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1. Ui x 1 0 U2 U3 T Table A 4 Vertex coordinates of the reference quadrilateral 67 FFC User Manual Anders Logg 0 1 0 0 1 0 Figure A 3 The reference quadrilateral A 4 The reference tetrahedron The reference tetrahedron is shown in Figure A 4 and is defined by its four vertices with coordinates as specified in Table A 5 Vertex Coordinate vo x 0 0 0 v z 0 0 v x 0 1 0 Ua 0 0 1 Table A 5 Vertex coordinates of the reference tetrahedron 68 FFC User Manual Anders Logg 0 0 1 0 0 0 1 0 0 Figure A 4 The reference tetrahedron Vertex Coordinate Vertex Coordinate Up x 0 0 0 v4 x 0 0 1 Ui x 1 0 0 Us x 1 0 1 V2 qc Ug z 1 11 V3 x 0 1 0 Ur g 0 1 1 Table A 6 Vertex coordinates of the reference hexahedron A 5 The reference hexahedron The reference hexahedron is shown in Figure A 5 and is defined by its eight vertices with coordinates as specified in Table A 6 69 FFC User Manual Anders Logg 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 Figure A 5 The reference hexahedron 70 Appendix B Numbering of mesh entities The numbering of mesh entities used in FFC follows the UFC specification 7 The UFC specification dictates a certain numbering of the vertices edges etc of the cells of a fi
2. h MeshSize shape Note that it is the responsibility of the user or the system for which the code is generated to map this function to a function coefficient that interpolates the mesh size onto piecewise constants FacetNormal The function FacetNormal is a predefined Function that may be used to represent the unit normals of mesh facets n FacetNormal shape Note that it is the responsibility of the user or the system for which the code is generated to map this function to a function coefficient that interpolates the facet normals onto vector valued piecewise constants 37 FFC User Manual Anders Logg 5 4 Scalar operators The basic operators used to define a form are scalar addition subtraction and multiplication Note the absence of division which is intentionally left out but is supplied for Functions see below 5 4 1 Scalar addition Scalar addition is supported for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types In addition unary plus is supported for all basic data types 5 4 2 Scalar subtraction Scalar subtraction is supported for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types In addition unary minus is supported for all basic data types 5 4 3 Scalar multiplication Scalar multiplication is supp
3. The JIT compiler caches generated modules such that if a Python script including a call to the JIT compiler is run twice in the same directory the Python module is only generated once The generated modules are stored in a cache directory defined by Instant To clean the cache run the command instant clean 24 Chapter 5 Form language FFC uses a flexible and extensible language to define and process multilinear forms In this chapter we discuss the details of this form language In the next section we present a number of examples to illustrate the use of the form language in applications 5 1 Overview FFC compiles a given multilinear form a Vi xV x xV oR 5 1 into code that can be used to compute the corresponding tensor A a j di aed Qi 5 2 In the form language a multilinear form is defined by first specifying the set of function spaces V V2 Vf and then expressing the multilinear form in terms of the basis functions of these function spaces A function space is defined in the form language through a FiniteElement and a corresponding basis function is represented as a BasisFunction The 25 FFC User Manual Anders Logg following code defines a pair of basis functions v and u for a first order La grange finite element on triangles element FiniteElement Lagrange triangle 1 v BasisFunction element u BasisFunction element The two basis functions can now be used to
4. 2 1 Downloading and installing FFC The latest version of FFC can be found on the FENICS web page http www fenics org The following commands illustrate the installation process assuming that you have downloaded release x y z of FFC tar zxfv ffc x y z tar gz cd ffc x y z sudo python setup py install 13 FFC User Manual Anders Logg Make sure that you download the latest release You may also need to install the Python packages FIAT and NumPy See Appendix C for detailed instructions 2 2 Compiling Poisson s equation with FFC The discrete variational finite element formulation of Poisson s equation Au f reads Find u V such that a v u L v Vv Vi 2 1 with Vi Vp a pair of suitable function spaces the test and trial spaces The bilinear form a V x V R is given by a v un f Vv Vu dz 2 2 Q and the linear form L V R is given by L v fota 2 3 To compile the pair of forms a L into code that can called to assemble the linear system Az b corresponding to the variational problem 2 1 for a pair of discrete function spaces specify the forms in a text file with extension form e g Poisson form as follows element FiniteElement Lagrange triangle 1 v TestFunction element u TrialFunction element f Function element a dot grad v grad u dx L v f dx 14 FFC User Manual And
5. FFC User Manual Anders Logg 5 3 11 Index The data type Index represents an index used for subscripting derivatives or taking components of vector valued functions If an Index is declared without any arguments i Index a free Index is created representing an index range determined by the con text if used to subscript a vector valued BasisFunction or a Function the range is given by the number of vector dimensions n and if used to subscript a derivative the range is given by the dimension d of the underlying shape of the finite element space As we shall see below indices can be a powerful tool when used to define forms in tensor notation An Index can also be fixed meaning that the value of the index remains constant i Index 0 5 3 12 Built ins FFC declares a set of built in variables and constructors for convenience as outlined below Predefined indices FFC automatically declares a sequence of free indices for convenience i j k 1 m n Note however that a user is free to declare new indices with other names or even reuse these variables for other things than indices 36 FFC User Manual Anders Logg Identity The data type Identity represents an n x n unit matrix of given size n An Identity is declared by specifying the dimension n I Identity n MeshSize The function MeshSize is a predefined Function that may be used to repre sent the size of the mesh
6. UL V2 V3 U4 U5 V6 U7 80 Appendix C Installation The source code of FFC is portable and should work on any system with a standard Python installation Questions bug reports and patches concerning the installation should be directed to the FFC mailing list at the address ffc devOfenics org FFC must currently be installed directly from source but Debian Ubuntu packages will be available in the future for FFC and other FENICS compo nents C 1 Installing from source C 1 1 Dependencies and requirements FFC depends on a number of libraries that need to be installed on your system These libraries include FIAT and the Python NumPy module In addition you need to have a working Python installation on your system 8l FFC User Manual Anders Logg Installing Python FFC is developed for Python 2 5 but should also work with Python 2 3 and 2 4 To check which version of Python you have installed issue the command python V python V Python 2 5 1 If Python is not installed on your system it can be downloaded from http www python org Follow the installation instructions for Python given on the Python web page For Debian Ubuntu users the package to install is named python Installing NumPy In addition to Python itself FFC depends on the Python package NumPy which is used by FFC to process multidimensional arrays tensors Python NumPy can be dow
7. the default location for user installed Python packages usually something like usr lib python2 5 site packages In addition the compiler exe cutable ffc a Python script will be installed in the default directory for user installed Python scripts usually in usr bin To see a list of optional parameters to the installation script type python setup py install help If you don t have root access to the system you are using you can pass the home option to the installation script to install FFC in your home directory mkdir local python setup py install home local This installs the FFC package in the directory local lib python and the FFC executable in local bin If you use this option make sure to set the environment variable PYTHONPATH to local lib python and to add local bin to the PATH environment variable C 1 4 Compiling the demos To test your installation of FFC enter the subdirectory src demo and com pile some of the demonstration forms With FFC installed on your system just type 84 FFC User Manual Anders Logg ffc Poisson form to compile the bilinear and linear forms for Poisson s equation This will generate a C header file called Poisson h containing UFC code that can be used to assemble the linear system for Poisson s equation It is also possible to compile the forms in src demo without needing to install FFC on your system In
8. the values of v Similarly if iis an Index free or fixed then v i represents a function corresponding to component i of the values of v 39 FFC User Manual Anders Logg 5 5 2 Inner product dot v w The operator dot accepts as arguments two logically vector valued or matrix valued expressions In the case of two vector valued epxressions it returns the inner product dot product of the two vectors n 1 dot v w o v w vw 5 5 i 0 Note that this operator is only defined for vectors of equal length For two matrix valued expressions it returns the Frobenius inner product m 1mn 1 i 0 j 0 5 5 3 Vector product cross v w The operator cross accepts as arguments two logically vector valued expres sions and returns a vector which is the cross product vector product of the two vectors cross v w DUXW vw U311 V2W0 UYWa VOWI1 vwo 5 7 Note that this operator is only defined for vectors of length three 5 5 4 Matrix product mult v w The operator mult accepts as arguments two matrices or more generally tensors and returns the matrix tensor product 5 5 5 Transpose transp v The operator transp accepts as argument a matrix and returns the transpose of the given matrix transp v i j e v Jij vy 5 8 40 FFC User Manual Anders Logg 5 5 6 Trace trace v The operator trace accepts as argument a square matrix v and returns its trace that is the sum o
9. using selling offering for 105 FFC User Manual Anders Logg sale or importing the Program or any portion of it 11 Patents A contributor is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based The work thus licensed is called the contributor s contributor version A contributor s essential patent claims are all patent claims owned or controlled by the contributor whether already acquired or hereafter acquired that would be infringed by some manner permitted by this License of making using or selling its contributor version but do not include claims that would be infringed only as a consequence of further modification of the contributor version For purposes of this definition control includes the right to grant patent sublicenses in a manner consistent with the requirements of this License Each contributor grants you a non exclusive worldwide royalty free patent license under the contributor s essential patent claims to make use sell offer for sale import and otherwise run modify and propagate the contents of its contributor version In the following three paragraphs a patent license is any express agreement or commitment however denominated not to enforce a patent such as an express permission to practice a patent or covenant not to sue for patent infringement To grant such a patent license to a party means to make such an ag
10. where form_representation is a list that holds a particular internal repre sentation of each input form The form representation depends on the chosen representation mode Accessing this data is mainly intended for developers 4 1 3 Compiling finite elements The compile function may also be used to compile finite elements directly without associated forms The following example demonstrates how to generate code for a fifth degree Lagrange finite element on tetrahedra from ffc import element FiniteElement Lagrange tetrahedron 5 compile element P5 4 2 Just in time JIT compiler jit The jit function expects a single form as its first argument It also accepts up to three additional optional arguments jit form representation language options However instead of generating code the jit function returns the compiled form as a Python object It does this by generating code compiling it by calling the C compiler and wrapping it as a Python module by calling Instant SWIG The jit function returns a tuple 23 FFC User Manual Anders Logg compiled_form compiled_module form_data where compiled_formis the compiled form a Python wrapper for ufc form compiled_module is a Python module containing the compiled form finite elements dof maps etc a Python wrapper for the complete set of generated code and form data is form metadata generated from the input form
