UMesh User Manual
Contents
1. backgrid points together with a specified metric and also a list of points Steiner points without any explicit metric In the second case the metric at the Steiner points is generated automatically by an interpolation on the background triangles they belong to Positioning the backgrid points and choosing the values of their correspond ing metric is an effective mean for modifing the grid layout which is otherwise influenced only by the metric on the boundary together with its interpolation The data for controlling backgrid and Steiner points are specified according the following input n number of backgrid or Steiner points x y Cartesian coordinates of backgrid or Steiner points h metric except for Steiner nstep lsmooth 1 0 1000 1 n backgrid points 1 x y hdata 0 80 0 80 0 10 3 n Steiner points 1 x y 0 50 0 30 Table 3 The domain two circles file Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 11 4 Output files of UMesh 4 1 Intermediate files knots grid name geometry grid name The intermediate file knots grid name contains the points of every boundary edge n curv number of edges dim number of spatial dimensions always 2 points number of points idc 0 for line and data 2 for ellipse and circle X y point coordinates The intermediate file geometry grid name contains the data of the interpolant curve associated with the boundary edge2. S REBAY A GUARDONE D DUSSIN 2 1 Introduction UMesh is a program for generating 2D unstructured grids of triangles The generation produces a Delaunay triangulation 2 see the example in Figure 1 The program implements the Delaunay advancing method developed by Stefano Rebay which is centered around an original and smart use of Bowyer Watson algorithm Figure 1 Typical mesh produced by UMesh 2 Overview of the program In this section the main features of the UMesh program are described to pro vide an overview of the mesh generation process 2 1 Input files The program UMesh can be run from any directory by issuing the command umesh2d exe During its execution the program asks for the name of the prob lem for instance grid name and assumes that the two files boundary grid name and domain grid name exist prepared according to the correct format see sec tion 3 The extension grid name indicates the name chosen by the user to identify his grid together with all the associated files The two input files con tain the data defining the domain to be triangulated and the characteristics of the mesh that the user wants to produce The file boundary grid_name consists of two parts The first part contains the data for the geometrical description of the boundary which in general may be composed of several edges i e portions of the boundary The second part of the file accounts for the con
3. listed in the second Table Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN boundary grid name Error messages Possible cause of er ror Possible solution Segmentation fault wrong number of ver tices check the number of vertices Loading topology Invalid input wrong number of edges or wrong number of edges and number of vertices check the number of edges and or number of vertices command not known invalid geometry op tion only line cir cle ellipse and data al lowed watch out for blank line After number of seg blank line after param watch out for blank ments command not eter s description line known simplex center error in topology sec check in topology warning tion Stitch points simplex center warning intersection between edges plot your boundary and see Problem between element n and m the boundary is not closed move the position of some vertices nv ne domain grid name Error messages Possible cause of er ror Possible Solution Backgrid points Invalid Input for integer editing wrong number of points check the file Background grid Invalid Input for integer editing blank line s check for blank line s References 1 S REBAY Efficient unstructured mesh generation by means of Delaunay t
