either one uses higher order finite elements or special structural elements for beams and shells We follow the second approach and define a new ma terial Sil iconShel1 inheriting from Silicon all physical properties but using shell elements of type Elast icShe11J to compute the mechanical displacement Actually since we are solving a 2D problem the elements will be beam elements MaterialSpec SiliconShell From Silicon Equation ElasticShellJ Enable 44 SESES Tutorial September 2012 Next we define the material properties of the aluminum layer For this conducting ma terial we solve also for the electric field responsible for heating the cantilever The dis sipated Joule s is defined as the source of thermal heat with the statement Parameter Heat JouleHeat The mismatch in the thermal expansion coefficient with respect to silicon of one order of magnitu
290. ollowing important parts e Solution procedure The computation of numerical solutions e Accuracy Adaptively refine the computational mesh to obtain solutions with en ough accuracy e Post processing Writing of computed fields into a graphics data file Data for visualization inside the Front End program and generation of relevant data for further post processing tasks This typical structure of the input files is displayed with the help of comments in the tutorial examples included within the distribution The input files are written in an input description language whose keywords are highlighted in blue and comments in violet inside the text editor In case of syntax errors the first invalid input line is highlighted in red For detailed informations on the input syntax please consult the manual explaining the syntax requirements with the help of railway diagrams Starting the Front End program The Front End program can be launched by clicking the executables g2d or g3d inside a browser or by entering in a shell the command g2d lt file gt org3d lt file gt with an optional file name If the input file is not specified you can open one by selecting the open amp button However if the associations for the file endings s2d s3d and the executables g2d g3d are set appropriately the most common way to start the Front End program is by clicking on the input file inside a browser To illustrate the navigation simulation and visuali
291. ollowing two step procedure first for all repetitive struc tures detailed 3D FE simulations are performed to obtain effective parameters for the transport equations The complex structural information is thereby cast into effective material properties In a second step these averaged quantities are used to simulate the original 3D structure but now without all fine details thereby decreasing the com putational cost In this tutorial the volume averaging method will be illustrated for the thermal insula tion structure shown in Fig 2 147 It consists of two insulation walls with air channels and an air gap between the walls An external pressure gradient causes an air flow from the left to the right In addition there is an external temperature gradient caus ing a heat conduction flow in the opposite direction In the most simple case i e when the inertia terms in the momentum balance are negligible the internal resistance of a SESES Tutorial September 2012 201 solid structure with respect to fluid flow is described by Darcy s law 4 Vp w 2 172 Darcy s law states that the average fluid velocity in a given direction is proportional to the external pressure gradient that causes the fluid motion The proportionality constant is given by the ratio of the shear viscosity p to the permeability k The shear viscosity is a property of the considered fluid whereas the permeability takes into account geometrical details of th
292. on is also a pulldown menu allowing the select the initial SESES Tutorial September 2012 19 drawing of a B zier or NURBS curve Due to symmetries we just draw 90 arcs so that only two controlling points are necessary here These are the starting and end arc points which are defined with a double click By default these two points are connected by a straight line and to define a circular arc open the arc panel 4 and set an angle of 90 Be sure you have selected a NURBS and not a B zier curve since this latter cannot exactly represents a circular arc If not the case select the two controlling points and apply the rational segment option 4 After a curve has been defined one has to attach macro element nodes to the curve With the attach task Pf pick a macro element node drag it over the curve until a snap takes place and move the node along the curve at the correct position Attached nodes are marked afterwards with a thick blue point At the end one can quickly perform a uniform refinement with the split task I Se lect the whole mesh and pick a single edge of the selection These simple shapes are easily combined together to form more complex meshes Here it is better to work with unrefined and coarse meshes and to perform the uniform refinement just as last operation on the whole mesh The construction process is almost the same as for the basic shapes With copy cpaste operations selections are duplicated and with the join task
293. on linear solid shell elements enabled with the equation ElasticShel1SingleK and the mo del MechNonLin Since these mixed FEs use internal variables and the Green Lagran ge strain used to evaluate material laws is not directly computed from the deformation gradient the quasi positivity condition gt 1 of the Green Lagrange strain eigenval ues is not given The principal log strain values cannot always be computed and NaN values may be returned In general this just happen by taking too large steps so that the condition of invalid log strain values is checked within our material routine J2FlowPrincipal if isnan STRAIN 0 isnan STRAIN 1 isnan STRAIN 2 failure LOGSTRAIN For Nan log strain values a failure message is fired and the solution step will be re peated with a smaller step value Friction less contacts are path independent and therefore do not destroy the symme try of the system In particular the derivative of the traction 2 121 is symmetric so that the global system is symmetric as well This condition is not detected auto matically and therefore we use the declaration LinearSolver Symmetric to force a symmetric linear solver We then solve a series of stationary problems by changing the position of one stamp and at each step a Newton iteration is performed up to con vergence of the displacement s increments see Fig 2 103 The step length is chosen adaptively based on the convergence rates of the
294. onging almost exclusively to the class of partial differential equations PDEs In light of the specific geometries and boundary conditions that appear in realistic problems and the complexity of material laws it is no surprise that analytical solutions to these problems can rarely be found This in turn motivates the use of numerical tools In particular the SESES software package allows for the numerical simulation of complex physical problems in various fields of applications as e Micro Opto Electro Mechanical systems Magnetic field and eddy current simulations e Piezoelectric actuators and sensors Electrochemical processes fuel cells batteries e Microfluidics microreactors membranes flow sensors Thermal transport by convection conduction and radiation e Semiconductor devices Almost all of these equations can be coupled and solved together thus allowing the modeling and analysis of complex multiphysics problems A rather complex example for an application of SESES is the numerical simulation of fuel cells where the Navier Stokes equations the kinetics of the electro chemical reactions and heat generation are all coupled together The major effects here are the production and consumption of species and heat on the one hand and the separation of charges across the electro chemical double layer at the triple phase boundary on the other hand This conversion process from chemical into thermal and electric energy strong
295. or six consecutive linear steps Fig 2 43 shows a solution of the magnetic potential and the magnetic field is shown in Fig 2 44 Afterwards we compare several methods to obtain the magnetic force acting on the bendable lamella The first method is the Maxwell s stress tensor method of 2 27 the second method is a numerical improvement of the first one and can be used whenever the integrand function is divergence free which is always the case for the Maxwell s stress tensor in vacuum The third method is the energy method of 2 24 requiring the magnetic en ergy density 2 23 and the fourth and fifth methods are the energy methods of 2 25 2 26 All energy methods require a second solution computed for a slight change of the distance parameter The fourth method requires to backup the magnetic field and volume weights used for numerical integration and the numerical integration must be self programmed with the help of the built in routine t raverse The fifth method is a hybrid between the third and fourth methods and does not require to backup the volume weights since in vacuum we work with the magnetic energy density Solve Stationary Define Force0 integrate Bound SurfaceReed MagnTraction Forcel residual Bound SurfaceReed MagnStress ZeroC0O integrate Bound SurfaceCheck MagnTraction ZeroCl residual Bound SurfaceCheck MagnStress Energy0 integrate MagnEnergy Energyl integrate Domain material Vacuum MagnEnerg
296. or the voltage and Hall contacts 2 3 correspond to Dirichlet and Float ing BCs BC Voltl 0 0 Type ny Dirichlet Psi 0 V BC Volt2 nx 0 0 n J JType ny Dirichlet Psi vapplied V BC Halll nx0 IT BC Hall2 nx0 IT Type nxl Floating Psi 0 A Type nx1 Floating Psi 0 A This single field boundary value problem is linear with respect to the dof field Psi and a solution is computed by selecting an automatic mesh refinement as shown in Fig 2 7 For a non vanishing magnetic field or a misalignment the solution is sin gular at the corner of the Hall contacts Here the automatic mesh refinement would 42 SESES Tutorial September 2012 M 1 0E 00 95E 01 9 0E 01 85E 01 Psi 7 5E 01 7 0E 01 65E 01 6 0E 01 55E 01 5 0E 01 45E 01 4 0E 01 3 5E 01 3 0E 01 25E 01 2 0E 01 1 5E 01 1 0E 01 5 0E 02 2 2E19 Figure 2 7 Adaptively refined computational mesh showing field singulariries at the con tact s corners timeo Figure 2 8 Example of a calculated electro static potential field in the Hall sensor never stop to refine due to the singularities However the local singular behavior has little influence on the overall device performances and we can stop refining around the singularities In practice refinement of singularities is characterized by a small number of new elements each time we refine and the parameter MinFractionElmt is used to stop refinement when the fraction of new and actual elements undergoes
297. ore let Q be a domain of R and assume in 2 we have N closed and disjoint conductors Am C Q mN Nn 0 form 4 nand m n 1 N Just within the conductor domain Owire UmQm we have a non zero conductivity For time varying currents up to some upper limit frequency the eddy current model represents a valid submodel of Maxwell s equation see 1 for a mathematical background which is given by solving the following equations in Q VB 0 VxE 0 B 0 Vx H J cE 2 52 with E H B the electric magnetic and induction field u the conductivity and permeability and the material laws B uH J oE By taking the divergence of V x H J we also have V J 0 In order to simplify a little bit the matter and so the computational work too we assume a linear dynamical system Since the above equations are all linear with respect to the electro dynamical fields this implies linear material laws and thus conductivity o and permeability u can only be functions of the coordinates For the numerical solution of the above equations several formulations are available and the one adopted in SESES is based on the computation of the vector potential A and the time integral Y f W dt of the scalar potential Y as primary fields In this case we have B V x A VY E 0 A and the following equations are left to solve Vx lV xA 0 cA oVV 0 amp V cA oVV 0 2 53 although the second one is implied by the first one by taking the divergence In prac ti
298. orm and so we may use a bisection or Newton s algorithm to find the solutions What is needed as input to SESES are the current values and species production rates at the borders of both GDLs as function of the parameters p together with all non zero partial derivatives But since all production rates are directly pro portional to the current j we see that actually all what we need is the current j and the derivatives 0j 0j The equation r j P together with the partial derivatives Or O j 7 can be conveniently assembled symbolically and some Maple code is provided within the input If one uses a Newton s algorithm to find solutions then one just performs the iteration j j r j Or 0j up to convergence of j and at the end we have 0j Op Or Op Or 07 In our case this Newton s iteration is initialized with the analytical solution available when species diffusion is neglected and by taking asinh x z in 2 181 What is left to do is the definition of the production rates and partial derivatives re quired by SESES as function of the computed values j 0j 0p This step only de pends on the stoichiometry of the electro chemical reaction and it is therefore inde pendent from the particular 1D model used However we now come to an impor tant fact which absolutely cannot be neglected and it is often a source of troubles for beginners The problem lies in the fact that oxygen in the cathode air channel disappears in order to rea
299. ormulations thus allowing to devise better optimized solution al gorithms Here the price to pay is a more complex problem specification and this contribution aims at explaining the different available choices The scalar formulation is generally the preferred one but has the drawback that one cannot directly specify the current source Jo Instead one has first to define the associated Biot Savart field Hgs x f l xa Yay 2 44 and then solve the equation V uVOrea Hs 0 2 45 for the reduced scalar potential 0 q with B u Hps VOrca Rare are the cir cumstances when the Biot Savart field Hgs is known analytically and so one has to evaluate the integral 2 44 numerically which may be critical and imprecise for x supp Jo Also this integral over the volume of supp Jo has to be evaluated on many points of Q generally taken as the element integration points of the mesh which therefore can be computationally quite expensive Another drawback of the scalar formulation is its instability for non linear material laws u p H stemming from the subtraction of the term wHps in 2 45 forcing the user to combine a total and reduced potential formulation These drawbacks are partially eliminated by the vector formulation solving the equation Vx tVxA Jo Vx Ho 2 46 82 SESES Tutorial September 2012 for the vector potential A with B V x A The additional numerical work required to compute the vector field A instead of
300. ot really necessary There is a switch NoP si to turn off solving for the electric potential and in order to compute the electric current we define a user element field ElmtFieldDef LocCur DofField 02 on the cathode where just the species O is defined Each time the model Coupling1D is called we then store in the element field LocCur the current value and at the end of a computation a user integral over the cathode s domain Write AtStep 1 Voltage f V Current e A Power e W n double totcur If WithPsi Apply Psi Flux Depth Else integrate Domain domain dm_Cathode LocCur return Voltage totcur totcur Voltage will return the total current together with the fuel cell electrical power as shown in Fig 2 156 and displaying a linear I V characteristic If the computation of the potential field is enabled the total current is available as the BC characteristic stored in the variable Apply Psi Flux which however has been multiplied internally by the virtual thickness of the device and need to be amended
