We describe some algorithms and tools that have been developed to generate composite overlapping grids on geometries that have been defined with computer aided design (CAD) programs. This process consists of five main steps. Starting from a description of the surfaces defining the computational domain we (1) correct errors in the CAD representation, (2) determine topology of the patched-surface, (3) build a global triangulation of the surface, (4) construct structured surface and volume grids using hyperbolic grid generation, and (5) generate the overlapping grid by determining the holes and the interpolation points. The overlapping grid generator which is used for the final step also supports the rapid generation of grids for block-structured adaptive mesh refinement and for moving grids. These algorithms have been implemented as part of the Overture object-oriented framework

Henshaw, W.D.

We describe some algorithms and tools that have been developed to generate com-posite overlapping grids on geometries that have been defined with computer aided design (CAD) programs. This process consists of five main steps. Starting from a description of the surfaces defining the computational domain we (1) correct er-rors in the CAD representation, (2) determine the topology of the patched-surface, (3) build a global triangulation of the surface, (4) construct structured surface and volume grids using hyperbolic grid generation, and (5) generate the overlapping grid by determining the holes and the interpolation points. The overlapping grid generator which is used for the final step also supports the rapid generation of grids for block-structured adaptive mesh refinement and for moving grids. These algorithms have been implemented as part of the Overture object-oriented frame-work

William D. Henshaw

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Generating composite overlapping grids on CAD geometries

