MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1992 Lecture Notes Series on Computing  
We survey the computational geometry relevant to nite element mesh generation. We especially focus on optimal triangulations of geometric domains in two-and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a xed set of vertices and for the placement of new vertices (Steiner
more » ... . We brie y survey the heuristic algorithms used in some practical mesh generators. remove the rst edge e from the queue. If Q e is not reversed, we simply continue to the next edge. But if Q e is reversed, we remove e from the triangulation, replacing it with the other diagonal of Q e . This ip might change the status of some of the four outside edges of the quadrilateral, so we add the changed ones to the queue if not there already. When the queue is empty, we stop. Lemma 4. The ip algorithm terminates after O(n 2 ) ips. Proof: We use the lifting relation between DTs and convex hulls. Under the mapping that takes (x; y) to (x; y; x 2 + y 2 ), the DT of the input vertices lifts to the lower convex hull, and|due to the input edges|the CDT lifts to a surface above the lower convex hull. An arbitrary triangulation including the input edges lifts to
doi:10.1142/9789814355858_0002 fatcat:zjlfgakee5df7nihosmpeuhco4