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THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS

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

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THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS

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

  2. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Introduction • Mesh generation strategy • Back-tracking strategy • Validation studies • Future directions • Conclusions

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

  4. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  5. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  6. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  7. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  8. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  9. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Octree generation: Initial boundary mesh

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

  11. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Octree generation: Refinement for maximumcell size Cell refined

  12. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Octree generation: Refinement for minimum sizedisparity Cell refined

  13. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

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

  15. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Advancing-front procedure: Geometry-based element generation

  16. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  17. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Advancing-front procedure: Back-tracking element generation

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

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

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

  21. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Mesh generation strategy • Local mesh improvement: Laplacian smoothing with checks

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

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

  24. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Back-tracking strategy • General description (general algorithm): • Determine region R(Ek) for given element Ek R(Ek)

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

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

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

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

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

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

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

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

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

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

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

  36. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • 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

  37. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Validation studies • Quality measures: •  = 3Ri / Rc • Ri=> radius of inscribed sphere • Rc=> radius of circumscribed sphere • Optimal value => 1.0 • Range => [ 0, 1 ] • 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

  38. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Validation studies Example 1: A portion of a housing

  39. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Validation studies Example 1: A portion of a housing Crack

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

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

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

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

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

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

  46. THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Validation studies Example 3: A portion of a turbofan hub

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

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

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

  50. Example Histogram #Elements avg min max Housing Before 16463 0.675 0.025 0.729 Housing After 17043 0.696 0.023 0.740 Gear Before 17386 0.684 0.025 0.738 Gear After 16990 0.699 0.033 0.742 Turbofan Before 9628 0.668 0.018 0.733 Turbofan After 10046 0.692 0.022 0.741 THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKSJoaquim Bento Cavalcante-Neto Agosto,2007 • Validation studies Table for comparison

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