1. June 23, 2008 1 Variational tetrahedral meshing of mechanical models for FEA Matthijs Sypkens Smit Willem F. Bronsvoort CAD ’08 Conference, Orlando, Florida Faculty of Electrical Engineering, Mathematics and Computer Science

2. June 23, 2008 CAD’082 Outline Research motivation Variational tetrahedral meshing (VTM) The shortcomings of VTM for mechanical models Improvements / recommendations Results Conclusions

3. June 23, 2008 CAD’083 Research motivation (1) Analysis / product simulation: Reduces need to construct real world test models Decreases length of product development cycle Increases quality/safety Lowers total cost Most popular method: finite element analysis Requires a mesh / decomposition of geometry Can work with many types of meshes

4. June 23, 2008 CAD’084 Research motivation (2) Meshes and mesh quality: Zero bad quality elements Near regular elements in computational space Accurate representation of model boundary Efficient variation in sizing w.r.t. to accuracy Quality is context dependent. Generally: A higher quality mesh results in a quicker, more accurate, and more reliable analysis.

5. June 23, 2008 CAD’085 Research motivation (3) Variational tetrahedral meshing (VTM) offers: Exceptional quality distribution: Majority near regular elements Effectively no bad quality elements Mesh sizing VTM was not conceived for mechanical models.  We have investigated this application  We have made several improvements

6. June 23, 2008 CAD’086 Variational tetrahedral meshing (1) Central ideas: Delaunay mesh Optimisation through connectivity and node locations Boundary conformance as continuous process simultaneously executed with optimisation Delaunay property:

7. June 23, 2008 CAD’087 Variational tetrahedral meshing (2) Delaunay optimisation: Delaunay connectivity globally optimal Node relocation improves local quality  Combine these two into an optimisation procedure 1: 2: 3:

8. June 23, 2008 CAD’088 Variational tetrahedral meshing (3) Achieving boundary conformance: “Shape” the mesh by pulling the outer nodes towards the boundary:  Use boundary samples to perform pulling Separate treatment of corner, edge and face samples

9. June 23, 2008 CAD’089 Variational tetrahedral meshing (4) The role of boundary samples:

10. June 23, 2008 CAD’0810 Variational tetrahedral meshing (5)  Interior mesh optimising  Boundary shaping The algorithm: Initialise data structures Spread out nodes Optimisation loop Extract mesh

11. June 23, 2008 CAD’0811 Variational tetrahedral meshing (6) Mesh extraction: Delaunay mesh covers the convex hull Remove elements that are not part of the intended shape

12. June 23, 2008 CAD’0812 The weaker points of VTM (1) Points of attention for mechanical models: Boundary conformance  An accurate representation of the boundary should be present in the mesh Mesh extraction  An accurate representation of the boundary should result from Delaunay mesh extraction

13. June 23, 2008 CAD’0813 The weaker points of VTM (2) Boundary conformance: There is a risk that no set of tetrahedra exists in the mesh that is acceptable to represent the model boundary 3D: Tetrahedron “crossing” the boundary 2D: correct vs. “crossing”

14. June 23, 2008 CAD’0814 The weaker points of VTM (3) Mesh extraction: The heuristics for mesh extraction frequently fail to deliver an acceptable result Excess and missing tetrahedra after extraction:

15. June 23, 2008 CAD’0815 Improving the results (1) To get better results applying VTM on mechanical models, we must first understand the cause(s) of the defects. Boundary conformance: Why do we expect boundary conformance in our mesh? How can it go wrong? What can be done about it?

16. June 23, 2008 CAD’0816 Improving the results (2) Why do we expect boundary conformance? The boundary samples always pull on their closest node.  Result: (near) Gabriel edges and triangles everywhere on the boundary.

17. June 23, 2008 CAD’0817 Improving the results (3) How can boundary conformance fail? If the number of boundary samples is low, less of the boundary will be Gabriel.

18. June 23, 2008 CAD’0818 Improving the results (4) What can be done about it? The risk of failing boundary conformance can be reduced by increasing the number of boundary samples From experience: 10 samples per node on the boundary makes failure rare Caveat: Small dihedral angles

19. June 23, 2008 CAD’0819 Improving the results (5) Lack of resolution A geometry needs a certain amount of nodes for an accurate representation by a conforming Delaunay triangulation  Even with ample boundary samples, boundary conformance can still fail due to lack of nodes

20. June 23, 2008 CAD’0820 Improving the results (6) Detect lack of resolution and locally fix it Indication of lack of nodes:  Samples from different geometrical features are pulling on the same node Fix: split the node (= add a new node right next to it)

21. June 23, 2008 CAD’0821 Improving the results (7) Detect lack of resolution and locally fix it Example of how effective splitting is:  Start with one interior node and 54 corner nodes: continue splitting until no more splits occur Fix is a local solution; With many nodes missing better to start with more nodes

22. June 23, 2008 CAD’0822 Improving the results (8) Mesh extraction Improved mesh extraction logic: Project boundary nodes to model boundary Tetrahedra with an interior node are inside the model  Only tetrahedra with 4 boundary nodes left to consider If centroid of a tetrahedron falls outside, then tetrahedron outside the model Else tetrahedron inside the model

23. June 23, 2008 CAD’0823 Improving the results (9) Mesh extraction 2D example: Location of centroid is decisive Works for both concave and convex regions

24. June 23, 2008 CAD’0824 Results (1) Gear mesh

25. June 23, 2008 CAD’0825 Results (2) Gear volume-length ratio

26. June 23, 2008 CAD’0826 Results (3) Gear min/max dihedral angle

27. June 23, 2008 CAD’0827 Conclusions Number of boundary samples and number of nodes are important for success Node splitting is effective at enforcing boundary conformance Improved mesh extraction recovers the intended boundary Improved VTM can be used for the construction of meshes of mechanical models Results in high quality meshes of mechanical models

28. June 23, 2008 CAD’0828 Credits Research supported by NWO: Computational Geometry Algorithms Library (CGAL) was used in the creation of some of the illustrations