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Shape measures, which provide an effective quantitative mean of comparison
of element shapes in a mesh, are of great relevance in many fields related
to finite element analysis, but particularly in mesh adaptation. Still,
while most serious works in the field of mesh adaptation directly make use
of shape measures, very little work has been devoted to the actual
comparison of shape measures, with the notable exceptions of Liu and Joe
(1994) who have thoroughly analyzed a set of a few selected measures.
While the published works present some of the standard shape measures in
current use, new shape measures steadily appear in recent literature for
which no analysis is available. Furthermore, no classification scheme has
been proposed, and fitness of new measures is often not assessed. This
lecture aims to survey a wider range of shape measures in general use, to
define validity criteria for those measures and to classify then in broad
categories, beginning with valid vs. invalid shape measures. The lecture
also addresses issues regarding the use of shape measures in non-Euclidean
spaces, such as the use of shape measures in Riemannian spaces for
anisotropic mesh adaptation.
The lecture summarizes important properties of simplices and introduces a
classification of simplex degeneracies in two and three dimensions. I will
present a wide range of shape measures, introduce shape measures validity
criteria, and present a visualization scheme that helps analyze and compare
shape measures to one another. Shape measures are then classified, and
conclusions are drawn on the pertinence of developing new shape measures or
choosing one among the currently existing ones.
Mesh adaptivity is a process that generates a sequence of meshes and
numerical solutions on these meshes such that the sequence converges to some
goal which usually is error equirepartition whilst minimizing the
computational effort by minimizing the number of vertices of the mesh. For
unstructured meshes, the process of computing a mesh in the sequence can be
decomposed in two steps: first, a size specification map is computed by
analyzing the numerical solution; second, a mesh is computed that satisfies
this size specification map.
The subject of the present lecture is to offer a measure of the degree to
which a mesh satisfies it\'s size specification map.
More than ten years ago, Marie-Gabrielle Vallet (1990, 1991, 1992) showed
that giving the size specification map using a metric tensor representation
eased the generation of adapted and anisotropic meshes by combining the
desired size and stretching into a single mathematical concept. Metric
tensors modify the way distances are measured. The adapted and anisotropic
mesh in the real Euclidean space is constructed by building a regular,
isotropic and unitary mesh in the metric tensor space.
The use of a metric tensor representation for the size specification map is
now a widely used tool for the generation and adaptation of anisotropic
meshes. It has been used in two and three dimensions, for various PDE
simulations with finite element and finite volume methods, for surface
discretization, graphic representation, etc. The most complete references
are George and Borouchaki (1997) and Frey and George (1999) the references
However, the issue of metric conformity is still not clear. There is no well
defined way to measure the degree to which a mesh satisfies a size
specification map given in the form of a field of metric tensors.
Most authors rely on two competing measures to assess the quality of their
meshes with respect to a size specification map. One measure compares the
simplex shape with the specified stretching. This is usually done by
computing a shape criterion