301. out 92 C As we can easily see the temperature gradients in the center part of the disk are really into the direction of the optical axis The temperature gradient becomes strongly radial in the border range of the pumped region This fact is also confirmed by the direction of the heat flow shown in Fig 2 84 In the center part of the disk the direction of the heat flow is parallel to the optical axis and in the border range the di rection of the heat flow becomes more and more radially The temperature raise on a radial straight line starting in the center of the disk and located on the front and back surface and in the middle of the disk are shown in Fig 2 85 The corresponding curves are very flat up to a radius of about 0 75mm which corresponds to 3 4 of the pump spot radius From a radius of 0 75 mm to 1 5 mm they drop significantly down and are again flat for higher radii Therefore we can expect a very weak thermal lensing in the central part of the disk up to a radius of 0 75 mm The optical path difference OPD induced by the thermal dispersion and the end effect is shown in Fig 2 86 As expected the OPD is very flat up to r 0 75mm 124 Temp K 3 6E 02 3 6E 02 3 6E 02 3 6E 02 3 5E 02 3 5E 02 3 5E 02 3 4E 02 3 4E 02 3 4E 02 3 3E 02 3 3E 02 3 3E 02 3 3E 02 3 2E 02 3 2E 02 3 2E 02 3 1E 02 3 1E 02 3 1E 02 3 1E 02 time 0 Figure 2 83 Temperature distribution in the disk and its central pumped with 100 W
302. ows to compute the mechanical tractions on any portion of the lamella boundary whereas the first method just yields total forces for a given and prescribed displacement It is therefore possible to perform a coupled magnetostatic mechanical analysis and to fol low in details the dynamic of the switch process The magnetic force per unit volume in the ferromagnetic material is J x B V x H x B integration over the lamella s volume Qz application of Gauss theorem and after some algebra we obtain F 3xBav f 7 dn 2 27 Qz oL with 7 BiH 1 26 B H the Maxwell stress tensor for the magnetostatic case and a scalar permeability The total force acting on a lamella is now given by a surface integral over the lamella boundary 00 and therefore the term 7 n with n the outward normal can be interpreted as the mechanical tractions acting locally on the surface It is to be noted that if only the total force is of interest then the volume Qz can be taken as any volume around the lamella since in vacuum J 0 and V r 0 and therefore the integral s value is invariant It is possible to get a rough estimate of the magnetic force by considering a magnetic ring core of cross section A length L and with an air gap of d see Fig 2 41 By dis 64 SESES Tutorial September 2012 carding all volume effects and by considering just the fields along the centerline of the torus from the relation V B 0 follows a constant magnetic field B
303. p lc micron Tamb S 3e n v0 Solve Unstationary At 0 Step tStop 50 automatic_step Until tStop Failure step 2 where we have used a loop for the different flow rate levels and the unstationary temperature solutions The average temperature at the heater boundary is calculated using the integral of the temperature at that boundary condition The transient sensor temperatures are written to the file pdyn txt The resulting transient temperature difference AT gt _ is depicted in Fig 2 128 and resembles the curve in Fig 2 125 as expected At the first flow rate step the temperature field changes from a symmetric to an unsymmetrical field as a function of time in a similar manner as depicted in Fig 2 124 for the different flow rates For the constant heater temperature mode the sensor temperature transient A7 gt _ is shown in Fig 2 129 In this plot the approach to steady state upon each flow rate step is faster than for the constant power mode As discussed above in the stationary simulation shown in Fig 2 125 the heater tem perature can be used as the sensor signal in constant power mode The transient response of this signal is depicted in Fig 2 130 with an monotonic decrease of the steady state values that is consistent with Fig 2 125 Finally we may also use the heater power as the sensor signal if the sensor is operated in constant heater tempera ture mode In this case the power increases to the steady state values as can be seen
304. parameters xoff and yoff The solution procedure is as before but since with the ForSimPar parameter sweep option of the Solve Stationary statement only one parameter can be swept we have used two preprocessor nested loops and the statement Define to update the simulation variables xof f and yoff as follows For i From 0 To nsteps For j From 0 To nsteps Define xoff ixsweep lx nsteps yoff j sweep xly nsteps Solve Stationary Write File tip_2Dscan txt Append Lattice tip 3e 3e 3e n xoff yoff Efield x separates data blocks for gnuplot x Write File tip _2Dscan txt Append n At the end of the inner loop we add a line feed to the data file in order to separate row data The exported simulation data is easily visualized using the Gnuplot software and the necessary files and commands are also included in the example The resulting graph for a 2D scan within a 400 x 400 nm region is shown in Fig 2 30 SESES Tutorial September 2012 55 R dx JZ J x dx t G dx U x dx t Figure 2 31 Circuit diagram of the voltage U x t and current J x t along a wire 2 6 Cross Talk and Telegraphy Electric signals running along parallel wires cross talk to each other i e part of the signal in one wire is transferred to the other wire The phenomenon might be conduc tive capacitive or inductive in nature and is treated in text books under the name of equation of telegraphy Fig 2 31 shows the situation and
305. pecies are only functions of concentrations cg pa p and Ja simplifies to v l1 Ja gt gt pDagV p 2 132 B 1 with Dag the diffusion coefficient of a in 6 5 For a binary mixture of species A and B we have v 2 and the diffusive flux J 4 follows from 2 132 as Ja p DapV ca DapV pa 2 133 Since Ja 0 the diffusive flux of A is balanced by a reverse flux of B i e Jg J 4 Taking into account 2 133 the mass balance of A follows from 2 130 as pA Ot For mass transport under steady state conditions and in the absence of any chemical reactions eq 2 134 further simplifies to V pa W DapV pa 2 135 V pa W DapV7pa a 2 134 By noting that for p const the overall mass balance reduces to V W 0 eq 2 135 takes its final form as W Vo4 DaB V pa and division by p leads to the diffusion convection equation in terms of concentrations W Vea DapV ca 2 136 170 SESES Tutorial September 2012 Diffusion Convection Equation for a Plug Flow Consider a two dimensional system in the Cartesian coordinates x y and assume a straight fluid flow in x direction with a uniform velocity V const i e a plug flow In this case eq 2 136 simplifies to Oca yo o Fe Dan O7c4 Gar 2 137 Now suppose the transport of A in x direction is mainly by convection the term 0 c 4 0x can then be neglected with the result that OcA O7c py 2 138 Ox
306. portional to the velocity By consid ering a vanishing body force and electric bulk charge we have to solve the coupled equations pu yu Vs pu yu u 0 ep o 0 J ae SESES Tutorial September 2012 105 with D2 a differential second order operator acting on the solution u In order to get a solution of 2 75 for the material laws 2 74 and a harmonic voltage applied to an electric contact we use a technique based on the decomposition into homogeneous eigen solutions and an inhomogeneous stationary solution which prevent us from explicitly integrating 2 75 with respect to the time As first step we look for solutions of 2 75 with a time harmonic dependency of the form exp iw t by using uniquely homogeneous BCs for the mechanical and electrical problem and by considering y 0 Insertion of the generic solution un x n x exp iwnt into 2 75 leads to the solution of the following eigen problem Da Sr tag Pa o 2 76 which has eigen solutions un dn w2 i 1 00 with positive eigenvalues w form ing a base in the solution space of D2 By choosing the orthogonality relation with the density p as weight we can therefore write lt Um Un gt PUm UndV bmn 2 77 Q Itis to be noted that the electric eigen solutions do no enter this orthogonality relation As second step les us compute the following inhomogeneous solution uo Da as 0 2 78 with the same type of BCs
307. ppear at the anode side in form of O ions before the electrochemical reaction takes place Nothing special but now the sums of the sin gle production rates at the cathode Teathode Ho Iy and at the anode Ianode Ily 0y 0 11y Ico Uco is not anymore zero in other words mass disappears at the cathode Teathode lt 0 in order to reappear at the anode Hanode gt 0 however we still have Ieathode Hanode 0 This is an inconsistency for the single 2D transport prob lems with the result that the constraint for the sum of the mole fractions at the cathode LO y 1 and at the anode zy TH 0 N FCO LCOy 1 cannot be any more obtained by the numerical solutions which would require Ieathode anode O In order to proceed we need to amend the transport equations and in order to see how this is done we state the governing balance equations for the stationary mass conservation and species transport V pv Ilo V Pav V ja Ma with pa the species mass density p gt Pa the total mass density ja the species diffusion currents II the species production rates and Ilo Ha the total mass production rate In SESES the above species transport equations are not solved in this form but they are transformed with the help of the first equation and by considering the usual case IIo 0 Without this assumption we can repeat this algebraic step to 214 SESES Tutorial September 2012 SOFC Electric Characterist
308. prove the equivalency an w2A and therefore the impedance for the piezo crystal is given by abu Win Z w t iw ag S iMmAanw Yn w 2 84 n 1 In summary in order to compute the impedance for the piezo crystal one has first to compute the inhomogeneous solution yielding the charge ay Then one computes a representative number of eigenpairs of the coupled system 2 76 and for each pair one evaluates either the volume integrals A or the surface integral n With the spec ification of the empirical damping constants yn the impedance 2 84 is fully specified and can be plotted Near to a resonance frequency w wy in 2 84 we may just keep the dominating n th term By considering w w 1 and a zero damping Yn 0 the electric impedance is given by an equivalent circuit with an LC series in parallel with a capacitor Co and values Co ao L a7 LC w The impedance is therefore given by Zs 1 e t e _ 1 w2 w Hl C Co age a2 2 iwCo Toce Tw iwao w w Computational example In this example we are going to compute the resonance frequency spectra of a PZT 5A piezoceramic disk as published in 1 and the SESES input file can be found at example PiezoDisk s2d The paper also presents the same impedance theory al though from the computational finite element point of view In a first step we define the material PZT5A For this material we have to enable the computation of the dis plac
309. py DTemp Density Val veloc W m 2 K Parameter ThermStab zero The parameters stated above correspond to water In order to consider the convec tive heat transport we have to define the term pOrhv with the material parameter ThermConv with h the enthalpy p the density and v the flow velocity The user ele ment field veloc is used here to store the velocity profile Since for this flow sensor the convective flow is small with respect to the diffusive one there is no need to use streamline diffusion stabilized finite elements and therefore we set to zero the stabi lization term For the boundary of the simulation domain we formulate a heat transfer boundary condition not shown in the screenshot of Fig 2 123 BC Transfer OnChange 1 OnBC Heater Disable Neumann Temp D_Temp alphax Temp Tamb alpha W cm 2 W cm 2 K with a the heat transfer coefficient The heater is specified either with a Dirichlet or a Neumann type boundary condition depending on whether we wish to operate the sensor in constant heater temperature or power mode also compare with Table 2 4 To implement the constant power mode we write BC Heater nnx3 nytot IType nnx4 nnx3 Neumann Temp Power lc xhcb W um 2 SESES Tutorial September 2012 183 Figure 2 124 Steady state temperature fields for different flow rates at identical power of the heater The upper left shows the temperature field when the fluid is at rest followed by exam ples of increasing flow ra
310. r Note also that there is some similarity between the simulated flow field and the experimental one shown in Fig 2 118 The same good agreement can be seen by comparing the temperature distributions obtained from SESES and CFX TASCflow simulations shown in Fig 2 121 Finally Table 2 3 com pares the values for the heat flux in horizontal direction obtained from the simula tions with the calculated value as obtained from 2 155 Again there is satisfying agreement between the different solutions Figure 2 120 Calculated velocity field in cavity with SESES left and with CFX TASCflow 4 right for Ra 10 180 SESES Tutorial September 2012 Figure 2 121 Calculated temperature field in cavity with SESES left and with CFX TASCflow 4 right for Ra 10 SESES CFX TASCflow 4 VDI Warmeatlas 2 Ra 10 23 8 23 7 21 0 W m Table 2 3 Comparison of the horizontal heat flux qwan in W m for Ra 10 References 1 S J M LINTHORST W M M SCHINKEL C J HOOGENDOORN Flow Struc ture with Natural Convection in Inclined Air Filled Enclosures Journal of Heat Transfer Transactions of the ASME Vol 103 No 3 pp 535 539 1981 2 VDI WARMEATLAS Berechnungsbliitter fiir den Warmeiibergang Hrsg Verein Deutscher Ingenieure Springer Verlag 2002 3 YUNUS A CENGEL Heat Transfer A Practical Approach Mc Graw Hill 2003 4 CFX TASCFLOw CFD Tool ANSYS Inc Canonsburg PA USA http www waterloo ansys co
311. r and the total amount of current is prescribed This value is to be considered for a default device s depth of 1m For this device the source of heat is the dissipated Joule s heat in the aluminum layer and the sink of heat is represented by the thermal bad attached to the bulk silicon Black body radiation all along the device s boundary is also a sink of heat and may be defined with a Neumann BC but due to the limited temperature variation can be neglected BC Plus nx ny0 JType 1 Floating Psi cur 1E6 A BC Minus nx0 ny0 JType 1 Dirichlet Psi 0 V BC Support 0 0 JType ny 0 0 IType nx 0 ny IType nx Dirichlet Disp 0 0 m Dirichlet Temp 300 K Although this example involves a coupled computation with the thermal electric and mechanical fields a closer analysis shows that the coupling scenario is unidirectional and the fields can be computed sequentially If the electric conductivity of aluminum is considered independent from the temperature and the mechanical displacement the electric potential can be exactly computed in a first step without knowledge of the other fields Note that if this assumption is reasonable for aluminum it may not hold for semiconductors where the electric conductivity can be strongly temperature s de pendent Analogously if the thermal conductivity is independent from the displace ment the temperature can be exactly computed as a second step Here we use the SESES Tutorial September 2012 45 Disp m
312. r expansion of the discrete eddy current model The input file for this model can be found at example Eddy s3d The initial sec tion is divided in two parts in the first one we have the geometry definition One defines here the number of wires with the macro NWIRE and for each i wire the three domains Wire lt i gt CutLeft lt i gt CutRight lt i gt and a macro Corner lt i gt The do main Wire lt i gt defines the wire itself The domains Cut Left lt i gt CutRight lt i gt are used to define a wire cut section in order to apply a potential jump and to compute the current The cut section must be defined by the intersection of the two domains The macro Corner lt i gt must define a point within the wire and is used to ground the potential at that point This geometry section is the only one which needs to be changed in case of other geometrical choices For this example we have chosen a simple geometry with two coils as displayed in Fig 2 58 2 59 The second part of the initial section constructs the computational domain and define two materials Wi re AirEddy The Wire material is the copper conductor with a con ductivity of o 5 8x10 A V m Solving the harmonic eddy formulation 2 53 with nodal elements is done by defining the material equation EddyCurrentHarmonic Nodal The AirEddy material is defined for solving 2 53 outside the conductors where we have o 0 Here we can assume V 0 and therefore we select the mate rial equa
