surface meshing material tret de: s. j. owen, "a survey of unstructured mesh generation...
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Surface Meshing
Material tret de:
S. J. Owen, "A Survey of Unstructured Mesh Generation Technology", Proceedings 7th International Meshing Roundtable, 1998.
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Surface Meshing
Direct 3D Meshing Parametric Space Meshing
u
v
•Elements formed in 3D using actual x-y-z representation of surface
•Elements formed in 2D using parametric representation of surface•Node locations later mapped to 3D
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Surface Meshing
A
B
3D Surface Advancing Front•form triangle from front edge AB
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A
B
Surface Meshing
C
NC
3D Surface Advancing Front•Define tangent plane at front by averaging normals at A and B
Tangent plane
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Surface Meshing
A
B
C
NC
D
3D Surface Advancing Front•define D to create ideal triangle on tangent plane
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A
B
C
NC
D
3D Surface Advancing Front•project D to surface (find closest point on surface)
Surface Meshing
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Surface Meshing
3D Surface Advancing Front•Must determine overlapping or intersecting triangles in 3D. (Floating point robustness issues)•Extensive use of geometry evaluators (for normals and projections)•Typically slower than parametric implementations•Generally higher quality elements•Avoids problems with poor parametric representations (typical in many CAD environments)•(Lo,96;97); (Cass,96)
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Surface Meshing
Parametric Space Mesh Generation•Parameterization of the NURBS provided by the CAD model can be used to reduce the mesh generation to 2D
u
v
u
v
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Surface Meshing
Parametric Space Mesh Generation•Isotropic: Target element shapes are equilateral triangles
•Equilateral elements in parametric space may be distorted when mapped to 3D space.•If parametric space resembles 3D space without too much distortion from u-v space to x-y-z space, then isotropic methods can be used.
u
v
u
v
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Surface Meshing
•Parametric space can be “customized” or warped so that isotropic methods can be used.•Works well for many cases.•In general, isotropic mesh generation does not work well for parametric meshing
u
v
u
v
Parametric Space Mesh Generation
Warped parametric space
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Surface Meshing
u
v
u
v
•Anisotropic: Triangles are stretched based on a specified vector field
•Triangles appear stretched in 2d (parametric space), but are near equilateral in 3D
Parametric Space Mesh Generation
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Surface Meshing
•Stretching is based on field of surface derivatives
Parametric Space Mesh Generation
u
v
x
y
z
z
u
y
u
x
u
,,
z
v
y
v
x
v
,,
z
v
y
v
x
v
,,v
z
u
y
u
x
u
,,u
uu E vu F vv G
GF
FE)(XM
•Metric, M can be defined at every location on surface. Metric at location X is:
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Surface Meshing
•Distances in parametric space can now be measured as a function of direction and location on the surface. Distance from point X to Q is defined as:
XQXQXQl T )()( XM
u
v
x
y
z
Parametric Space Mesh Generation
X
Q
)(XQl
u
v
X Q)(XQl
M(X)
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Surface Meshing
Parametric Space Mesh Generation•Use essentially the same isotropic methods for 2D mesh generation, except distances and angles are now measured with respect to the local metric tensor M(X).•Can use Delaunay (George, 99) or Advancing Front Methods (Tristano,98)
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Surface Meshing
Parametric Space Mesh Generation
•Is generally faster than 3D methods•Is generally more robust (No 3D intersection calculations)•Poor parameterization can cause problems•Not possible if no parameterization is provided
•Can generate your own parametric space (Flatten 3D surface into 2D) (Marcum, 99) (Sheffer,00)
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Smoothing
Topological Improvement
Mesh Post-Processing
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Post-Processing
Smoothing
Topological Improvement
Adjust locations of nodes without changing mesh topology (element connectivity)
Change connectivity without affecting node locations
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Post-Processing
Smoothing•Averaging Methods•Optimization Based•Distortion Metrics•Combined: Laplacian/Optimization based smoothing
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Smoothing
P1
P2
P3
P4
P5
P
Laplacian
Averaging Methods
(Field, 1988)
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Smoothing
P1
P2
P3
P4
P5P
Laplaciann
n
ii
1
P
P
Averaging Methods
Centroid of attached nodes
Can create inverted elements
(Field, 1988)
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Smoothing
C1
P
Area Centroid Weighted
Averaging Methods
Weighted average of triangle centroids
C2
C3
C4C4
A1
A2
n
ii
n
iii
A
A
1
1
C
P
Ai=area of triangle i
Ci=centroid of triangle i
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Smoothing
Radius Ratiocirc
in
r
r3
rin
rcirc
A
C
B
D
4
1
2216
i is
v
))()()(( acbbcacbacba
v = volume of tetsi = areas of four faces of tet
a,b,c = products of the lengths of opposite edges of tet
BCADcCDABbBDACa ,,
(Liu, Joe, 1994)
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Post-Processing
Topological Improvement•Triangles•Tetrahedra•Quads
Smoothing•Averaging Methods•Optimization Based•Distortion Metrics•Combined Laplacian/Optimization based smoothing
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Topology Improvement
Diagonal FlipPossible Criteria
Delaunay criterion
Distortion metrics
Node valence
Deviation from surface
Comparison before/after:
(Canann, Muthukrishnan, Phillips, 1996)
(Edelsbrunner, Shah, 1996)
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Topology Improvement
Diagonal FlipPossible Criteria
Delaunay criterion
Distortion metrics
Node valence
Deviation from surface
Comparison before/after:
(Edelsbrunner, Shah, 1996)
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Topology Improvement
Node valence = number of edges connected to node
7
7
Ideal node valence for triangle mesh is 6
edges/node
6
6
Swapping can improve node valence
Allowing Smoothing to do a better job
(Canann, Muthukrishnan, Phillips, 1996)