11. 9 4 Combining operators 22 024 48 0 10 Ingex DOLO ouod dex Re o9 ii dicis 48 5 11 User defined operators len 49 Examples 51 Dl Themas DADO 2E dar ras FAA 51 C2 Poisons equation uuu orn EX we a es 52 6 3 Vector valued Poisson a aoaaa or REOR E G 53 6 4 The strain strain term of linear elasticity 53 6 5 The nonlinear term of Navier Stokes 54 6 6 The heat equation cens 55 6 7 Mixed formulation of Stokes uso eR 56 6 8 Mixed formulation of Poisson lt 2 lt 2 2 57 6 9 Poisson s equation with DG elements 58 6 10 Quadrature Clemente co nh ees kon m 9 iaa EES Reference cells A 1 The reference interval o rice ta roras ea k 0000 eee A 2 The reference triangle 20 0 000 4 A 3 The reference quadrilateral A A The reference tetrahedron 00004 A 5 The reference hexahedron o Numbering of mesh entities bli Basic concepts eu n cw a oum as ere B 2 Numbering of vertices o B 3 Numbering of other mesh entities B 3 1 B 3 2 Relative orderne Suo y we wk a a D oane G LIMAS ies ei ew Rhee we ERS B 4 Numbering schemes for reference cells B 4 1 B 4 2 B 4 3 B 4 4 B 4 5 Numbering of mesh entities on intervals Numbering of mesh entities on triangular cells
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13. Numbering of mesh entities on quadrilateral cells Numbering of mesh entities on tetrahedral cells Numbering of mesh entities on hexahedral cells 65 66 66 67 68 69 71 71 72 73 75 TT 78 78 78 79 79 80 C Installation Col Installing Io BOUPOB o ea eh dra de C 1 1 Dependencies and requirements C 1 2 Downloading the source code Calar DO FFC lt lt cos es RR eS Ee ESSE Od CLA Compiling the demo lt cce ecc Oe hx RD es C 1 5 Verifying the generated code C 2 Debian Ubuntu package o o D Contributing code D 1 Creating bundles patches o o D 1 1 Creating a Mercurial hg bundle D 1 2 Creating a standard diff patch file D 2 Sending bundles patches o D 3 Applying changes D 3 1 Applying a Mercurial bundle 2 D 3 2 Applying a standard patch file D 4 License agreement E License 81 81 81 83 84 84 85 85 87 87 87 89 90 91 91 91 92 95 About this manual Since this manual was written FFC has moved to using the UFL form language for expression of variational forms As a consequence this manual does not accurately describe the form language used by FFC For information about the UFL form lan guage refer to the UFL manual Intended audience Thi
14. Q Note that an enriched element is not a finite element in the Ciarlet sense in particular the basis is not a nodal basis 5 3 5 QuadratureElement The data type QuadratureElement is different from the FiniteElement in the sense that it represents discrete function values rather than a finite ele ment space i e it can be used to define functions that do not come from a finite element space Apart from this the class QuadratureElement can be used in the same way as the class FiniteElement with the only difference that it is not possible to take derivatives of a function which is defined on a QuadratureElement A QuadratureElement is declared by specifying the underlying shape and the number of values integration points in each dimension 3l FFC User Manual Anders Logg element QuadratureElement shape num_points The argument shape is a string and possible values include e interval representing an interval in R e triangle representing a triangle in R e tetrahedron representing a tetrahedron in R3 The argument num points is an integer specifying the number of values in each dimension Thus the following element element QuadratureElement triangle 3 returns an element with 3 x 3 values on a triangle The coordinates of these values coincide with the coordinates of the 3 x 3 integration scheme returned by FIAT This element is useful in cases where q
15. QuadratureElement in the other forms Typical values of the relative residual for each Newton iteration for all 3 approaches are shown in Table It is noted that the convergence rate is quadratic as it should be for all 3 methods Iteration REF1 FE1 QE1 1 6 342e 02 6 342e 02 6 342e 02 2 5 305e 04 5 305e 04 5 305e 04 3 3 699e 08 3 699e 08 3 699e 08 4 2 925e 16 2 925e 16 2 475e 16 Table 6 1 Relative residuals for each approach for linear elements However if quadratic elements are used to interpolate the unknown field u the order of all elements in the above forms is increased by 1 This influ ences the convergence rate as seen in Table Clearly using the standard 62 FFC User Manual Anders Logg FiniteElement leads to a poor convergence whereas the QuadratureElement still leads to quadratic convergence Iteration REF2 FE2 QE2 1 2 637e 01 3 910e 01 2 644e 01 2 1 052e 02 4 573e 02 1 050e 02 3 1 159e 05 1 072e 02 1 551e 05 4 1 081e 11 7 221e 04 9 076e 09 63 Table 6 2 Relative residuals for each approach for quadratic elements Appendix A Reference cells The definition of reference cells used in FFC follows the UFC specification 7 The following five reference cells are covered by the UFC specification the reference interval the reference triangle the reference quadrilateral the ref erence tetrahed
16. Vu 7 h vuds Lo f ofast f vods The corresponding finite element variational problem for discontinuous first order elements may be implemented as follows 58 FFC User Manual Anders Logg DG1 FiniteElement Discontinuous Lagrange triangle 1 v TestFunction DG1 u TrialFunction DG1 Function DG1 Function DG1 FacetNormal triangle MeshSize triangle aD BO h ll a dot grad v grad u dx dot avg grad v jump u n dS dot jump v n avg grad u dS A alpha h dot jump v n jump u n dS dot grad v jump u n ds dot jump v n grad u ds gamma h v u ds L v f dx v g ds This example is implemented in the file PoissonDG form in the collection of demonstration forms included with the FFC source distribution 6 10 Quadrature elements We consider here a nonlinear version of the Poisson s equation to illustrate the main difference between the FiniteElement and QuadratureElement The strong equation looks as follows V 1 u Vu f 6 23 The linearised bilinear and linear forms for this equation ou fo ua Vu Vuda ove Vuodz 6 24 Lw fosas Jo u2 Vu Vuo dz 6 25 99 FFC User Manual Anders Logg can be implemented in a single form file as follows NonlinearPoisson form element FiniteElement Lagrange triangle 1 v TestFunction element u TrialFunction element u0 Function el
17. agree on the orientation of incident subsimplices Thus two incident triangles will agree on the orientation of the common edge and two incident tetrahedra will agree on the orientation of the common edge s and the orientation of the common face if any This is illustrated in Figure B 5 for two incident triangles sharing a common edge 75 FFC User Manual Anders Logg U3 0 v2 Vi Figure B 4 Mesh entities are ordered based on a lexicographical ordering of the corresponding ordered tuples of non incident vertices The first edge ey is non incident to vertices vo and v Figure B 5 Two incident triangles will always agree on the orientation of the common edge 76 FFC User Manual Anders Logg B 3 2 Limitations The UFC specification is only concerned with the ordering of mesh entities with respect to entities of larger topological dimension In other words the UFC specification is only concerned with the ordering of incidence relations of the class d d where d gt d For example the UFC specification is not concerned with the ordering of incidence relations of the class 0 1 that is the ordering of edges incident to vertices TT FFC User Manual Anders Logg B 4 Numbering schemes for reference cells The numbering scheme is demonstrated below for cells isomorphic to each of the five reference cells B 4 1 Numbering of mesh entities on intervals Entity Incident vertices Non inc
18. defined as the cartesian product of the function spaces of a given list of ele ments A MixedElement is declared by specifying a list of FiniteElements mixed_element MixedElement e0 el Alternatively a MixedElement can be created as the product of a pair of FiniteElements The following example illustrates how to create a Taylor Hood element quadratic velocity and linear pressure P2 VectorElement Lagrange triangle 2 P1 FiniteElement Lagrange triangle 1 TH P2 P1 Elements may be mixed at arbitrary depth so mixed elements can be used as building blocks for creating new mixed elements In fact a VectorElement just provides a simple means to create mixed elements Thus a Taylor Hood element may also be created as follows Note that multiplying more than two elements will create a nested mixed element For example e eO el e2 will correspond to e MixedElement MixedElement eO e1 e21 30 FFC User Manual Anders Logg P2 FiniteElement Lagrange triangle 2 P1 FiniteElement Lagrange triangle 1 TH P2 P2 P1 5 3 4 EnrichedElement The data type EnrichedElement represents the vector sum of two or more finite elements Example The Mini element can be constructed as P1 VectorElement Lagrange triangle 1 B VectorElement Bubble triangle 3 Q FiniteElement Lagrange triangle 1 Mini Pi B
19. functions TestFunctions and TrialFunctions can be used to construct tuples of TestFunctions and TrialFunctions as illustrated here for a mixed Taylor Hood element TestFunctions TH TrialFunctions TH v q u p 5 3 8 Function The data type Function represents a function belonging to a given finite element space that is a linear combination of basis functions of the fi 33 FFC User Manual Anders Logg nite element space A Function must be declared for a previously declared FiniteElement f Function element Note that more than one function can be declared for the same FiniteEle ment The following example declares two BasisFunctions and two Functions for the same FiniteElement v BasisFunction element u BasisFunction element f Function element g Function element Function is used to represent user defined functions including right hand sides variable coefficients and stabilization terms FFC treats each Function as a linear combination of basis functions with unknown coefficients It is the responsibility of the user or the system for which the form is compiled to supply the values of the coefficients at run time In the case of DOLFIN the coefficients are automatically computed from a given user defined function during the assembly of a form In the notation of the UFC interface Functions are referred to as coefficients Note that the order in which Fun
20. multilinear forms We assume that dx has been declared as an integral over the interior of Q and that both i and j have been declared as a free Index The examples presented below can all be found in the subdirectory src demo of the FFC source tree together with numerous other examples 6 1 The mass matrix As a first example consider the bilinear form corresponding to a mass matrix aliai Jonas 6 1 which can be implemented in FFC as follows element FiniteElement Lagrange triangle 1 TestFunction element TrialFunction element V u 51 FFC User Manual Anders Logg a v u dx This example is implemented in the file Mass formin the collection of demon stration forms included with the FFC source distribution 6 2 Poisson s equation The bilinear and linear forms form for Poisson s equation 2e Sg 2 VL I ve Vudz 6 2 Q v f dz 6 3 Q m 2N e NS I can be implemented as follows element FiniteElement Lagrange triangle 1 v TestFunction element u TrialFunction element f Function element a dot grad v grad u dx L v f dx Alternatively index notation can be used to express the scalar product a D v i D u i dx This example is implemented in the file Poisson form in the collection of demonstration forms included with the FFC source distribution 52 FFC User Manual Anders Logg 6 3 Vector valued P