4. n curv nd ns S x1 x2 xsi xs2 number of edges number of spatial dimensions always 2 number of curve points normalized curvilinear abscissa x and y point coordinates x and y derivatives with respect to s 4 2 Files of the final mesh nodes grid name grid grid name The file nodes nd np_D np_B j rr 1 2 j jB jD_jB jB integer grid name contains data of the domain and boundary points number of spatial dimensions always 2 number of points belonging to the domain number of points belonging to the boundary domain point index 1 j lt np D Cartesian coordinates of each point boundary point index 1 lt jB lt np B domain point index corresponding to boundary point index irrelevant for FEM FE The file grid grid name contains data of the domain and boundary elements ne D number of domain elements ne B number of boundary elements m index of domain element 1 m lt ne D type element type 2 for triangle j_m 1 3 m connectivity matrix nodes element ma m 1 3 m connectivity matrix adjacent elements element mB global index of boundary element 1 lt mB lt ne B type mB element type 1 for segment side mB index of the side the boundary element belongs to jB_mB 1 2 mB boundary connectivity matrix boundary nodes boundary element mBa mB 1 2 mB boundary connectivity matrix adjacent boundary elements boundary element Revised by N CALOSSO and L QUA
5. 3423920 0 1000000000 0 0000000000 Table 2 The small_circle data file translation Cartesian coordinates of a displacement vector to be applied to all points rotation rotation angle degrees around the origin to be applied to all points after the translation The parameters and the points of file data must be given respecting the assumed positive orientation of the boundary as illustrated in Figure 4 Figure 4 Positive orientation of the boundary Input of the topological and metric informations for the boundary This part of file boundary grid_name is necessary to connect the edges defined before and to impose the boundary metric along every edge This is simply obtained by specifying the length of the triangle s base in a few points identified by the values of a normalized arc length parameter curvilinear abscissa relative to the edge they belong to the growth direction of the abscissa is a consequence of the geometry definition The metric is then interpolated choosing between three options and the program works in such a way so as to spread the metric information inside the domain and to obtain almost equilateral triangles of the proper size everywhere in the mesh Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 bv ev S REBAY A GUARDONE D DUSSIN 9 edge number number of points where the metric is imposed on edge e note that this value must be less than or equal to number of po
6. B 1 1 1 1 2 2 0 2 1 1 3 4 3 1 3 1 1 5 6 151 1 6 301 302 152 150 152 1 6 303 304 0 151 Table 5 The grid two circles fie Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 15 5 Messages and errors on using UMesh P 5 1 Echoes on the screen For every boundary edge the program shows the number of points resulting from the imposition of the metric For the background grid it shows the number of boundary points and segments included in the following number of domain points and triangles A part of the displayed output pertains to the advancing front step step number nfro number of proposed new points nadd number of added new points max length the maximum side length between domain elements In case the value of nstep chosen is not enough to complete the mesh the pro gram shows the percentage of triangles that do not satisfy the metric specified If the Laplacian smoothing is present the number of iterations and the residual are shown The number of domain triangles is written after the label int head 5 2 Frequent errors When UMesh encounters an error in one of the input files it alerts the user by issuing some error messages The list of possible error messages with their respective possible cause is given in the following Tables Messages of errors in the input file boundary grid name are listed in the first Table while those in the input file domain grid name are
7. RTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN Figure 7 Global and local indices see table 5 Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 12 S REBAY A GUARDONE D DUSSIN 13 TEETHETHEIEHETHEIETHETHEIETHEIE HE HEIETHEIEHETHEIETHEEEHIETHEIEHETHEIETHEIHEIEHEHETHEIEHHHHENHE NAME two_circles AHBHEHEBBEBHEHEIHHEBRHBRHEBHEHEBRRHERBHHRHEHBRHEHEHBRHEHHEHEBHEBHEHHHHEHBHENE nd np_D np_B 2 915 304 TEETHETHEIETHETHEIETHETHEIETHEIETHEEEIETHEIEEHETHEIETHEIEHETHEIEHETHEETHEIHEIEHHEHEIHHHHHERNHE id KKK kK k k k kk DOMAIN KKK kK K kk kK AHHEHBHBHBHHHEHBHBHBHHHEHBHBHBHHHHEBHBHBHHHHEBHBHBHHHHEHHHHHHBHHHHHHHHE do NODES j rr 1 2 j 1 0 218614037E 01 0 218693044E 01 2 0 147814262E 01 0 218775883E 01 3 0 147810419E 01 0 978162969E 00 4 0 218773470E 01 0 978142850E 00 5 0 878989450E 00 0 687758689E 00 914 0 107071034E 01 0 229289218E 00 915 0 109239125E 01 0 261730356E 00 TEHETHETHEIETHETHEIETHETHEIETHEIEHETHEIETHEIEEHETHEIETHEIEEHIETHEIEEHETHEETHEIHEIEHHETHEIEHHHHER HE id 2 OK K K k k kk BOUNDARY 2K 2 kK kK K kk kK THEIEIEHEIEIEEHHHIEHERBIBHHEBBRHEBE HERE HH HBBEHBHEHHHBRHHHBBHHHBHBSBE NODES jB jD_jB jB integer 1 764 1 2 770 1 3 770 1 4 771 1 5 771 1 6 772 1 303 915 6 304 769 6 Table 4 The nodes two_circles file 4 3 Plotting files gridplot grid name backplot grid name The file gridplot grid name contains the data to