313. r is large and of the order 10 so that one needs stabilized finite elements in order to compute numerical solutions Here streamline diffusion is automatically ap plied when defining the thermal convection with the material parameter ThermConv Without stabilization in order to obtain smooth solutions a factor 10 larger artificial diffusion would be otherwise required ElmtFieldDef velocity DofField Temp 3 MaterialSpec air Equation ThermalEnergy Parameter Viscosity 0 0000186 Pars Parameter Kappalso 0 0262 W Kxm cp 29 0 029 J K xkg density 1 16 kg m x3 x Parameter ThermConv velocity 29 0 029 1 16 W m 2 K In the BC definition section we define boundary conditions The hot air flow is set to a constant temperature of 305 K at the inlet and to a constant but yet unknown temperature at the outlet A constant temperature profile at the boundary is attached to the cold air flow as well The temperature at the inlet is set to 280 K Results and Discussion The right of Fig 2 150 shows the calculated temperature distribution for a constant air velocity of 2 5 m s and for incoming air temperatures of 305 K and 280 K respectively The temperature change of the air flow at the end of each channel yields the following 206 SESES Tutorial September 2012 Oxygen Fuel Cell Potential V 1 2 i i i l 0 005 O1 015 02 025 03 y position mm Hydrogen fuel Figure 2 151 A schemati
314. r temperature and humidity are key parameters for comfort as well as for many fabrication processes The heat exchange between cold and hot air flow plays an im portant role in temperature control Fig 2 149 displays a drawing of a heat exchanger which is commercially available These heat exchangers consist of hundreds of crossed channels for the hot and cold air flows separated by thin metal sheets The chan nels lay horizontally and have heights between 3 and 15mm The efficiency of the exchanger is expressed with the averaged ratio of hot and cold flows at the in and outlet Tcold out Teold in a ea Or hot in cold in where values of 0 5 are considered as good For ISO certification the efficiency of heat exchangers needs to be measured Efficiency predictions by model calculations are difficult due to the complex and irregular channel shapes and turbulent air flows Air conditioning engineers however suggest that the presence of turbulence plays a minor role for the value of It seem to be the channel structure and dimensions that govern the value When modeling just two channels it is possible to calculate as a function of channel thickness and compare the values to measurements Implementation of the Model In order to simply the matter we have modeled two air channels only with a given rectangular flow profile Implementing a more realistic parabolic flow profile in volves either coupled velocity calculatio
315. rative solver to the system S x b and since for the new condition number we have p S 1 we expect a fast conver gence Since iterative solvers are generally initialized with x 0 we do not have to perform the initial operation with the inverse x Po and at the end after having computed the solution z we obtain our solution as z P ign x This precondition ing technique is quite general but clearly the success lies in the availability of cheap preconditioner matrices Pier Prignt and a good approximation property S Id Running an iterative solver directly on a linear system obtained by the discretiza tion of some PDES should never be done Instead one should always use a diagonal preconditioner which is very cheap to compute does not require additional memory and is always superior than no preconditioner at all The preconditioner is given by SESES Tutorial September 2012 101 Pet Pign D with D a diagonal matrix defined by Dj Sii and has the only requirement 5 4 0 which is always satisfied when discretizing PDEs In our example Table 2 1 shows quite an improvement for this diagonal preconditioner but although this preconditioner can help it is still not very robust and we have a large variation on the number of iterations necessary to reach convergence Another very popular preconditioner is the incomplete LU factorization without fill in in short notation ILU 0 It is defined as left precondi
316. re i 1 N so that the wire current is equal to T Differently from before we now prescribe the driving current instead of the voltage and without loss of generality we may assume Ip to be real If we assume the driving electric field Eo to be constant over the i th wire cross section then to a first order approximation the total voltage can be expressed as a sum of path integrals along the center of the rings with radius R N V 20 gt RiEge 2 34 l Clearly by taking another path within the rings other results are obtained since the path length is different This is slightly incorrect and a better model is to consider the driving voltage Vo e t constant on a wire cross section instead of the electric field Egie which results in an electric field Eo Voi 27r inversely proportional to the radius coordinate r So the native variables to be determined are now the wire driving potentials Vo and the question remains how to compute them Since the whole formulation is linear the eddy currents induced by a wire into the other ones add linearly with all other contributions and the fields Vo can be obtained by solving the following N problems For each i th problem with i 1 N set the following probe voltage Voi 6 on each wire j 1 N and compute the resulting wire current Gj for 7 1 N The potential fields Vo to be applied to obtain the current SESES Tutorial September 2012 69 Io in each wire are then given by
317. re is that thin mechani cal structures tend to bend and the resulting displacement is a third order function of 132 SESES Tutorial September 2012 the coordinates badly approximated by first or low order finite elements Because the analytical solution of the 3D continuum mechanical equations is seldom possible except for some trivial cases there has always been interest in obtaining more tractable governing equations by model reduction For shell structures and based on physical sounded arguments stemming from the shallowness hypothesis one can make kinematic assumptions on the solution The most prominent one is that straight vertical fibers in the undeformed configuration remain straight during any deforma tion By considering the now constrained displacement integration of the balance laws in the transverse direction yields shell models and as particular cases plate and beam models With this approach many analytical solutions have been obtained see for example 10 for linear axis symmetric shells of revolution Numerical methods for shells may use the underlying shell models for their discretiza tion At the present time these shell models are mostly linear models where the dis placement just enters linearly into the governing equations Among the most promi nent models we have the linear model of Koiter for thin shells with zero transverse shear strain and the linear Nagadi model for thick shells 2 5 6 For numerical appli
318. rmal 1 A 2 Normal 0 A 0 Normal 2 Normal 2 0 Normal 0 A O0 Normal 1 A 1 Normal 0 Normal 1 Normal 0 0 BC bc2 Restrict material Scalar OnChangeOf 2 3xmaterial Scalar material Vector tmaterial VectorJ Neumann Afield D_MagnPotGrad SetAlways A_CROSS_N__DA MagnPotGrad SIunit Symmetric Linear HideFlux The Command section solves the preprocessing step to compute the current Jo and various magnetostatic formulations to compute the magnetic induction B Numerical results In order to compare the numerical solutions we use a large empty space u p as our domain Q so that for given a current Jo the solution is given by the Biot Savart inte gral 2 44 The integral is computed by the built in routine Biot Savart by evaluat ing the integrand at the integration points of supp Jo so that the result is just accurate for points outside supp Jo Fig 2 55 shows the five computational domains Qvyec used SESES Tutorial September 2012 87 amp amp amp amp amp Figure 2 56 The magnetic field Euclidian norm B at the wire s boundary and on its interior together with directional cones 1 66405 ae iot Savart 1 40405 dug Case T 1 2e 05 cases 1 0e 05 3 Case 4 8 0e 04 6 0e 04 4 0e 04 2 0e 04 0 0e 00 H field A m Z coord mm Figure 2 57 The z component of the magnetic induction H along the symmetry axis for the vector formulation The domain size for the vector
319. rongly coupled to the N3 species and ne glecting this dependency will in general prevent convergence of Newton s iteration However we know that solutions are invariant by addition of any constant row to the diffusion matrix Dag So the trick is to add those constant rows resulting in a zero column Dan 0 which further reduce the Nz dependency of the residual equations This trick is enabled with the option ZeroEntry N2 in the StefanMaxwellDiff model and can only by used if we do not directly solve for Nz otherwise numerical instabilities arise In fact the flexibility available in the choice of the diffusion matrix is used to obtain stable numerical computations when all species are solved together For this particular SOFC problem the application of these two ideas lead to very ac ceptable convergence rates even without amending the user defined derivatives but we remind that in a very general case it may be necessary to compute all species to gether For this problem a possible solution strategy is to solve for all dof fields together ex SESES Tutorial September 2012 217 cept for the Nz mole faction with a fully coupled Newtow s algorithm The solutions are quite smooth and non linearities are weak so that we can speed up the Newton s iteration by using a direct solver and reusing the factorized stiffness matrix for sub sequent iterations This behavior is obtained with the option ReuseFact oriz in the Increment statement and in
320. roperty V Jo 0 at the discrete level is well defined although it only holds in a weak form Therefore the problem specification starts by defining the domain supp Jo to compute the current defined by the material law Jo cE with the electric conductivity and E the static electric field In the electrostatic we have V x E 0 and so we can define the electric potential Y with E VW and from V Jo we are left to solve the Laplace equation V cVW 0 by choosing BCs for W i e the applied voltage at the electrical contacts Model specification When solving the vector formulation 2 46 in SESES we have the choice of either us ing the nodal elements of H Q or the edge elements of H curl Q Edge elements directly discretize the curl curl operator of 2 46 which has a large null space and so without an additional fixing reducing the number of redundant dofs the discretized linear system is singular In order to have solutions the right hand side must be con sistent and must lie in the image space of the linear map a failure condition detected e g by the non convergence of iterative linear solvers For edge elements this implies that the current computed by the preprocessing step must be first projected on a well defined functional space and at the present this rather technical step must be done by the user On the other side the usage of nodal elements with the correct choice of BCs results in a regular linear system These elements
321. rt of the command section Here the solution of the Navier Stokes equations is defined with the dof fields for the pressure p and velocity v com puted all together in a single block The Navier Stokes equations are non linear how ever the Hagen Poiseuille we are solving is a linear problem and we just need to perform a single block iteration BlockStruct Block Pressure Velocity Convergence 1 Solve Stationary Since for the Hagen Poiseuille problem the analytical solution is known it is of interest to check the numerical error affecting the solution with the statement Write Error for Velocity e e n maxvalue Nodal Velocity 0 dp width xwidth 4 x x 8 viscoxlength By starting the computation the textual output will look similar to SESES Tutorial September 2012 165 gt STATIONARY SOLUTION gt SOLVING AT time 0 0000e 00 STEP 0 0 000000e 00 FIELD Pressure AbsResidNorm 1 13e 10 FIELD VelocityX AbsResidNorm 0 00e 00 FIELD VelocityY AbsResidNorm 4 71le 06 lt BC_data gt lt BC Name Inlet Type Basic gt lt Field Pressure Flux MassFlux TypeBC Dirichlet gt 0 500000002 kg s lt Field VelocityX Flux ForceX TypeBC Dirichlet gt 2 81167505e 16 N lt BC Name Outlet Type Basic gt lt Field Pressure Flux MassFlux TypeBC Dirichlet gt 0 500000002 kg s lt Field VelocityX Flux ForceX TypeBC Dirichlet gt 4 96040282e 15 N lt BC Name NoSlip Type Basic gt lt Field VelocityX Flux ForceX TypeBC Dirich
322. ry we have the following scenario Our fuel cell consists of three materials the cathode the electrolyte and the anode stacked above of each other Each layer has a thickness of 0 1mm and the width and depth of the cell is 1mm Within the cathode we have the diffusion of oxygen so that we need to solve for the oxygen mass flow there At the top of the cathode we supply oxygen and since we suppose there is no shortage we can assume a constant oxygen concentration corresponding to a Dirichlet BCs The oxygen diffuses towards the interface with the electrolyte where the oxygen s reduction acts as a sink with a rate given by 2 177 and modeled by a Neumann BCs depending on the normal electrical current At the anode we have a similar situation for the hydrogen at the bottom of the anode we prescribe a fixed hydrogen concentration with a Dirichlet BC and at the interface with the electrolyte we have a sink of hydrogen modeled by a Neumann BCs with a rate of 2 177 At the 208 SESES Tutorial September 2012 anode electrolyte interface we have additionally the production of water steam but we assume the steam can freely escape through the anode and therefore does not need to be considered Over the whole device we solve for the electric potential and set the top of the cathode at ground By the generated electrical field electrons drift towards the electrolyte Here the potential undergoes a first jump due to the Nernst potential of the oxygen re
323. s Transport0O2 TransportN2 and at the an ode for TransportN2 TransportH20 TransportH2 TransportCO Transport CO2 Species convection is automatically considered by solving for the velocity v and for species diffusion we use the Stefan Maxwell diffusion law available in SESES with the built in model Ste fanMaxwe11Diff In order to work with the Stefan Maxwell model we need to work with species mole fractions x for the dof fields and the con straint a 1 Numerical solutions will satisfy this latter condition by a proper setting of the BCs and if the sum of the species production rates II is everywhere zero Xa Ila 0 a fact proven in SESES manual Actually these 2D transport equations should not only consider the transport in the air and gas channels but also the lateral transport in the underlying GDL of cathode and anode However being these GDL layers made of solid matter their contributions to transport may be neglected Due to the high temperature the hydrogen at the anode may burn out if there are some oxygen s leftovers but since we are excluding oxygen as species at the anode this case does not apply However as an example we will consider the so called CO shift reaction CO H20 CO 2 H with the following kinetic model relying on phenomenological formulas For species compositions far from thermodynamic equilibrium the forward rate expression is given by Ea 2 17 RgasT zco 7 9 Tred ko exp 212 S