21. revision control system used by FFC Follow the procedure described below to create your bundle 87 FFC User Manual Anders Logg 1 Clone the FFC repository hg clone http www fenics org hg ffc 2 If your contribution consists of new files add them to the correct loca tion in the FFC directory tree Enter the FFC directory and add these files to the local repository by typing hg add lt files gt where lt files gt is the list of new files You do not have to take any action for previously existing files which have been modified Do not add temporary or binary files 3 Enter the FFC directory and commit your contribution hg commit m lt description gt where lt description gt is a short description of what your patch accom plishes 4 Create the bundle hg bundle ffc lt identifier gt lt date gt hg http www fenics org hg ffc written as one line where lt identifier gt is a keyword that can be used to identify the bundle as coming from you your username last name first name a nickname etc and lt date gt is today s date in the format yyyy mm dd The bundle now exists as ffc lt identifier gt lt date gt hg When you add your contribution at point 2 make sure that only the files that you want to share are present by typing hg status 88 FFC User Manual Anders Logg This will produce a list of files Those marked
22. runs or a compiler used to produce the work or an object code interpreter used to run it The Corresponding Source for a work in object code form means all the source code needed to generate install and for an executable work run the object code and to modify the work including scripts to control those activities However it does not include the work s System Libraries or general purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work For example Corresponding Source includes interface definition files associated with source files for the work and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require such as by intimate data communication or control flow between those subprograms and other parts of the work The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source The Corresponding Source for a work in source code form is that same work 98 FFC User Manual Anders Logg 2 Basic Permissions A11 rights granted under this License are granted for the term of copyright on the Program and are irrevocable provided the stated conditions are met This License explicitly affirms your unlimited permission to run the unmodified Program The output from running a covered work is covered by this
23. v lt zi Il 5 10 Index notation FFC supports index notation which is often a convenient way to express forms The basic principle of index notation is that summation is implicit over indices repeated twice in each term of an expression The following 48 FFC User Manual Anders Logg examples illustrate the index notation assuming that each of the variables i and j have been declared as a free Index n 1 v il w i e vw 5 20 i 0 d 1 Div 1 xD w i e 3 SEDE W Vu 5 21 V Ov D v i i e 2 V 0 5 22 ip Ai n 1l d 1 Bau D v il D w il j e LZ 5 23 2 2 s Ox Index notation is used internally by FFC to represent multilinear forms and FFC will try to simplify forms by replacing sums with index expressions 5 11 User defined operators A user may define new operators using standard Python syntax As an example consider the strain rate operator e of linear elasticity defined by e v Vo Vv 5 24 This operator can be implemented as a function using the Python def key word def epsilon v return 0 5 grad v transp grad v Alternatively using the shorthand lambda notation the strain operator may be defined as follows epsilon lambda v 0 5 grad v transp grad v 49 Chapter 6 Examples The following examples illustrate basic usage of the form language for the definition of a collection of standard
24. with a question mark are not tracked by Mercurial You can track them by using the add command as shown above Once you have added these files their status changes form to A D 1 2 Creating a standard diff patch file The tool used to create a patch is called diff and the tool used to apply the patch is called patch Here s an example of how it works Start from the latest release of FFC which we here assume is release x y z You then have a directory structure under ffc x y z where you have made modifications to some files which you think could be useful to other users 1 Clean up your modified directory structure to remove temporary and binary files which will be rebuilt anyway make clean 2 From the parent directory rename the FFC directory to something else mv ffc x y z ffc x y z mod 3 Unpack the version of FFC that you started from tar zxfv ffc x y z tar gz 4 You should now have two FFC directory structures in your current directory ls ffc Xx y z ffc x y z mod 5 Now use the diff tool to create the patch 89 FFC User Manual Anders Logg diff u new file recursive ffc x y z ffc x y z mod gt ffc lt identifier gt lt date gt patch written as one line where lt identifier gt is a keyword that can be used to identify the patch as coming from you your username last name first name a nickname etc and lt date gt is tod
25. wm x n 38 56 123 Scalar subtraction S e e cas eee SRE RRR 38 5 4 3 Scalar multiplication lt 1 6 ss zb 38 5 4 4 Sealar division 0 0 20 00 ee 39 5 9 5 6 5 7 5 8 Vector Operators o e oa o FG coh DEES ROO 39 5 5 1 Component access v i 2 2 99 9 x es 39 50 2 Inner product dot v W coccion rs 40 Soo Vector product crosety W e cr ner xe 40 5 5 4 Matrix product mult v w 2 24252 24824 40 5 5 5 Transpose transp v aa aaau aaa aan 40 So Trace tracey 224450844244 22245 o 41 mo Vector length len y 26 2446644405844 41 900 Rank Pankey soos ewe Re ae EE o 41 5 5 9 Vectorization vec v o 41 Differential operators pra e A 42 5 6 1 Scalar partial derivative D v i 42 5 6 2 Gradient grad iv is 664 444 24 Se GeO Aw 42 D63 Divergence diviv ee uo 9 o 8s 4566845 43 DOA CUE svi uode o RD RA 43 los MD 43 5 7 1 Cell integrals 4d 2 2 2229 x mox 43 5 7 2 Exterior facet integrals ds 44 5 5 9 Interior facet integrales d5 2 2 222222 44 5 7 4 Integrals over subsets lll rss 45 DO operators IE 45 5 8 1 Restriction v and vO 45 58 2 Jump UN casos era bk age SERS 46 58 3 Averaper AVECV sce bd esri e o9 ee 47 5 9 Special operators 222r o o m o 3o RR ss 4T 59l Tnyerse T y omis om EI 47 59 2 Modulus modulus v 2 4 2 2 2 Sek x 48 ee DOUATCLDON EE eoo ko Ry ook o et 48 5
26. 7 This requirement modifies the requirement in section 4 to keep intact all notices c You must license the entire work as a whole under this License to anyone who comes into possession of a copy This License will therefore apply along with any applicable section 7 additional terms to the whole of the work and all its parts regardless of how they are packaged This License gives no permission to license the work in any other way but it does not invalidate such permission if you have separately received it d If the work has interactive user interfaces each must display Appropriate Legal Notices however if the Program has interactive interfaces that do not display Appropriate Legal Notices your work need not make them do so A compilation of a covered work with other separate and independent works which are not by their nature extensions of the covered work and which are not combined with it such as to form a larger program in or on a volume of a storage or distribution medium is called an aggregate if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation s users beyond what the individual works permit Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate 100 FFC User Manual Anders Logg 6 Conveying Non Source Forms You may convey a covered work in object code form under the terms of s
27. FFC User Manual March 22 2010 Anders Logg www fenics org Visit http www fenics org for the latest version of this manual Send comments and suggestions to ffc dev fenics org Contents About this manual 9 1 Introduction 11 2 Quickstart 13 2 1 Downloading and metalungFFC 2 29 9 13 2 2 Compiling Poisson s equation with FFC 14 3 Command line interface 17 4 Python interface 21 Al Compiling forme compile scenar 2 4822 28442224 22 ALI Input arguments uuo no bee eM eee e 22 11 2 Output arguens 2222542 e mone poeyi 3 9 ek 22 4 1 3 Compiling finite elements 23 42 Just in time JIT compile jit 2 252024 0 ns 23 3 5 Form language 25 eto o0 e as eh PR e a ir 25 5 2 The form language as a Python extension 2T Do dose datatype ers ea etr A ee a Eee esc as 28 DOLI PiniteElement as ica aa 28 2 4 VectorElement 2 o x EERE DS 29 5 3 3 MixedElement 2222 3 bo x oko x 9d 30 5 3 4 EnrichedElement cisco 31 5 3 9 QuadratureElement 31 S0 BasisPnDBCe blUM lt 22 629249 be eo Ele ee we 32 5 3 7 TestFunction and TrialFunction 2 60 amp 25 33 Do DUIMCUTOR ou ceeded sche dE ae AAA ER 33 D39 DODSDHNR acs ex ad ee Roe RS ESE ADRS od 30 et Vectot onsbtanb s ew mo pe bee ek Ee A 30 edd SHOR eos ros Cheb eee ne ee ee A 36 E eu uox woe ck Se gs RD Yes 36 5 4 Scalar operators 2 o Seda esas ent y ORC 38 5 4 1 Scalar addition ico cnc
28. HE PROGRAM AS PERMITTED ABOVE BE LIABLE TO YOU FOR DAMAGES INCLUDING ANY GENERAL SPECIAL INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES 17 Interpretation of Sections 15 and 16 If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new program and you want it to be of the greatest possible use to the public the best way to achieve this is to make it free software which everyone can redistribute and change under these terms To do so attach the following notices to the program It is safest to attach them to the start of each source file to most effectively State the exclusion of warranty and each file should have at least the copyright line and a pointer to where the full notice is found one line to give the p
29. actions infringe copyright if you do not accept this License Therefore by modifying or propagating a covered work you indicate your acceptance of this License to do so 10 Automatic Licensing of Downstream Recipients Each time you convey a covered work the recipient automatically receives a license from the original licensors to run modify and propagate that work subject to this License You are not responsible for enforcing compliance by third parties with this License An entity transaction is a transaction transferring control of an organization or substantially all assets of one or subdividing an organization or merging organizations If propagation of a covered work results from an entity transaction each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party s predecessor in interest had or could give under the previous paragraph plus a right to possession of the Corresponding Source of the work from the predecessor in interest if the predecessor has it or can get it with reasonable efforts You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License For example you may not impose a license fee royalty or other charge for exercise of rights granted under this License and you may not initiate litigation including a cross claim or counterclaim in a lawsuit alleging that any patent claim is infringed by making
30. ation of Poisson s equation as a pair of first order equations for c H div and u La o Vu 0 6 17 Veo f 6 18 We multiply the two equations by a pair of test functions 7 and w and integrate by parts to obtain the following variational problem Find o u V H div x L such that a r w c u L 7 w V r w V 6 19 where a r w c u fro rutu ode 6 20 Li rou INES 6 21 We may implement the corresponding forms in the FFC form language using first order BDM H div conforming elements for and piecewise constant Ls conforming elements for u as follows 97 FFC User Manual Anders Logg BDM1 DGO FiniteElement Brezzi Douglas Marini triangle 1 FiniteElement Discontinuous Lagrange triangle 0 element BDM1 DGO tau w TestFunctions element sigma u TrialFunctions element f Function DGO dot tau sigma div tau u w div sigma dx L w f dx p Il This example is implemented in the file MixedPoisson form in the collection of demonstration forms included with the FFC source distribution 6 9 Poisson s equation with DG elements We consider again Poisson s equation but now in an interior penalty dis continuous Galerkin formulation Find u V L such that a v u L v Vv V where a v u f Vv Vudzx Q TY f Vv u s v Vu o h v s un dS 2 6 22 I Vo ula o
31. ay s date in the format yyyy mm dd 6 The patch now exists as ffc lt identifier gt lt date gt patch and can be distributed to other people who already have ffc x y z to easily create your modified version If the patch is large compressing it with for example gzip is advisable gzip ffc lt identifier gt lt date gt patch D 2 Sending bundles patches Patch and bundle files should be sent to the FFC mailing list at the address ffc devOfenics org Include a short description of what your patch bundle accomplishes Small patches bundles have a better chance of being accepted so if you are making a major contribution please consider breaking your changes up into several small self contained patches bundles if possible 90 FFC User Manual Anders Logg D 3 Applying changes D 3 1 Applying a Mercurial bundle You have received a patch in the form of a Mercurial bundle The following procedure shows how to apply the patch to your version of FFC 1 Before applying the patch you can check its content by entering the FFC directory and typing hg incoming p bundle lt path gt ffc lt identifier gt lt date gt hg written as one line where lt path gt is the path to the bundle lt path gt can be omitted if the bundle is in the FFC directory The option p can be omitted if you are only interested in a short summary of the changesets found in the bundle 2 To apply the pa
