1. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-Neto Department of Computing Federal University of Ceará, Brazil CRAbUFC PUC-Rio

2. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Introduction Mesh generation strategy Back-tracking strategy Validation studies Future directions Conclusions

3. Introduction There is a wide variety of techniques in the literature for mesh generation: Watson, 1981 Joe, 1990 Many others Delaunay THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

4. Introduction There is a wide variety of techniques in the literature for mesh generation: Watson, 1981 Joe, 1990 Many others Yerry and Shephard, 1984 Perucchio et al., 1989 Many others Delaunay Octree THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

5. Introduction There is a wide variety of techniques in the literature for mesh generation: Watson, 1981 Joe, 1990 Many others Yerry and Shephard, 1984 Perucchio et al., 1989 Many others Peraire et al., 1988 Lohner and Parikh, 1988 Moller and Hansbo, 1995 Many others Delaunay Octree Advancing-front THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

6. Introduction Objective of the work: Propose a new strategy to generate meshes in 3D Strategy is based on a back-tracking idea It should deal with cracks THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

7. Introduction Objective of the work: Propose a new strategy to generate meshes in 3D Strategy is based on a back-tracking idea It should deal with cracks Requirements: Produces well shaped elements, avoiding elements with poor aspect ratios Generates a mesh that conforms to an existing boundary mesh of the model Has the ability to transition well between regions of highly varying size elements Deals with cracks in a robust and efficient way THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

8. Mesh generation strategy Octree generation: Initialization based on boundary mesh Refinement for maximum cell size Refinement for minimum size disparity Advancing-front procedure: Geometry-based element generation Topology-based element generation Back-tracking element generation Local mesh improvement: Laplacian Smoothing with checks Back-tracking optimization THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

9. Mesh generation strategy Octree generation: Initial boundary mesh THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

10. Mesh generation strategy Octree generation: Initialization based on boundary mesh THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Region with many cells

11. Mesh generation strategy Octree generation: Refinement for maximum cell size THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Cell refined

12. Mesh generation strategy Octree generation: Refinement for minimum size disparity THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Cell refined

13. Mesh generation strategy Advancing-front procedure: Geometry-based element generation Base face chosen from the list of active faces Candidate vertices found Optimal region for a vertex defined Search for existing vertices in the optimal region performed New vertex is created if none is found Additional geometric checks are performed Cracks are treated List of faces is updated THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

14. Mesh generation strategy Advancing-front procedure: Geometry-based element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Optimal vertex N1

15. Mesh generation strategy Advancing-front procedure: Geometry-based element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

16. Mesh generation strategy Advancing-front procedure: Topology-based element generation Base face chosen from the list of faces that failed Candidate vertices found Any vertex is considered Proximity checks for optimization performed Additional geometric checks performed Cracks are treated List of faces is updated THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

17. Mesh generation strategy Advancing-front procedure: Back-tracking element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

18. Mesh generation strategy Advancing-front procedure: Back-tracking element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Element to be deleted (a,b,c,d) Unmeshable region

19. Mesh generation strategy Advancing-front procedure: Back-tracking element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Element to be deleted (a,b,c,d) Meshable region Unmeshable region

20. Mesh generation strategy Advancing-front procedure: Back-tracking element generation THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Element to be deleted (a,b,c,d) Meshable region Region meshed Unmeshable region

21. Mesh generation strategy Local mesh improvement: Laplacian smoothing with checks THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

22. Mesh generation strategy Local mesh improvement: Back-tracking optimization THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Bad element (in gray)

23. Back-tracking strategy General description (definitions): E k => element to be considered R(E k ) => local region in the neighborhood of E k M(E) => set of all mesh elements D(E k ) => set of deleted elements for E k (during back-tracking) F(E k ) => set of new generated elements for R(E k ) THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

24. Back-tracking strategy General description (general algorithm): Determine region R(E k ) for given element E k R(E k ) THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

25. Back-tracking strategy General description (general algorithm): Determine region R(E k ) for given ill-shaped element E k Perform visibility tests for region R(E k ) Check if R(E k ) is a star-shaped region R(E k ) Star-shaped? THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

26. Back-tracking strategy General description (general algorithm): Determine region R(E k ) for given ill-shaped element E k Perform visibility tests for region R(E k ) Check if R(E k ) is a star-shaped region If there are intersections: Update R(E k ) by R(E k ) new = R(E k ) old – D(E k ) Update M(E) by M(E) new = M(E) old – D(E k ) R(E k ) Star-shaped? Update R(E k ) Update M(E) No THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

27. Back-tracking strategy General description (general algorithm): Determine region R(E k ) for given ill-shaped element E k Perform visibility tests for region R(E k ) Check if R(E k ) is a star-shaped region If there are intersections: Update R(E k ) by R(E k ) new = R(E k ) old – D(E k ) Update M(E) by M(E) new = M(E) old – D(E k ) Repeat the process until a kernel is found. Then: Find the set of new elements F(E k ) Update M(E) by M(E) new = M(E) old + F(E k ) R(E k ) Star-shaped? Find F(E k ) Update M(E) End Update R(E k ) Update M(E) Yes No THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

