Construction of polyhedral finite element meshes based upon marching cube algorithm

Highlights

• An automatic algorithm for generating polyhedral meshes was systematically developed.

• Hexahedral elements near the given STL surface were categorized into polyhedral volume topologies based on the MC algorithm.

• Topology ambiguities of the MC algorithm were effectively resolved to provide an appropriate mesh for complex geometry.

• The effectiveness was demonstrated through numerical analysis using the smoothed finite element method.

Abstract

Hexahedral meshes that exhibit the superiority in terms of solution accuracy and convergence rate are preferred to other types of meshes in the finite element analysis. However, the construction of the hexahedral meshes for complex geometries is still considered troublesome due to their poor geometric adaptability. This paper presents an efficient grid-based scheme to automatically generate polyhedral meshes including the hexahedral elements, and thus to provide hexahedral-dominant meshes for three-dimensional geometry with complex shapes. On the basis of the marching cube algorithm with a background grid composed of a regular arrangement of cubes, surface topologies for the background cubes are defined to represent the three-dimensional boundaries of a given domain. Then, in order to generate a three-dimensional finite element mesh, the surface topologies of the marching cube algorithm are systematically expanded to polyhedral volume topologies. Meanwhile, a topology ambiguity problem inherent in the marching cube algorithm is effectively resolved to generate an appropriate polyhedral mesh even for an arbitrary complex geometry. Several examples including biostructure modeling demonstrate that the proposed mesh generation scheme can easily discretize complex three-dimensional domains with hexahedral-dominant meshes, which are composed of the polyhedral elements near the domain boundaries and the hexahedral elements that come from the background cubes inside the domains. Furthermore, to show the applicability and effectiveness of polyhedral meshes in the finite element analysis, some structural analyses are performed using the smoothed finite element method that can be straightforwardly adapted to polyhedral elements of arbitrary shape.

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.