324. s file contains the definition of a rotation function called rot and rotating mesh elements about the origin Thus in the input file of the example of Fig 1 10 one may simply use the statements Include Homotopic sfc Coord x 1 y block 1 2 1 2 Coord 1x ly 2 rot x 1x 2 y ly 2 angle block 5 7 1 3 to obtain the same geometry The next example is a little more involved and uses the function bump for graded translation on the first 10 x 10 block and the function sphere on the second 10 x 10 block to create the shapes of Fig 1 12 with the following statements Include Homotopic sfc QMET 10 dx 1 1 QMEJ 10 dy 1 1 Coord coord 1 2 bump x 1 dx 4 dx 6 dx 9 dx 1 xbump y lxdy 4 dy 6 dy 9 dy 1 block Block 0 QMEI Start 12 10 dx 0 9 QMEJ Start 1 10 dy 0 9 Coord sphere coord 5 dx 12 5 dy 1 1 block Block 1 In the remainder of this section we will describe the actions of some functions defined inside Homotopic sfcand apply them to a 3D rectangular 10x 10x 10 macro element mesh These are the statements to start with Include Homotopic sfc QMET 10 1 QMEJ 10 1 QMEK 10 1 The mesh distortion is then performed with the Coord statement and the functions of Homotopic sfc It is surprising that with only a few well defined functions com bined together many different geometries are easily obtained 22 SESES Tutorial September 2012 Figure 1 13 Direct scaling rotation with rot and combination with a
325. s is that the cost of solving a second linear system with the same matrix as the first system but a different right hand side vector can be done with much less effort then when 30 SESES Tutorial September 2012 solving for the first time This is because the numerical factorization of the system matrix is by far much more expensive than the final backward forward substitution which has to be done for each different right hand side vector In the first computed non linear example we have seen that with a wrong derivative the convergence rate of Newton s algorithm is slow and generally linear but the iteration may soon or later converges towards the convergence criterion Fast algorithms are therefore obtained by updating the derivative and its inverse just when the convergence rate is slow and this implies avoiding factorizing the system matrix We just keep the matrix computed in one of the previous steps At each linear iteration step we just update the residual values and this only implies changing the right hand side vector of the linear system to be solved In SESES this behavior is achieved by defining fast increments meaning that the linear system matrix of the previous step is not to be changed In order for this example to converge by computing the derivative just in the first step we need to cut the increments in the first step In general the increments com puted by the Newton algorithm far from the solution are overestimated and can e
326. s projection is computed by the routine RigidIntersection however before calling it one has to declare the rigid bodies with the statement Misc RigidBody StampDown Down StampUp Up Smooth The values St anpDown St ampUp are the block ME numbers for the two stamps and the Down Up specifiers determine the contact surface of the hexahedral ME block The Smooth option activates cubic interpolation so that normal tangents and curvatures are continuous functions In this example both the analytical and numerical approach are used and compared For the former a series of input routines ParabolaProj Profile StampCoord StampP ro j is defined evaluating the projection as described previously Typo errors in these user routines are not so easily discovered and within comments some additional testing code that has been used is provided The selection of one of the two approaches is simply determined by the setting of a global variable At the input s beginning we have defined routines to be used by the penalty and al gebraic contact approach Since they are pretty generic they can be placed once inside some input files to be included For the penalty method the routine Penalt yContact is used to define the Neumann BCs and takes as input the data structure CONTACT 156 SESES Tutorial September 2012 specifying the closest point projection This data needs to be set by the user or by calling the routine RigidIntersection Other ramp functions tha
327. s set with the parameter period and the turn off time toff is set to half this time interval The time dependent simulation is defined with the statement Dump AtStep 1 Temp Title TIME g n time TempDiff Title TIME g n time Solve Unstationary At 0 Step period npts Until period The first line above makes sure that the temperature and heat flux fields are stored in the Data file at each time step For a heating parameter of h 160 W cm some time dependent temperature fields during heating are shown in Fig 2 14 The lower most distribution corresponds to the steady state In Fig 2 15 some temperature fields during cooling t gt t are shown Note that the color scale is normalized for each instance in time and therefore cannot be compared directly A 8 SESES Tutorial September 2012 Figure 2 14 Temperature fields at different times after turn on during heating temper ature scale not shown Steady state Temperature Profile experiment 1 F 350 L iment 2D simulation h 130 id 2D simulation h 160 x 300 1D model 20 65 sinh 0 021 x rd temperature K DS 100 0 20 40 60 80 100 120 distance cm Figure 2 16 Steady state temperature profile in the stick for two distinct heating powers Figure 2 15 Temperature fields at different times after turn off during cooling temper ature scale not shown Temperature Transients at 4 Positions T10 exp a
328. s slowly cooled in the furnace whereas by normalization the cooling process takes place in the 72 SESES Tutorial September 2012 Figure 2 46 Computational mesh lo cally adapted to resolve the boundary layer Figure 2 47 Heat boundary layer in duced by the skin effect air Hardening involves heating the steel to its normalizing temperature followed by a rapid cooling in a fluid as oil or water In this way the steel becomes very hard but also brittle By tempering the steel is reheated to remove the brittleness created by the hardening process If a furnace with controlled temperature is generally used for these thermal treatments for small volumes an alternative is represented by a coil with AC current surrounding the steel object to be treated By induction eddy currents are generated and their power dissipation will heat up the steel From a numerical model as presented here one can expect to answer questions as the temperature distribution in the steel the dynamic behavior or the ability to optimize the coil geometry towards some given design goals In this example in order to re duce the complexity and to avoid 3D computations we are going to assume rotational symmetry of the steel object and steady state conditions Due to the elicity of the wire windings this symmetry never holds for the coil so that we make use of the proximity approximation as discussed and applied in the previous example The computation of the eddy c
329. s with local refinement and with a circular shape The construction process is shown in Fig 1 9 from left to right As first by selecting the insert task 4 a rectangular quad with just few macro elements is defined Start drawing the quad with a double click defining the first boundary node the other boundary nodes are defined by a single click and the last boundary node with a final double click If something goes wrong click on the undo button or press of the ctrl z key combination and start again As second some nodes need to be moved at the correct position Either with the insert task or with the affine task amp and option ff click and hold down the first mouse button to select a node and drag it to its new position As next one has to join together macro element edges to define more generic shapes than simple quads With the join task E and the join always option Pf click and hold down the first mouse button on a edge and drag it over the edge to be joined until a snap takes place Correctly joined edges are marked afterwards with a thick blue line Be sure that edges drawn with a thick red line are just shown on the boundary and not inside the mesh In this latter case the two macro elements with a common thick red edge have not been joined and therefore the two macro elements are not topological neighbors but there is a split hole in between To create a mesh with local refinement as shown in the first three rows of Fig 1
330. s1 The family of solutions computed by the load control algorithm is shown in Fig 2 96 up to the first warp solution with the applied load as function of the displacement L norm A reference solution has been computed with second order elements and a homogeneous refinement level of 1 and the other curves are for solid shell and first order solid elements with a homogeneous refinement level of 0 and 1 The solid shell elements are in very good agreement with the most precise solution and the first order SESES Tutorial September 2012 135 solid elements are not so dramatically worse This is not so a critical example for nu merical locking since the elements are well shaped and the shell is rather thick How ever first order elements are by far less performant as soon as the thickness size ratio of the elements is getting worse The solid shell elements specified by the equation ElasticShellSingleK always use displacement degree of freedom defined just at the top and bottom of the shell independently from the number of elements defined through the thickness Therefore the time spent within the linear solver is the same as when using first order solid elements with a single element through the thickness which is the cheapest way to compute a solution However the overall solution time is larger since solid shell elements are more expensive to assemble If not tremen dously slow the time spent to compute elements is in general of little concern
331. scosities v layer 0 01 10 4 10 1 One of them is the 2D Blasius boundary layer flow over a flat plate at zero angle of incidence with respect to a uniform stream of velocity U at infinity Although just an approximation of the Navier Stokes equations 2 127 this solution can be used to test and calibrate Navier Stokes solvers which is what we are going to present in this example Without going into the details of the Prandtl boundary layer theory the Blasius solu tion for a plate located along the x axis can be obtained as follow For the Blasius prob lem we have f 0 and by assuming a priori 0 p 0 just the incompressibility condi tion 0 u 0y v 0 and balance of momentum in x direction u0 u vdyu v Ozu 07u need to be solved together uin order to the obtain velocity v u v with v u p the kinematic viscosity Within the thin boundary layer we have u lt 0 u and therefore the boundary layer approximation consists in neglecting the term 32u The continuity equation is automatically satisfied by working with a stream function y x y where we have u Oyw and v 0 and hence we are left to solve the single equation Oy POny Ox pO voy Blasius found a separation of variables by introducing the similarity variable n yV U vx and working with the Ansatz z n Us vxf n By considering the new independent variables x n substitution yields u Ux f v Uxv 4x nf f and Us MY s
332. se cross is shown in Fig 2 87 We can see that depolarization is almost not existing inside the pump spot area and be comes much higher outside We can determine the relative amount of depolarization in a homogenous beam of radius r as follows We summarize the relative depolar ization of each lattice inside the beam cross section and divide the obtained result by the number of lattices inside this cross section When we perform this for a varying radius r we obtain the relative amount of depolarization as a function of the beam radius r A laser beam in the resonator passes the disk twice because the back sur face of the disk acts as high reflecting mirror Therefore we have to use a double forth and back path to estimate the real depolarization of the disk If the resonator contains polarizing elements this relative amount of depolarization can be interpreted as the depolarization loss of the disk for this kind of resonators The obtained result is shown in Fig 2 88 As we can see the depolarization losses are extremely low in the range of 0 01 0 05 When we chose a beam radius smaller than 0 75 mm the depolariza tion loss only amounts to 0 002 whereas the highest loss of 0 05 is obtained for a beam with radius r 1 5mm In summary we can say that the thin disk laser represents a laser concept which can really overcome the drawback of high radial temperature gradients We have obtained an extremely weak thermal lensing and almost
333. sed by the derivative of an elastic potential w Fe with respect to the elastic strain Fe By considering objectivity one necessarily has a dependency of the potential from the symmetric right Cauchy strain C FT Fe and so we have 20w 0C Ow J0E with the Green Lagrange strain defined by 142 SESES Tutorial September 2012 E Id 2 If we further restrict our attention to isotropic materials then the potential w can only be a function of the invariants of C which for this example are taken to be their eigenvalues Since C is positive definite the eigenvalues are real and positive and we can write the spectral decomposition in the form C gt gt N N with 0 00 the eigenvalues and N the orthonormal eigenvectors The potential can now be expressed in the functional form w w A In order to evaluate material laws expressed with respect to principal values quite some numerical work is involved in computing the eigenpair decomposition at every integration point and so it may seem over killing to use the spectral formulation for isotropic materials However as we will seen this is the only additional step required to upgrade an infinitesimal plastic model to the finite strain case For this goal we introduce the log strains log A so that the stress is expressed by 3 woe H TN 2 109 with 7 w The push forward of S on Q defines the Kirchhoff stress r Fe S Js given by T 9 min Q n
334. should not be used when on some regions the permeability is different from jo since they enforce an unphysical conti nuity of A This will not be the case for our example and also we like the possibility to have unique solutions in order to work with a robust direct linear solver an appre ciated feature when doing numerical experiments on small sized problems Therefore in this example we are going to use nodal elements for the discretization of A and a direct solver but we will notice the necessary changes when using an iterative solver The SESES input file for this example can be found at example ScalVecFormul 3d The Initial section is divided in two parts the first one defining the geome try The user has to define the ME domain Vect ord representing supp Jo and where the electrical current Jo is computed by the electrostatic model In addition the user SESES Tutorial September 2012 85 defines the ME domain Vector used either to make the remaining domain Qfree sim ply connected or as a cladding layer around supp Jo This domain is then added to Vectord to form the domain Qyec where we solve for the vector formulation 2 46 with nodal elements For large problems requiring an iterative solver you should use edge elements instead With the macro NWIRE the user defines the number of closed wires and for each 7 wire i 1 NWIRE one needs to define four ME domains Cut LJ lt i gt CutRJ lt i gt Cut LP lt i gt CutRP lt i gt T