28. Two-dimensional Back-tracking R(E k ) => P(T k ) P => polygon T => triangle M(E) => M(T) D(E k ) => D(T k ) F(E k ) => F(T k ) P(T k ) Star-shaped? Find F(T k ) Update M(T) End Update P(T k ) Update M(T) Yes No THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

29. Two-dimensional Back-tracking Element to be deleted (in gray) THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

30. Two-dimensional Back-tracking Element to be deleted (in gray) New region THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

31. Two-dimensional Back-tracking Element to be deleted (in gray) New elements New region THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

32. Three-dimensional Back-tracking R(E k ) => P(T k ) P => polyhedron T => tetrahedron M(E) => M(T) D(E k ) => D(T k ) F(E k ) => F(T k ) P(T k ) Star-shaped? Find F(T k ) Update M(T) End Update P(T k ) Update M(T) Yes No THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

33. Three-dimensional Back-tracking Element to be deleted (a,b,c,d) THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

34. Three-dimensional Back-tracking Element to be deleted (a,b,c,d) New region THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

35. Three-dimensional Back-tracking Element to be deleted (a,b,c,d) New elements New region THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

36. Validation studies Quality measures: V => volume of a tetrahedron Si => length of an edge Optimal value => 8.5 Range => [ 1,  ] Threshold: Lower bound => 5.0 Upper bound => 8.5*factor(30) Use of metric (Parthasarathy et al., 1993): Sensitive to small changes Computationally efficient Any metric can be used Definition of ill-shaped element THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

37. Validation studies Quality measures:  = 3R i / R c R i => radius of inscribed sphere R c => radius of circumscribed sphere Optimal value => 1.0 Range => [ 0,  ] Threshold: Lower bound => 0.2 Upper bound => 0.5 Use of metric: More intuitive interpretation Widely used in the literature Any metric can be used Evaluation of element quality THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

38. Validation studies Example 1: A portion of a housing THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

39. Validation studies Example 1: A portion of a housing THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Crack

40. Validation studies Example 1: A portion of a housing Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

41. Validation studies Example 1: A portion of a housing Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After Very poor elements Best elements THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

42. Validation studies Example 2: A portion of a spiral-bevel gear THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

43. Validation studies Example 2: A portion of a spiral-bevel gear THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Crack

44. Validation studies Example 2: A portion of a spiral-bevel gear Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

45. Validation studies Example 2: A portion of a spiral-bevel gear Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Very poor elements Best elements

46. Validation studies Example 3: A portion of a turbofan hub THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

47. Validation studies Example 3: A portion of a turbofan hub THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007 Crack

48. Validation studies Example 3: A portion of a turbofan hub Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

49. Validation studies Example 3: A portion of a turbofan hub Element quality % of elements 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  Before After Very poor elements Best elements THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

50. Validation studies Table for comparison ExampleHistogram#Elements  avg  min  max HousingBefore164630.6750.0250.729 HousingAfter170430.6960.0230.740 GearBefore173860.6840.0250.738 GearAfter169900.6990.0330.742 TurbofanBefore96280.6680.0180.733 TurbofanAfter100460.6920.0220.741 THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

51. Validation studies Table for comparison ExampleHistogram#Elements  avg  min  max HousingBefore164630.6750.0250.729 HousingAfter170430.6960.0230.740 GearBefore173860.6840.0250.738 GearAfter169900.6990.0330.742 TurbofanBefore96280.6680.0180.733 TurbofanAfter100460.6920.0220.741 Increases THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

52. Validation studies Table for comparison ExampleHistogram#Elements  avg  min  max HousingBefore164630.6750.0250.729 HousingAfter170430.6960.0230.740 GearBefore173860.6840.0250.738 GearAfter169900.6990.0330.742 TurbofanBefore96280.6680.0180.733 TurbofanAfter100460.6920.0220.741 Increases Decreases THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

53. Future directions Back-tracking strategy More tests Convergence studies Implementation of the strategy in parallel Based on the octree cells Does not need domain subdivision Automatic modification of boundary meshes Have to meet advancing-front requirements Mesh generation for multi-domains models Treatment of multi-connected regions THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007

54. Conclusions Strategy proposed is based on a Back-tracking idea It is an iterative process than can converge or not Original configuration for each region is “saved” for recuperation Mesh improvement attested by validation studies Input mesh has already good quality % of very poor elements drops significantly (to 0.49% - 0.90%) % of best elements increases (around 2% to 4%) In large models, this % can be very representative Convergence studies would be welcomed THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento Cavalcante-NetoAgosto,2007