32. bal vertex numbers In particular a tuple of increasing local vertex numbers corresponds to a tuple of increasing global vertex numbers This is illustrated in Figure B 1 for a mesh consisting of two triangles For non simplicial cells quadrilaterals and hexahedra the numbering is arbitrary as long as each cell is isomorphic to the corresponding reference cell by matching each vertex with the corresponding vertex in the reference cell This is illustrated in Figure B 2 for a mesh consisting of two quadrilaterals 72 FFC User Manual Anders Logg 1 Figure B 1 The vertices of a simplicial mesh are numbered locally based on the corresponding global vertex numbers B 3 Numbering of other mesh entities When the vertices have been numbered the remaining mesh entities are num bered within each topological dimension based on a lexicographical ordering of the corresponding ordered tuples of non incident vertices As an illustration consider the numbering of edges the mesh entities of topological dimension one on the reference triangle in Figure B 3 To number the edges of the reference triangle we identify for each edge the corresponding non incident vertices For each edge there is only one such vertex the vertex opposite to the edge We thus identify the three edges in the reference triangle with the tuples vo v1 and v2 The first of these is edge eo between vertices v and va opposite to vertex vo the second is edge e bet
33. ce you may not convey it at all For example if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program 13 Use with the GNU Affero General Public License Notwithstanding any other provision of this License you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work and to convey the resulting work The terms of this License will continue to apply to the part which is the covered work 107 FFC User Manual Anders Logg but the special requirements of the GNU Affero General Public License section 13 concerning interaction through a network will apply to the combination as such 14 Revised Versions of this License The Free Software Foundation may publish revised and or new versions of the GNU General Public License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns Each version is given a distinguishing version number If the Program specifies that a certain numbered version of the GNU General Public License or any later version applies to it you have the option of following the terms and conditions either of that numbered version or of any
34. code However the improvements depends on the GCC compiler options as well as the characteristics of the variational form f precompute_ip_const Like the precompute_basis_const option with the only difference that code will be generated to compute terms which are constant in the loops involving the integration points only f quadrature_degree n Will generate a quadrature rule accurate up to degree n regardless of the polynomial degree of the form This option is only valid for UFL forms and the specified degree will apply to ALL terms of the given form for which no degree has been specified through metadata As default FFC will determine the degree automatically from the form f split Generate separate files for declarations and the implementation 0 optimize Generate optimized code with a lower operation count compared to non optimized code for the assembly of the local element tensor This will in general increase the run time performance of the code If the representation see r option is tensor then FFC will use FErari optimizations However this option is currently ignored This option requires FErari and should be used with caution since it may be very costly at compile time for other than simple forms If the representation is quadrature the compile time increase tends to be much less drastic compared to FErari for very complex forms o directory output directory directory Specify the directory where the g
35. ction Ov Ox i 5 10 D v i e Alternatively the member function dx can be used For v an expression the two expressions D v i and v dx i are equivalent but note that only the operator D works on vector valued expressions that are defined in terms of Python lists 5 6 2 Gradient grad v The operator grad accepts as argument an expression v and returns its gra dient If v is scalar the result is a vector containing the partial derivatives in the coordinate directions v Ov Ov xo Ori Oxa 1 l grad v gt grad v Vv 5 11 If v is logically vector valued the result is a matrix with rows given by the gradients of each component grad o ilj gt grad v Vo 2 5 12 J Ox Thus if v is scalar valued then grad grad v returns the Hessian of v and if v is vector valued then grad v is the Jacobian of v 42 FFC User Manual Anders Logg 5 6 3 Divergence div v The operator div accepts as argument a logically vector valued expression and returns its divergence dd div v o divv V v Ovi i 0 Ox i 5 13 Note that the length n of the vector v must be equal to the dimension d of the underlying shape of the FiniteElement defining the function space for v 5 6 4 Curl curl v The operator curl accepts as argument a logically vector valued expression and returns its curl 1 n Veo ee hoa PE v A dd a Ox Ox Oxo Oxo An Note that this operator is only
36. ctions are declared is important The code generated by FFC accepts as arguments a list of functions that should correspond to the Functions appearing in the form in the order they have been declared For a MixedElement the function Functions can be used to construct tuples of Functions as illustrated here for a mixed Taylor Hood element f g Functions TH 34 FFC User Manual Anders Logg 5 3 9 Constant The data type Constant represents a constant scalar value that is unknown at compile time A Constant is declared for a given cell shape interval triangle or tetrahedron c Constant shape Constants are automatically replaced by discontinuous piecewise constant Functions The following two declarations are thus equivalent DGO FiniteElement Discontinuous Lagrange triangle 0 cO el Constant triangle Function DGO 5 3 10 VectorConstant The data type VectorConstant represents a constant vector value that is unknown at compile time A VectorConstant is declared for a given cell shape interval triangle or tetrahedron c VectorConstant shape VectorConstants are automatically replaced by discontinuous vector valued piecewise constant Functions The following two declarations are thus equiv alent DGO VectorElement Discontinuous Lagrange triangle 0 c0 VectorConstant triangle c1 Function DGO 35
37. d Before creating the patch please update the author and date information of the file s you have modified according to the following example 92 FFC User Manual Anders Logg __author__ Anders Logg logg simula no __date__ 2004 11 17 2005 09 09 __copyright__ Copyright C 2004 2005 Anders Logg __license__ GNU GPL Version 3 or any later version Modified by Foo Bar 2007 As a rule of thumb the original author of a file holds the copyright 93 Appendix E License FFC is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 3 of the License or at your option any later version The GNU GPL is included verbatim below GNU GENERAL PUBLIC LICENSE Version 3 29 June 2007 Copyright C 2007 Free Software Foundation Inc lt http fsf org gt Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed Preamble The GNU General Public License is a free copyleft license for software and other kinds of works The licenses for most software and other practical works are designed to take away your freedom to share and change the works By contrast the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program to make sure it remains free software for all its users W
38. define a bilinear form a v D u 0 dx corresponding to the mathematical notation a v u v dz 5 3 Note that the order of the argument list of the multilinear form is deter mined by the order in which basis functions are declared not by the order in which they appear in the form Thus both a v D u 0 dx and a D u 0 v dx define the same multilinear form The arity number of arguments of a multilinear form is determined by the number of basis functions appearing in the definition of the form Thus a v u dx defines a bilinear form namely a v u f vudz whereas L v dx defines a linear form namely L v f v da In the case of a bilinear form the first of the two basis functions is referred to as the test function and the second is referred to as the trial function One may optionally use the keywords TestFunction and TrialFunction to specify the test and trial functions This has the advantage that the order of specification of the two functions does not matter the test function will always be the first argument of a bilinear form and correspond to a row in the corresponding assembled matrix Thus the example above may optionally be specified as follows 26 FFC User Manual Anders Logg element FiniteElement Lagrange triangle 1 v TestFunction element u TrialFunction element Not every expression is a valid multilinear form The following list explains some of th
39. defined for vectors of length three Alternatively the name rot can be used for this operator 5 7 Integrals Each term of a valid form expression must be a scalar valued expression integrated exactly once Integrals are expressed through multiplication with a measure representing either an integral over the interior of the domain Q cell integral the boundary OQ of Q exterior facet integral or the set of interior facets interior facet integral 5 7 1 Cell integrals dx A measure for integration over the interior of is created as follows 43 FFC User Manual Anders Logg dx Integral cell For convenience FFC automatically declares the measure dx which can be used to define cell integrals If v is a scalar valued expression then the integral of v over the interior of 2 is written as v dx 5 7 2 Exterior facet integrals ds A measure for integration over the boundary of 2 is created as follows ds Integral exterior facet For convenience FFC automatically declares the measure ds which can be used to define cell integrals If v is a scalar valued expression then the integral of v over the boundary of 2 is written as v ds 5 7 3 Interior facet integrals dS A measure for integration over the set of interior facets of is created as follows dS Integral interior facet For convenience FFC automatically declares the measure dS which can be used to define ce
40. e input ufl DESCRIPTION Compile multilinear forms into efficient low level code The FEniCS Form Compiler FFC accepts as input one or more files each specifying one or more multilinear forms and compiles the given forms into efficent low level code for automatic assembly of the tensors rep resenting the multilinear forms In particular FFC compiles a pair of bilinear and linear forms defining a variational problem into code that can be used to efficiently assemble the corresponding linear system By default FFC generates code according to the UFC specification ver sion 1 0 Unified Form assembly Code see http www fenics org but 17 FFC User Manual Anders Logg OPTIONS this can be controlled by specifying a different output language option 1 It is also possible to add new output languages to FFC For a full description of FFC including a specification of the form language used to define the multilinear forms see the FFC user manual available on the FEniCS web page http www fenics org h help Display help text and exit V version Display version number and exit d debuglevel debug debuglevel Specify debug level default is O s silent Silent mode no output is printed same as debuglevel 1 l language language language Specify output language one of ufc default or dolfin UFC with a small layer of DOLFIN specific bindings r representation representatio
