This paper reviews geometric measures used to assess the shape of finite elements in two- and three-dimensional meshes Measures have been normalized and made scale invariant whenever possible This paper also introduces a Universal Similarity Region that enhances comparisons of triangles and their measures As a byproduct, the USR provides a dynamic way to compare improved triangular meshes Copyright © 2000 John Wiley & Sons, Ltd

Qualitative measures for initial meshes

Fast Tree-Based Redistancing for Level Set Computations

This book describes in detail the key algorithms needed for computing with spline functions and illustrates their use in solving several basic problems in numerical analysis, including function approximation, numerical quadrature, data fitting, and the numerical solution of PDEs. The focus is on computational methods for bivariate splines on triangulations in the plane and on the sphere, although both univariate and tensor-product splines are also discussed. The book contains numerous examples and figures to illustrate the methods and their performance. All of the algorithms in the book have been coded in a separate MATLAB package available for license. The package can be used to run all of the examples in the book and also provides readers with the essential tools needed to create software for their own applications. In addition to the included bibliography, a list of over 100 pages of additional references can be found on the book's website. Audience: This book is designed for mathematicians, engineers, scientists, and anyone else wanting to make use of spline functions for numerical computation.

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Spline Functions: Computational Methods

It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n/sup 2/). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modifications the algorithm works for arbitrary normed metrics. >

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A quadratic time algorithm for the minmax length triangulation

It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$. The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modifications the algorithm works for arbitrary normed metrics.

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One way of evaluating a triangulation method is to study how well it controls error bounds and stability in computations. Lawson’s method, which maximizes the smallest angle over all triangulations, controls error bounds for linear interpolates as well as for certain derivative estimates. In particular, no other triangulation improves these bounds by more than a factor of 2. Although Lawson’s triangulation method does not minimize the largest edge over all triangulations, it nearly does; this edge is no longer than ${2 / {\sqrt 3 }}$ times the longest edge in any other triangulation. Thus, Lawson’s triangulation method controls error bounds in polynomial interpolation schemes.

Wendy Spratte

Papers

Lawson's triangulation is nearly optimal for controlling error bounds