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The paper discusses the problem of subdividing unstructured mesh topologies containing hexahedra, prisms, pyramids and tetrahedra into a consistent set of only tetrahedra, while preserving the overall mesh topology. Efficient algorithms for volume rendering, iso-contouring and particle advection exist for mesh topologies comprised solely of tetrahe...
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... The strategy adopted in the implementation consists in taking the diagonal that start from the node with the smallest identifier. Conformity is always ensured thanks to this subdivision rule [2]. Once subdivided the faces, each element is split into tetrahedra, two for the pyramid, three for the prism, five or six for the hexahedron. ...
Particle tracking of a dispersed phase within an underlying flow field is routinely used to analyse both industrial processes and natural phenomena. The efficiency of parallel algorithms to simulate fluid-particle systems is strongly influenced by the different evolution of the flow and the particles dynamics. In unsteady simulations, the parallel efficiency is even more critical because the flow solution changes over time. Indeed, a domain partitioning based on particle workload is possibly sub-optimal in terms of the number of fluid volume elements associated to each process. An efficient mesh partitioning based on graph representation is implemented. It can handle unstructured hybrid meshes composed by triangles and quadrilaterals in two spatial dimensions, and by tetrahedra, hexahedra, prisms, and pyramids in three dimensions. In order to obtain a domain decomposition to efficiently follow the particle trajectories, a preliminary solution is computed to suitably tag the fluid domain cells. The obtained weights represent the element probabilities to be crossed by particles and are used by the load-balancing algorithm to partition the mesh. Another challenging aspect of a scalable implementation is the initial particle location due to the arbitrary shapes of each subdomain. An innovative parallel raytracing particle location algorithm is presented. It takes advantage of a global identifier for each particle, resulting in a significant reduction of the overall communication among processes. In addition, the parallel particle evolution and collection efficiency computation are described. The proposed approach is tested against reference cases for the coupled flow-particle simulation of ice accretion over 2D and 3D geometries. Two different cloud droplet impact testcases have been simulated: a NACA 0012 wing section and a NACA 64A008swept horizontal tail. Furthermore, a cloud droplet impact test case starting from an unsteady flow around a 3D cylinder has been simulated to evaluate the code performances.
... By adding the hexahedron centroid as a data point, we can generate a subdivision into 12 tetrahedra, all of them Sommerville tetrahedra number 3 (ST3) similar to one another, where each face is still split by a single diagonal. From here, we can progressively add face centroids, dividing a face into four triangles to produce, 14, 16, 18, 20, 22 or 24 isosceles trirectangular tetrahedra [1]. Finally, a further subdivision is carried out by adding new vertices on the cube's edges to obtain 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46 or 48 regular right-type tetrahedra [3]. ...
... Each of these is then subdivided into two tetrahedra by adding an arbitrary diagonal across the face (Face Divided), generating 12 Sommerville tetrahedra number 3 (ST3) [3], and all of them similar to each other, see Fig. 1a. A futher subdivision is carried out by adding progressively face centroids (Face Centered) [1], splitting every single triangle into two sub-triangles generating up to 24 isosceles trirectangular tetrahedra congruent to one another, see Fig. 1b. Finally, the complete barycentric partition of a 3D-cube is achieved by adding progressively new edge centroids (Edge Divided), dividing each triangle into two sub-triangles generating up to 48 regular right-type tetrahedra similar between them, see Fig. 1c, and it is equivalent to the Freudenthal partition of a 3D-cube. ...
The barycentric partition of a 3D-cube into tetrahedra is carried out by adding a new node to the body at the centroid point and then, new nodes are progressively added to the centroids of faces and edges. This procedure generates three types of tetrahedra in every single step called, Sommerville tetrahedron number 3 (ST3), isosceles trirectangular tetrahedron and regular right-type tetrahedron. We are interested in studying the number of similarity classes generated when the 8T-LE partition is applied to these tetrahedra.
... Therefore, the tetrahedralization leads to a larger number of elements (from five to six times). The subdividing algorithms are similar to the algorithms described in [17][18][19]. The adaptive grids retain adaptation grid lines to borders of a geometry and allow one to obtain solutions of better quality than with grids generated by ordinary finite element mesh generators. ...
https://download.atlantis-press.com/article/25902006.pdf
... For example, many finite element solvers, such as adaptive mesh refinement in scientific computation, are designed for tetrahedral meshes but not prismatic elements [8,9]. Many efficient computer graphics algorithms, such as those for volume rendering, iso-contouring and particle advection, only work for tetrahedral meshes [10,11]. ...
We consider the tetrahedral subdivision problem for a polygonal prismatic mesh with prescribed boundary constraints and without Steiner points. We prove the necessary and sufficient conditions for the existence of solutions, and also provide algorithms to compute such a constrained subdivision if there exists one. The result applies to arbitrary k-gonal prismatic meshes and even mixed prismatic meshes, allowing arbitrary topology for the base mesh and arbitrary constraints on the boundary.
... In this manner, no information is needed from the neighboring elements, and the resulting mesh will still be conforming. For more details on the construction of tetrahedra from hexahedral elements, the reader is referred to [1,13]. It is also necessary to map face boundary conditions which are applied to the hexahedral elements to the proper faces of the newly created tetrahedral elements. ...
