Balanced Octree for Tetrahedral Mesh Generation

Nowadays, with the advances of Finite Element Analysis (FEA) packages, some of the engineering and design problems such as stress or thermal deformation can be successfully solved. These are convenient for better incorporating the design constraints of various tasks such as injection molded parts, or rapid prototyping and tooling. Mesh generation is the major step of finite element method for numerical computation. Common types of mesh include triangulation or tetrahedralization. During the mesh generation process, we always find difficulty in the formation of a uniform, non-conformal mesh. The undesirable mesh will adversely influence the accuracy and meshing time of the model. This paper will, thus, propose an effective approach to extend to threedimensional (3D) mesh generation by octree balancing method so as to adjust the mesh pattern. In this paper, the implementation of octree balancing will be explained and illustrated with real life example. The proposed method includes three main steps. Problematic unbalanced octants will be detected and Steiner points will be added as appropriate before the tetrahedral mesh generation. The balanced octree will form good tetrahedral meshes for further analysis. Then the balanced and unbalanced meshes will be compared for efficiency and accuracy for mesh generation. Introduction Spatial enumeration method [1] such as quadtree, octree or regular grid has been contributed to mesh generation of models for design analysis such as FEA. Some researches have proposed balancing quadtree [2] for mesh generation. This method can provide the mesh with smoothing and conformal approach for FEA analysis. The quadtree balancing technique has been successfully applied to 2D mesh generation which can solve the above mentioned difficulty. This technique is proposed by generating adaptively non-uniform triangular mesh. In 3D mesh generation, octree representation is a good approximation method than similar regular grids for 3D model subdivision. It can recursively subdivide the model based on their boundary if necessary. Thus, the number of grids generated by octree is less than the regular grid method. It can reduce the computation storage space. However, the non-uniform tetrahedralization of octree will lead to mesh that would no longer be conforming or topologically consistent. By definition, the mesh must be conformed, non-uniform and well-shaped. Otherwise, the processing time required for the mesh generation will increase and numerical calculation will be unstable. Related Work. Kwak and Im [3] have proposed an effective and accurate remeshing scheme by using octree based refinement. This method makes use of a simple geometric model which is enclosed by hexahedral mesh generation with refined octree. The appealing of the mesh pattern of the models induces with non-conformal hexagonal meshes. Although, the number of elements decreases, the global relative error (%) of the analysis decreases less sharply than uniform meshing model. Jung and Lee [4] proposed a tetrahedral-based octree encoding algorithm for the generation of finite element meshes automatically. This technique provides a numbering scheme to relocate the model vertices without topologic transformation. Perucchio and Saxena [5] have proposed an automatic meshing algorithm for FEA. The analyzed solid model is partitioned by recursive spatial decomposition and approximated into finite element meshes with only tetrahedral type. Materials Science Forum Vols. 471-472 (2004) pp 608-612 online at http://www.scientific.net © (2004) Trans T ch P blications, Switzerland Online available since 2004/Dec/15 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.34-14/04/08,10:51:19) Materials Science Forum Vols. *** 609 Overview of Mesh Generation for FEA During the finite element analysis of complex 3D geometric models, mesh generation is implemented to subdivide a model called domain (Ω) into various smaller topologic elements (K.) The triangular elements formation is called tessellation. The domain is defined as the finite union of elements. Triangulation can be illustrated in 2D or 3D which depended on the input of topological data. For 2D triangulation, the input domain is a polygonal region of the plane and triangles intersect between each other by sharing edges and vertices. The representation of 3D meshing is called tetrahedralization. Each tetrahedron must share the vertices with each of its adjacent tetrahedrae. Meshing is a collection of triangular contiguous, non-overlapping faces joined together along their edges. It is an irregular network. A mesh contains vertices, edges and faces. The easiest representation is a single face. Finite element methods are generally used to generate a surface mesh. Meshing Features. During the tetrahedral mesh generation process, smoothing of the mesh model can be acquired by the fulfillment of the following features. 1 Well-shaped element mesh: The angles of the mesh triangle should not be too large or too small. The angles should be maintained within 45o to 90o. 2 Non-uniform: The mesh distribution should have a high density near the boundary region. It should be coarse inside or outside the boundary of the model. 