1. Computational Modeling Sciences jfs Hexahedral Sheet Insertion Jason Shepherd October 2008

2. Computational Modeling Sciences jfs Motivation … create an alternate geometric representation consisting of hexahedral elements… Given a geometric representation of an object, G,… Model Creation 25% of DTA time* Decomposition for Hex Meshing 32% of DTA time* Hex Meshing 14% of DTA time* However, decomposition for pave-sweep is more art than science… *timings based on a Design-thru-Analysis study conducted by the DART project Unfortunately, there is only a limited class of geometries for which hexahedral meshing (pave and sweep) can be automated with current tools/algorithms. So…

3. Computational Modeling Sciences jfs 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. + = =+

4. Computational Modeling Sciences jfs Outline Definitions Framework description Recent efforts Conclusion

5. Computational Modeling Sciences jfs Fundamental Hexahedral Meshes Definition: A fundamental mesh (M f ) 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 (M c ) 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. Boundary SheetsFundamental Sheet Fundamental MeshNon-Fundamental Mesh

6. Computational Modeling Sciences jfs Definitions Sheet conformation affects quality*… …and each hex derives from three sheets. Conformations of each of the sheets affect overall hexahedral quality. (Three planar sheets have highest potential for quality. Feasible regions for optimimum quality are reduced when sheet curvature and orthogonality are reduced.) *See J. F. Shepherd, C. R. Johnson, “Hexahedral Mesh Generation Constraints,” Engineering with Computers, Vol. 24, No. 3, March 2008.

7. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min 1. 2. 3. *From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,” published Doctoral Dissertation, University of Utah, May 2007.

8. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min 1. 2. 3. From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,” published Doctoral Dissertation, University of Utah, May 2007.

9. Computational Modeling Sciences jfs -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→MfMc→Mf

10. Computational Modeling Sciences jfs 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)

11. Computational Modeling Sciences jfs -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 A DB C E B F

12. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min 1. 2. 3. Mesh Matching Coarsening

13. Computational Modeling Sciences jfs M c → M* c Mesh Matching - Matt Staten, et al., Poster at the 16 th International Meshing Roundtable. Hexahedral Coarsening – Adam Woodbury, et al., Paper at the 17 th International Meshing Roundtable

14. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min 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.

15. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min Definition: A hexahedral mesh is minimal (M min ) 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: M f →M min - Appears to hold true, except when thin regions are present in the mesh…

16. Computational Modeling Sciences jfs M f → M min All Hexahedral Meshes in G All Fundamental Meshes in G Minimal Mesh in G Fundamentality is not a requirement for quality hexahedral meshes, but is useful as a tool for landmarking meshes with respect to geometry. Characterizing a mesh as fundamental is testable. A mesh that is not a fundamental mesh can be converted to fundamental using sheet insertion (and/or other) algorithms. Showing that the minimal mesh is also related to the fundamental mesh can be used to reduce the complexity of an all-hex algorithm (i.e., if I can prove that my algorithm satisfies the fundamental mesh requirements, I can also prove that it will generate a mesh in G).

17. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min M nc →M c A nc →A c A non-conforming mesh (M nc ) is defined as a ‘topologically equivalent’ and ‘geometrically similar’ mesh to a given geometry, G. (Note: The base quality of M nc and the degree of ‘geometric similarity’ of M nc has a great impact on the final quality of M c.) An assembly mesh (A x ) is simply of collection of geometries meshed contiguously.

18. Computational Modeling Sciences jfs Technical Framework Mc→MfMc→Mf Mc→M*cMc→M*c M f →M min M nc →M c A nc →A c A non-conforming mesh (M nc ) is defined as a ‘topologically equivalent’ and ‘geometrically similar’ mesh to a given geometry, G. (Note: The base quality of M nc and the degree of ‘geometric similarity’ of M nc has a great impact on the final quality of M c.) An assembly mesh (A x ) is simply of collection of geometries meshed contiguously.

19. Computational Modeling Sciences jfs M nc → M c 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 M nc to G This transformation can be accomplished by embedding the geometric-topology boundary ‘graph’ of G into the mesh-topology boundary ‘graph’ of M nc –(Some embeddings may require mesh-enrichment.)

20. Computational Modeling Sciences jfs M nc → M c

21. Computational Modeling Sciences jfs Demos Sbase1 UCP5

22. Computational Modeling Sciences jfs Examples

23. Computational Modeling Sciences jfs 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.