41. e the Free Software Foundation use the GNU General Public License for most of our software it applies also to 95 FFC User Manual Anders Logg any other work released this way by its authors You can apply it to your programs too When we speak of free software we are referring to freedom not price Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software and charge for them if you wish that you receive source code or can get it if you want it that you can change the software or use pieces of it in new free programs and that you know you can do these things To protect your rights we need to prevent others from denying you these rights or asking you to surrender the rights Therefore you have certain responsibilities if you distribute copies of the software or if you modify it responsibilities to respect the freedom of others For example if you distribute copies of such a program whether gratis or for a fee you must pass on to the recipients the same freedoms that you received You must make sure that they too receive or can get the source code And you must show them these terms so they know their rights Developers that use the GNU GPL protect your rights with two steps 1 assert copyright on the software and 2 offer you this License giving you legal permission to copy distribute and or modify it For the developers and authors protection t
42. e basic rules that must be obeyed in the definition of a form e A form must be linear in each of its arguments otherwise it is not a multilinear form Thus a v v u dx is not a valid form since it is quadratic in v e The value of a form must be a scalar Thus if v is a vector valued basis function see below then L v dx is not a valid form since the value of the form is not a scalar e The integrand of a form must be integrated exactly once Thus neither a v u nor a v u dx dx are valid forms 5 2 The form language as a Python extension The FFC form language is built on top of Python This is true both when calling FFC as a compiler from the command line or when calling the FFC compiler from within a Python program Through the addition of a collection of basic data types and operators FFC allows a form to be specified in a language that is close to the mathematical notation Since the form language is built on top of Python any Python code is valid in the definition of a form but not all Python code defines a multilinear form In particular comments lines starting with and functions keyword def see Section 5 11 below are allowed in the definition of a form 27 FFC User Manual Anders Logg 5 3 Basic data types 5 3 1 FiniteElement The data type FiniteElement represents a finite element on an interval triangle or tetrahedron A FiniteElement is declared by specifying the finite element family the underlyi
43. eclared and used in the definition of a form The following example declares two elements one linear and one quadratic Lagrange finite element P1 P2 FiniteElement Lagrange tetrahedron 1 FiniteElement Lagrange tetrahedron 2 5 3 2 VectorElement The data type VectorElement represents a vector valued element Vector valued elements may be created by repeating any finite element scalar vector valued or mixed a given number of times The following code demon strates how to create a vector valued cubic Lagrange element on a triangle element VectorElement Lagrange triangle 3 This will create a vector valued Lagrange element with two components If the number of components is not specified it will automatically be chosen to be the equal to the cell dimension Optionally one may also specify the number of vector components directly element VectorElement Lagrange triangle 3 5 29 FFC User Manual Anders Logg Note that vector valued elements may be created from any given element type Thus one may create a nested vector valued element with four com ponents where each pair of components is a first degree BDM element as follows element VectorElement BDM triangle 1 2 5 3 3 MixedElement The data type MixedElement represents a mixed finite element on an inter val triangle or tetrahedron The function space of a mixed finite element is
44. ections 4 and 5 provided that you also convey the machine readable Corresponding Source under the terms of this License in one of these ways a Convey the object code in or embodied in a physical product including a physical distribution medium accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange b Convey the object code in or embodied in a physical product including a physical distribution medium accompanied by a written offer valid for at least three years and valid for as long as you offer spare parts or customer support for that product model to give anyone who possesses the object code either 1 a copy of the Corresponding Source for all the software in the product that is covered by this License on a durable physical medium customarily used for software interchange for a price no more than your reasonable cost of physically performing this conveying of source or 2 access to copy the Corresponding Source from a network server at no charge c Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source This alternative is allowed only occasionally and noncommercially and only if you received the object code with such an offer in accord with subsection 6b d Convey the object code by offering access from a designated place gratis or for a charge and offer equivalent access to the Corresponding S
45. ed then avg v will also be vector valued 5 9 Special operators FFC provides a set of special operators for taking the inverse absolute value and square root of an expression These operators are interpreted in a special way and should be used with care Firstly the operators are only valid on monomial expressions that is expressions that consist of only one term Secondly the operators are applied directly to the coefficients of the basis function expansion of the expression on which the operators are applied Thus if v 5 vii then op v is evaluated by op v 2 op vj 5 19 5 9 1 Inverse 1 v The inverse of a monomial expression for example a product of one or more functions may be evaluated in the sense described above as follows w 1 v 47 FFC User Manual Anders Logg 5 9 2 Modulus modulus v The modulus i e the absolute value of a monomial expression for example a product of one or more functions may be evaluated in the sense described above as follows w modulus v 5 9 3 Square root sqrt v The square root of a monomial expression for example a product of one or more functions may be evaluated in the sense described above as follows w sqrt v 5 9 4 Combining operators The special operators may applied successively and repeatedly on any mono mial expression Thus the following expression is valid Function element sqrt abs 1
46. ement f Function element a 1 u0x u0 dot grad v grad u dx 2 uO u dot grad v grad u0 dx L v f dx 1 u0 u0 dot grad v grad u0 dx Here uo represents the solution from the previous Newton Raphson iteration The above form will be denoted REF1 and serve as our reference implemen tation for linear elements A similar form REF2 using quadratic elements will serve as a reference for quadratic elements Now assume that we want to treat the quantities C 1 u and ey 14 u2 V ug as given functions to be computed elsewhere Substituting into bilinear linear forms we obtain a v u ove Vudz f 2v Vug dz 6 26 f v f dz Vv 09 dz 6 277 o Q Then two additional forms are created to compute the tangent C and the gradient of uy This situation shows up in plasticity and other problems where certain quantities need to be computed elsewhere in user defined functions The 3 forms using the standard FiniteElement linear elements can then be implemented as tS t c cea Il FE1NonlinearPoisson form element FiniteElement Lagrange triangle 1 DG FiniteEFlement Discontinuous Lagrange triangle 0 60 FFC User Manual Anders Logg sig VectorElement Discontinuous Lagrange triangle 0 V TestFunction element TrialFunction element u0 Function element C Function DG sig0 Function sig f Function eleme
47. enerated files should be written to The default output directory is the current directory q rule quadrature rule rule Specify the quadrature rule that should be used when integrating the forms This will affect both tensor and quadrature representation Currently no quadrature rules has been implemented so the default from FIAT will be used BUGS Send comments questions bug reports etc to ffc dev fenics org Written by Anders Logg logg simula no with help from Kristian lgaard Marie Rognes Garth N Wells and many others FFC 1 19 Chapter 4 Python interface FFC provides a Python interface in the form of a standard Python module The following example demonstrates how to define and compile the varia tional problem for Poisson s equation in a Python script from ffc import element FiniteElement Lagrange triangle 1 v TestFunction element u TrialFunction element f Function element a dot grad v grad u dx L v f dx compile a L Poisson At the basic level the only difference between the command line interface and the Python interface is that one must add the import statement of the FFC module and that the function compile must be called when using the Python interface 21 FFC User Manual Anders Logg 4 1 Compiling forms compile The compile function expects a form see Section 5 or a list of forms as its first argument It also accepts up to fou
48. ers Logg The example is given for piecewise linear finite elements in two dimensions but other choices are available including arbitrary order Lagrange elements in one two and three dimensions To compile the pair of forms implemented in the file Poisson form call the compiler on the command line as follows ffc Poisson form This will generate the file Poisson h containing low level C code in the UFC Unified Form assembly Code format The generated code can be used by any UFC based assembler such as DOLFIN to assemble the discrete representations the matrix A and vector b of the bilinear form a and linear form L of Poisson s equation Note that by adding the flag 1 dolfin additional DOLFIN specific wrap pers are added to the generated code which simplifies the use of the generated code with DOLFIN In particular the handling of forms depending on coef ficients like f in Poisson s equation is simplified For further help on the ffc command and available command line options refer to the FFC man page man ffc 15 Chapter 3 Command line interface The command line interface of FFC is documented by the FFC man page man ffc A copy of this documentation is included below for convenience NAME FFC the FEniCS Form Compiler SYNOPSIS ffc h v d debuglevel s 1 language r representation f option 0 o output directory q quadrature rul
49. es not include within the scope of its coverage prohibits the exercise of or is conditioned on the non exercise of one or more of the rights that are specifically granted under this License You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software under which you make payment to the third party based on the extent of your activity of conveying the work and under which the third party grants to any of the parties who would receive the covered work from you a discriminatory patent license a in connection with copies of the covered work conveyed by you or copies made from those copies or b primarily for and in connection with specific products or compilations that contain the covered work unless you entered into that arrangement or that patent license was granted prior to 28 March 2007 Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law 12 No Surrender of Others Freedom If conditions are imposed on you whether by court order agreement or otherwise that contradict the conditions of this License they do not excuse you from the conditions of this License If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations then as a consequen
50. f its diagonal elements n 1 trace v e trace v vs 5 9 i 0 5 5 7 Vector length len v The operator len accepts as argument a logically vector valued expression and returns its length the number of vector components 5 5 8 Rank rank v The operator rank returns the rank of the given argument The rank of an expression is defined as the number of times the operator can be applied to the expression before a scalar is obtained Thus the rank of a scalar is zero the rank of a vector is one and the rank of a matrix is two 5 5 9 Vectorization vec v The operator vec is used to create a Python list object from a logically vector valued expression This operator has no effect on expressions which are already lists Thus if vis a vector valued BasisFunction then vec v returns a list of the components of v This can be used to define forms in terms of standard Python list operators or Python NumPy array operators The operator vec does not have to be used if the form is defined only in terms of the basic operators of the form language 41 FFC User Manual Anders Logg 5 6 Differential operators 5 6 1 Scalar partial derivative D v i The basic differential operator is the scalar partial derivative D This dif ferential operator accepts as arguments a scalar or logically vector valued expression v together with a coordinate direction i and returns the partial derivative of the expression in the given coordinate dire
