Analysis of triangle quality measures

Several of the more commonly used triangle quality measures are analyzed and compared. Proofs are provided to verify that they do exhibit the expected extremal properties. The asymptotic behavior of these measures is investigated and a number of useful results are derived. It is shown that some of the quality measures are equivalent, in the sense of displaying the same extremal and asymptotic behavior, and that it is therefore possible to achieve a concise classification of triangle quality measures. Introduction Assessment of mesh quality is an important requirement in both the selection of a finite element mesh and the evaluation of meshes that have undergone adaptation [2, 5]. Several measures of element quality have been proposed [1, 8, 11] based on the dimensionless ratios of various geometric parameters. Apart from the work of [11], there appears to be almost no discussion in the literature on the relative merits of these particular quality measures. More recently, alternative quality measures have been suggested [3, 7, 9, 10]. These alternative measures are derived from the singular values of a matrix whose columns are formed by the edge vectors of the mesh element. An element is said to be degenerate if its volume is zero. Let Q be a quality measure defined for any nondegenerate simplex t and let the range of Q be the real interval [1,+∞[. It is assumed that Q satisfies the following extremal properties: (i) Q attains its minimum value of 1 if and only if t is a regular simplex; (ii) Q has no other extrema. In many cases, these extremal properties have been assumed, or stated, without proof. Although the extremal behavior of these quality measures might appear obvious, we believe that this behavior should be established rigorously and precise bounds should be found. In this paper we examine the triangular case since properties of the triangle are particularly amenable to analysis. We consider several of the more commonly used triangle quality measures, provide proofs of their extremal properties, and examine their asymptotic behavior. Our goal is to provide a number of useful results on triangle quality measures that may lead to a better assessment of both planar triangulations and triangulated surfaces. Received by the editor July 9, 2001 and, in revised form, December 21, 2001. 2000 Mathematics Subject Classification. Primary 32B25, 65M50; Secondary 51N20.