THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS Joaquim Bento...
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THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY
DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto
Department of Computing
Federal University of Ceará, Brazil
CRAb
UFC PUC-Rio
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Introduction
• Mesh generation strategy
• Back-tracking strategy
• Validation studies
• Future directions
• Conclusions
• 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-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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Mesh generation strategy
Octree generation: Initial boundary mesh
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Mesh generation strategy
Octree generation: Initialization based on boundary mesh
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Region with many cells
• Mesh generation strategy
Octree generation: Refinement for maximum cell size
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Cell refined
• Mesh generation strategy
Octree generation: Refinement for minimum size disparity
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Cell refined
• 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-Neto Agosto,2007
• Mesh generation strategy
Advancing-front procedure: Geometry-based element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Optimal vertex N1
• Mesh generation strategy
Advancing-front procedure: Geometry-based element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Mesh generation strategy
Advancing-front procedure: Back-tracking element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Mesh generation strategy
Advancing-front procedure: Back-tracking element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Element to be deleted (a,b,c,d)
Unmeshable region
• Mesh generation strategy
Advancing-front procedure: Back-tracking element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Element to be deleted (a,b,c,d)
Meshable regionUnmeshable region
• Mesh generation strategy
Advancing-front procedure: Back-tracking element generation
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Element to be deleted (a,b,c,d)
Meshable region Region meshedUnmeshable region
• Mesh generation strategy
Local mesh improvement: Laplacian smoothing with checks
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Mesh generation strategy
Local mesh improvement: Back-tracking optimization
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Bad element (in gray)
• 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)
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Back-tracking strategy
General description (general algorithm):
• Determine region R(Ek) for given element Ek
R(Ek)
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
R(Ek)
Star-shaped?
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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)
R(Ek)
Star-shaped?
Update R(Ek)
Update M(E)
No
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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)
R(Ek)
Star-shaped?
Find F(Ek)
Update M(E)
End
Update R(Ek)
Update M(E)
Yes
No
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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)
P(Tk)
Star-shaped?
Find F(Tk)
Update M(T)
End
Update P(Tk)
Update M(T)
Yes
No
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Two-dimensional Back-tracking
Element to be deleted (in gray)
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• 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-Neto Agosto,2007
• 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-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)
P(Tk)
Star-shaped?
Find F(Tk)
Update M(T)
End
Update P(Tk)
Update M(T)
Yes
No
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• 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-Neto Agosto,2007
• 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-Neto Agosto,2007
• Three-dimensional Back-tracking
Element to be deleted (a,b,c,d)
New elementsNew region
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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
V
Sγ rms
3
5
0
2
6
1i irms SS
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim 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, ]• 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-Neto Agosto,2007
• Validation studies
Example 1: A portion of a housing
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Example 1: A portion of a housing
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Crack
• Validation studies
Example 1: A portion of a housing
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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-Neto Agosto,2007
• Validation studies
Example 1: A portion of a housing
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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-Neto Agosto,2007
• Validation studies
Example 2: A portion of a spiral-bevel gear
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Example 2: A portion of a spiral-bevel gear
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Crack
• Validation studies
Example 2: A portion of a spiral-bevel gear
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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-Neto Agosto,2007
• Validation studies
Example 2: A portion of a spiral-bevel gear
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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-Neto Agosto,2007
Very poor elements
Best elements
• Validation studies
Example 3: A portion of a turbofan hub
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Example 3: A portion of a turbofan hub
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
Crack
• Validation studies
Example 3: A portion of a turbofan hub
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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-Neto Agosto,2007
• Validation studies
Example 3: A portion of a turbofan hub
Element quality
% o
f el
emen
ts
0 2
0
40
60
80
10
0
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 elementsBest elements
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Table for comparison
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 CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Table for comparison
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
Increases
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• Validation studies
Table for comparison
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
Increases
Decreases
THREE-DIMENSIONAL UNSTRUCTURED MESH GENERATION FOR ARBITRARY DOMAINS WITH CRACKS
Joaquim Bento Cavalcante-Neto Agosto,2007
• 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-Neto Agosto,2007
• 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-Neto Agosto,2007