8. UMesh User Manual S Rebay A Guardone and D Dussin October 15 2013 Abstract This manual explains how to use program UMesh P to generate two di mensional unstructured isotropic grids of Delaunay type by means of the Delaunay advancing front method of Rebay 1 A detailed description of the input files necessary to define and triangulate arbitrary computa tional regions in two dimensions is given The program can deal with multiply connected domains and allows for specifying the local dimen sion of the triangles at various locations on its boundary and within the domain A complete example is provided to illustrate the structure and format of the input data as well as the form of the output files Revised by N Calosso and L Quartapelle Contents 1 Introduction 2 Overview of the program 2 1 Inputhles 4 5 uv kms Ae eee uec eee ra 2 2 Grid generation process e 2 3 Output files ue Pus lah soe eas Be Aedes e Input file for UMesh 3 1 Thefileboundary grid name less 3 2 Thefiledomain grid name Output files of UMesh 4 1 Intermediate files knots grid name and geometry grid name 4 2 Files of the final mesh nodes grid name and grid grid name 4 3 Plotting files gridplot grid name and backplot grid name Messages and errors on using UMesh 5 1 Echoes on the screen 222r 5 2 Ereg ent errors sora e uq ie E E woe Tek Ge eae ee ed S OOo PO b o A 11 11 11 13
9. acters Records beginning with the character contain comments Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 3 iHHHHHHEHE CHOICE BETWEEN FINITE ELEMENT vs VOLUME BOUNDARY NODES 1HHEHHHHHE 3 FE HHHHH GLOBAL DATA OF THE BOUNDARY GEOMETRY nv ne 6 6 HHHHHE DESCRIPTION OF EACH BOUNDARY PART i i BEGIN 1 left side line 61 0 00 0 0 0 1 0 END 1 BEGIN 2 top side line 61 0 0 1 0 1 5 1 0 END 2 BEGIN 5 larger circle circle 100 0 5 0 5 2 0 360 END 5 BEGIN 6 smaller circle data small_circle_file_name translation 1 0 0 3 rotation 0 0 END 6 HHHHH TOPOLOGY AND METRIC i BEGIN 1 e n bv ev 1 2 1 2 i s h 1 0 0 0 05 1 1 0 0 05 END 1 BEGIN 2 e n bv ev 2 2 2 3 i s h 1 0 0 0 05 1 1 0 0 05 END 2 BEGIN 5 larger circle e n bv ev 5 4 5 5 i s h 1 0 0 0 02 1 0 3 0 05 1 0 7 0 04 1 1 0 0 02 END 5 BEGIN 6 smaller circle e n bv ev 6 2 6 6 i s h 1 0 0 0 04 1 1 0 0 04 END 6 Table 1 The boundary two_cicles file Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 7 Input of the geometry of the boundary Every edge begins at a vertex and ends in a vertex For a closed boundary the two vertices belonging to any pair of consecutive edges must be coincident In the file boundary grid name every set of input lines is regrouped within the BEGIN and END st
10. atements the former possibly containig some comments written at a complete discretion of the user Each edge is defined by specifying either its geometrical character together with a set of suitable parameters or alternatively the name of a data file which contains the list of the Cartesian coordinates of the points The possible geometrical choices for an edge and for the associated param eters are xf Yt J Xi Yi segment circular arc arc of an ellipse data line number of points for spline approximation which must at least exceeds the number of points where the metric along the boundary is imposed see below coordinates of the begin point coordinates of the end point ellipse number of points for spline approximation coordinates of the center length of the x and y semiaxis begin angle end angle degrees circle number of points for spline approximation coordinates of the center ra dius begin angle end angle degrees data name of the file containing the Cartesian coordinates of a set of points Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 8 the first record must contain the number of points and each subsequent record contains the two Cartesian coordinates of the point see Table 2 100 0 1000000000 0 0000000000 0 0997986676 0 0063423920 0 0991954813 0 0126592454 0 0991954813 0 0126592454 0 0997986676 0 006
