Optimal Point Placement for Mesh Smoothing

We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another? We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straight-line segments). We highlight the similarities and differences of the two settings, describe many extensions and generalizations, highlight algorithmic issues, outline several applications and mention open problems.

Flips in planar graphs

The issue of mesh quality for unstructured triangular and tetrahedral meshes is considered. The theoretical background to finite element methods is used to understand the basis of present-day geometrical mesh quality indicators. A survey of more recent research in the development of finite element methods describes work on anisotropic meshing algorithms, and on providing good error estimates that reveal the relationship between the error and both the mesh and the solution gradients. The reality of solving complex three-dimensional problems is that such indicators are presently not available for many problems of interest. A simple tetrahedral mesh quality measure using both geometrical and solution information is described. Some of the issues in mesh quality for unstructured tetrahedral meshes are illustrated by means of two simple examples.

Mesh Quality: A Function of Geometry, Error Estimates or Both?

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.

/pdf/optimal-point-placement-for-mesh-smoothing-2ajl7owp6c.pdf

Optimal point placement for mesh smoothing

We consider the family of finite element spaces of fixed dimension, defined over all triangulations of a given polygonal domain in R2, based on a common set of nodes (vertices). A minimum energy triangulation, relative to a given elliptic problem, is a triangulation for which the finite element solution has the minimal energy. The minimum energy triangulation can be considered optimal, as it minimizes the solution error in the natural norm associated with the problem. It is shown in this paper that the well-known Delaunay triangulation of a set of points in R2 is a minimum energy triangulation for the energy functional associated with the nonhomogeneous Laplace equation. Minimum energy triangulations may be very expensive to compute. In this paper, therefore, we present algorithms for constructing locally minimal energy triangulations and outline efficient schemes for computing sub-optimal triangulations. In both cases the basic idea is to improve an initial triangulation by using local operations on the edges of the triangulation. It is shown for several model problems that such sub-minimal energy triangulations can significantly improve the quality of the approximate solution.

S. Rippa

Papers

Minimum energy tiangulations for elliptic problems