three-dimensional unstructured mesh generation for arbitrary domains with cracks joaquim bento...

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



• Introduction



• Yerry and Shephard, 1984• Perucchio et al., 1989 • Many others

Delaunay

Octree



• Introduction



• 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



• 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



• 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




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




Octree generation: Initial boundary mesh




Octree generation: Initialization based on boundary mesh



Region with many cells


Octree generation: Refinement for maximum cell size



Cell refined


Octree generation: Refinement for minimum size disparity



Cell refined


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







Optimal vertex N1


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




Advancing-front procedure: Back-tracking element generation







Element to be deleted (a,b,c,d)

Unmeshable region






Meshable regionUnmeshable region






Meshable region Region meshedUnmeshable region


Local mesh improvement: Laplacian smoothing with checks




Local mesh improvement: Back-tracking optimization



Bad element (in gray)


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)




General description (general algorithm):

• Determine region R(Ek) for given element Ek

R(Ek)





• 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?







• 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







• 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



• 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




Element to be deleted (in gray)





New region





New elements

New region



• 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





New region





New elementsNew region




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




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




Example 1: A portion of a housing







Crack



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





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




Example 2: A portion of a spiral-bevel gear







Crack



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





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


Example 3: A portion of a turbofan hub







Crack



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





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




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






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






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



• 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



• 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...

Documents

arbitrary domains

boundary mesh refinement

existing boundary mesh

cracks requirements

new strategy

d strategy

cell refined slide

optimal vertex n1 slide