335. single wire one is limited at computing the internal inductance by excluding the contribution from the free space This is also easily done analytically Substitution of 2 30 into 2 29 and by considering rotational symmetry we obtain the Bessel s eq 02 r718 a J 0 with a iwpgc The solutions are Bessel s functions and with the BC J ro Ege we have J o Eo Jo ar iwt 3 ao Do with Jo x 2 Eo Computing the integral 2 31 and insertion into 2 32 yields the values Jo aro RNA a F Oe ae eee ae with 6 2 wji97 the skin depth and Jj the derivative of Jo The DC values are R nor2 and L po 87 Now that we have seen how to compute resistance and inductance values of a straight wire we want to do the same for an air coil with N windings Here the matter is a little bit more involved since only 3D computations can truly compute the magnetic field in a coil however these computations are too expensive for our purposes and we therefore look for valid 2D alternatives The main idea is to consider the coil as a rotational symmetric structure by neglecting the spiral nature of the wire and by considering the windings as separate independent wires In doing this we assume each wire to be a ring carrying the same total current I Ipe The solution of our coil problems then consists in determining which cross sectional electric field Ep e need to be applied in each wi
336. singular points and on which solution branch we land afterwards In particular the warp solutions of Fig 2 95 have been computed with first order solid elements As a rule of thumbs low order elements with few degree of freedoms are more robust when solving non linear problems and in this case they can easily jump over the singularity This is not the case for the 2nd order solid elements or the solid shell elements with many internal degree of freedoms The geometry of the conical shell is easily defined as well as the definition of the linear isotropic elastic law and BCs To use a load control algorithm we need to specify the derivative of the global residuals with respect to the load parameter in our case the pressure load By default a difference method is implemented to compute this derivative however in our example the load is applied as a Neumann Pres BCs which enters linearly into the formulation In this special case as alternative to the numerical difference there is the possibility to exactly specify the derivative One has to define a second Neumann Pres1 BCs with a value representing the derivative with respect to the load parameter The boundary for this second BC is not used and can be empty BC Press 0 0 0 IKType nx nz Neumann Disp 0 0 load Pa BC Pressl Neumann Disp 0 0 1 Pa Disable The declaration of the exact derivative is done by associating both BCs with the state ment Increment DR_DSimPar Linear BC Press Pres
337. son is that for a given current Jo the computation Biot Savart integral 2 40 has to be per formed at each integration point of each element belonging to the meshing of 2 and this is a very CPU intensive computation which is better avoided And indeed this is possible if one chooses Qreq as the minimal domain containing the support of Jo and so that Qot Q Qrea is simply connected Then Ho and thus the Biot Savart integral must only be evaluated within Qreq and on OQinter 2 13 The scalar and vector formulation of magnetostatic The magnetostatic solution of Maxwell s equation is an important topic in the design and development of electrical machineries driven by magnetostatic forces Basically this problem formulation computes on a domain 2 C R the magnetic induction B and the magnetic field H starting from a given time constant current Jo having the property V Jo 0 and a given permeability u specifying the material law B pH It is interesting to know that this static model may also be used to find low frequency approximations of the computational more expensive time dependent eddy current model and therefore good optimized solution algorithms are of large interest The magnetostatic model can either be solved by computing a scalar or a vector potential both determining the magnetic field by differentiation and both having their draw backs and advantages However a considerable increase in flexibility is first obtained by combining both f
338. son for using here the exponential approximation Now that we have performed the time integration we can change the formulation to a spatial description which is the preferred one to formulate the plastic laws From the relation 7F N Fe C N F F2 F N we note that the elastic left Cauchy strain be Fe F F C Fr a A n n has the same eigenvalues of C and is coaxial with 7 As for the infinites imal case for the time integration of the Kuhn Tucker equations 2 112 we apply an operator split based on a first trial step followed by a return to yield step for details see 5 Let the deformation together with the internal variables F n Coni be given at time tn together with a time step At At time t 41 tn At the deformation F is known and we are therefore seeking En 1 C gt 7 41 If the trail principal stresses Tirail i computed with the eigenvalues A2 of the trial strain be trial Frit Cor FT 41 are trial i in the elastic range f Tira lt 0 then Fn41 241 En C on ad is the solution at t 41 Otherwise we have to solve the following system to determine be n 1 n 1 and the plastic multiplier AX AAt be n 1 exp 2A0 fn 1 S De trial Enti En or Abg fn 1 fn 1 f T Be wii a n 1 0 and update the new values of 41 and Co n F ben 1 Eii At this point we use the coaxiality of be T and 0 f stemming from the isotropy assumption to proceed further in th
339. sport mechanism in the cross direction We first present the stationary diffusion convection equations for a binary mixture and solve them analytically for our geometry Thereafter a SESES model solving these equations by the Finite Element FE method will be set up and the numerical results will be compared with the analytical solution SESES Tutorial September 2012 169 Diffusion Convection Equation Consider first the mass balance for some species a ba V pa Wa Ia 2 128 with pa Ma V the partial mass density W the local velocity and IIa the produc tion or consumption rate of a respectively 5 Since in general the transport of a happens by convection as well as by diffusion we split W into its diffusive and con vective contributions To specify the latter consider the local mass average velocity V w wa 2 129 a 1 p with the total mixture density given by p _ Pa and v the number of species Wa can now be expressed relative to W as Wa W Ja Pa with Ja the diffusion flux of a and its insertion into 2 128 leads to 2 V pa W VJe Ta 2 130 From irreversible thermodynamics we know that J can be written as v 1 B tee 2 FE V H Ha 2 131 with Ha and ug the chemical potentials of species a and 8 and Bag a phenomenological coefficient 5 Consider now the particular case of an isothermal incompressible fluid mixture for which T p const Then the chemical potentials of all s
340. st asserts the lin earity of the BC thus helping SESES to select the correct solution algorithm and the Symmetric option asserts the symmetry of the resulting matrix From 2 49 2 50 86 SESES Tutorial September 2012 Figure 2 55 The computational domains Qyec solving for the vector potential A we know that the overall contributions from the interface BCs are anti symmetric al though the single element contributions are not In order to have an overall symmetric system and thus less computational work the following symmetrization is possible The linear system arising from this scalar vector formulation can be characterized by the following block structure A oO at 4 BY psn The diagonal blocks A B are symmetric positive definite matrices stemming from the discretization of the vector and scalar potentials A The coupling between both formulations is uniquely defined by the D matrix stemming from the interface BCs 2 49 2 50 By changing the sign of the second block row we obtain a symmetric matrix and this is accomplished with the following statement GlobalSpec Parameter MagnPotEqScaling 1 This symmetrization just works for a direct solver but may lead to an unacceptable slow down of an iterative solver and therefore is not used per default and must to be enabled In a similar way we define the second interface Neumann BCs for A Macro A_CROSS_N__DA A A 1 Normal 2 A 2 Normal 1 0 Normal 2 No
341. statement Four domain s boundaries Inlet Outlet SymmetryWalls Cylinder are defined representing the flow inlet the flow outlet the surface of the cylinder and the upper lower sides of the rectangular domain This initial macro element is refined manually around the cylinder and the result is shown in Fig 2 132 With the next statement the material model for solving the stationary incompressible Navier Stokes equations is defined MaterialSpec m_AIR Equation CompressibleFlow Enable Parameter PressPenalty EpsP s Parameter FlowStab zero Parameter Density Val DensityAIR kg m 3 Parameter Viscosity ViscosAIR Pa s Before we start our simulation the BC values must defined The BCs for the side walls of our channel are so called symmetry conditions no fluid can cross the boundary and the fluid can move along the boundary without experiencing friction slip conditions The BCs for the cylinder surface are no slip conditions with a zero flow velocity At the inlet Dirichlet conditions are used to specify the flow velocity Furthermore the total mass flux must be specified through a pressure floating condition The pressure relative to the reference state is specified as a Dirichlet condition at the outlet BC SymmetryWalls Dirichlet Velocity Y 0 m s BC Cylinder Dirichlet Velocity 0 0 m s BC Inlet Dirichlet Velocity Veloc 0 m s Floating Pressure Veloc DensityAIR Area kg s BC Outlet Dirichlet Pressure 0 Pa Dirichlet Velocity Y 0 m s
342. step one solves the electrostatic problem 2 61 by applying the driving voltage and the computed current Jo is used to define a stiff contribution to the current J cE Jo If the electric potential is removed from the solution then it is also possible to symmetrize the linear system thus obtaining a fur ther speed up Given the field approximation A gt H A with H the real shape functions and A C the complex unknowns for nodal shape functions one solves SESES Tutorial September 2012 93 the complex linear equations R 1 V x H u V x An uz V H V An iowH A JdV 0 2 67 Q with u a constant average of u on Q By splitting real and imaginary parts and by assuming is real we have the following real linear system RA SA RA A B SA amp d 2 68 with the blocks Aij fal V x H y7 i V x H u V H V H dV and Bij Jo owH HjdV being symmetric matrices This global non symmetric matrix with positive definite diagonal blocks is the default assembled matrix but we see that if we change the sign of the second row block a symmetric matrix is left to be solved Although we can apply these two optimization techniques to our example our advice is to keep the electric potential otherwise there is no exact current conservation within the conductors and the computed admittance or impedance matrices are in general far less precise In particular the low frequency approximation does not agree with a Taylo
343. t and just change the upper tip and lower substrate domain boundary by applying a sine function to distort the regular geometry The result is shown in Fig 2 28 together with an example of a computed electrostatic potential Let us now consider a unidirectional scan across the surface during which we shall record the electric field at the tip end We can perform such a scan by moving the sub strate in x direction relative to the tip A simulation parameter xof f was introduced for this purpose The command section of the input file thus reads as follows Dump AtStep 1 Phi Efield Write File tip_1Dscan_s tshape n nsteps txt Text speriod 2f sheight 2f tshape 2f theight 2f n speriod sheight tshape theight xoff nm Efield V m Phi V n Write AtStep 1 File tip_1Dscan_s tshape n nsteps txt Append Lattice tip 3e 3e 3e n xoff lx Efield Phi Solve Stationary ForSimPar xoff At 0 Step 0 0l nsteps The file name contains the tip shape parameter t shape and the number of incre mental scan steps nsteps The 3 column data file obtained can now be plotted with Gnuplot In scanning probe experiments the acquired images are subject to artefacts caused by the non ideal tip geometry In general the acquired signal is a convolution of the tip geometry and the substrate topography This effect is schematically shown in Fig 2 27 in a constant distance acquisition mode Depending on the sharpness of the tip
344. t case of longitudinal pumping the pump radiation is absorbed along the optical z axes in the rod Further assuming the paraxial approx imation the absorbed part only depends on the distance z to the rod surface and the absorption coefficient a of the rod These simplifications lead to the heat source qlr z n Po p r z a e 2 94 with Fo the total incident pump power and 7 the fraction of pump power converted to heat here In this example we consider 7 0 4 For a focused laser beam the radius at the distance z from the beam waist w reads 2 w z wo4 1 ar 2 95 T Wo n with the pump beam wavelength M the beam quality factor and n the refractive index of the crystal For simplicity we assume the pump distribution to be Gaussian shaped so that p r z reads 2 ar P r z 7 exp z 2 96 with w z defined by 2 95 114 SESES Tutorial September 2012 Folding Figure 2 71 Pumping of the laser rod through the two folding mirrors Figure 2 70 Laser rod in a x folded resonator Model Specification To keep the example as simple as possible we perform a 2D rotational symmetric simulation which is enabled with the statement GlobalSpec Model AxiSymmetric Enable Afterwards some geometrical parameters for the laser rod are defined followed by the relevant pump beam parameters relating to the models 2 94 2 95 and 2 96 Define P 20 pump power W w0 0 250
345. t radius x pe dP_dy 2 PI xr rx l eta power density on rod axes As we have a unpumped and pumped Nd YAG part we define the two materials UP_NdYAG and P_NdYAG MaterialSpec UP_NdYAG Equation ThermalEnergy Elasticity Enable Parameter StressOrtho LinElastIso Emodule 307 GPa PoissonR 0 3 AlphalIso alpha Tair SIunit Parameter Refrac 1 82 Parameter DRefrac 7 3e 6 Temp Tair Parameter KappaIso D_Temp kappa Dkappa W m K W m Kx 2 MaterialSpec P_NdYAG From UP_NdYAG Equation ThermalEnergy Elasticity Enable Parameter Heat conv p_cx 1 1 eta x x r r W m 3 For the pumped part we only have to add the heat distribution therefore we derive the material P_NdYAG from UP_NdYAG and additively define the heat distribution We further assume Sylgard 184 as compensator material and dedfine the following mate rial properties MaterialSpec Compensator Equation ThermalEnergy Elasticity Enable Parameter Refrac 1 43 Parameter DRefrac 3 0e 4 Temp Tair Parameter Kappalso 0 15 W m K Parameter StressOrtho LinElastIso Emodule 20 kPa PoissonR 0 45 AlphalIso 0 25e 4 Temp Tair STIunit SESES Tutorial September 2012 129 After the definition of the material properties we define the mesh and maps the ma terials to to mesh QMEI nr r nr QMEJ ny1 dyl nyl ny2 dy2 ny2 ny3 dy3 ny3 ny4 dy4 ny4 ny5 dy5 ny5 ny6 dy6 ny6 ny7 dy7 ny7
346. t statements BlockStruct Block H2 H20 02 N2 Temp Convergence MaxIncr H2 lt le 7 amp amp MaxIncr O2 lt le 7 amp amp MaxIncr Temp lt le 4 Increment Standard 2 ReuseFactoriz 10 Standard Control 1 Solve Stationary defines the fields to be solved for sets convergence criteria an optimized solution strategy and evokes the actual calculation Numerical Results and Discussion Fig 2 144 shows the axial and radial velocity ditributions within the reactor as pre dicted from the CFX 5 simulations At the inlet of the free flow region we have a Figure 2 144 Velocity distributions in axial a and radial b directions along the reactor the lighter the color the higher the velocity uniform axial velocity of 0 040m s Due to the no flow condition at the boundary between the free and porous regions the fluid slows down near that boundary Con sequently the continuity of mass leads to an increase of the velocity near the center with a maximum velocity of 0 075 m s at the outlet Since the considered burning of hydrogen is strongly exothermic a substantial gener ation of heat occurs This leads to a temperature increase within the reactor Within SESES Tutorial September 2012 199 the prorous region the heat transfer is rather slow since it only happens by con duction and diffusion Consquently the thermal conductivity value of the porous material has a large impact on the observed temperature distribution This is illus
