1. Filling Arbitrary Holes in Finite Element Models 17 th International Meshing Roundtable 2008 Schilling, Bidmon, Sommer, and Ertl

2. Motivation Source: Schilling et al. Computer-Aided Engineering (CAE) –Preprocessing for Finite Element analysis e.g. computational fluid dynamics simulation for acoustics thousands of holes

3. Related Work Volumetric Approaches –Split region around a hole into inner and outer parts Boundary layer defines missing surface –[Davis et al.]: Signed distance function is “extended through the volume” until it spans all holes Resulting quality not sufficient for CAE –[Podolak et al.] Spatial decomposition for complex holes Hard to treat many independent meshes Geometry-Based Techniques (more promising) –[Wang et al.]: Moving least squares (MLS) to reconstruct locally smooth surface MLS: Evaluate vertices’ neighborhood using weighted least squares Unsuitable for CAE general hole shapes –[Tekumalla et al.]*: Advancing front method to close holes incrementally –[Liepa et al.]*: Dynamic programming to create minimal surfaces –Holes need not be planar –* basis of this new work Source: Schilling et al.

4. Types of Holes Source: Schilling et al.

5. Hole Patching Requirements Preserve existing geometric features. Provide acceptable results even when neighborhood information is absent. Interactive influence on shape of patching mesh. Source: Schilling et al.

6. Setting up Semantic Holes across Multiple Meshes Balance user effort and computation speed for complex semantic holes. v s, v t are selected by user (could be from different meshes) Use a bounding volume hierarchy to calculate distances and find connecting node pairs. Connection criteria (see Figure 3): 1)Minimize . 2) Minimize || d j || 3) Minimize Source: Schilling et al. vj1vj1

7. Setting up Semantic Holes across Multiple Meshes (continued) Source: Schilling et al.

8. Patching Holes: Advancing Front Adapted from [Tekumalla et al.]* Edge (by default) is part of advancing front. Source: Schilling et al. makeConvex addVertices bisector of edge uses MLS *DP approach from [Liepa et al.] is adapted to fill small, simple holes with minimal surfaces.

9. Patching Holes: Potential Advancing Front Problems Enclosed feature lines can be removed by makeConvex. MLS in addVertices can extend patching mesh too far (see Figure 6.) Source: Schilling et al. [11]= Tekumalla et al. [12] = Wang et al.

10. Patching Holes: Fixing Advancing Front Problems In makeConvex, test & avoid cutting feature lines (Fig. 7a). Instead of MLS, place new node in plane of adjacent element. User-defined bending angle is enforced later. Source: Schilling et al.

11. Patching Holes: User-Defined Bending Source: Schilling et al.

12. Patching Holes: Collinearity Issue Collinear neighboring nodes can prevent a node from moving, hindering optimization. –Moving might increase distance to neighbors. Source: Schilling et al.

13. Patching Holes: Fronts Missing Each Other Fronts not oriented towards each other can miss each other as they grow. Source: Schilling et al. consider as a neighbor

14. Patching Holes: Bending Enforced by User Sometimes only C 0 continuity is desired. Initial maximum bend is allowed. Source: Schilling et al.

15. Patching Holes: Edge Lengths of New Triangles Two user-controllled edge length options: Source: Schilling et al. leaves gap between meshes (a) (b)

16. Results Hole detection, hole meshing and volume mesh generation can be done now in hours rather than days. Hole classification filters help reduce number of holes. –Car model: 463 parts and 3359 initial holes, reduced to 248 holes. 201 holes closed automatically. Source: Schilling et al.

17. Results Source: Schilling et al.

18. Results Source: Schilling et al.