Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM

High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems

Adaptive Meshing Techniques for Viscous Flow Calculations on Mixed Element Unstructured Meshes

An adaptive refinement strategy based on hierarchical element subdivision is formulated and implemented for meshes containing arbitrary mixtures of tetrahendra, hexahendra, prisms and pyramids. Special attention is given to keeping memory overheads as low as possible. This procedure is coupled with an algebraic multigrid flow solver which operates on mixed-element meshes. Inviscid flows as well as viscous flows are computed an adaptively refined tetrahedral, hexahedral, and hybrid meshes. The efficiency of the method is demonstrated by generating an adapted hexahedral mesh containing 3 million vertices on a relatively inexpensive workstation

On a general implementation of hh- and pp-adaptive curl-conforming finite elements

Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, specially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR. We cover into detail how to implement arbitrary order (i.e., $p$-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation

Combinatorial Methods in Grid based Meshing

This paper describes a novel method of generating hex-dominant meshes using pre-computed optimal subdivisions of the unit cube in a grid-based approach. Our method addresses geometries that are standard in mechanical engineering and often must comply with the restrictions of subtractive manufacturability. A central component of our method is the set of subdivisions we pre-compute with Answer Set Programming. Despite being computationally expensive, we obtain optimal meshes of up to 35 nodes available to our method in a template fashion. The first step in our grid-based method generates a coarse Precursor Mesh for meshing complete parts representing the bar stock. Then, the resulting mesh is generated in a subtractive manner by inserting and fitting the pre-generated subdivisions into the Precursor Mesh. This step guarantees that the elements are of good quality. In the final stage, the mesh nodes are mapped to geometric entities of the target geometry to get an exact match. We demonstrate our method with multiple examples showing the strength of this approach

Three-dimensional unstructured gridding for complex wells and geological features in subsurface reservoirs, with CVD-MPFA discretization performance

Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The features to be honoured are classified into two groups; (1) involving layers, faults, pinchouts and fractures, and (2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point (defined as the centroid of the control-volume) well aligned grids (WAGs) are required. Depending on discretization method type and formulation, a choice of control-point and control-volume type is made, i.e. for a cell-centered method the primal grid cells act as control-volumes, otherwise for a vertex-centered method the dual-grid cells act as control-volumes. Novel three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to complex wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both grid types use tetrahedra, pyramids, prisms and hexahedra as grid elements. Primal-cell feature aligned grids are generated using special boundary surface protection techniques together with constrained cell-centered well trajectory alignment. Dual-cell feature aligned grids are generated from underlying primal-meshes, whereby features are protected such that dual-cell control-volume faces are aligned with interior feature boundaries, together with protected vertex-centered (control point) well trajectory alignment. The novel methods of grid generation presented enable practical application of both method types in 3-D for the first time. The primal and dual grids generated here demonstrate the gridding methods, and enable the first comparative performance study of cell-vertex versus cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent mesh resolution on challenging problems in 3-D. Pressure fields computed by the cell-centered and vertex-centered CVD-MPFA schemes are compared and contrasted relative to the respective degrees of freedom employed, and demonstrate the relative benefits of each approximation type. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency. Numerical results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA method

High-order finite elements on pyramids: approximation spaces, unisolvency and exactness

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial approximation properties.Comment: 37 pages, 3 figures. This work was originally in one paper, then split into two; it has now been recombined into one paper, with substantial changes from both of its previous form