An Introduction to Unstructured Mesh Generation

1
An Introduction to Unstructured Mesh Generation

• Material tret de
• S. J. Owen,
• "A Survey of Unstructured Mesh Generation
  Technology",
• Proceedings 7th International Meshing Roundtable,
  1998.

2
Tri/Tetrahedral Meshing

• Triangle and tetrahedral meshing are by far the
• most common forms of unstructured mesh
• generation.
• Most techniques currently in use can fit into one
  of
• three main categories
• Delaunay
• Quadtree/Octree
• Advancing Front.

3
Delaunay

• A typical approach is to first mesh the boundary
  of the geometry to
• provide an initial set of nodes.
• The boundary nodes are then triangulate according
  to the Delaunay
• criterion.
• Nodes are then inserted incrementally into the
  existing mesh,
• redefining the triangles or tetrahedra locally as
  each new node is
• inserted to maintain the Delaunay criterion.
• It is the method that is chosen for defining
  where to locate the interior
• nodes that distinguishes one Delaunay algorithm
  from another.

4
Delaunay

• Begin with Bounding Triangles (or Tetrahedra)

5
Delaunay

• Insert boundary nodes using Delaunay method
  (Lawson or Bowyer-Watson)

6
Delaunay

• Insert boundary nodes using Delaunay method
  (Lawson or Bowyer-Watson)

7
Delaunay

• Insert boundary nodes using Delaunay method
  (Lawson or Bowyer-Watson)

8
Delaunay

• Insert boundary nodes using Delaunay method
  (Lawson or Bowyer-Watson)

9
Delaunay

• Insert boundary nodes using Delaunay method
  (Lawson or Bowyer-Watson)

10
Delaunay

• Recover boundary
• Delete outside triangles
• Insert internal nodes

11
Delaunay
h

• Grid Based
• Nodes introduced based on a regular lattice
• Lattice could be rectangular, triangular,
  quadtree, etc
• Outside nodes ignored

Node Insertion
12
Delaunay

• Grid Based
• Nodes introduced based on a regular lattice
• Lattice could be rectangular, triangular,
  quadtree, etc
• Outside nodes ignored

Node Insertion
13
Delaunay

• Centroid
• Nodes introduced at triangle centroids
• Continues until edge length,

Node Insertion
14
Delaunay
l

• Centroid
• Nodes introduced at triangle centroids
• Continues until edge length,

Node Insertion
15
Delaunay

• Circumcenter (Guaranteed Quality)
• Nodes introduced at triangle circumcenters
• Order of insertion based on minimum angle of any
  triangle
• Continues until minimum angle gt predefined
  minimum

Node Insertion
(Chew,Ruppert,Shewchuk)
16
Delaunay

• Circumcenter (Guaranteed Quality)
• Nodes introduced at triangle circumcenters
• Order of insertion based on minimum angle of any
  triangle
• Continues until minimum angle gt predefined
  minimum

Node Insertion
(Chew,Ruppert,Shewchuk)
17
Delaunay
A
B
C

• Advancing Front
• Front structure maintained throughout
• Nodes introduced at ideal location from current
  front edge

(Marcum,95)
Node Insertion
18
Delaunay

• Advancing Front
• Front structure maintained throughout
• Nodes introduced at ideal location from current
  front edge

(Marcum,95)
Node Insertion
19
Delaunay

• Voronoi-Segment
• Nodes introduced at midpoint of segment
  connecting the circumcircle centers of two
  adjacent triangles

(Rebay,93)
Node Insertion
20
Delaunay

• Voronoi-Segment
• Nodes introduced at midpoint of segment
  connecting the circumcircle centers of two
  adjacent triangles

(Rebay,93)
Node Insertion
21
Delaunay
h
h
h

• Edges
• Nodes introduced at along existing edges at lh
• Check to ensure nodes on nearby edges are not too
  close

(George,91)
Node Insertion
22
Delaunay

• Edges
• Nodes introduced at along existing edges at lh
• Check to ensure nodes on nearby edges are not too
  close

(George,91)
Node Insertion
23
Delaunay

• Boundary Intersection
• Nodes and edges introduced where Delaunay edges
  intersect boundary

Boundary Constrained
24
Delaunay

• Boundary Intersection
• Nodes and edges introduced where Delaunay edges
  intersect boundary

Boundary Constrained
25
Delaunay

• Local Swapping
• Edges swapped between adjacent pairs of triangles
  until boundary is maintained

Boundary Constrained
26
Delaunay

• Local Swapping
• Edges swapped between adjacent pairs of triangles
  until boundary is maintained

Boundary Constrained
27
Delaunay

• Local Swapping
• Edges swapped between adjacent pairs of triangles
  until boundary is maintained

Boundary Constrained
28
Delaunay

• Local Swapping
• Edges swapped between adjacent pairs of triangles
  until boundary is maintained

Boundary Constrained
29
Delaunay

• Local Swapping
• Edges swapped between adjacent pairs of triangles
  until boundary is maintained

(George,91)(Owen,99)
Boundary Constrained
30
Octree/Quadtree

• Define intial bounding box (root of quadtree)
• Recursively break into 4 leaves per root to
  resolve geometry
• Find intersections of leaves with geometry
  boundary
• Mesh each leaf using corners, side nodes and
  intersections with geometry
• Delete Outside
• (Yerry and Shephard, 84), (Shepherd and Georges,
  91)

31
Octree/Quadtree
QMG, Cornell University
32
Octree/Quadtree
QMG, Cornell University
33
Advancing Front

• Begin with boundary mesh - define as initial
  front
• For each edge (face) on front, locate ideal node
  C based on front AB

34
Advancing Front
D
A
B

• Determine if any other nodes on current front are
  within search radius r of ideal location C
  (Choose D instead of C)

35
Advancing Front
D

• Book-Keeping New front edges added and deleted
  from front as triangles are formed
• Continue until no front edges remain on front

36
Advancing Front

• Book-Keeping New front edges added and deleted
  from front as triangles are formed
• Continue until no front edges remain on front

37
Advancing Front

• Book-Keeping New front edges added and deleted
  from front as triangles are formed
• Continue until no front edges remain on front

38
Advancing Front

• Book-Keeping New front edges added and deleted
  from front as triangles are formed
• Continue until no front edges remain on front

39
Advancing Front
r
C
A
B

• Where multiple choices are available, use best
  quality (closest shape to equilateral)
• Reject any that would intersect existing front
• Reject any inverted triangles (AB X AC gt 0)
• (Lohner,8896)(Lo,91)