Hexahedral mesh generation is still an ongoing research, and a major conclusion so far is that the generation of full-hex conforming meshes on arbitrary domains is beyond our reach nowadays. In fluid dynamics, boundary layers made of hexahedra are effective for capturing large gradients and resolving viscous flows near the boundary, and semi-structured boundary-layer meshes attract significant interest (see, e.g. In solid mechanics, hexahedra exhibit higher accuracy than tetrahedra, which are plagued by locking problems. For the same number of vertices, hex meshes have fewer elements, which speeds up the matrix/residual assembly. A number of arguments can indeed be stated in favor of hex-meshing. Yet, it is a fact that a large number of finite element users would highly appreciate having automatic hex-meshing procedures for general 3D domains. This paper does not aim at deciding on that issue. Whether hex-meshing or tet-meshing is better for finite element computations is a long-standing controversy. However, non-conformal quadrilateral faces adjacent to triangular faces are present in the final meshes. Non-uniform mixed meshes obtained following our approach show a volumic percentage of hexahedra that usually exceeds 80%. A maximum number of tetrahedra are then merged into hexahedra using the algorithm of Yamakawa-Shimada. Once the vertex placement process is completed, the region is tetrahedralized with a Delaunay kernel. The quality of the vertex alignment inside the volumes relies on the quality of the alignment on the surfaces. This can be achieved with a frontal algorithm, which is applicable to both the two- and three-dimensional cases. In order to reach larger ratios, the vertices of the initial mesh need to be anticipatively organized into a lattice-like structure. If the vertices are placed at random, less than 50% of the tetrahedra will be combined into hexahedra. We show that the percentage of recombined hexahedra strongly depends on the location of the vertices in the initial 3D mesh. Instead, part of the remaining tetrahedra are combined into prisms and pyramids, eventually yielding a mixed mesh. Contrary to the 2D case, a 100% recombination rate is seldom attained in 3D. In this paper, a similar indirect approach is applied to the three-dimensional case, i.e., a method to recombine tetrahedra into hexahedra. This way, high-quality full-quad meshes suitable for finite element calculations can be generated for arbitrary two-dimensional geometries. So called triangle-merge techniques are then used to recombine the triangles of the initial mesh into quadrilaterals. Indirect quad mesh generation methods rely on an initial triangular mesh.
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