11. f the points of each edge of the boundary specified by the user in the file boundary grid_name geometry grid name which contains some data about the interpolant curves associated with each boundary edge The files collecting informations for visualization purposes are backplot grid name which contains informations about the background grid gridplot grid name which contains informations needed to plot the gen erated grid The background and final grids can be viewed by means of the program Plotmtv by issuing respectively the commands plotmtv backplot grid name and plotmtv gridplot grid name from the directory in which the two files are stored All the data which constitue the final generated mesh are contained in the last two output files nodes grid name which contains the Cartesian coordinates of the nodes of the mesh grid grid name which contains the connectivity matrices for the element to node correspondence connectivity matrices of domain elements and of boundary elements are provided separately 3 Input file for UMesh To better explain the grid generation process we will refer to an example We will describe how to produce the grid shown in figure 1 Figures and tables in the next sections refer to this example The tables contain the input data or part of thereof To produce a grid the user must provide the two files boundary grid name and domain grid name written according to the correct syntax as illu
12. ints used for spline ap proximation on the line ellipse or circle begin vertex for edge e end vertex for edge e coincident with bv for a closed edge type of interpolation for the metric along the normalized curvilinear ab scissa of the edge e 1 linear 2 sinusoidal 3 geometrical increasing curvilinear abscissa normalized to 1 for imposing the metric normalized length of the boundary element Figure 6 Boundary metric Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 10 For an explanation of the main errors that the user could make see section 5 2 It is important to remember that no blank record is allowed between two input lines and that the last record of the input must be END 3 2 The file domain grid name The second input file of UMesh P is domain grid name It contains a few addi tional data to control the size of the triangles inside the domain The first two data must be always provided by the user and are nstep is the maximum number of subsequent steps allowed to the program in its generation process of points necessary to improve the grid quality and to match the metric requirements If the number indicated is not sufficient a message on the screen will show the percentage of triangles larger than requested by the input lsmooth is a boolean integer for including 1 or avoiding 0 the Laplacian smoothing It is possible to insert a list of points
13. nection between the extremities of the edges i e the boundary topology Moreover it contains the data for controlling the size of the triangles on the boundary namely the boundary metric Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 3 The file domain grid name allows the user to control the size of finite ele ments triangles inside the computational domain by specifying their typical dimension in a selected number of internal points This set of data will be denoted as the domain metric 2 2 Grid generation process The algorithm for grid generation proceeds basically as follows A set of points is generated on the boundary with a spacing local dimension determined by the boundary metric These points are then connected to each other and to any additional point backgrid point see page 10 specifying the domain metric to form a background grid called also backgrid Fig 2 0 1 5 Figure 2 The background grid with one backgrid point inserted The background grid is then used to produce on the whole domain the metric information needed by the mesh generation process basically the metric as signed on the boundary is interpolated inside all the triangles of the backgrid Figure 3 Schematic description of Bowyer Watson algorithm The background grid is taken as the initial Delaunay triangulation to which internal grid points are added one by one by means of
