An Introduction to Unstructured Mesh Generation Material tret

Tri/Tetrahedral Meshing Triangle and tetrahedral meshing are by far the most common forms of

Delaunay A typical approach is to first mesh the boundary of the geometry to

Delaunay • Begin with Bounding Triangles (or Tetrahedra)

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

Delaunay • Recover boundary • Delete outside triangles • Insert internal nodes

Delaunay h Grid Based • Nodes introduced based on a regular lattice • Lattice

Delaunay Grid Based • Nodes introduced based on a regular lattice • Lattice could

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

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

Delaunay Circumcenter (“Guaranteed Quality”) • Nodes introduced at triangle circumcenters • Order of insertion

Delaunay A B C Advancing Front • “Front” structure maintained throughout • Nodes introduced

Delaunay Advancing Front • “Front” structure maintained throughout • Nodes introduced at ideal location

Delaunay Voronoi-Segment • Nodes introduced at midpoint of segment connecting the circumcircle centers of

Delaunay h h h Edges • Nodes introduced at along existing edges at l=h

Delaunay Edges • Nodes introduced at along existing edges at l=h • Check to

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

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

Octree/Quadtree • Define intial bounding box (root of quadtree) • Recursively break into 4

Advancing Front C A B • Begin with boundary mesh - define as initial

Advancing Front r C A • Determine if any other nodes on current front

Advancing Front D • Book-Keeping: New front edges added and deleted from front as

Advancing Front • Book-Keeping: New front edges added and deleted from front as triangles

Advancing Front r C A B • Where multiple choices are available, use best

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

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.

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.

Delaunay • Begin with Bounding Triangles (or Tetrahedra)

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

Delaunay • Recover boundary • Delete outside triangles • Insert internal nodes

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

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

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

Delaunay l Centroid • Nodes introduced at triangle centroids • Continues until edge length, Node Insertion

Delaunay Circumcenter (“Guaranteed Quality”) • Nodes introduced at triangle circumcenters • Order of insertion based on minimum angle of any triangle • Continues until minimum angle > predefined minimum (Chew, Ruppert, Shewchuk) Node Insertion

Delaunay A B C Advancing Front • “Front” structure maintained throughout • Nodes introduced at ideal location from current front edge (Marcum, 95) Node Insertion

Delaunay Advancing Front • “Front” structure maintained throughout • Nodes introduced at ideal location from current front edge (Marcum, 95) Node Insertion

Delaunay Voronoi-Segment • Nodes introduced at midpoint of segment connecting the circumcircle centers of two adjacent triangles (Rebay, 93) Node Insertion

Delaunay h h h Edges • Nodes introduced at along existing edges at l=h • Check to ensure nodes on nearby edges are not too close (George, 91) Node Insertion

Delaunay Edges • Nodes introduced at along existing edges at l=h • Check to ensure nodes on nearby edges are not too close (George, 91) Node Insertion

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

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

Delaunay Local Swapping • Edges swapped between adjacent pairs of triangles until boundary is maintained (George, 91)(Owen, 99) Boundary Constrained

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)

Octree/Quadtree QMG, Cornell University

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

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

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

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

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| > 0) • (Lohner, 88; 96)(Lo, 91)