In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM
gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM
gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.
... It has been shown in [7] that such a prismal mesh needs to be split into a tetrahedral mesh to adapt existing unstructured tetrahedral solvers. It has also been shown in [8,9] that such a conversion is necessary in computer graphics, especially when one wants to apply efficient algorithms for volume rendering and iso-contouring that exist for purely tetrahedral meshes. ...
... The conversion from a prismal mesh to a tetrahedral mesh has been addressed in a lot of work, such as [8,9,10,11,12]. However, none of these methods allows the user to control the triangulation on the boundary of the output volumetric mesh. ...
In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a two-dimensional simplicial complex), assign each triangle with a triple of ±1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) and consistent (i.e. every edge shared by two triangles has opposite signs in these triangles). We show that this problem is the major challenge in converting a volumetric mesh consisting of prisms into a mesh consisting of tetrahedra, where each prism is cut into three tetrahedra. In this paper we provide a complete solution to this problem for topological disks under various boundary conditions ranging from very restricted one to the most flexible one. For each type of boundary conditions, we provide efficient algorithms to compute valid assignments if there is any, or report the obstructions otherwise. For all the proposed algorithms, the convergence is validated and the complexity is analyzed.
... A cube can be decomposed into five or six tetrahedra without inserting additional points, or even more tetrahedra with insertion of cube and face centroids [10]. Carr et al. [7] reported a visualization result and visual artifacts for each decomposition method. ...
... A trilinear isosurface inside a cube can have complex topology and geometry that requires sophisticated procedures for correct reconstruction. Marching Cubes (MC) [10] is the most popular method to extract a piecewise linear approximation of an isosurface. It visits each cubical cell and performs local triangulation depending on a sign configuration of eight vertices of the cell 1 . ...
We describe a method to decompose a cube with trilinear interpolation into a collection of tetrahedra with linear interpolation, where the isosurface topology is preserved for all isovalues during decomposition. Visualization algorithms that require input scalar data to be defined on a tetrahedral grid can utilize our method to process 3D rectilinear data with topological correctness. As one of many possible examples, we apply the decomposition method to topologically accurate tetrahedral mesh extraction of an interval volume from trilinear volumetric imaging data. The topological correctness of the resulting mesh can be critical for accurate simulation and visualization.
... The tetrahedral walk method, which locates a particle based on a tetrahedron enclosed by triangular plane surfaces, needs not to consider the non-planarity of grid. But the method involves a sophisticated tetrahedral decomposition in structured curvilinear grids containing hexahedral cells or unstructured hybrid grids containing mixed element types (Albertelli and Crawfis, 1997; Sadarjoen et al., 1998; Dompierre et al., 1999). On the contrary, the face-by-face search method approximately treats non-planar faces as plane surfaces . ...
a b s t r a c t When particles are traced individually in a numerical simulation of particle-fluid flow, a point-locating scheme is necessary to relate the coordinates of a given point to the grid cell containing it. A new point-locating scheme combining the advantages of the existing ones is presented in this paper. The new method is first explained in detail. Then its performance under three-dimensional hybrid meshes is compared with the existing approaches in the literature. The results show that the proposed method is efficient, multifunctional and easy to implement in a structured/unstructured fluid flow solver.
... With the octree finished we can use several templates to generate the tetrahedra [19]. The four main differentiating characteristics are the number of elements generated, its quality, the distribution of connectivities and whether it is necessary to check the state of the neighbouring octants [20]. ...
One of the most important steps in the process of semiconductor device simulation, or any other numerical simulation based on finite elements, finite differences or similar standard techniques, is the discretization of the domain of the problem. A mesh must be generated, and its properties determine the stability of the numerical solver, computational time and quality of the solution.
In this paper an octree-based mesh generator is presented. The classical model for octree generation have been modified to optimize the programme for special regions of interest in the semiconductor device problem, Manhattan structures with very narrow layers. Using this technique, several meshing patterns have been tested and compared. Numerical results of the generation of general meshes are presented to demonstrate the efficiency of the algorithms from two points of view: mesh quality and computational effort.
It has been successfully applied to the modelling and simulation of different transistors, High Electron Mobilty Transistors (HEMTs) and Metal Oxide Semiconductor Field Effect Transistors (MOSFETs). Copyright
... II. PREVIOUS WORK Previous analysis of simplicial subdivision of cubic cells [8], [9], [10] focusses on the number of triangles generated [10], and on the topological consistency [8], [10] and correctness [10] of the isosurfaces. ...
... II. PREVIOUS WORK Previous analysis of simplicial subdivision of cubic cells [8], [9], [10] focusses on the number of triangles generated [10], and on the topological consistency [8], [10] and correctness [10] of the isosurfaces. ...
... The simplicial subdivision most often used is the minimal subdivision (Sec. IV-B) of 5 tetrahedra per cube [2], [5], [8], [9], [10], [11], [12]. Also reported [8], [10], [11], [13], [14], [15] is the Freudenthal subdivision (Sec. ...
We review schemes for dividing cubic cells into simplices (tetrahedra) for interpolating from sampled data to IR3, present visual and geometric artifacts generated in isosurfaces and volume renderings, and discuss how these artifacts relate to the filter kernels corresponding to the subdivision schemes.