347. t stream during execution of the kernel program Then several one dimensional lattices for the OPD evaluation are defined with a For loop statement Here we define straight lines along the pump lasing direc tion where the local OPD will be integrated 116 SESES Tutorial September 2012 For k From 0 To nrOPD Lattice OPD_ k Index i 0 nlOPD k rOPD nrOPD 1OPDO 0 10PD i nlOPD The lattice names are numbered with the loop index k by OPD_ k afterwards follows the definition of the lattice index used to define the lattice points as equally distributed along the lattice line The corresponding parameters nrOPD and n1lOPD were previ ously defined in the initial section Before starting the simulation the field Temp has to be set to the temperature value of the surrounding air Without the statement Solve Init Temp Tair the SESES kernel will initialize the temperature to 0 K which is disadvantageous for the simulation time and convergence The next section specifies the solution algo rithm In this example we perform a parameter study for different pump powers and the solution procedure looks as follows For i From 1 To P_n while 1 Solve Stationary ForSimPar P At i xP_Max P_n Store StressAver Continuous 1 FreeOnRef Stress TempDiffAver Continuous 1 FreeOnRef TempDiff Remesh Refine ErrTemp TErr Global Refine ErrDisp MErr Global if REMESH break Write File OPD_ i dat Text For k From 0 To nrOPD
348. t the Hagen Poiseuille flow in a straight channel as displayed in Fig 2 110 This laminar viscous flow problem is one of the few known analytical solutions of the incompressible Navier Stokes equations The flow is characterized by a constant mass density p a constant viscosity u an inlet outlet at constant pressure Pin Pout with zero tangential flow v 0 and no slip conditions v Oat the other boundaries The 2D solution for an infinitely deep channel is given by the flow velocity Pa ia ut _ 2 yr W v Sal wW Ax rel 5 n 2 125 with L the channel length and W the channel width The total mass flow at the in let outlet correspond to the mass flow integral over the channel area Ww 2 p P m out W3 M 2 12 n pvdz TZL 2 126 The SESES input file for this Hagen Poiseuille problem can be found at example PoiseuilleFlow s2d The initial section starts with the definition of the channel dimensions viscosity density and pressure drop as user variables in order to parame terize the computation These values correspond to an air flow channel of dimension 0 25 x 0 05 m Define length 0 25 m width 0 05 m visco 0 0008 Paxs x density 1000 kg m 3 x dp 0 01 x Pa x massflow density dpxwidth xwidth width 12 viscoxlength 164 SESES Tutorial September 2012 The simple rectangular channel geometry is defined with the next statements defining a macro element mesh of 25 x 10 el
349. t to flow situations with Reynolds num bers ranging from 5 to 48 We will compare the calculated force the length of the bubbles and the pressure distribution on the cylinder surface to data obtained by ex periments and taken from 2 The equations to be solved are the continuity and the Navier Stokes equations For a steady incompressible flow the continuity equation can be simplified as V v 0 2 158 SESES Tutorial September 2012 187 Figure 2 133 The computed velocity field be hind the cylinder for Re 46 with the cones Figure 2 132 Generated grid around the showing the direction of flow thus indicating cylinder the region of recirculating flow and the Navier Stokes equation as ov V v Vp uV 2 159 with p the constant density v the velocity p the pressure and u the viscosity These equations can not be solved analytically for the flow field at hand For our computa tions we will consider a cylinder of radius 0 05 m and infinite length and can therefore assume a 2D flow situation We then define a domain of rectangular shape and di mensions 0 7 x 1 4m with the center of the cylinder placed at 0 55 m behind the flow entrance and in th
350. tained by storing instead of the single complex coefficient G the 2 x 2 real matrix and for a complex vector component v the two real components Rv Sv RGi SGi Sy RG Define double R 0 wL 0 InvMatDotVec MAT RHS SOL For i To nWind 1 write Driving voltage 0f e i SOL 2 i SOL 2 i 1 write R e Ohm L e H n R Se V n R R SOL 2 i wL wLt SOL 2 itl wL 2 PI frequency The effective resistance and inductance are given by 2 36 as the sum of the real and imaginary part of the driving voltages since our computations are done with Jp 1 SESES Tutorial September 2012 71 These values are written to the output together with the single driving voltages for each wire and the following is the result Driving voltage 0 2 006826e 02 9 018853e 03 V Driving voltage 1 2 001835e 02 9 859638e 03 V Driving voltage 2 2 006826e 02 9 018853e 03 V Driving voltage 3 2 214240e 02 1 120095e 02 V Driving voltage 4 2 214240e 02 1 120095e 02 V Driving voltage 5 2 431755e 02 1 057987e 02 V Driving voltage 6 2 425840e 02 1 153813e 02 V Driving voltage 7 2 431755e 02 1 057987e 02 V R 1 773332e 01 Ohm L 1 320940e 07 H For visualization purposes in a final linear step we compute the solution when all driving voltages Vo act together MaterialSpec Coil Parameter Current0 Z For i To nWind 1 SOL 2 i 0 domain Wire i 0 sigma 2 PI x A mx 2
351. te element mesh and the chosen degree of the approximation for u X Other FEs may add other dofs to determine the displacement as for example rotational dofs for shell elements Here the numeri cal implementation of contact algorithms will be more complex and not so general so that it will be not considered here Penalty method One advantage of the penalty approach to contact mechanics is that it does not neces sarily require any modifications of the underlying finite element implementation and it is solely based on the specification of contact forces in form of tractions By com paring the known stamp s position with the actual computed displacements as soon the impenetrability condition of both bodies is violated the penalty method adds stiff forces T in the direction of the rigid body normal n in order to restore the impene trability condition Actually this latter condition can only be satisfied approximately and the degree of approximation is controlled by a penalty parameter The dilemma is that the better the impenetrability approximation is the worse is the ill conditioness of the system to be solved and computing solutions will be slower and harder For a generic boundary point x X u with rigid body projection x9 let be the actual value of penetration defined by xo x n The penalty method defines the trac tions as T h d n with h 6 a stiff function having the properties h lt 0 0 and h 5 gt 0 gt 0
352. tem must be computed in a cheap way and one major drawback is that optimized and very effective preconditioners are problem de pendent and need to be investigated and found for each application But also within a single application the best known preconditioner may not be very robust and the convergence rate may heavily depend on material parameters and the shape of the elements both having a strong impact on the condition of the linear system In this example we are going to study preconditioning techniques available for solv ing the harmonic eddy current problem both with nodal and edge elements This model is obtained from the Maxwell s equations by dropping the displacement cur rent term by using the time dependency e t e e complex field values the linear material laws J cE and B uH and is given by VB 0 VxE iwB 0 V xp B Jo oE 2 69 with the electric conductivity u the permeability w the frequency and Jo a possible stiff driving current From the property V B 0 we can define the vector potential A with B V x A From the second equation in 2 69 we obtain the integrability condition V x E iwA 0 for the scalar potential Y with VY E iwA For a piecewise constant conductivity one can show that it makes sense to assume a globally zero scalar potential Y 0 so that one is left with the single linear equation Vxp iV x A iwcA Jo 2 70 to be solved for the complex vector potential A Sinc
353. tes where the negative sign prescribes the inward direction of the heat flux The temper ature sensors have been defined as boundary conditions of Float ing type BC SensUp nnx1 1 nytot IType nnx2 nnx1 2 Floating Temp 0 W BC SensDown nnx5 1 nytot IType nnx6 nnx5 2 Floating Temp 0 W and are visible in the graphics window The temperature and the thermal flux power at these boundaries will be exported for postprocessing In the command section we first initialize the velocity and we specify the temperature field as the only dof field to be calculated since the velocity field is prescribed and it is stored in the user element field veloc Store veloc v0x 1 4 yxy h xh 0 Constant Heating Power Mode For designing a realistic flow sensor the geometry is optimized such that the sensor characteristics are as desired for the application of interest In particular the relation between the flow rate and the sensor temperatures for a given heater power range needs to be analyzed This relation will then be used for calibration in practice First let us consider the situation when the fluid is not flowing for reference For a non zero heating power and a zero flow one expects a symmetric temperature field as shown in the left corner of Fig 2 124 Steady state temperature fields for increasing flow rate are shown in the other examples of Fig 2 124 The difference in the thermal conduc tivity is apparent in the contour lines trav
354. the acquired image reveals more or less structure of the surface topography The simplest simulation scan is performed in constant height mode The shape of the tip can be 54 SESES Tutorial September 2012 Tip Sample Interaction Tip Sample Interaction 2 5e 07 l i d 125 d 166 z 2e 07 4250 gt a ge ee gt S 5 1 5e 07 a 2 2 texO7 oO De a a 5e 06 t 0 1 1 1 1 1 1 L 0 50 100 150 200 250 300 350 400 position nm Figure 2 30 Plot of the calculated tip sample interaction using exported 2D simulation Figure 2 29 Simulation of the image acqui gt 7 data Contour lines are drawn in the xry sition artefact caused by non ideal tip shape aa with d the width of the tip projection plane varied by the t shape parameter For a small t shape parameter the tip is sharp and the resulting field correspondingly high Thus if the electric field is taken as a measure for the substrate topography one gets a stronger apparent topography modulation This effect is shown in Fig 2 29 for three different t shape values As a next step we discuss the extension example SpmSine s3dof the above model to 3D in order to perform a 2D surface scan in constant height acquisition mode The 2D geometry has been extended to 3D in a simple way The shape of the tip is cho sen rotation symmetric around the z axis its radial shape is determined by a cosine function and its lateral xy position has been parameterized with two user
355. the averaged thermal lens by a parabolic fit The fit was performed over the extension from r 0 to r 1 5mm and the corre sponding results are shown in Fig 2 94 The dioptric power of the averaged thermal lens has a linear dependence from the thickness of the compensating disk The linear regression to this data leads to the optimal thickness of the compensating element of dopt 0 86 mm for a dioptric power of 0 In summary the thermo optically self compensated amplifier is a very promising de SESES Tutorial September 2012 131 Parameter Study Thermal Lensing Averaged thermal Lens Linear Averaged thermal Lens Dioptric Power 1 m ly 8 4707x 7 2539 R 0 9965 OPD mm eee 90000cces eee doceo T 5 0 r mm Figure 2 94 Dioptric power of the averaged Figure 2 93 OPD distributions as a function thermal lens as a function of the thickness of of the thickness of the compensating disk the compensating disk dots and the corre sponding linear fit solid line sign It overcomes the drawback of strong thermal lensing and takes benefit of the easy to manage and not costly technique of transversal diode pumping In addition the design rests easy scalable for higher output powers and is therefore very interest ing for industrial application References 1 TH GRAF E Wyss M ROTH H P
356. the initial guess system Q and with an additional initial strain eo V o V o 2 Id to be added to e and representing the deformation do Qo gt Q go Qo A possible solution approach for 2 124 consists in keeping the time derivative 0 Ot e and by computing a series of mechanical solutions with the plastic strain ep directly evaluated by the method of characteristics until a steady solution emerges The draw backs are that convergence to the steady solution is slow and evaluating the charac teristics works well for analytical functions not however for discretized ones expe cially discontinuous ones Here one needs some sort of artificial diffusion in upwind downwind direction and a careful analysis for a stable evaluation since one has also to integrate the plastic strain rates Another solution approach advocated here consists in defining the plastic strain as dof fields and solve the steady transport equations Vep v H with stabilized finite elements To find a solution of the non linear sys tem of equations 2 124 we can have a fully coupled algorithm solving all equations together or a uncoupled algorithm solving iteratively for the displacement u and the plastic strain p A priori one cannot say which method will be faster so we give all partial derivatives associated with material laws required for a coupled solution with optimal convergence rate In our example the body force f is a dead load and so the
357. the square pulse 1 0 lt lt 1 2 0 Z A Fourier series analysis of c leads to 1 2 C2 Hs and C3 sin 2 147 2 iT 2 and insertion of 2 147a and 2 147b into 2 144 gives the final result 2 9 h Dre word sin 2 costing 2 148 sin in calZ 9 5 2 8 z costing Fig 2 114 shows predictions of 2 148 for the concentrations along our straight chan nel For the parameter value K 0 1m t chosen here the initial square pulse is transformed into the uniform concentration of c4 0 5 during a distance of about x 2 0 m Since the local cross flow diffusion flux is proportional to the local concentration gradient the homogenization process is much stronger at distances closer to the inlets Model Specification The input file example DiffusionDuct s2dstarts with the statement Species AB defining the names for the considered species Next the geometry is defined QMEI nx 40 1x 2 0 nx QMEJ ny 21 ly 1 0 ny followed by the statements 172 SESES Tutorial September 2012 K 0 1 K D Vx ly 2 in 1 m density le3 kg m 3 x diffCoeff 1 0e 9 m x 2 s x velocity diffCoeff K ly 2 m s mmolA 1E 3 kg m 3 x mmolB 1E 3 kg m 3 x MaterialSpec m Fluid Equation CompressibleFlow TransportA TransportB Enable Parameter Density Val density kg m 3 Parameter Mmol A mmolA kg mol Parameter Mmol B mmolB kg mol Parameter Diff
358. thod is justified by arguing that heat transport happens on a time scale much longer than the fluidic mass flow Therefore we simply increase the magnitude of the velocity field at certain instants in time and calculate how the temperature field evolves and approaches a new steady state For instance in constant power mode we perform a calculation of the transient tem perature field for a series of flow rate steps The flow rate staircase and the correspond ing unstationary solution of the temperature field is carried out with the statements For i From 0 To nFlowSteps SESES Tutorial September 2012 185 1 0e 03 2 1 0e 03 1 8 8 0e 04 _ 1 6 8 0e 04 _ g 1 4 g g 6 000 04 E 412 6 0ce 04 E 5 2 y 1 2 4 0e 04 8 E08 4 0e 04 8 lt 2 lt 06 2 2 0e 04 04 2 06 04 02 0 0e 00 0 0 0e 00 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Time s Time s Figure 2 128 Dynamic response of the sen Figure 2 129 Dynamic response of the sen sor temperature difference AT gt _ red toa sor temperature difference A7 gt _ red to a series of velocity vo steps blue in constant series of velocity vo steps blue in constant power mode p 10 mW temperature mode ATy 10K Define v0 i nFlowSteps v0max Store veloc v0x 1 4 y y h h 0 Write AtStep 1 File pdyn txt Append S12 10 time ixtStop 12 10 SensDown Temp Shift Tamb 12 10 SensUp Temp Shift Tamb S12 10f integrate Bound Heater Tem
