In this paper, we extend a simultaneous untangling and smoothing technique previously developed for triangular and tetrahedral meshes to quadrilateral and hexahedral ones. Specifically, we present a technique that iteratively untangles and smooths a given quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a modification of an algebraic quality measure. The proposed method optimizes the mesh quality by a local node relocation process. That is, without modifying the mesh connectivity. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when we start from tangled meshes.

Discretization techniques such the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in applied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to generate. However, in many applications such as boundary layers in computational fluid dynamics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.

One of the most used algorithms to generate hexahedral meshes for extrusion volumes is the multi-sweeping method. The algorithm decomposes the geometry into many-to-one sub-volumes and then meshes each sub-volume separately.However, the quality of the final mesh depends on the decomposition process. First, the location of inner nodes created during the decomposition process may induce bad quality elements. To avoid this drawback, we propose a three-stage decomposition process to locate those nodes. Second, the imprinting process is not robust when dealing with non-planar surfaces. For this reason, we introduce the new concept of the computational domain. The computational domain is a planar representation of the levels of the geometry. In this way, we improve the operations needed to perform imprints.