51. he GPL clearly explains that there is no warranty for this free software For both users and authors sake the GPL requires that modified versions be marked as changed so that their problems will not be attributed erroneously to authors of previous versions Some devices are designed to deny users access to install or run modified versions of the software inside them although the manufacturer can do so This is fundamentally incompatible with the aim of protecting users freedom to change the software The systematic pattern of such abuse occurs in the area of products for individuals to use which is precisely where it is most unacceptable Therefore we have designed this version of the GPL to prohibit the practice for those products If such problems arise substantially in other domains we stand ready to extend this provision to those domains in future versions of the GPL as needed to protect the freedom of users Finally every program is threatened constantly by software patents States should not allow patents to restrict development and use of software on general purpose computers but in those that do we wish to 96 FFC User Manual Anders Logg avoid the special danger that patents applied to a free program could make it effectively proprietary To prevent this the GPL assures that patents cannot be used to render the program non free The precise terms and conditions for copying distribution and modification foll
52. iability to the recipient for any liability that these contractual assumptions directly impose on those licensors and authors A11 other non permissive additional terms are considered further restrictions within the meaning of section 10 If the Program as you received it or any part of it contains a notice stating that it is governed by this License along with a term that is a further restriction you may remove that term If a license document contains a further restriction but permits relicensing or conveying under this License you may add to a covered work material governed by the terms of that license document provided that the further restriction does not survive such relicensing or conveying If you add terms to a covered work in accord with this section you must place in the relevant source files a statement of the additional terms that apply to those files or a notice indicating where to find the applicable terms Additional terms permissive or non permissive may be stated in the form of a separately written license or stated as exceptions the above requirements apply either way 8 Termination You may not propagate or modify a covered work except as expressly provided under this License Any attempt otherwise to propagate or modify it is void and will automatically terminate your rights under this License including any patent licenses granted under the third paragraph of section 11 However if you cease all vi
53. ident vertices vo 0 0 vo vi vi 0 1 v1 vo Co 1 0 vo v1 0 B 4 2 Numbering of mesh entities on triangular cells Entity Incident vertices Non incident vertices vo 0 0 vo v1 V2 vi 0 1 v1 vo v2 va 0 2 v2 vo U1 eg 1 0 v1 V2 vo e 1 1 vo v2 v1 eg 1 2 vo v1 uz co 2 0 vo V1 V2 0 FFC User Manual Anders Logg B 4 3 Numbering of mesh entities on quadrilateral cells Entity Incident vertices Non incident vertices vo 10 0 vo v1 V2 v3 v 0 1 v1 vo V2 V3 ve 0 2 v2 vo v1 V3 v3 0 3 uz vo V1 V2 eo 1 0 V2 U3 Up U1 e 1 1 v1 V2 vo U3 es 1 2 Up vs v1 V2 es 1 3 vo v1 V2 vs co 2 0 Uo v1 V2 U3 0 B 4 4 Numbering of mesh entities on tetrahedral cells Entity Incident vertices Non incident vertices vo 0 0 vo v1 V2 V3 v1 0 1 vi to V2 U3 v 0 2 v2 Vo U1 V3 vs 0 3 v3 Vo U1 V2 eo 1 0 v2 v3 vo v1 amp 1 1 v1 v3 vo V2 amp m 1 2 v1 v3 vo v3 amp 1 3 vo v3 v1 V2 e4 1 4 vo V2 vi v3 es 1 5 vo v1 vz V3 fo 2 0 v1 V2 V3 vo f 2 1 vo Va U3 v1 f 2 2 vo t1 V3 va fs 2 3 vo U1 v2
54. later version published by the Free Software Foundation If the Program does not specify a version number of the GNU General Public License you may choose any version ever published by the Free Software Foundation If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used that proxy s public statement of acceptance of a version permanently authorizes you to choose that version for the Program Later license versions may give you additional or different permissions However no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version 15 Disclaimer of Warranty THERE IS NO WARRANTY FOR THE PROGRAM TO THE EXTENT PERMITTED BY APPLICABLE LAW EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND OR OTHER PARTIES PROVIDE THE PROGRAM AS IS WITHOUT WARRANTY OF ANY KIND EITHER EXPRESSED OR IMPLIED INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU SHOULD THE PROGRAM PROVE DEFECTIVE YOU ASSUME THE COST OF ALL NECESSARY SERVICING REPAIR OR CORRECTION 16 Limitation of Liability IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER OR ANY OTHER PARTY WHO MODIFIES AND OR CONVEYS 108 FFC User Manual Anders Logg T
55. ll integrals If v is a scalar valued expression then the integral of v over the interior facets of Q is written as v dS 44 FFC User Manual Anders Logg 5 7 4 Integrals over subsets Integrals over multiple disjoint subdomains of 2 may be defined by specifying an additional argument for the number of the subdomain associated with each integral The different measures may then be combined to express a form as a sum of integrals over the different subdomains dxO Integral cell 0 dx1 Integral cell 1 dsO Integral exterior facet 0 dsi Integral exterior facet 1 ds2 Integral exterior facet 2 dSO Integral interior facet 0 dxO dx1 dsO dsi ds2 ds0 p Il 5 8 DG operators FFC provides operators for implementation of discontinuous Galerkin meth ods These include the evaluation of the jump and average of a function or in general an expression over the interior facets edges or faces of a mesh 5 8 1 Restriction v and v When integrating over interior facets dS one may restrict expressions to the positive or negative side of the facet element FiniteElement Discontinuous Lagrange tetrahedron 0 45 FFC User Manual Anders Logg TestFunction element u TrialFunction element lt Il Il Function element p Il f dot grad v grad u dS Restriction ma
56. m for fixed point iteration on the nonlinear term of the in compressible Navier Stokes equations alu u f w V u dz 6 8 with w the frozen velocity from a previous iteration can be conveniently implemented using index notation as follows 54 FFC User Manual Anders Logg element VectorElement Lagrange tetrahedron 1 v TestFunction element u TrialFunction element w Function element a vlil w jl D ulil j dx This example is implemented in the file NavierStokes form in the collection of demonstration forms included with the FFC source distribution 6 6 The heat equation Discretizing the heat equation V cVu f 6 9 in time using the dG 0 method backward Euler we obtain the following variational problem for the discrete solution up u x t Find u un tn with up 1 gt up tn 1 given such that MS Up up dr ovo Vugdo f vfrdz 6 10 for all test functions v where k t t _ denotes the time step In the example below we implement this variational problem with piecewise linear test and trial functions but other choices are possible just choose another finite element Rewriting the variational problem in the standard form a v un L v for all v we obtain the following pair of bilinear and linear forms alu up foide ets eve Yuzaz 6 11 Lw IS 6 12 Q Q which can be implemented as follows 95 FFC User Manual Ande
57. mployer if you work as a programmer or school if any to sign a copyright disclaimer for the program if necessary For more information on this and how to apply and follow the GNU GPL see lt http www gnu org licenses gt The GNU General Public License does not permit incorporating your program into proprietary programs If your program is a subroutine library you may consider it more useful to permit linking proprietary applications with the library If this is what you want to do use the GNU Lesser General Public License instead of this License But first please read lt http www gnu org philosophy why not lgpl html gt 110
58. n representation Specify representation for precomputation and code generation one of tensor default or quadrature experimental f option Specify code generation options The list of options available depends on the specified language format Current options include fblas fno foo fprecision n fprecompute basis const fprecompute ip const fquadrature degree n and fsplit described in detail below f blas Generate code that uses BLAS to compute tensor products This option is currently ignored but can be used to reduce the code size when the BLAS option is re implemented in future versions f no foo Don t generate code for UFC function with name foo Typical options include fno evaluate_basis and fno evaluate basis derivatives to reduce the size of the generated code when these functions are not needed f precision n Set the number of significant digits to n in the generated code The default value of n is 15 f precompute basis const Additional optimisation option for quadrature representation This option is ignored if optimisation is not used see 0 option and it also implies the precompute ip const option This option will generate code that precompute terms 18 FFC AUTHOR User Manual Anders Logg which are constant in the loops involving basis indices This can result in a reduction of the operation count and thereby improve the runtime efficiency of the generated
59. ng shape and the polynomial degree element FiniteElement family shape degree The argument family is a string and possible values include e Lagrange or CG representing standard scalar Lagrange finite ele ments continuous piecewise polynomial functions e Discontinuous Lagrange or CG representing scalar discontinu ous Lagrange finite elements discontinuous piecewise polynomial func tions e Crouzeix Raviart or CR representing scalar Crouzeix Raviart elements e Brezzi Douglas Marini or BDM representing vector valued Brezzi Douglas Marini H div elements e Brezzi Douglas Fortin Marini or BDFM representing vector valued Brezzi Douglas Fortin Marini H div elements e Raviart Thomas or RT representing vector valued Raviart Thomas H div elements e Nedelec representing vector valued Nedelec H curl elements of the first kind The argument shape is a string and possible values include 28 FFC User Manual Anders Logg e interval representing an interval in Rt e triangle representing a triangle in R e tetrahedron representing a tetrahedron in R The argument degree is an integer specifying the polynomial degree of the finite element Note that the minimal degree for Lagrange finite elements is one whereas the minimal degree for discontinuous Lagrange finite elements is Zero Note that more than one FiniteElement can be d
60. nite element mesh First an ad hoc numbering is picked for the vertices of each cell Then the remaining entities are ordered based on a simple rule as described in detail below B 1 Basic concepts The topological entities of a cell or mesh are referred to as mesh entities A mesh entity can be identified by a pair d i where d is the topological dimension of the mesh entity and 7 is a unique index of the mesh entity Mesh entities are numbered within each topological dimension from 0 to ng 1 where n4 is the number of mesh entities of topological dimension d For convenience mesh entities of topological dimension 0 are referred to as vertices entities of dimension 1 as edges entities of dimension 2 as faces entities of codimension 1 as facets and entities of codimension 0 as cells These concepts are summarized in Table B 1 71 FFC User Manual Anders Logg Entity Dimension Codimension Vertex 0 Edge 1 Face 2 Facet il Cell 0 Table B 1 Named mesh entities h2 Thus the vertices of a tetrahedron are identified as vo and va 0 2 the edges are eg 1 0 e 1 1 es e4 1 4 and es 1 5 the faces facets are fo 2 fo 2 2 and fs 2 3 and the cell itself is cy 3 0 B 2 Numbering of vertices For simplicial cells intervals triangles and tetrahedra of a finite element mesh the vertices are numbered locally based on the corresponding glo