359. tic strain according to SirialMn 1 2u l If the step is elastic ftria lt 0 we just define the isotropic and linear elastic law 2 107 The geometry and BCs are defined according to the reference P oP En 1 En a Numerical Results Before computing a solution we must allocate the plastic strain field with the state ment 140 SESES Tutorial September 2012 Disp Norm m 7 0E 02 6 8E 02 6 SE 02 6 3E 02 6 0E 02 5 7E 02 5 5E 02 5 2E 02 5 0E 02 4 7E 02 4 5E 02 4 2E 02 4 0E 02 3 7E 02 3 5E 02 3 2E 02 3 0E 02 2 7E 02 2 5E 02 2 2E 02 2 0E 02 time 0 07 Yield 1 0E 00 9 5E 01 9 0E 01 8 5E 01 8 0E 01 7 5E 01 7 0E 01 6 5E 01 6 0E 01 5 5E 01 5 0E 01 4 5E 01 4 0E 01 3 5E 01 3 0E 01 2 5E 01 2 0E 01 1 5E 01 1 0E 01 5 0E 02 0 0E 00 time 0 07 Stress XY Pa 6E 04 4 3 9E 04 3 2E 04 2 5E 04 1 8E 04 1 0E 04 3 3E 03 3 8E 03 1 1E 04 1 8E 04 2 5E 04 3 2E 04 3 9E 04 4 7E 04 5 4E 04 6 1E 04 6 8E 04 7 SE 04 8 2E 04 8 9E 04 9 7E 04 time 0 07 Figure 2 97 Displacement shear stress and yield region for a vertical displacement of 0 007 m Store StrainP RestoreOnFailure zero otherwise its value will always be zero The BCs have been defined to be time depen dent and the solution is computed in a series of stationary steps The convergence of Newton s algorithm is optimal but differently from the reference we need far more steps to avoid d
360. ting the numerical solutions In general by transport problems with more than one species one solves for all species together in a single dof field block and here as first step one just drops the equation for the Na dof field However without any other contrivance the constraint 5 a 1 valid after initialization will be immediately violated after the first Newton s update Therefore at the end of each 216 SESES Tutorial September 2012 Newton s step we set up an off block dof field correction for N to restore this condi tion This is done with the Increment directive and the built in function set dof as follow Increment Standard If WithN2 setdof N2 1 Dof 02 Incr 02 Dof H2 Incr H2 Dof H20 Incr H20 Dof CO Incr CO Dof CO2 Incr CO2 3 The preprocessor switch If WithN2 is used to enable disable the direct computa tion of the N dof field within this example and the variable Wit hN2 is defined at the beginning of the input It is to be noted that in order to compute this new value we need both the values of the other dof fields and their block increments since dof field values within setdof are values before the block dof increments are subtracted to the dof fields Also such an amendment makes only sense if without it a solution is computed satisfying the constraint 5 a 1 Otherwise nothing will converge since the correction is actually inconsistent with the solution Assuming that New ton
361. tion EddyF reeHarmonicNodal not solving for the electric potential Y The problem description is concluded by defining BCs To compute the electric potential we define for each wire a point BC to fix and ground the potential and a section where we apply a potential jump This is the most neutral method to apply a driving volt age since it does not define any equipotential surface within the wire At the exterior boundary OQ we define the Dirichlet BC A 0 representing an infinitely extended domain The command section solves several linear problems in order to compute the impedan ce matrix 2 56 and the low frequency approximation 2 59 Coefficients of the linear system matrix 2 68 which are known in advance to be always zero are not defined 94 SESES Tutorial September 2012 within the sparse matrix format used by the linear solver However the zeroness of many matrix coefficients actually depend on the user specifications in which case they are always defined independently from their numerical value In our case the ma terial AirEddy has a zero conductivity o 0 so that the off diagonal block matrix B of 2 68 has many zero entries This default behavior is important for non linear problems since the sparse format is build once at the beginning of the non linear block solution and here it is inconvenient to make the format dependent on the zeroness of the coefficients in the first iteration which may not be zero in later iterations
362. tioner Pet ILU 0 with Prignt Id On a copy of the system matrix S one performs the LU factorization by discarding the fill in i e if a coefficient of the spare matrix S is zero it will be zero also in the factorized form S LU ILU 0 One knows that for an M matrix the incomplete LU factorization cannot fail 1 i e we will never get a zero pivot but the discretization of PDEs does not always yield M matrices Since the amount of fill in in the LU factorization is dependent on the ordering of the equations permutation of the matrix can have a strong impact on the effectiveness of the ILU 0 preconditioner In our example for nodal elements the system matrix is not an M matrix but we are lucky and the ILU 0 preconditioner works quite well for first order elements with good convergence rates for reasonable slenderness factors and frequencies but only if the real and imaginary dofs of a complex dofs are clustered together in the global ordering of the dofs Without this clustering the ILU 0 preconditioner easily fails and this is also the case when starting e g from an initial Cuthill McKee ordering and by widely and randomly permuting the dofs but by keeping the pairs together In summary for first order nodal elements we have found a robust preconditioner it is very good in reducing the convergence s dependency on the frequency and we just have a noticeable dependency from the slenderness factor see Table 2 1 The conver gence r
363. total force acting on the lamellas times the displacement and so ow F 2 24 T 2 24 SESES Tutorial September 2012 63 0 9 i 7 La 0 E ALES UTR Sos oo A 0 6 et E i x d L 0 4 i P i03 e 2 0 1 0 0 200 400 600 800 1000 H Field A m Figure 2 42 Saturation of the magnetic field B for large Figure 2 41 Ring coil with air gap values of the H field can be interpreted as the component of this force along the displacement This method requires to compute two solutions for small different values of d and obtains the force by numerical difference of the magnetic energy If the magnetic energy density E B is too complex to compute there is an alternative based on computing the B and H field only Taking the derivative of 2 22 2 23 we obtain AW fo H B ABdV fo E B Adv 2 25 Jo H B ABdV A fojo 23H BAV 2 26 Since the parameter d changes the domain s shape we have to consider also the pos sible change in the integration domain formally denoted by AdV It seems that still we need to know the energy density E B however we may well assume that the ferromagnetic material just undergoes rigid body deformations without mechanical strain and therefore AdV 0 on the lamella s domain Qz Just in vacuum 9 Qz there is a change in the domain but there we have E B H B 2 The second method is based on Maxwell s stress tensor it is more general and all
364. tric and so the matrices G and Z as well The matrix G has a block diagonal format and if this is also the case for 0 f then Z is block diagonal too By letting Z Z be the two diagonal blocks of Z and since 0 oq n n 0 we arrive at OTn 0 f Z2 0 f Tod Noe te 2 116 OE trial o 3 f Z 3 f T af Zala f For our strain driven formulation we assume that at the beginning of a time step tn 1 the data is given Fn 1 fn C51 and one looks for Fn41 En 1 Ora How ever we know from objectivity arguments that material laws can only depends from right Cauchy strain C FT F and therefore a formulation working with the data Crntl En Ch must be equivalent The point here is that the strain C or E C Id 2 is always given when evaluating material laws not necessarily the deforma tion gradient F since numerically the symmetrized strain forms may not be directly obtained from F 0 0X but instead from internal variables The eigen prob lem F C21 FT n rn is equivalent to C 1 FTF F n F n or C C N A N Indeed the eigenvalues and eigenvectors N is all what is needed to evaluate any isotropic plastic laws but now they are determined by a non symmetric eigen problems By considering that C is positive definite the nu merical work is at most doubled One can compute C by spectral decomposition by running a first symmetric eigen problem and then use C to evaluate the second symmetr
365. tted in Fig 2 76 The figure indicates that the focal length of the averaged thermal lens has a hyperbolic dependence on the pump power of the form fav Fsp Ppump with Fip the specific focal length For this example the specific focal length is Fsp 615mmW References 1 W KOECHNER Solid State Laser Engineering Springer Verlag 5 Ed pp 407 468 1999 2 TH HALLER H U SCHWARZENBACH G STEINER Finite Element Model Based Product Development with NM SESES Orell Fiissli Verlag AG pp 71 102 2002 3 J EICHLER H J EICHLER Laser Springer Verlag 374 Ed 1998 4 W KOECHNER Absorbed Pump Power Thermal Profile and Stresses in a cw Pumped Nd YAG Crystal Appl Opt Vol 9 No 6 pp 1429 1434 1970 5 R WEBER B NEUENSCHWANDER M Mac DONALD M B Roos H P WE BER Cooling Schemes for Longitudinally Diode Laser Pumped Nd YAG Rods IEEE J Quantum Electron Vol 34 No 6 pp 1046 1053 1998 6 D C BROWN Nonlinear Thermal and Stress Effects and Scaling Behavior of YAG Slab Amplifiers IEEE J Quantum Electron Vol 34 No 12 pp 2393 2402 1998 2 18 Thin Disk Laser Most diode pumped solid state lasers DPSSL suffer from strong thermal lensing ef fects especially at high pump powers The strong thermal lens is mainly induced by high radial temperature gradients normal to the optical axis in the laser crystal A very efficient way to overcome this drawback is offered by the thin disk laser concept
366. uc 1mm We vary the thickness of the compensating disk in 0 1mm steps from 0 6 mm up to 1 3mm The absorbed pump power per length amounts 10 W mm which corresponds to total pump power of 400 W Fig 2 92 shows the temperature distribu tion in the center part of the device for thicknesses of the compensating disk of 0 6 mm and 1 3mm We can see that especially in the outer parts of the rod the tempera ture distribution of the laser rods is not exactly transferred into the compensating disk and decreases in the disk This deviation becomes more pronounced for thicker disks From this we can expect that the compensation of the thermal lens will not be optimal in the outer part of the device This assumption is confirmed by the OPD s as shown in Fig 2 93 The curvature of the OPD near the optical axis is concave for small thick nesses of the disk and becomes convex when the thickness is increased Therefore we have first a positive focal length of the thermal lens which becomes subsequent negative with increasing thickness of the disk For suitable thicknesses of the disk the OPD near the optical axis is almost constant and the thermal lens in this range therefore vanishes The OPD in the outer part of the device is not influenced by the compensating disk this is due to the fact the the outer part of the disk adopts the tem perature of the cooling water as we have previously seen in Fig 2 92 From the OPD data we can determine the focal length of
367. ues of the applied pressure on the roof i e a family of solutions parame terized by the pressure representing our simulation parameter We assume longitudi nal invariance of the mechanical structure so that beam elements can be used and we perform the analysis in a small strains but large displacements framework This re sults in a non linear problem that must be solved for each applied load By zero load 32 SESES Tutorial September 2012 we have zero mechanical displacements and by increasing the load the roof starts to bend inwards By further increasing the load we reach the snap through point char acterized by the fact that locally no other solution exists for higher values of the load If we want to find solutions with still larger displacements we have to reduce the load but we also have to avoid turning back on our solution path At this limit point the mechanical structure becomes instable and will snap through the next stable solution with the same load value We will not perform here a dynamic analysis but limit our self to compute static solutions although some of them may be unstable We will see how limit points are hard to pass by and present methods to successfully compute the load displacement curve Without discussing the details of non linear mechanical models not so important for an understanding of this example the input file defines the roof geometry material properties and models to enable a non linear mechanical an
368. unique To find the single positive solution of f x 0 we may use a simple Newton Raphson iteration but we have to avoid getting close to a possible local minimum d dx f x 0 for x gt 0 Initialization with zo 4 F a fot P gt aP and zo P otherwise always avoids this critical region and results in a robust monotone decreasing sequences 41 in f Xn d dx f xn with f p41 gt 0 Tny1 lt Tn and tea gt 0 For this analytical example we can define a unique projection for any point x x y z By computing the radial value r x y and depending on which region displayed in Fig 2 102 the point r z lies into the point r z is orthogo nally projected onto the radial shape either trivially for the line segments or with the algorithm presented above for the two parabola segments The projection is al ways unique but is C just for points close enough to the contact surface Once the radial projection has been computed we additionally need to provide the or thonormal triads of tangent vectors and normal as well as the main curvatures By parametrizing the surface as s r rcos rsin g r with g r the radial shape of Fig 2 102 the two unnormalized tangent vectors are given by t 0 8 cos sin 0 g and t2 ps rsin rcos 0 The normal is given by n t x t2 t x te cos 0 g sin 0 g 1 1 0 92 The tangent vec tors are orthogonal and since 0 0ys n
369. upled among each other If yes decide on the solution algorithms e Calculation Launch the calculation and inspect the convergence and accuracy of the results For validation check if similar but simpler problems with analytical solutions exist e Postprocessing Post process the numerical results for data extraction and visual ization e Optimization Modify the problem specification in order to find a design leading to improved device performances This solution approach is reflected in the structure of the input file Namely the initial section is structured in parts as follows 10 SESES Tutorial September 2012 e Parameter definition Parameters can be defined for use in the material laws and geometry definitions e Material definition Materials are defined by specifying material laws and param eters that are responsible for the physical properties e Geometry definition The problem geometry and dimension is specified e Material mapping Materials are assigned to local or global domains of the defined geometry e Boundary condition BC definition The boundary conditions for the problem domain need to be specified which basically consider the interactions with the outside world i e where no modeling is performed e Minimum Refinement Level The refinement level selects the number of refine ment steps starting from the initially coarse mesh A minimum refinement level can be set The command section contains the f
370. urrents is done in the same way hence we quickly review this matter and mainly present the thermal problem To solve the eddy current problem we assume an harmonic excitation and a linear system therefore allowing us to work with complex variables and fields all with an implicit time dependency of e t with w the harmonic frequency By the rotational symmetry we use cylinder coordinates p z and the problem consists in computing the azimuthal component A of the vector potential A given a driving electric field E The current is then given by jg o Eg o E4 E with g 0 Ag iwAg the induced electric field and the electric conductivity By symmetry the other vector components of A E j are all zero For a coil with N winding by neglecting their elic ity we consider separately the N rings and for each ring we assume a constant voltage on each radial cross section Hence for generic applied voltages V along the whole cir cumference with i 1 N the wire index approximately on each wire cross section we have the driving electric field E V 27p These fields induce currents and the wire currents are given by evaluating the integrals I Jo j dA over the wire SESES Tutorial September 2012 73 cross sections 2 However a consistent solution is first given when there is the same current in each wire thus constraining the possible values of V Such a solution can be computed by first solving N problems with j
