computational modeling sciences jfs hexahedral sheet insertion jason shepherd october 2008
TRANSCRIPT
jfs
Computational Modeling Sciences
Hexahedral Sheet Insertion
Jason Shepherd
October 2008
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Computational Modeling Sciences
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.
+ = = +
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Computational Modeling Sciences
Outline
• Definitions• Framework description• Recent efforts• Conclusion
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Computational Modeling Sciences
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.
Boundary SheetsFundamental Sheet
Fundamental Mesh Non-Fundamental Mesh
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin1.
2.
3.
*From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,”published Doctoral Dissertation, University of Utah, May 2007.
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin1.
2.
3.
From J. Shepherd “Topologic and Geometric Constraint-Based Hexahedral Mesh Generation,”published Doctoral Dissertation, University of Utah, May 2007.
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Computational Modeling Sciences
-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
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Computational Modeling Sciences
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)
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Computational Modeling Sciences
-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
D B
C
E
B
F
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin1.
2.
3.
•Mesh Matching•Coarsening
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Computational Modeling Sciences
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
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin
•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.
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin
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 object2. 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…
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin
Mnc→Mc
Anc→Ac
•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.
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Computational Modeling Sciences
Technical Framework
Mc→Mf
Mc→M*c
Mf→Mmin
Mnc→Mc
Anc→Ac
•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.
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Computational Modeling Sciences
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.)
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Computational Modeling Sciences
Mnc → Mc
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Computational Modeling Sciences
Demos
• Sbase1• UCP5
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Examples
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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.