61. nloaded from http www scipy org For Debian Ubuntu users the package to install is python numpy Installing FIAT FFC depends on the latest version of FIAT which can be downloaded from 82 FFC User Manual Anders Logg http www fenics org FIAT is used by FFC to create and evaluate finite element basis functions and quadrature rules The installation instructions for FIAT are similar to those for FFC given in detail below C 1 2 Downloading the source code The latest release of FFC can be obtained as a tar gz archive in the down load section at http www fenics org Download the latest release of FFC for example ffc x y z tar gz and unpack using the command tar zxfv ffc x y z tar gz This creates a directory ffc x y z containing the FFC source code If you want the very latest version of FFC it can be accessed directly from the development repository through hg Mercurial hg clone http www fenics org hg ffc This version may contain features not yet present in the latest release but may also be less stable and even not work at all 83 FFC User Manual Anders Logg C 1 3 Installing FFC FFC follows the standard installation procedure for Python packages Enter the source directory of FFC and issue the following command python setup py install This will install the FFC Python package in a subdirectory called ffc in
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63. nt a v dx i C u dx i dx v dx i 2 u0 u u0 dx i dx L v f dx dot grad v sig0 dx FElTangent form element FiniteElement Lagrange triangle 1 DG FiniteElement Discontinuous Lagrange triangle 0 v TestFunction DG u TrialFunction DG u0 Function element a v u dx L v 1 0 u0 u0 dx FElGradient form element FiniteElement Lagrange triangle 1 DG VectorElement Discontinuous Lagrange triangle 0 v TestFunction DG u TrialFunction DG u0 Function element a dot v u dx L dot v grad u0 dx The 3 forms can be implemented using the QuadratureElement in a similar fashion in which only the element declaration is different 61 FFC User Manual Anders Logg QEiNonlinearPoisson form element FiniteElement Lagrange triangle 1 QE QuadratureElement triangle 2 sig VectorQuadratureElement triangle 2 QE1Tangent form element FiniteElement Lagrange triangle 1 QE QuadratureElement triangle 2 QElGradient form element FiniteElement Lagrange triangle 1 QE VectorQuadratureElement triangle 2 Note that we use 2 points when declaring the QuadratureElement This is because the RHS of the Tangent form is 2 order and therefore we need 2 points for exact integration Due to consistency issues when passing func tions around between the forms we also need to use 2 points when declaring the
64. oisson The bilinear and linear forms for a system of independent Poisson equa tions alu u f Vu Vudz 6 4 L v fv tae 6 5 with v u and f vector valued can be implemented as follows element VectorElement Lagrange triangle 1 v TestFunction element u TrialFunction element f Function element a dot grad v grad u dx L dot v f xdx Alternatively index notation may be used p Il D vli j D ulil j dx v i f i dx Il This example is implemented in the file PoissonSystem form in the collec tion of demonstration forms included with the FFC source distribution 6 4 The strain strain term of linear elasticity The strain strain term of linear elasticity a v u f e v e u dz 6 6 Q 93 FFC User Manual Anders Logg where e v Vo Vv 6 7 can be implemented as follows element VectorElement Lagrange tetrahedron 1 v TestFunction element u TrialFunction element def epsilon v return 0 5 grad v transp grad v a dot epsilon v epsilon u dx Alternatively index notation can be used to define the form a 0 25 D v i j D v j i O u il j D j i dx This example is implemented in the file Elasticity form in the collection of demonstration forms included with the FFC source distribution 6 5 The nonlinear term of Navier Stokes The bilinear for
65. olation of this License then your license from a particular copyright holder is reinstated a provisionally unless and until the copyright holder explicitly and finally terminates your license and b permanently if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation Moreover your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the 104 FFC User Manual Anders Logg violation by some reasonable means this is the first time you have received notice of violation of this License for any work from that copyright holder and you cure the violation prior to 30 days after your receipt of the notice Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License If your rights have been terminated and not permanently reinstated you do not qualify to receive new licenses for the same material under section 10 9 Acceptance Not Required for Having Copies You are not required to accept this License in order to receive or run a copy of the Program Ancillary propagation of a covered work occurring solely as a consequence of using peer to peer transmission to receive a copy likewise does not require acceptance However nothing other than this License grants you permission to propagate or modify any covered work These
66. orted for all scalar valued basic data types thus including BasisFunction Function Constant and expressions involving these data types 38 FFC User Manual Anders Logg 5 4 4 Scalar division Division is not allowed for BasisFunctions and thus not for TestFunctions and TrialFunctions in the definition of a form This is because division by a BasisFunction in the definition of a form does not result in a valid multi linear form since a multilinear form must be linear in each of its arguments However division is allowed for Functions and is applied to the coefficients of its nodal basis expansion Thus 1 f for a Function f corresponds to the operation 1 f gt 0 4 Qi 5 4 See also Section 5 9 5 5 Vector operators Vectors are defined in the form language using Python s built in list type This means that all list operations such as slicing list comprehension etc are supported There is one exception to this rule namely vector valued BasisFunctions and Functions which are not lists but can be made into lists using the operator vec discussed below The operators listed below support all objects which are logically vectors thus including both Python lists and vector valued expressions 5 5 1 Component access v i Brackets are used to pick a given component of a logically vector valued expression Thus if v is a vector valued expression then v 0 represents a function corresponding to the first component of
67. ource in the same way through the same place at no further charge You need not require recipients to copy the Corresponding Source along with the object code If the place to copy the object code is a network server the Corresponding Source may be on a different server operated by you or a third party that supports equivalent copying facilities provided you maintain clear directions next to the object code saying where to find the Corresponding Source Regardless of what server hosts the Corresponding Source you remain obligated to ensure that it is available for as long as needed to satisfy these requirements e Convey the object code using peer to peer transmission provided 101 FFC User Manual Anders Logg you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d A separable portion of the object code whose source code is excluded from the Corresponding Source as a System Library need not be included in conveying the object code work A User Product is either 1 a consumer product which means any tangible personal property which is normally used for personal family or household purposes or 2 anything designed or sold for incorporation into a dwelling In determining whether a product is a consumer product doubtful cases shall be resolved in favor of coverage For a particular product received by a particular user
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71. r additional optional arguments compile forms prefix representation language options 4 1 1 Input arguments The prefix argument can be used to control the prefix of the file containing the generated code which we in the above example set to Poisson The suffix h will be added automatically The representation argument can be used to control the form represen tation used for precomputation and code generation The default value is tensor which indicates that the code should be generated based on a tensor representation of the multilinear form as described in Alterna tively quadrature may be used to specify that code should be generated based on direct quadrature at run time experimental The language option can be used to control the output language for the generated code The default value is ufc which indicates that code should be generated in the UFC format Alternatively dolfin may be used to generate code according to the UFC format with a small set of additional DOLFIN specific wrappers The compile function accepts a dictionary of special code generation options The default values for these options may be accessed through the variable FFC_OPTIONS available in FFC 4 1 2 Output arguments The compile function returns a tuple 22 FFC User Manual Anders Logg form_data form_representation where form_data is a list of metadata extracted for each input form and
72. reement or commitment not to enforce a patent against the party If you convey a covered work knowingly relying on a patent license and the Corresponding Source of the work is not available for anyone to copy free of charge and under the terms of this License through a publicly available network server or other readily accessible means then you must either 1 cause the Corresponding Source to be so available or 2 arrange to deprive yourself of the benefit of the patent license for this particular work or 3 arrange in a manner consistent with the requirements of this License to extend the patent license to downstream recipients Knowingly relying means you have actual knowledge that but for the patent license your conveying the covered work in a country or your recipient s use of the covered work in a country would infringe one or more identifiable patents in that country that you have reason to believe are valid If pursuant to or in connection with a single transaction or 106 FFC User Manual Anders Logg arrangement you convey or propagate by procuring conveyance of a covered work and grant a patent license to some of the parties receiving the covered work authorizing them to use propagate modify or convey a specific copy of the covered work then the patent license you grant is automatically extended to all recipients of the covered work and works based on it A patent license is discriminatory if it do
73. rogram s name and a brief idea of what it does Copyright C year name of author This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 3 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program If not see lt http www gnu org licenses gt 109 FFC User Manual Anders Logg Also add information on how to contact you by electronic and paper mail If the program does terminal interaction make it output a short notice like this when it starts in an interactive mode lt program gt Copyright C lt year gt lt name of author gt This program comes with ABSOLUTELY NO WARRANTY for details type show w This is free software and you are welcome to redistribute it under certain conditions type show c for details The hypothetical commands show w and show c should show the appropriate parts of the General Public License Of course your program s commands might be different for a GUI interface you would use an about box You should also get your e
74. ron and the reference hexahedron see Table A 1 The UFC specification assumes that each cell in a finite element mesh is always isomorphic to one of the reference cells Reference cell Dimension Vertices Facets The reference interval il 2 2 The reference triangle 2 3 3 The reference quadrilateral 2 4 4 The reference tetrahedron 3 4 4 The reference hexahedron 3 8 6 Table A 1 Reference cells covered by the UFC specification 65 FFC User Manual Anders Logg oT 0 1 Figure A 1 The reference interval Vertex Coordinate VO xz 0 Ui a 1 Table A 2 Vertex coordinates of the reference interval A 1 The reference interval The reference interval is shown in Figure A 1 and is defined by its two vertices with coordinates as specified in Table A 2 A 2 The reference triangle The reference triangle is shown in Figure A 2 and is defined by its three vertices with coordinates as specified in Table A 3 Vertex Coordinate Vo x 0 0 V1 x 1 0 V2 x 0 1 Table A 3 Vertex coordinates of the reference triangle 66 FFC User Manual Anders Logg 0 1 0 0 1 0 Figure A 2 The reference triangle A 3 The reference quadrilateral The reference quadrilateral is shown in Figure A 3 and is defined by its four vertices with coordinates as specified in Table A 4 Vertex Coordinate Vo x 0 0