371. urve using 1 3 title y displacement with lines A PostScript file named curveplot ps is then created by running the command gnuplot script gnu Here the second and third column of the data file curve is plotted against the first column The resulting graph is shown in Fig 1 21 SESES allows to specify the content of a command line to be executed in line with the numerical computations so that it is possible to launch postprocessing tasks as soon as the numerical data is available The command line can extend over multiple lines with the help of the backslash escape character To launch the previous plotting command within SESES one uses the statement System gnuplot script gnu Plotting with Mathematica Mathematica is a mathematical software by Wolfram Research Inc useful for linear algebra analysis graphical visualization and programming The content of our data file curve can be read with the ReadList command and stored in a Mathematica variable of the same name and plotted with the following statements file OpenRead curve Skip file String SESES Tutorial September 2012 37 curve ReadList file Number RecordLists gt True Close file ListPlot Map 1 2 amp curve Frame gt True PlotJoined gt True FrameLabel gt current A displacement m PlotLabel gt Cantilever Microactuator PlotStyle gt Hue 1 GridLines gt Automatic DefaultFont gt Helvetica The Skip comm
372. us flow the shape of this initial domain Qo is generally straight but there are molds specially devised to give the strand an initial constant curvature Knowing the stress free referential system Qo we can push back the spatial formula tion and start computing solutions by applying all mechanical loads present in the casting process to obtain the deformation Qo gt Q Qo or equivalently the shape of the strand Depending on the layout of the casting machine and es pecially for strands with initial curvature the referential domain Qo and the spatial one Q may not be close and large deformations will result This fact complicates the numerical analysis a lot since a geometric non linear pushed back formulation 160 SESES Tutorial September 2012 of the governing equation 2 124 needs to be used However with the assumption of small strains the formulation can be simplified by considering that the several rolls of the casting machine set sharp bounds on the possible shape of the strand In other words from the geometrical layout of the rolls we can make a good guess Q on the final shape of the strand and we can assume 2 Q The deformation 9 u now consist of a known contribution o from the stress free referential system to this initial guess system 9 Qo gt 91 9 Qo and an unknown but small displacement u Q gt Q x u Q1 Therefore we are left to solve the same equations of 2 124 now in
373. utorial September 2012 79 the divergence operator of 2 41 and this shortcoming will be even more pronounced for non linear problems with u u H For computing this analytical solution the obvious result is to use the total formulation however in practice seldomly we have Jo 0 on all Q and a reduced formulation is therefore required The solution to this dilemma is given by combining both the reduced and total formulations and by noting that in empty space and generally also in conductors we have Hre 1 whereas for the magnetic materials with urea gt gt 1 we generally have Jo 0 However the reduced formulation with 4ye 1 is not at all critical and yields stable numerical results so that we can continue to use it just within the magnetic material we have to use the total formulation thus avoiding the before mentioned problem related to numerical cancellation In order to proceed we split our domain in two parts Q Qrea U Qtot assume Jo 0 on Qio and first compute the reduced magnetic potential O q in all Q Within Qt and because Jo 0 the Ho field is a gradient field and so locally there exists a potential field O with Ho VOo If we additionally assume the domain iot to be simply connected or has simply connected components Nf with Oot U Mot we can write a f Hod 2 43 xxi with x x 0 and y x x any path on 1 joining x to the fixed point xi In summary we can write V UV Orea HHo V
374. velocities within the free flow region Instead of giving their functional forms the velocity data is read from external files using the FromData statement In this example the data has been generated from CFX 5 simulations of a developing laminar flow The so defined routines u veloc and w_veloc linearly interpolate between the given discrete data points In the next section various op erational parameters are defined reference temperature and pressure as well as the mole fractions of all species at the air and burning gas inlets respectively The next section defines global and material parameters As first we define the production rates for the burning reaction of hydrogen H2 O2 gt 2H20 with the help of several macros following a common scheme used to define chemical reactions This model relies on a phenomenological kinetics formula for the species reaction rates IT PHP b Ig kMaVa Til TO TiO exp ref Ah TAS RT yo with k the reaction rate Va the stoichiometric coefficient for a Ma the species molar mass Ah the reaction enthalpy AS the reaction entropy Prep a reference pressure parameter and R the universal gas constant Next the global definitions GlobalSpec Model AxiSymmetric Enable Parameter AmbientTemp TO K are used to employ the rotational symmetric model description as well as the ambient temperature As next we define the material parameters and models specifying the physical chemical model of the free f
375. vely We start our problem specification by writing the initial section of input file which has the same structure as explained in the previous example Although the syntax statements may follow in arbitrary order we will consistently use the sequence introduced in the earlier section about the modeling strategy Since we will not define parameters for this example the first statement is the definition of the material between the tip and the sample surface namely air MaterialSpec Air Equation ElectroStatic Parameter EpsIso 1 We name our material Air and specify it as a dielectric material with an isotropic dielectric permittivity EpsIso of value 1 as in vacuum and correspondingly enable the electrostatic equation computing the potential field Phi The unknown Phi is called a degree of freedom field in short a dof field Next we define a 2D domain consisting of a mesh with 2 x 2 macro elements with the statements QMEIT 2 1E 6 QMEJ 2 1E 6 The QMEI and QMEJ keywords are followed by the number of elements in the rect angular mesh in x and y direction respectively and the size of each mesh element in these directions The syntax is QMEI n dx with n being the number of elements and dx the size of each element Frequently one defines the total length parameter xlen and substitutes x len n for dx Moreover the 14 SESES Tutorial September 2012 rectangular mesh can be transformed to other geometries with the help of math
376. x9 which is represented by the linear kinematic constraint n x xo 0 2 120 Since the projection values n and xo will generally change at each iteration step in order to compute correct derivatives we also require from the closest point projection the surface characterization at xo in form of two orthonormal tangent vectors tq ta 1 a 1 2 with n t x t2 and the two main curvature coefficients ka with respect to the tangent vectors An exact algebraic or inexact penalty method are two possible choices to enforce these linear kinematic constraints The algebraic method is more complex to implement than the penalty one however the penalty method is ill conditioned in its solution is far less precise than the one of the exact approach Both problems can be elegantly studied within a continuum formulation without yet considering any numerical dis SESES Tutorial September 2012 151 cretization of the governing equations see 1 2 However for simplicity and suc cinctness we will assume the governing equations have already been discretized and we are looking for methods applying the contact conditions directly on the discretized form Many discretizations of the governing equations of elasticity use pure displace ment Lagrange FEs where at some selected sampling points X i 1 N three degree of freedoms u u X R determine the displacement u X at X In turn the set of sampling points is determined by the fini
377. y a 3D model is derived with only minor corrections from the 2D one For our cylindrical stick we keep the x axis parallel to the stick and choose the z axis perpendicular to x and y By use of the cylinder homotopy routine we can thus define the 3D stick geometry and the resulting geometry is illustrated in Fig 2 20 QMEI nx xlen nx QMEJ ny ylen ny QMEK nz ny zlen ylen ny Coord cylx coord ylen 2 zlen 2 1 1E 2 1 With the heat flux now chosen as 60 W m we calculate the stationary distribution shown in Fig 2 21 The heated spot on the stick is pointing upward in this figure With the refined 3D model we expect the data to be reproduced more accurately which is confirmed in the comparison of Fig 2 22 The curvature of the temperature profile is now matched more closely to the measured data Note that the temperature drops at distances exceeding 102 5 cm since the stick is heated at this location As for the tran sient behavior the 3D model gives comparable results as the 2D model see Fig 2 23 With the 3D model the temperature decay is matched somewhat closer The last simulation aspect to be discussed here is the thermal expansion resulting from the heat distribution calculated above For this purpose the thermal expansion coef ficient Alphalso and the displacement field Disp needs to be defined and enabled respectively The following statement inserted in the command section defining the calculation sequence is also required
378. y the chemical composition of the melt but the whole casting process has a strong impact on the quality of the steel As first one has to avoid the breakout of the melt due to a broken shell at the exit of the mold then a key point is the minimization of the residual stresses in the strand due to thermal strains and mechanical contacts by the rolls which may lead to cracks and so to a poor steel qual ity Direct quality measurements in the production line are inherently difficult due to the high temperatures and the moving strand Therefore there is a major interest for a numerical simulation of the casting process which should give qualitative answers on temperatures and stresses of the strand We present here a numerical model for the continuous casting process from the exit of the mold up to the cut of the strand possessing the major ingredients of a full fea tured simulation In particular we use simplified material laws simple support for mechanical contacts and a 2D modeling domain but otherwise the model is pretty complete As first we note that within or after the mold the melt in the strand can be a turbulent fluid due to the presence of magnetic stirrers However if we abstain ourself from such complex situations and if the strand does not get in mechanical resonance the continuous casting of steel is a quasi stationary or steady process with the shape temperature and stresses of the strand at a given spatial point being constant in time
379. yLin Store BfieldBack Bfield WeightBack VolWeight Solve Stationary ForSimPar LDisp At LDisp LEps Write Bexternal e n LDisp e um n BExt LDisp ForceStressO E N Accuracy E n Force0 2 ZeroCc0 2 ForceStressl SE N Accuracy E n Forcel 2 ZeroC1 2 ForceEnergy0 E N n 1E6 Energy0 integrate MagnEnergy LEps ForceEnergyl E N n ForceZ traverse MagnDeltaEnergy return 1E6 ForceZ LEps ForceEnergy2 E N n ForceZ Energyl integrate Domain material Vacuum MagnEnergyLin t traverse Domain material Vacuum MagnDeltaEnergyMat return 1E6 ForceZ LEps The accuracy of the stress tensor methods can be approximately obtained by inte grating the divergence free integrand over the vacuum surface SurfaceCheck The results for a single run are as follows ForceStressO 1 073258E 06 N Accuracy 7 838263E 08 ForceStressl 1 402041E 06 N Accuracy 6 623098E 08 ForceEnergyO 1 432143E 06 N ForceEnergyl 1 432908E 06 N ForceEnergy2 1 428980E 06 N The most inaccurate method is generally the first one whereas the third fourth and fifth methods are in general the most accurate and stable ones they all should give the same results up to some numerical cancellation The discrepancy here is caused by the fact the our deformation is not actually a truly rigid body deformation since some lamella s elements are actually str
380. ying FE software will then integrate these tractions over the contact boundary and add the contributions to the dof residual equations and map the traction derivatives to dof derivatives This latter task will be more complex if the FE discretization does not solely use displacement dofs For FEs with just nodal displacement dofs the above relations can be directly used to amend the nodal values at X Here the tractions T T x with x X u will be added to the right hand side vector and the derivatives OT Ou to the matrix of the linearized system to be solved The two meth ods yield slight different numerical results which however can be compensated by a little change of the penalty parameter To explain the ill conditioness of the penalty approach for simplicity let us consider the direct nodal approach for the boundary node X and associated three degree of freedom u with zero curvatures ka 0 If gt 0 in practice one adds the 3 x 3 matrix n n 0sh as diagonal subblock at the row and columns associated with u and h d n on the right hand side If the normal n has just a single non zero component e g n ez by just considering large values we have uj O5h h d This linear system with a single unknown is regular and its solution uis h d 05h dominates the global solution Here the problem is not ill conditioned at all since the penalty method is equivalent to setting a priori Dirichlet BCs for uix so that the pe
381. zation we shall next consider a simulation example of a lake with a dam The situation is depicted in Fig 1 2 and the input file is found at example Dam s2d For this problem we want to compute and visualize the mechanical stress and displacement in the 2D cross section of the dam SESES Tutorial September 2012 11 3 SESES 2D Dam s2d mi Blslale ajaj sjef gt lofe Ea Dam h o tf Figure 1 2 Schematic overview of the intro ductory example of a dam exposed to the wa ter pressure of a lake Figure 1 3 The Front End program showing the dam example example Dam s2d caused by the water pressure of the lake Upon opening the input file the Front End program will look like Fig 1 3 showing the mesh geometry of the problem with a legend of assigned materials on the lower left corner Viewing and Editing the Input We may want to inspect the content of the input file namely the problem description and commands for the solution algorithm by starting the text editor with the edit button has shown in Fig 1 4 Within the editor syntax highlighting causes syntax key words to appear in blue comments delimited by x and in violet and the remaining text in black The syntax examples in this tutorial however are reproduced without syntax highlighting The input file of our first example is a container file starting with cat gt Seses lt lt EOF ending with Finish EOF an

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