CILAMCE November, 2004

1
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES Joaquim B. Cavalcante-Neto
(CRAb/UFC) Luiz Fernando Martha
(Tecgraf/PUC-Rio) Paul A. Wawrzynek (CFG/Cornell
University) Anthony R. Ingraffea (CFG/Cornell
University)
CILAMCE November, 2004
2
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Back-tracking strategy
• Two-dimensional Back-tracking
• Three-dimensional Back-tracking
• Validation studies
• Conclusion

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
3
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• There is a wide variety of techniques in the
  literature for optimization of meshes
• Field, 1988
• Hansbo, 1995
• Freitag and Ollivier-Gooch, 1997
• Cavalcante-Neto, 2001
• Many others

Mesh Smoothing
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
4
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• There is a wide variety of techniques in the
  literature for optimization of meshes
• Field, 1988
• Hansbo, 1995
• Freitag and Ollivier-Gooch, 1997
• Cavalcante-Neto, 2001
• Many others
• Freitag and Ollivier-Gooch, 1997
• Many others

Mesh Smoothing
Mesh Swapping
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
5
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• There is a wide variety of techniques in the
  literature for optimization of meshes
• Field, 1988
• Hansbo, 1995
• Freitag and Ollivier-Gooch, 1997
• Cavalcante-Neto, 2001
• Many others
• Freitag and Ollivier-Gooch, 1997
• Many others
• Chan and Anastasiou, 1997
• Rassineux, 1998
• Many others

Mesh Smoothing
Mesh Swapping
Mesh Deletion
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
6
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Some characteristics of the strategies
• Local modifications to improve quality
• Works very well in 2D
• Can generate negative volumes in 3D
• Variations consider weights
• Need for validity checking

Mesh Smoothing
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
7
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Some characteristics of the strategies
• Local modifications to improve quality
• Works very well in 2D
• Can generate negative volumes in 3D
• Variations consider weights
• Need for validity checking
• Uses swapping of entities
• Entities are usually faces and edges
• Local changes performed

Mesh Smoothing
Mesh Swapping
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
8
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Some characteristics of the strategies
• Local modifications to improve quality
• Works very well in 2D
• Can generate negative volumes in 3D
• Variations consider weights
• Need for validity checking
• Uses swapping of entities
• Entities are usually faces and edges
• Local changes performed
• Uses removal of bad tetrahedra
• Reconstruction of sub-volumes

Mesh Smoothing
Mesh Swapping
Mesh Deletion
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
9
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Objective of the work
• Propose a new strategy to improve mesh quality
  in 2D and 3D
• Strategy is based on a back-tracking idea

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
10
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Introduction
• Objective of the work
• Propose a new strategy to improve mesh quality
  in 2D and 3D
• Strategy is based on a back-tracking idea
• Characteristics
• Based on the deletion of ill-shaped elements in
  the mesh
• Work on local regions for remeshing
• Back-tracks until an optimal region is found
• Tries to find a visible kernel for local
  region
• Saves original configuration for failed cases

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
11
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Back-tracking strategy
• General description (definitions)
• Ek gt ill-shaped element of the generated mesh
• R(Ek) gt local region for remeshing in the
  neighborhood of Ek
• M(E) gt set of all mesh elements
• D(Ek) gt set of deleted elements for Ek (during
  back-tracking)
• F(Ek) gt set of new generated elements for R(Ek)

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
12
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Back-tracking strategy
• General description (general algorithm)
• Determine region R(Ek) for given ill-shaped
  element Ek

R(Ek)
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
13
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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)
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
14
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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)
No
Update R(Ek) Update M(E)
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
15
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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)

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
16
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Two-dimensional Back-tracking
• R(Ek) gt P(Tk)
• P gt polygon
• T gt triangle
• M(E) gt M(T)
• D(Ek) gt D(Tk)
• F(Ek) gt F(Tk)

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
17
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Two-dimensional Back-tracking

Bad element in gray
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
18
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Two-dimensional Back-tracking

Improved region
Bad element in gray
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
19
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Two-dimensional Back-tracking

Improved region
Bad element in gray
Better elements
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
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A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Three-dimensional Back-tracking
• R(Ek) gt P(Tk)
• P gt polyhedron
• T gt tetrahedron
• M(E) gt M(T)
• D(Ek) gt D(Tk)
• F(Ek) gt F(Tk)

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
21
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Three-dimensional Back-tracking

Bad element (a,b,c,d)
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
22
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Three-dimensional Back-tracking

Improved region
Bad element (a,b,c,d)
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
23
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Three-dimensional Back-tracking

Improved region
Bad element (a,b,c,d)
Better elements
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
24
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies
• Quality measures
• Si gt length of an edge
• V gt volume of a tetrahedron
• Optimal value gt 8.5
• Range gt 1, ?
• Threshold
• Lower bound gt 5.0
• Upper bound gt 8.5factor(30)
• Use of metric (Parthasarathy et al., 1993)
• Sensitive to small changes
• Computationally efficient
• Any metric can be used

Definition of ill-shaped element
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
25
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies
• Quality measures
• ? 3Ri / Rc
• Ri gt radius of inscribed sphere
• Rcgt radius of circumscribed sphere
• Optimal value gt 1.0
• Range gt 0, 1
• Threshold
• Lower bound gt 0.2
• Upper bound gt 0.5
• Use of metric
• More intuitive interpretation
• Widely used in the literature
• Any metric can be used

Evaluation of element quality
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
26
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 1 A portion of a housing
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
27
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 1 A portion of a housing
Crack
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
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A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
29
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
30
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 2 A portion of a spiral-bevel gear
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
31
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 2 A portion of a spiral-bevel gear
Crack
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
32
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 2 A portion of a spiral-bevel gear
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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
33
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Best elements
Example 2 A portion of a spiral-bevel gear
Very poor elements
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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
34
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 3 A portion of a turbofan hub
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
35
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Example 3 A portion of a turbofan hub
Crack
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
36
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
37
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• 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
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
38
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Table for comparison
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
39
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Table for comparison
Increases
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
40
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Validation studies

Table for comparison
Increases
Decreases
CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil
41
A BACK-TRACKING PROCEDURE FOR OPTIMIZATION OF
MESHES

• Conclusion
• 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
  (Cavalcante-Neto et al., 2001)
• 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

CAVALCANTE-NETO et al., November 2004 CILAMCE,
Recife, Brazil