ABSTRACT

In this chapter, regularizing geometric transformations for polygonal elements with any number of nodes and the most common polyhedral finite elements are presented. If applied iteratively, these transformations convert arbitrarily shaped elements into their regular counterparts. A mathematical proof for the regularizing effect of the polygonal transformations leads to a full classification with respect to both their parameters and the resulting limit shapes. Although primarily focusing on regular limits, a scheme to derive custom transformations for prescribed non-regular limit polygons is also given. For polyhedral elements, evidence for the regularizing property is given by numerical tests. The regularizing transformations obtained in both cases will be the driving force of the GETMe algorithms for mesh smoothing presented in the subsequent chapter.

5.1 Transformation of polygonal elements 5.1.1 Classic polygon transformations Laplacian smoothing on the one hand and global optimization-based approaches on the other somehow represent the poles of the smoothing world. Whereas Laplacian smoothing is a simple geometry-based and computationally inexpensive scheme, it is not quality driven. In contrast, global optimization approaches use the full mathematical power of numerical optimization to focus entirely on mesh quality in terms of the involved quality criterion and target function. This comes at the expense of a significantly increased computational complexity.