Research paperConstruction of polyhedral finite element meshes based upon marching cube algorithm
Introduction
Although finite element method (FEM) has been widely utilized to facilitate numerical solutions for various engineering problems, the trade-off of finite elements (FEs) between geometric adaptability and fast convergence is yet to be addressed. A typical example can be found in the use of tetrahedral and hexahedral elements. A tetrahedral element retains excellent geometric adaptability, and most of the mesh generators rely on triangulation to yield FE meshes composed of tetrahedral elements. However, in general, the tetrahedral element shows a relatively poor performance in terms of solution convergence, resulting in a significant increase in the degrees of freedom to provide accurate solutions [1]. On the other hand, a hexahedral element shows a very poor geometric adaptability, but it provides a faster solution convergence than the tetrahedral element. If a good mesh generation scheme is devised for the hexahedral element, the hexahedral element can be dominantly utilized even for domains with geometric complexity, replacing the tetrahedral element, which is currently used widely. For this reason, some authors became interested in discretization techniques by use of the hexahedral element [2–8].
For example, Cook and Oakes [2] proposed a mapping method to generate hexahedral elements by mapping two-dimensional rectangular elements in three dimensions. Taniguchi et al. [3] created hexahedral elements by using the indirect method of dividing a tetrahedron into four hexahedra. Schneiders [4] constructed hexahedral elements using the grid-based method. Li et al. [5] proposed a medial surface method to construct hexahedral elements by dividing the edges of a three-dimensional (3D) geometry. Canann [6] extended the paving algorithm to three dimensions to construct hexahedral elements. The Whisker Weaving method [7] and the Hex-Dominant method [8] were also reported. Although these techniques have been successful in automatic hexahedral mesh generation for specific targets, they need to be verified in terms of robustness and reliability to find applications in various problems. In particular, it is still not guaranteed to generate proper hexahedral meshes for complex geometries, such as biostructures, where the curvature of the surface varies rapidly.
To address the above-mentioned issues, attempts have been made to discretize a domain using polyhedral elements with good geometric adaptability. Liu et al. [9] produced octree-based polyhedral mesh. In the stereolithography (STL) model, the hexahedral mesh was subdivided to trim the mesh that intersected the STL model surface. In order to connect the hexahedra of different sizes, the surface mesh was reconstructed by triangulating the surfaces of the hexahedral elements and the scaled boundary FEM [10] was applied. Trimmed hexahedral elements with curved surface were developed by Kim et al. [11,12] which were subdivided into tetrahedral sub-domains to define shape functions using the moving least square approximation. Based on the grid-based method, Sohn et al. [13,14] proposed the carving technique to generate hexahedral-dominant mesh whose outer boundary, corresponding to the STL model surface, was discretized using poly-pyramid elements or polyhedral elements with the triangulated surface. The interior region of the STL model was discretized with hexahedral elements since the carving technique was based on the marching cube (MC) algorithm [15]. The generated polyhedral mesh was then applied to cell-based smoothed finite element method (CS-FEM), which is one type of the smoothed finite element methods (S-FEMs) [16], [17], [18], [19].
The purpose of the present study is to generate polyhedral FE meshes for 3D domains with complex geometry based upon the MC algorithm. Considering polyhedral topologies based on the MC algorithm, a systematic procedure to generate hexahedral-dominant meshes has been presented from the viewpoint of practical implementation and applications. Especially, topology ambiguity has been considered which reveals a serious problem of the MC algorithm that cannot guarantee the continuity for some specific geometry [20,21], thereby affecting geometric adaptability. In contrast to the previous carving technique [13,14], the current proposed scheme does not perform triangulation on the model surface in order to reduce computational cost. Therefore, the 3D geometry can be represented by the lower number of faces compared to the previous study. Finally, the polyhedral element is numerically implemented with the aid of edge-based smoothed finite element method (ES-FEM) [22], which is known to provide the best performance among the S-FEMs [23]. To achieve the effective use of polyhedral elements in the FE analysis, several factors should be considered such as the simplicity in formulating the shape functions, the accuracy of numerical integration, and the robustness to mesh distortion. Since the S-FEMs can meet these requirements [16,23], in current work, the FE analyses using the polyhedral meshes are conducted in the S-FEM framework. The present paper demonstrates the superiority of geometric adaptability, efficiency, and performance via comparison with the previous studies including the carving technique [14].
The rest of the paper is organized as follows: Section 2 elaborates the procedure of polyhedral mesh generation, and special emphasis has been placed on the polyhedral topology to improve geometric adaptability. The proposed method will be verified through numerical examples with complex geometry and various topologies in Section 3. Finally, Section 4 presents some concluding remarks.
Section snippets
Polyhedral mesh generation procedure
The 3D geometry is discretized into polyhedral meshes based on the MC algorithm [15]. The mesh generation procedure proposed in the current study is as follows: (1) extraction of surface information of 3D target geometry, (2) construction of background hexahedral mesh, (3) categorization of the state of every background node inside the 3D model, (4) calculation of intersection point between the background mesh and the surface of the 3D target geometry, and (5) construction of the polyhedral
Numerical implementation and examples
This section presents numerical implementation and examples of the proposed polyhedral mesh generation algorithm. The characteristics of the generated polyhedral meshes were investigated when the density of background hexahedral mesh or STL triangular mesh was changed. The polyhedral meshes were constructed for some 3D models with complex geometries and various topologies, such as biostructures and Stanford bunny. Furthermore, the polyhedral mesh was utilized for S-FEM analysis in order to
Conclusions
In the current work, a polyhedral mesh generation scheme was systematically developed based on the MC algorithm and the detailed procedure was elaborated. In order to guarantee geometric adaptability of the polyhedral mesh, surface topologies of the MC algorithm were expanded to polyhedral volume topology and the ambiguity problem inherent in the MC algorithm was resolved as well. The procedure of the proposed algorithm was performed automatically so that it can enable CAE engineers to save
Acknowledgments
This work was supported by a grant from the National Research Foundation of Korea (NRF) funded by the Korean Government (MSIP) (No. NRF-2015R1A2A1A15056263).
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