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U.S. Department of EnergyLawrenceLivermoreNationalLaboratoryPreprintUCRL-JC-147161Generating CompositeOverlapping Grids on CADGeometriesW.D. HenshawThis article was submitted to8th International Conference on Numerical Grid Generation inComputational Field Simulations, Honolulu, HI, June 2-6, 2002February 7, 2002Approved for public release; further dissemination unlimitedDISCLAIMERThis document was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United states Government nor the University of California nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately owned rights. 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Since changes may bemade before publication, this preprint is made available with the understanding that it will not be citedor reproduced without the permission of the author.This report has been reproduced directly from the best available copy.Available electronically at http: //www.doe.gov/bridgeAvailable for a processing fee to U.S. Department of Energyand its contractors in paper fromU.S. Department of EnergyOffice of Scientific and Technical InformationP.O. Box 62Oak Ridge, TN 37831-0062Telephone: (865) 576-840iFacsimile: (865) 576-5728E-mail: repor ts@adonis.osti.govAvailable for the sale to the public fromU.S. Department of CommerceNational Technical Information Service5285 Port Royal RoadSpringfield, VA 22161Telephone: (800) 553-6847Facsimile: (703) 605-6900E-mail: orders@ntis.fedworld.govOnline ordering: http://www.ntis.gov/ordering.htmORLawrence Livermore National LaboratoryTechnical Information Department’s Digital Libraryhttp: //www.llnl.gov / tid/Library.htmlGenerating Composite Overlapping Gridson CAD GeometriesWilliam D. HenshawCentre for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermore, California, 94551 USAhenshawl @llnl.govAbstractWe describe some algorithms and tools that have been developed to gener-ate composite overlapping grids on geometries that have been defined withcomputer aided design (CAD) programs. This process consists of five mainsteps. Starting from a description of the surfaces defining the computationaldomain we (1) correct errors in the CAD representation, (2) determine topology of the patched-surface, (3) build a global triangulation of the sur-face, (4) construct structured surface and volume grids using hyperbolicgrid generation, and (5) generate the overlapping grid by determining theholes and the interpolation points. The overlapping grid generator whichis used for the final step also supports the rapid generation of grids forblock-structured adaptive mesh refinement and for moving grids. These al-gorithms have been implemented as part of the Overture object-orientedframework.IntroductionWe briefly describe some algorithms and tools that have been developedto generate overlapping grids on geometries that have been defined withcomputer aided design (CAD) programs. The fundamental steps in this"process axeCAD fixup : When the CAD geometry is defined from a file format, suchas IGES, it is usually necessary to repair mistakes in the representa-tion.CAD connectivity : Since an IGES file usually contains no topologyinformation it is necessary to determine how the trimmed surfacepatches connect to one another. We construct a global triangulationthrough an edge matching algorithm.Surface and Volume Grid Generation : We typically use a hyperbolicgrid generator to btiild both the structured surface grids and thevolume grids.Overlapping Grid Generation : We have developed a new overlappinggrid generator, Ogen , that determines the overlapping grid (holecutting, interpolation information) from a collection of overlappinggrids.These algorithms have been implemented as part of the Overture object-oriented framework. Overture is a toolkit that can be used to develop PDEsolvers on overlapping grids and includes the grid generation tools that wedescribe here. The OverBlown Navier-Stokes flow solver which is build us-ing Overture is also available from this web page. OverBlown can be usedto solve the time-dependent incompressible and compressible Navier-Stokesequations and supports moving grids and adaptive mesh refinement. Over-ture and OverBlown are freely, available from www. llnl.gov/ChSC/-Overture.Preparation of the CAD geometryThe geometry is usually defined as a patched-surface consisting of a set ofsub-surfaces (or patches), see for example figure (1). A sub-surface may defined in a variety of ways such as with a spline, B-spline or non-uniform-rational-b-spline (NURBS). In general the sub-surface will be trimmed, which case only a portion of the surface will be used, the valid regionis defined by trimming curves, see figure (2). The output from a CADprogram will often be saved in a standard file format such as IGES or thenewer STEP specification. Unfortunately the typical IGES output file doesnot include any connectivity (topology) information; that is there is information specifying how a given patch connects to other neighbouringpatches. To further complicate matters the trimmed patches will often beinaccurate, or contain mistakes, making it difficult to determine where twoneighbouring patches should be joined.CAD fixup: Errors in the geometry representation are fixed in stages.Gross errors in the trimming curves are detected when the geometry is firstread from the database file. Errors detected at this initial step includetrim curves that lie outside the unit square in parameter space, trim curvesthat don’t close on themselves (i.e. they should be periodic), and trimcurves that self-intersect. These gross errors should be fixed before pro-ceeding