This article deals with solving partial differential equations with the finite element method on hybrid non-conforming hexahedral-tetrahedral meshes. By non-conforming, we mean that a quadrangular face of a hexahedron can be connected to two triangular faces of tetrahedra. We introduce a set of low-order continuous (C0) finite element spaces defined on these meshes. They are built from standard tri-linear and quadratic Lagrange finite elements with an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to recover continuity. We consider both the continuity of the geometry and the continuity of the function basis as follows: the continuity of the geometry is achieved by using quadratic mappings for tetrahedra connected to tri-affine hexahedra and the continuity of interpolating functions is enforced in a similar manner by using quadratic Lagrange basis on tetrahedra with constraints at non-conforming junctions to match tri-linear hexahedra. The so-defined function spaces are validated numerically on simple Poisson and linear elasticity problems for which an analytical solution is known. We observe that using a hybrid mesh with the proposed function spaces results in an accuracy significantly better than when using linear tetrahedra and slightly worse than when solely using tri-linear hexahedra. As a consequence, the proposed function spaces may be a promising alternative for complex geometries that are out of reach of existing full hexahedral meshing methods.
The idea of using non-conforming hexahedral-tetrahedral meshes is not new and has been successfully developed in the context of the discontinuous Galerkin Method in electromagnetic in @cite_10 , @cite_17 and @cite_22 . For continuous Galerkin methods, constraints to ensure continuity of the divergence and of the rotational along non-conforming interfaces have been briefly proposed in @cite_18 . In a engineering approach, hexahedra-tetrahedra non-conforming junction have been firstly discussed in @cite_13 which proposes various multi-point constraints to ensure the function continuity or to minimize the error, depending of the finite element considered in their software. An extention of this approach @cite_6 discards non-conforming hexahedral-tetrahedral junctions in favor of pyramidal elements. Our contribution is to give a formal approach to this problem and to deal with the geometric discontinuity arising with non-planar hexahedra faces, which was not considered in previous work to our knowledge.
@cite_18 @cite_22 @cite_10 @cite_6 @cite_13 @cite_17
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We present a high-order discontinuous Galerkin (DG) method for solving the time-dependent Maxwell equations on non-conforming hybrid meshes. The hybrid mesh combines unstructured tetrahedra for the discretization of irregularly shaped objects with a hexahedral mesh for the rest of the computational domain. The transition between tetrahedra and hexahedra is completely non-conform, that is, no pyramidal or prismatic elements are introduced to link these elements. Within each mesh element, the electromagnetic field components are approximated by a arbitrary order nodal polynomial and a centered approximation is used for the evaluation of numerical fluxes at inter-element boundaries. The time integration of the associated semi-discrete equations is achieved by a fourth-order leap-frog scheme. The method is described and discussed, including algorithm formulation, stability, and practical implementation issues such as the hybrid mesh generation and the computation of flux matrices with cubature rules. We illustrate the performance of the proposed method on several two- and three-dimensional examples involving comparisons with DG methods on single element-type meshes. The results show that the use of non-conforming hybrid meshes in DG methods allows for a notable reduction in computing time without sacrificing accuracy. 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. A new method for constructing H ( curl ) or H ( div ) conforming hexahedral tetrahedral hybrid meshes of arbitrary discretisation order is presented. Tetrahedral elements that conform directly to hexahedral elements are constructed, avoiding the need for pyramidal or other joining element types. The effectiveness of this method is investigated in a finite-element framework, and some caveats pertaining to higher-order pyramidal elements are raised. 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.