Overview

In Chapter 18, we described several methods to optimize planar or volumic meshes based on criteria notably related to the shape and the size of the elements. We mentioned that these methods cannot generally be applied directly to surfaces. This is why we now deal with the optimization of surface meshes which, while using the same general principles as the methods described in Chapter 18, nevertheless presents numerous specificities.

Surface meshes play an important role in various numerical applications. Hence, for finite element methods, it is well established that the quality of the geometric approximation may affect the accuracy of the numerical results as well as the convergence of the computational scheme [Ciarlet-1991]. In this type of application, a surface mesh is conceived, in principle, as the description of the boundary of a computational domain in three dimensions (cf. Chapters 5 to 7). Actually, to be useful, these meshes must conform to certain criteria, related to the geometry of the surfaces they represent (we expect an element size variation based on the local curvature) or to the physical behavior of the problems studied (element density greater in regions where the gradient of the solution varies). However, in the last case, following a physical criterion does not mean excluding conformity to the geometric properties of the surface.

Moreover, it is frequent that a given surface mesh is not satisfactory, either because it corresponds to too coarse an approximation of the surface, or because it contains too many elements to be...