to the connectivity stage. In Overture we have the ability to editFigure 1. Top left: a CAD geometry represented as a composite surface ofmultiple trimmed surfaces. Top right: global triangulation for the geometry.Bottom left: a surface grid is generated with the hyperbolic grid generator.Bottom right: overlapping grid for the geometry.the trim curves to fix these types of errors. See Petersson and Chand [13]for more details of how we repair CAD geometries. Errors detected laterat the connectivity stage would include large gaps between patches or mul-tiple definition of patches. There are many approaches to fixing CAD andremoving unwanted etails, see for example [10][14].CAD connectivity: The topology of the CAD geometry is determinedusing an edge-curve matching algorithm whereby the boundary edges oftrimmed surfaces are merged with the boundary edges of neighbouringpatches. Our algorithm is similar to that of [17] but the idea has beenused elsewhere [10][14].This technique, shown in figure (3) begins by building curves (edge-curves)Figure 2. The trimmed patch (top) is formed from an untrimmed sur-face (middle) and a set of one or more trimming curves (bottom). untrimmed surface is a~ mapping from two-dimensional parameter spaceinto three-dimensional cartesian space. The trimming curves axe defined inparameter space, The trimming curve is defined in the CAD file as a set ofsub-curves (bottom).on the boundaries of all trimmed-patches and then attempts to identifywhere an edge-curve from one patch matches to the edge-curve of a neigh-bouring patch. It is usually necessary to split theedge-curves at appropri-ate locations in order to perform the matching. When two edge-curves areidentified to be the same we say the edges have been merged and’ chooseone edge-curve to define the boundary segment for both patches, By merg-ing edge-curves we effectively remove any gaps or overlaps present in theoriginal representation. See [7] for further details. °Global triangulation: A global triangulation can be built once the edge-curves have been merged. When two edge-curves are merged, one of thetwo curves is defined to be the true edge-curve. The trimmed-patches axeredefined to use the common edge. The global triangulation is formed byinitially triangulating each surface patch independently. The triangulationfor a patch uses pre-defined nodes on the edge-curves that form the bound-ary of the trimmed-patch and thus triangulations for adjacent patches willshare common odes. The surface patches axe triangulated in the parameterspace of the patch. This allows us to use fast two-dimensional triangulationalgorithms. We use the triangle program from Shewchuk [15] to compute aconstrained Delaunay triangulation. After the patch has been triangulatedin parameter space it is a simple matter to map the 2D parameter spacenodes to 3D. The triangulations for the patches must be stitched togetherto form a global triangulation. Figure (1) shows a global triangulation builtin this way.Figure 3. Figure showing the three stages of determining the connectivity.(1) Edge-curves are built on each side of each patch. (2) The edge-curves merged and then split and merged. (3) Triangulations are built separatelyfor each patch and then stitched together at the common boundary points(bottom).Projection algorithm: The global triangulation serves as a basis for a fastalgorithm for projecting points onto the patched surface. This projectionalgorithm is used by the hyperbolic surface grid generator. To projecta point onto the patched surface we start by projecting the point ontothe global triangulation. Finding the closest triangle is performed by.awalking-algorithm if an initial guess is known or otherwise by a global searchusingan alternating-digital-tree (ADT) tree. Since each triangle belongs just one sub-patch we can then project the point onto the sub-patch usingNewton’s method.Surface and Volume Grid GenerationStructured surface grids can be generated using hyperbolic grid generation.This approach was developed by Steger, Chan and Buning [3, 1, 2] andis also available in Gridgen, see Steinbrenner and Chawner [16]. We haveimplemented our own version within the Overture framework [6].The basic marching equations to determine the surface grid x(r, t), on surface C(x) = 0, are defined by the hyperbolic PDE, xt = S(r, t) n(r, t)x(r, O) = xo(r) , initial curveC(x(r, t)) = 0 , grid is constrained to C(x) B(x(r, t)) = 0 , boundary conditionswherexr x nsn(r, t) - [ixr ×ns][ ’ normal to the front,ns : normal to the surface C at x,S(r, t) : scalar speed function.There are a variety of ways to define the speed function S. See [6] for somepossible approaches. These equations march the grid in the direction locallyorthogonal to the current front, The parameter t is a time like variable.Figure (1) shows a surface grid generated on a geometry. Volume gridscan also be generated with hyperbolic grid generation. The algorithm isbasically the same as that described above.Overlapping Grid GenerationThe overlapping grid generation algorithm determines how the differentcomponent grids communicate with each other. The algorithm must alsodetermine those parts of component grids that are removed from the com-putation because that part of the grid either lies underneath another gridof higher priority or else that part Of the grid lies outside the domain. Thedetermination of the’ hole region is the critical and key step of the algorithm.Other overlapping grid generators include PEGSUS [18], Beggar [8], DCF[9]and xCog/Chalmesh [11][12].We have developed a newoverlapping grid generator called Ogen [5]. Ogenis based on the CMPGRD grid generator [4] but usually a substantiallydifferent algorithm resulting in a more robust approach. Some of the keyfeatures of Ogen arei. new hole cutting algorithm with a graceful failure mode.2. corrections for boundaries of grids that overlap but do not match;this is an especially troublesome problem when the grids are highlystretched.mummmmmmmmIIFigure 4. Steps in the overlapping grid algorithm, left to right, top tobottom. 