75. rs Logg element FiniteElement Lagrange triangle 1 v TestFunction element Test function ul TrialFunction element 4 Value at t n u0 Function element Value at t n 1 c Function element Heat conductivity f Function element Heat source k Constant triangle Time step a v ul dx k c dot grad v grad ul dx L vx xu0x xdx k v f dx This example is implemented in the file Heat form in the collection of demon stration forms included with the FFC source distribution 6 7 Mixed formulation of Stokes To solve Stokes equations A4 Vp Jg 6 13 Vou 0 6 14 we write the variational problem in standard form a v u L v for all v to obtain the following pair of bilinear and linear forms Vu Vu V v p aq V ujdz 6 15 a v q u p L v q f fdz 6 16 Using a mixed formulation with Taylor Hood elements this can be imple mented as follows P2 P1 VectorElement Lagrange triangle 2 FiniteElement Lagrange triangle 1 56 FFC User Manual Anders Logg TH P2 Pl v q TestFunctions TH u p TrialFunctions TH f Function P2 a dot grad v grad u div v p q div u dx L dot v f dx This example is implemented in the file Stokes form in the collection of demonstration forms included with the FFC source distribution 6 8 Mixed formulation of Poisson We next consider the following formul
76. s manual is written both for the beginning and the advanced user There is also some useful information for developers More advanced topics are treated at the end of the manual or in the appendix Typographic conventions e Code is written in monospace typewriter like this e Commands that should be entered in a Unix shell are displayed as follows configure make FFC User Manual Anders Logg Commands are written in the dialect of the bash shell For other shells such as tcsh appropriate translations may be needed Enumeration and list indices Throughout this manual elements x of sets x of size n are enumerated from i 0 toi n 1 Derivatives in R are enumerated similarly o 2 2 Oxo zi i Dan Contact Comments corrections and contributions to this manual are most welcome and should be sent to ffc devOfenics org 10 Chapter 1 Introduction This chapter has not yet been written In the meantime refer to where the algorithms that FFC is based on are described in detail 11 Chapter 2 Quickstart This chapter demonstrates how to get started with FFC including down loading and installing the latest version of FFC and compiling Poisson s equation These topics are discussed in more detail elsewhere in this man ual In particular see Appendix C for detailed installation instructions and Chapter 5 for a detailed discussion of the form language
77. tch to your version of FFC type hg unbundle lt path gt ffc lt identifier gt lt date gt hg followed by hg update D 3 2 Applying a standard patch file Let s say that a patch has been built relative to FFC release x y z The following description then shows how to apply the patch to a clean version of release x y z 1 Unpack the version of FFC which the patch is built relative to 91 FFC User Manual Anders Logg tar zxfv ffc x y z tar gz 2 Check that you have the patch ffc lt identifier gt lt date gt patch and the FFC directory structure in the current directory ls ffc x y z ffc lt identifier gt lt date gt patch Unpack the patch file using gunzip if necessary 3 Enter the FFC directory structure cd ffc x y z 4 Apply the patch patch p1 lt ffc lt identifier gt lt date gt patch The option p1 strips the leading directory from the filename references in the patch to match the fact that we are applying the patch from inside the directory Another useful option to patch is dry run which can be used to test the patch without actually applying it 5 The modified version now exists as ffc x y z D 4 License agreement By contributing a patch to FFC you agree to license your contributed code under the GNU General Public License a condition also built into the GPL license of the code you have modifie
78. that case you need to supply the path to the FFC executable bin ffc Poisson form C 1 5 Verifying the generated code To verify the output generated by the compiler enter the sub directory src test regression from within the FFC source tree and run the script test py python test py This script compiles all forms found in src demo and compares the output with previously compiled forms in src test regression reference C 2 Debian Ubuntu package In preparation 85 Appendix D Contributing code If you have created a new module fixed a bug somewhere or have made a small change which you want to contribute to FFC then the best way to do so is to send us your contribution in the form of a patch A patch is a file which describes how to transform a file or directory structure into another The patch is built by comparing a version which both parties have against the modified version which only you have Patches can be created with Mercurial or diff D 1 Creating bundles patches D 1 1 Creating a Mercurial hg bundle Creating bundles is the preferred way of submitting patches It has several advantages over plain diffs If you are a frequent contributor consider pub lishing your source tree so that the FFC maintainers and other users may pull your changes directly from your tree A bundle contains your contribution to FFC in the form of a binary patch file generated by Mercurial the
79. u may at your option remove any additional permissions from that copy or from any part of it Additional permissions may be written to require their own removal in certain cases when you modify the work You may place additional permissions on material added by you to a covered work for which you have or can give appropriate copyright permission Notwithstanding any other provision of this License for material you add to a covered work you may if authorized by the copyright holders of that material supplement the terms of this License with terms a Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License or b Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it or c Prohibiting misrepresentation of the origin of that material or requiring that modified versions of such material be marked in reasonable ways as different from the original version or d Limiting the use for publicity purposes of names of licensors or authors of the material or 103 FFC User Manual Anders Logg e Declining to grant rights under trademark law for use of some trade names trademarks or service marks or f Requiring indemnification of licensors and authors of that material by anyone who conveys the material or modified versions of it with contractual assumptions of l
80. uantities should be evaluated at quadrature points rather than interpolated at nodal points An example is given in Section 6 It is also possible to construct mixed elements from a QuadratureElement and for convenience the VectorQuadratureElement is available with the same basic functionality as the VectorElement The only difference is that the VectorQuadratureElement will return a vector valued element based on the QuadratureElement 5 3 6 BasisFunction The data type BasisFunction represents a basis function on a given finite element A BasisFunction must be created for a previously declared finite element simple or mixed 32 FFC User Manual Anders Logg v BasisFunction element Note that more than one BasisFunction can be declared for the same FiniteElement Basis functions are associated with the arguments of a multilinear form in the order of declaration For a MixedElement the function BasisFunctions can be used to construct tuples of BasisFunctions as illustrated here for a mixed Taylor Hood ele ment v q BasisFunctions TH u p BasisFunctions TH 5 3 7 TestFunction and TrialFunction The data types TestFunction and TrialFunction are special instances of BasisFunction with the property that a TestFunction will always be the first argument in a form and TrialFunction will always be the second ar gument in a form order of declaration does not matter For a MixedElement the
81. uz 3 0 Uo UL V2 V3 0 79 FFC User Manual Anders Logg B 4 5 Numbering of mesh entities on hexahedral cells Entity Incident vertices Non incident vertices vo 0 0 vo v1 V2 U3 V4 U5 U6 U7 v 0 1 v1 Vo V2 U3 V4 U5 UG U7 v2 0 2 v2 vo UL U3 V4 V5 UG U7 v3 0 3 v3 vo UL V2 V4 V5 UG U7 va 0 4 v4 Vo UL V2 U3 U5 UG U7 v5 0 5 vs vo V1 U2 U3 U4 V6 V7 ve 0 6 v6 vo UL V2 U3 U4 U5 U7 v7 0 7 v7 Ug UL V2 U3 V4 U5 Uc eo 1 0 vg v7 Ug V1 V2 U3 V4 U5 e1 1 T us vg Vo V1 U2 U3 V4 U7 en 1 2 va v7 Uo UL V2 U3 U5 Uc e3 1 3 v4 U5 to UL v2 U3 U6 U7 e4 1 4 va v7 Uo U1 V2 V4 U5 UG e5 1 5 v2 ve Vo UL U3 V4 U5 U7 es 1 6 v2 v3 Vo UL v4 U5 V6 U7 e7 1 7 vi vs vo v2 U3 V4 V6 UT eg 1 8 v1 v2 Vo U3 V4 U5 UG U7 eg 1 9 Vo v4 v1 V2 U3 U5 U6 U7 e10 1 10 vo v3 v1 Va V4 U5 V6 V7 e11 1 11 vo v1 U2 U3 V4 U5 Uo V7 fo 2 0 v4 U5 U6 U7 vo V1 V2 V3 f 2 1 v2 U3 U6 U7 Ug UL V4 U5 f 2 2 v1 V2 U5 U6 vo U3 V4 U7 f3 2 3 to U3 V4 U7 v1 v2 U5 Ug fa 2 4 vo V1 Va U5 v2 U3 U6 U7 fs 2 5 vo v1 V2 U3 v4 U5 U6 U7 co 3 0 vo
82. ween vertices Ug and v2 opposite to vertex v and the third is edge es between vertices Ug and v opposite to vertex vo Similarly we identify the six edges of the reference tetrahedron with the corresponding non incident tuples vo vi vo v2 vo v3 vi v2 vi va and va v3 The first of these is edge ey between vertices v2 and v3 opposite T3 FFC User Manual Anders Logg 0 1 2 Figure B 2 The local numbering of vertices of a non simplicial mesh is arbitrary as long as each cell is isomorphic to the reference cell by matching each vertex to the corresponding vertex of the reference cell 74 FFC User Manual Anders Logg to vertices vp and v as shown in Figure B 4 U2 o Vo Ui Figure B 3 Mesh entities are ordered based on a lexicographical ordering of the corresponding ordered tuples of non incident vertices The first edge ey is non incident to vertex vo B 3 1 Relative ordering The relative ordering of mesh entities with respect to other incident mesh entities follows by sorting the entities by their global indices Thus the pair of vertices incident to the first edge eo of a triangular cell is v1 v2 not vz v1 Similarly the first face fo of a tetrahedral cell is incident to vertices v1 V2 v3 For simplicial cells the relative ordering in combination with the convention of numbering the vertices locally based on global vertex indices means that two incident cells will always
83. xtent such circumvention is effected by exercising rights under this License with respect to the covered work and you disclaim any intention to limit operation or modification of the work as a means of enforcing against the work s users your or third parties legal rights to forbid circumvention of technological measures 4 Conveying Verbatim Copies You may convey verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and 99 FFC User Manual Anders Logg appropriately publish on each copy an appropriate copyright notice keep intact all notices stating that this License and any non permissive terms added in accord with section 7 apply to the code keep intact all notices of the absence of any warranty and give all recipients a copy of this License along with the Program You may charge any price or no price for each copy that you convey and you may offer support or warranty protection for a fee 5 Conveying Modified Source Versions You may convey a work based on the Program or the modifications to produce it from the Program in the form of source code under the terms of section 4 provided that you also meet all of these conditions a The work must carry prominent notices stating that you modified it and giving a relevant date b The work must carry prominent notices stating that it is released under this License and any conditions added under section
84. y be applied to functions of any finite element space but will only have effect when applied to expressions that are discontinuous across facets 5 8 2 Jump jump v The operator jump may be used to express the jump of a function across a common facet of two cells Two versions of the jump operator are provided If called with only one argument then the jump operator evaluates to the difference between the restrictions of the given expression on the positive and negative sides of the facet jump v e v 2 v v 5 15 If the expression v is scalar then jump v will also be scalar and if v is vector valued then jump v will also be vector valued If called with two arguments jump v n evaluates to the jump in v weighted by n Typically n will be chosen to represent the unit outward normal of the facet as seen from each of the two neighboring cells If v is scalar then jump v n is given by jump v n e v v n tun 5 16 If v is vector valued then jump v n is given by jump v n e v v n v 5 17 Thus if the expression v is scalar then jump v n will be vector valued and if v is vector valued then jump v n will be scalar 46 FFC User Manual Anders Logg 5 8 3 Average avg v The operator avg may be used to express the average of a function across a common facet of two cells e N ur 5 18 If the expression v is scalar then avg v will also be scalar and if v is vector valu

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