3.2.4 Mesh Quality Improvement

The main goal of many mesh adaptation algorithms is to improve the mesh element quality. Mesh element quality improvement algorithms are available for both simplex and all-quad/all-hex meshes. There are algorithms for simplex meshes, which generate meshes with guaranteed element quality. Usually, mesh quality improvement algorithms have a set of local operations which they test on a set of elements: If an operation locally increases the mesh element quality, it is applied. For simplex, all-quad, and all-hex meshes, it is often sufficient to perform a combination of mesh smoothing and flip operations to obtain meshes with a good quality [75][79]. However, for tetrahedral meshes, these operations might lead to poor local quality optima. More aggressive methods, like vertex insertion, are required to further increase the mesh element quality [32]. In contrast to triangular meshes, algorithms which optimize the radius-edge ratio of tetrahedral meshes still might generate slivers (cf. Section 2.2.2). Sliver exudation algorithms must be applied after mesh generation to remove this type of element [129]. Approaches for smoothing algorithms can be extended to various element types [76]. There are optimization algorithms for quadrilateral and hexahedral meshes operating on the dual mesh, which, however, might be non-local [108][115]. Many algorithms even consider complete mesh re-sampling to improve mesh element quality [139].