1) After interpolating boundaries of grids that share a edge withanother grid (interpolation points are shown small squares). 2) After cut-ting holes. Each physical boundary cuts hole points (large squares) in othergrids. 3)’The hole points are swept out starting from the holes computed inthe previous step. 4) All the possible interpolation points are found so thatexcess overlap can removed or so that better quality interpolation pointscan be used. 5) The final overlapping grid. The excess overlap has beenreduced to the (user specified) amount.3. support for higher order discretizations and’ interpolation.4. support for cell:centred and vertex-centered interpolation.5. optimized algorithms for moving grids.6. Support for block structured adaptive mesh refinement.The basic steps in the overlapping grid algorithm are shown in figure (4).Figure (5) shows an example of a moving grid computation and an adaptivemesh refinement computation. In these examples the grid generator mustbe called repeatedly to regenerate the grids. Optimized versions of theoverlap algorithm are used for this purpose. Further details will appear ina future paper.Acknowledgements: Thanks to the many people who have contributed toOverture in particular the current members of the Overture team, DavidBrown, Kyle Chand, Petri Fast, Brian Miller, Dan Quinlan, Bobby Phillip,¯ and Anders Petersson. This research was performed under the auspices ofthe U.S. Department of Energy by the University of California, LawrenceLivermore National Laboratory under contract No. W-7405-Eng-48.References[1] W. CHiN AND P. BUNING, A hyperbolic surface grid generationscheme and its applications, paper 94-2208, AIAA, 1994.[2] W. M. CHIN, Hyperbolic methods for surface and field grid generation,in Handbook of Grid Generation, J. F. Thompson, B. K. Soni, andN. P. Weatherill, eds., CRC Press, 1999, ch. 5, pp. 1-26.[3] W. M. CHiN AND J. L. STEGER, Enhancements of a three-dimensional hyperbolic grid generation scheme, Applied Mathematicsand Computation, 51 (1992), pp. 181-205.[4] G. CHESSHIRE AND W. HENSHAW, Composite overlapping meshes forthe solution of partial differential equations, J. Comp. Phys., 90 (1990),pp. 1-64.[5] W. HENSHAW, Ogen: An overlapping grid generator for Overture, Re-search Report UCRL-MA-132237, Lawrence Livermore National Lab-oratory, 1998.[6] , The Overture hyperbolic grid generator, user guide, Version 1.0,Research Report UCRL-MA-134240, Lawrence Livermore NationalLaboratory, 1999.[7] W. D. HENSHAW, An algorithm for projecting points onto a patchedCAD model, Research RepOrt UCRL-JC-144016, Lawrence LivermoreNational Laboratory; 2001. Submitted for publication.[8] R. MAPLE AND D. BELK, A new approach to domain decomposition,the beggar code, in Numerical Grid Genertati0n in Computational FluidDynamics and Related Fields, N. Weatherill, ed., Pineridge Press Lim-ited, 1994, pp. 305-314.[9] R. MEAKIN, A new method for establishing intergrid communicationamong systems of overset grids, AIAA paper 91-1586-CP, AmericanInstitute of Aeronautics and Astronautics, 199L[10]A. MEZENTSEV AND T. WOEHLER, Methods and algorithms of au-tomated CAD repair for incremental surface meshing, inSth Interna-tional Meshing Roundtable, 1999, pp. 299-309.[11] N. A. PETERSSON, An algorithm ]or assembling overlapping grid sys-tems, SIAM J. Sci. Comp., 20 (1999), pp. 1995-2021.[12] --, Hole-cutting for three-dimensional overlapping grids, SIAM J.Sci. Comp., 21 (1999), pp. 646-665.[13] N. A. PETERSSON A D K. K. CHAND, Detecting translation errorsin CAD SEt]aces\and preparing geometries ]or mesh generation, inProceeding of the 10th International Meshing Rountable, 2001.[14] A. SHEFFER, T. BLACKER, J. CLEMENTS, AND M. BERCOVIER,Virtual topology operators for meshing, in 6th International MeshingRoundtable, 1997, pp. 49-66.[15] J. R. SHEWCHUK, Triangle: Engineering a 2D Quality Mesh Genera-tor and Delaunay Triangulator, in Applied Computational Geometry:Towards Geometric Engineering, M. C. Lin and D. Manocha, eds.,vol. 1148 of Lecture Notes in Computer Science, Springer-Verlag, May1996, pp. 203-222. From the First ACM Workshop on Applied Com-putational Geometry.[16] J. STEINBRENNER AND J. CHAWNER, Gridgen’s implementation ofpartial differential equation based structured grid generation methods,in 8th International Meshing Rountab!e, 1998. www.andrew.cmu.edu/-user/sowen/imr8.html.[! 7] J. STEINBRENNER, N. WYMAN, AND Z. CHAWNER, Fast surfacemeshing on imperfect CAD models, in 9th International MeshingRountable, 2000. www.andrew.cmu.edu/user/sowen/imr9.html.[18] N. SUHS AND R. TRAMEL, Pegsus 4.0 user’s manual, Research ReportAEDC-TR-91-8, Arnold Engineering Development Center, ArnoldAFB, TN, 1991.0~75 Ir~cor’n~ress= NS,dt=5.56e_o3nU=5.00e-02 u0.500.250.000.250,50Figure 5. Ogen supports the rapid generation of block-structured adap-tive mesh refinement grids and moving grids. The figures show solutionscomputed by OverBlown, of the equations of gas dynamics (top) andindompressible Navier-Stokes equations (bottom).

Generating Composite Overlapping Grids on CAD Geometries

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Generating Composite Overlapping Grids on CAD Geometries

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