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Hexahedral Sheet Insertion. Jason Shepherd October 2008. Background. Hexahedral meshes are composed of layers of hexahedral elements. (These layers can also be thought of as manifold surfaces, referred to as sheets.)
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Hexahedral Sheet Insertion Jason Shepherd October 2008
Background • Hexahedral meshes are composed of layers of hexahedral elements. • (These layers can also be thought of as manifold surfaces, referred to as sheets.) • New layers can be inserted into existing meshes using sheet insertion techniques (i.e., pillowing, dicing, grafting, meshcutting, etc.) • The goal, then, is to 1. define minimal sets of layers that must be present to capture the geometric object, 2. constrain the topology and geometry of the layers to satisfy analytic, quality, and topologic constraints for the final hexahedral mesh, and 3. automate the process. = + = +
Outline • Definitions • Framework description • Recent efforts • Conclusion
Fundamental Sheet Boundary Sheets Fundamental Mesh Non-Fundamental Mesh Fundamental Hexahedral Meshes • Definition: A fundamental mesh (Mf ) is a hexahedral mesh that contains one sheet for every surface, at least one continuous two-sheet intersection (chord) for every curve, and (vertex valence - 2) triple-point intersections (centroids) for every geometric vertex. • Definition: A conforming mesh (Mc) is a hexahedral mesh that conforms to a given geometry. That is, every geometric surface corresponds to a topologically equivalent collection of mesh faces, every curve corresponds to a line of mesh edges, etc.
Technical Framework Mc→Mf Mf→Mmin 1. 3. Mc→M*c 2. *From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,” published Doctoral Dissertation, University of Utah, May 2007.
Technical Framework Mc→Mf Mf→Mmin 1. 3. Mc→M*c 2. From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,” published Doctoral Dissertation, University of Utah, May 2007.
Mc→Mf • -Mouse model is courtesy of Jeroen Stinstra of the SCI Institute at the University of Utah • Bumpy Sphere model is provided courtesy of mpii by the AIM@SHAPE Shape Repository • Brain and Hand Models are provided courtesy of INRIA by the AIM@SHAPE Shape Repository
Mc→Mf Created for S. Shontz's IVC Collaboration with F. Lynch, M.D. (PSU Hershey Medical Center), M. Singer (LLNL), S. Sastry (PSU), and N. Voshell (PSU)
C A E B D B F • -Models A, C, D, E are provided courtesy of ANSYS • Model B is provided courtesy of Tim Tautges by the AIM@SHAPE Shape Repository • Model F is provided courtesy of Inria by the AIM@SHAPE Shape Repository
Technical Framework Mc→Mf Mf→Mmin 1. 3. Mc→M*c 2. • Mesh Matching • Coarsening
Mc → M*c Mesh Matching - Matt Staten, et al., Poster at the 16th International Meshing Roundtable. Hexahedral Coarsening – Adam Woodbury, et al., Paper at the 17th International Meshing Roundtable
Technical Framework Mc→Mf Mf→Mmin Mc→M*c • Proofs for these two transformations available in: • F. Ledoux, J. Shepherd, “Topological and Geometrical Properties of Hexahedral Meshes,” to appear in Engineering with Computers. • F. Ledoux, J. Shepherd, “Topological Modifications of Hexahedral Meshes via Sheet Operations: A Theoretical Study” to appear in Engineering with Computers.
Technical Framework Mc→Mf Mf→Mmin Mc→M*c • Definition: A hexahedral mesh is minimal (Mmin) within a geometric object if: • 1. The mesh contains the fewest number of hexahedra for all sets of possible hexahedral meshes for a given object • 2. The mesh does not contain any doublets. • 3. The mesh does not contain any 'geometric' doublets (i.e. two adjacent faces on a hex cannot belong to a single surface, and two adjacent edges of a hex cannot belong to a single curve.) • Conjecture: Mf→Mmin • - Appears to hold true, except when thin regions are present in the mesh…
Technical Framework Anc→Ac Mc→Mf Mf→Mmin Mnc→Mc Mc→M*c • A non-conforming mesh (Mnc) is defined as a ‘topologically equivalent’ and ‘geometrically similar’ mesh to a given geometry, G. • (Note: The base quality of Mnc and the degree of ‘geometric similarity’ of Mnc has a great impact on the final quality of Mc.) • An assembly mesh (Ax) is simply of collection of geometries meshed contiguously.
Technical Framework Anc→Ac Mc→Mf Mf→Mmin Mnc→Mc Mc→M*c • A non-conforming mesh (Mnc) is defined as a ‘topologically equivalent’ and ‘geometrically similar’ mesh to a given geometry, G. • (Note: The base quality of Mnc and the degree of ‘geometric similarity’ of Mnc has a great impact on the final quality of Mc.) • An assembly mesh (Ax) is simply of collection of geometries meshed contiguously.
Mnc → Mc • Converting a non-conforming mesh to a conforming mesh requires assignment of topologically equivalent collections of mesh entities to appropriate geometric entities • i.e., a topologically equivalent collection of quadrilaterals for each surface, • A line of mesh edges for each curve, • A node for each vertex. • Optimally, reducing the distortion caused by the transformation is beneficial, and is largely controlled by the ‘geometric-similarity’ of Mnc to G • This transformation can be accomplished by embedding the geometric-topology boundary ‘graph’ of G into the mesh-topology boundary ‘graph’ of Mnc • (Some embeddings may require mesh-enrichment.)
Demos • Sbase1 • UCP5
Needed efforts • Algorithmic improvements • Automated guarantees on topology equivalence • Conflict-free network/graph searches • Geometric similarity (how similar is close enough?) • Using smoothing for non-uniform scaling • Getting the ‘right’ mesh • Alternative sheet insertions can produce better quality (although the current solution is generally applicable…) • Assemblies • Given a topologically-equivalent, geometrically-similar meshed assembly, the transformations work for multiple volumes • Geometric tolerance • Selective topology capture is feasible • Parallel meshing • Sheet insertions can be localized allowing for potential parallel application.