14. plot the domain grid while the file backplot grid name contains the data to plot the background grid These two files are saved by the program UMesh in plotmtv format The background and final grids can be viewed by means of the program Plotmtv by issuing respectively the commands plotmtv gridplot grid name and plotmtv backplot grid name from the directory in which the two files are stored Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN TEETHETHEIETHETHERETHETHEIETHEIEHETHEIETHEIEEHETHEIETHEIEEHIETHEIEHETHEIETHETHEIEHEHETHEIEHHHHHENHE NAME two_circles AHBHEHEIBBEHHEHEHHEBRHERERBHEHBBRRHEHHRHEBRHEHHBRHEHHEHEBHHBHEHEHHHHBHENER 4 ne D ne B 1680 152 AHEHEHEBHEBHEHEHHEHBRRHEBHEHEBRRHERBHERHEBHEHEHBHEHHEHBBHHBHEHHBHHBHENEN id KKK K k k k kk DOMAIN KKK OK k kk kK AHEHEHEBBEBHEBHEHRHEBHERHEBHEHEBRRHERBHRHEHBRRHEHEHBRHEBHEHEBHEBHEHEHHHEHBHEINHE ELEMENTS m type j m 1 3 m ma_m 1 3 m 4 1 2 764 1 770 153 0 100 2 2 770 646 771 154 0 155 3 2 771 644 772 156 0 154 10 2 83 31 45 75 33 13 1679 2 6 17 5 1678 1672 1680 1680 2 7 17 6 1679 1674 1677 TEETHETHEIETHETHEIETHETHEIETHEIEHETHEIETHETEEIETHEIETHEIEHIETHEIEHETHEIETHEHEIEHEHEHEIEHHBHENHE KKK K k k kk BOUNDARY 2K kk K K kk k THEHUHIHEHBHHBHBRHBRBRERBRBRBBRBBBBRRBRBERERIEEIEEE IEEE HE IH E IHEHHHHHEHH HHH ELEMENTS mB type mB side mB jB_mB 1 2 mB mBa_mB 1 2 m
15. riangulation and Bowyer Watson algorithm J Comput Phys 106 125 138 1993 2 P L GEORGE and H BonoucHAkI Delaunay triangulation and meshing application to finite elements Hermes 413 1998 Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 16
16. strated in Tables 1 and 3 Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 5 3 1 The file boundary grid name Input of the global data of the boundary The first information that must be provided to the mesh generator in the file boundary grid_name is a record for selecting between a grid suitable for Finite Element computation and a grid for Finite Volume computation In the latter case UMesh produces a grid with duplicated boundary nodes for the very complex treatment of the boundary conditions in hyperbolic problems The first record of boundary grid name must contain two characters either FE or FV to select one of the two possible aforementioned choices The boundary consists of different possibly curved parts linked to each other Each distinct part is called a boundary edge The boundary consists of a number ne of edges and a number nv of vertices The first two integer data of the file boundary grid name are the number of vertices nv and of edges ne nv total number of boundary vertices ne total number of boundary edges The input line containing the two input data must be preceded by a control record with symbols nv and ne written explicitly Similar control records must be written before other lines containing input data as shown in Table 1 A free format is allowed in the input lines i e one blank character is equivalent to any number of consecutive blank char
17. the Bowyer Watson al gorithm Fig 3 New points are chosen by Rebay s algorithm which is based on an advancing front strategy This method in its earlier stage selects one point to be added for every triangle which is adjacent to the boundary and is also of a low quality with respect to the interpolated metric As new points are added Revised by N CALOSSO and L QUARTAPELLE on October 15 2013 S REBAY A GUARDONE D DUSSIN 4 the grid is refined and the front that is the frontier between accepted triangles and low quality triangles moves stepwise towards the interior of domain In most cases the mesh obtained during the generation process is not yet fully acceptable due to the presence of too stretched triangles Steiner points namely fixed points which will belong to the final grid are possibly added How ever a Laplacian smoothing technique based on a spring analogy can be acti vated by the user to move the points automatically thus reducing the stretching and simultaneously achieving elements with the requested size 2 3 Output files The run of program UMesh produces six files Two of them are auxiliary files which summarize informations useful for the user Other two files collect informations for visualizing the background grid and the generated mesh The last two files contain all the geometrical and connectivity data of the final mesh The two auxiliary files are knots grid_name which contains the coordinates o
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