Surface Meshing

1. Surface Meshing Material tret de: S. J. Owen, "A Survey of Unstructured Mesh Generation Technology", Proceedings 7th International Meshing Roundtable, 1998.

2. Surface Meshing Direct 3D Meshing Parametric Space Meshing v u • Elements formed in 3D using actual x-y-z representation of surface • Elements formed in 2D using parametric representation of surface • Node locations later mapped to 3D

3. Surface Meshing A B • 3D Surface Advancing Front • form triangle from front edge AB

4. Surface Meshing NC Tangent plane A C B • 3D Surface Advancing Front • Define tangent plane at front by averaging normals at A and B

5. Surface Meshing NC A C D B • 3D Surface Advancing Front • define D to create ideal triangle on tangent plane

6. Surface Meshing NC A C B D • 3D Surface Advancing Front • project D to surface (find closest point on surface)

7. Surface Meshing • 3D Surface Advancing Front • Must determine overlapping or intersecting triangles in 3D. (Floating point robustness issues) • Extensive use of geometry evaluators (for normals and projections) • Typically slower than parametric implementations • Generally higher quality elements • Avoids problems with poor parametric representations (typical in many CAD environments) • (Lo,96;97); (Cass,96)

8. Surface Meshing Parametric Space Mesh Generation • Parameterization of the NURBS provided by the CAD model can be used to reduce the mesh generation to 2D v u v u

9. v u Surface Meshing Parametric Space Mesh Generation • Isotropic: Target element shapes are equilateral triangles • Equilateral elements in parametric space may be distorted when mapped to 3D space. • If parametric space resembles 3D space without too much distortion from u-v space to x-y-z space, then isotropic methods can be used. v u

10. v u Surface Meshing Parametric Space Mesh Generation • Parametric space can be “customized” or warped so that isotropic methods can be used. • Works well for many cases. • In general, isotropic mesh generation does not work well for parametric meshing v u Warped parametric space

11. v u Surface Meshing Parametric Space Mesh Generation • Anisotropic: Triangles are stretched based on a specified vector field • Triangles appear stretched in 2d (parametric space), but are near equilateral in 3D v u

12. v u y x z Surface Meshing Parametric Space Mesh Generation • Stretching is based on field of surface derivatives • Metric, M can be defined at every location on surface. Metric at location X is:

13. v u y v x z u Surface Meshing Parametric Space Mesh Generation • Distances in parametric space can now be measured as a function of direction and location on the surface. Distance from point X to Q is defined as: Q M(X) Q X X

14. Surface Meshing Parametric Space Mesh Generation • Use essentially the same isotropic methods for 2D mesh generation, except distances and angles are now measured with respect to the local metric tensor M(X). • Can use Delaunay (George, 99) or Advancing Front Methods (Tristano,98)

15. Surface Meshing Parametric Space Mesh Generation • Is generally faster than 3D methods • Is generally more robust (No 3D intersection calculations) • Poor parameterization can cause problems • Not possible if no parameterization is provided • Can generate your own parametric space (Flatten 3D surface into 2D) (Marcum, 99) (Sheffer,00)

16. Mesh Post-Processing Smoothing Topological Improvement

17. Post-Processing Smoothing Adjust locations of nodes without changing mesh topology (element connectivity) Topological Improvement Change connectivity without affecting node locations

18. Post-Processing Smoothing • Averaging Methods • Optimization Based • Distortion Metrics • Combined: • Laplacian/Optimization • based smoothing

19. Smoothing P4 Averaging Methods P3 P P5 P2 P1 Laplacian (Field, 1988)

20. Smoothing P4 Averaging Methods P3 P P5 Centroid of attached nodes Can create inverted elements P2 P1 Laplacian (Field, 1988)

21. Smoothing Averaging Methods Weighted average of triangle centroids C4 C4 P C3 C1 A1 C2 Ai=area of triangle i A2 Ci=centroid of triangle i Area Centroid Weighted

22. D B rin A rcirc C Smoothing Radius Ratio v = volume of tet si = areas of four faces of tet a,b,c = products of the lengths of opposite edges of tet (Liu, Joe, 1994)

23. Post-Processing Smoothing • Averaging Methods • Optimization Based • Distortion Metrics • Combined Laplacian/Optimization based smoothing Topological Improvement • Triangles • Tetrahedra • Quads

24. Topology Improvement Diagonal Flip Possible Criteria Comparison before/after: Delaunay criterion Distortion metrics Node valence Deviation from surface (Edelsbrunner, Shah, 1996) (Canann, Muthukrishnan, Phillips, 1996)

25. Topology Improvement Diagonal Flip Possible Criteria Comparison before/after: Delaunay criterion Distortion metrics Node valence Deviation from surface (Edelsbrunner, Shah, 1996)

26. Topology Improvement Node valence = number of edges connected to node 7 6 7 6 Ideal node valence for triangle mesh is 6 edges/node Swapping can improve node valence Allowing Smoothing to do a better job (Canann, Muthukrishnan, Phillips, 1996)