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 u v 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. A B Surface Meshing C NCNC 3D Surface Advancing Front Define tangent plane at front by averaging normals at A and B Tangent plane

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

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

7. 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 u v u v

9. 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. u v u v

10. Surface Meshing 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 u v u v Parametric Space Mesh Generation Warped parametric space

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

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

13. Surface Meshing 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: u v x y z Parametric Space Mesh Generation X Q u v X Q M(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. Smoothing Topological Improvement Mesh Post-Processing

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

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

19. Smoothing P1P1 P2P2 P3P3 P4P4 P5P5 P Laplacian Averaging Methods (Field, 1988)

20. Smoothing P1P1 P2P2 P3P3 P4P4 P5P5 P Laplacian Averaging Methods Centroid of attached nodes Can create inverted elements (Field, 1988)

21. Smoothing C1C1 P Area Centroid Weighted Averaging Methods Weighted average of triangle centroids C2C2 C3C3 C4C4 C4C4 A1A1 A2A2 A i = area of triangle i C i = centroid of triangle i

22. Smoothing Radius Ratio r in r circ A C B D v = volume of tet s i = areas of four faces of tet a,b,c = products of the lengths of opposite edges of tet (Liu, Joe, 1994)

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

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

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

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