Optimizing a mesh with respect to a given criterion is an operation that is frequently used with various goals in mind for a wide range of applications. First, optimization in itself is useful because the quality (the convergence of the computational schemes and the accuracy of the results) of the numerical solutions computed at the mesh nodes clearly depends on the quality of the mesh. In this respect, mesh generation methods usually include an optimization stage that takes place at the end of the entire mesh generation process. An optimization process may serve some more specific purposes such as the mesh adaptation, for instance, included in an adaptive computational procedure. Moreover, the tools involved in optimization methods can be also used in some particular applications (mesh simplification being a significant example).

The aim of this chapter is to introduce some methods designed for mesh optimization purposes. First, some information is given on how to compute element surface areas and volumes. Applications based on surface area and volume values are discussed, including localization and intersection problems, then we turn to the definition of mesh quality. Afterwards, we introduce some local tools for mesh optimization.

Having introduced these tools, and with regard to the given objectives, mesh optimization methods are discussed both in terms of strategies and computational aspects. Actually, mesh optimization can be considered as a step of a mesh generation method (in general, the last step of the method). It also can be seen as a stand-alone process.

This chapter...