In this work, we present a technique to map any genus zero solid object onto a hexahedral decomposition of a solid cube. This problem appears in many applications ranging from finite element methods to visual tracking. From this, one can then hopefully utilize the proposed technique for shape analysis, registration, as well as other related computer graphics tasks. More importantly, given that we seek to establish a one-to-one correspondence of an input volume to that of a solid cube, our algorithm can naturally generate a quality hexahedral mesh as an output. In addition, we constrain the mapping itself to be volume preserving allowing for the possibility of further mesh simplification. We demonstrate our method both qualitatively and quantitatively on various 3D solid models
In this section, we review related works focusing on hexahedral meshing algorithms with the understanding that there exists a wide variety of possible approaches to construct volumetric meshes @cite_4 @cite_19 . Generally, the hexahedral mesh generation algorithms are categorized by two classes: structured and unstructured methods. Strictly speaking, a structured mesh is characterized by all of the interior mesh nodes having an equal number of adjacent elements. On the other hand, unstructured meshes relax the nodal valence requirement, allowing any number of elements to meet at a single node. We note that the mesh we construct falls into the structured mesh category, which includes mapping techniques @cite_27 and submapping approaches @cite_12 . In addition to these frameworks, structured meshing algorithms have arisen in the form of octree approaches @cite_16 @cite_31 , multiblock methods @cite_35 @cite_7 , and sweeping algorithms @cite_24 @cite_37 @cite_40 . While it is beyond the scope of this note to detail all of these methods (and is by no means exhaustive), we refer the interested reader to a survey conducted by Owen @cite_17 . In what follows, we will further constrain our discussion to approaches related to mapping techniques.
@cite_35 @cite_37 @cite_4 @cite_7 @cite_24 @cite_19 @cite_27 @cite_40 @cite_31 @cite_16 @cite_12 @cite_17
1978623566 2069533434 77771864 2485446827
This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form non-manifold surfaces. In earlier papers, we developed an octree-based isocontouring method to construct unstructured 3D meshes for a single material (homogeneous) domain with manifold boundary. In this paper, we introduce the notion of a material change edge and use it to identify the interface between two or several different materials. A novel method to calculate the minimizer point for a cell shared by more than two materials is provided, which forms a non-manifold node on the boundary. We then mesh all the material regions simultaneously and automatically while conforming to their boundaries directly from volumetric data. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for proper hexahedral mesh construction. Finally, edge-contraction and smoothing methods are used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set of pillowing schemes is defined automatically as the boundary of each material region. Several application results of our multi-material mesh generation method are also provided. This paper describes an algorithm to extract adaptive and quality quadrilateral hexahedral meshes directly from volumetric data. First, a bottom-up surface topology preserving octree-based algorithm is applied to select a starting octree level. Then the dual contouring method is used to extract a preliminary uniform quad hex mesh, which is decomposed into finer quads hexes adaptively without introducing any hanging nodes. The positions of all boundary vertices are recalculated to approximate the boundary surface more accurately. Mesh adaptivity can be controlled by a feature sensitive error function, the regions that users are interested in, or finite element calculation results. Finally, a relaxation based technique is deployed to improve mesh quality. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations. A method is proposed whereby an existing non-conforming, mixed hexahedra-tetrahedra element mesh, is altered to conform by the insertion and formation of five-node or thirteen-node pyramids. Local tetrahedral transformations are performed to provide the topology enabling the merging of two adjacent tetrahedra into one pyramid. Local smoothing and cleanup operations improve the quality of the transition region. Other methods for the creation of transition pyramid elements are also discussed. Results show superior performance of the resulting elements in a commercial finite element code over non-conforming interface conditions. INTRODUCTION While many automated tetrahedra methods have now become commonplace in the industry, all-hexahedra free meshing methods have proven to be challenging research topics (Blacker, 1993; Price, 1995; Schneiders, 1996; Tautges, 1996). Some engineering disciplines have shown a preference for hexahedral elements over tetrahedra (Cifuentes, 1992; Benzley, 1995). While all-hexahedral element meshes may be preferred, some tetrahedra elements may be acceptable within the same model. For example, blocky or easily mapped regions of the solid may be first filled with hexahedra. Regions that may be more critical to the analysis, such as boundary layers or regions of high stress may also be better served by hexahedra. Tetrahedra elements may fill the remaining, more geometrically complex, or less critical regions of the solid. Since, geometrically, two tetrahedra faces are required to interface with a single hexahedron, discontinuities will arise at the boundary between the two element types. Traditional uses of the finite element method require that elements conform. In two-dimensions, this principle implies that no single element edge will have more than two elements adjacent. In three-dimensions, no single face will have more than two adjacent elements. Edge a-b and Face c-d-e-f in Figure 1 have more than the maximum two adjacent elements, thus rendering the mesh mathematically deficient for finite element analysis. This paper presents several solutions to the problem of non-conforming meshes that contain tetrahedra and hexahedra elements. Abstract This paper introduces developments in modified techniques for the generation of unstructured, non-uniform, dual orthogonal meshes which are suitable for use with co-volume solution schemes. Two new mesh generation techniques, a modified advancing front technique and an octree-Delaunay algorithm, are coupled with a mesh optimisation algorithm. When using a Delaunay–Voronoi dual, to construct mutually orthogonal meshes for co-volume schemes, it is essential to minimise the number of Delaunay elements which do not contain their Voronoi vertex. These new techniques provide an improvement over previous approaches, as they produce meshes in which the number of elements that do not contain their Voronoi vertex is reduced. In particular, it is found that the optimisation algorithm, which could be applied to any mesh cosmetics problem, is very effective, regardless of the quality of the initial mesh. This is illustrated by applying the proposed approach to a number of complex industrial aerospace geometries.