3 Conformance: The vertices of triangular or tetrahedral elements should not be bounded to the interior edge of another element according to the vertex to vertex rule. The difference between the non-conforming and conforming triangles is illustrated in Fig. 1.The vertices locate on the edge of adjacent triangle (Fig.1(a).) Thus, a degenerated facet will be formed. Fig.1 Triangulation of a domain in a planar a) Non-conforming mesh; b) Conforming mesh. Implementation An octree balancing technique is initialized in the mesh generation of the FEA. The proposed mesh generation technique to be described is divided into three steps. A real life model of a computer mouse is employed for analysis. The commercial FEA package employed is MSC/Patran for preprocessing and mesh generation. Fig. 2 Isometric view of computer mouse; a) Shaded model; b) Wireframe model Step 1: Model Discretization. The 3D CAD model is exported with the required file format such as Parasolid or IGES files. The CAD file is then transferred to the FEA business package of MSC/Patran. Under the pre-processing, the mouse model domain (Ω) is approximated into a (a) (b) (a) (b) Materials Science Forum Vols. 471-472 609 Advances in Materials Manufacturing Science and Technology 610 homogenous, heterogeneous and boundary regions (∂) by discretization. The shaded and wireframe mouse models are shown in Fig.2. Step 2: Spatial Enumeration and Balancing.The domain is decomposed into various sized octants L(Ω) in by recursive subdivision. The subdivision is terminated until a user satisfactory resolution (or depth) is reached. Then, a hierarchical representation is derived from these octants which cover the whole domain. Fig.3 shows the spatial enumeration of the domain. The domain is partitioned into three colours. The green colour represents the octants within the boundary. The red colour and the blue colour represent the boundary enclosed and the octants outside the boundary respectively. The octants enclosing domain is balanced by further subdividing the adjacent octants (inside or outside the boundary.) The subdivision reduces the size of octants at boundary which is based on the criterion that the difference is at most a factor of two or in one depth (Fig.4.) Fig.3 Spatial enumeration of the model; a) Top view of the Octree representation; b) 3D representation of octree subdivision (boundary in red colour) Fig.4 Balanced octree represented in; a) Top view; b) Isometric view (a) (b) Fig.5 Connectivity of two tetrahedrae; a) Edges connection without refinement; b) Insertion of Steiner points for well-shaped tetrahedral mesh formation Step 3: Meshing Generation Process. The balanced octree domain is then changed into tetrahedral mesh by tetrahedralization process. The connectivity between adjacent octants is based (a) (b) (b) 610 Advances in Materials Manufacturing Science and Technology Materials Science Forum Vols. *** 611 on the rules of vertex-to-vertex, face-to-face or edge-to-edge connections. In order to obtain wellshaped tetrahedrons, extra points called Steiner points (Fig.5) are allowed to insert into the domain for adjacent tetrahedrons to connect. Fig.6 and Fig.7 show the mesh generation of the domain. Fig.6 Tetrahedral mesh distribution; a) Balanced mesh; b) Unbalanced mesh Fig.7 Connectivity of tetrahedral mesh; a) Two tetrahedrons connected to perform a well-shaped tetrahedralization; b) Edges are connected (in blue and red colour) between two tetrahedrons Preliminary Results and Discussions A comparison of computation time under different meshing models is shown in Table 1 Table 1 Difference between computation time of mesh generation under unbalanced tetrahedral, balanced octree tetrahedral and regular tetrahedral meshes Meshing models Global edge length [mm] Number of nodes Number of elements Computation time [s] Global tolerance [mm] Unbalanced tetrahedral 1 1155 1309 7 0.00499 Balanced octree tetrahedral 1 1284 1477 9 0.00499 Regular tetrahedral 1 2650 3545 14 0.00499 The results in Table 1 illustrate that the balanced octree tetrahedral mesh model has smaller number of nodes and elements than the regular tetrahedral mesh one. The time required for numerical calculation for balanced octree tetrahedral mesh is reduced by 6s. The meshing efficiency increases by 35.7%. Besides, the proposed technique provides additional advantage such as mesh boundary smoothing for the mesh generation process. As the tetrahedral elements are mainly mapped to the boundary regions, the smaller sized tetrahedral elements enclose the model boundary better. This increases the accuracy of numerical calculation by FEA. (a) (b) Common edges shaded at the same facet (a) (b) Materials Science Forum Vols. 471-472 611 Advances in Materials Manufacturing Science and Technology 612 Conclusions An octree balancing technique is proposed in this paper. Case studies have been performed to verify the efficiency of the balanced tetrahedral mesh generation. The preliminary results of computation time for balanced octree tetrahedral mesh show a 35.7% decrease when compared to regular tetrahedral mesh. The most complex and essential region can be analyzed unambiguously. Extension to mesh refinement and optimization can be explored further. Acknowledgements The work described in this paper was supported by a grant from The Hong Kong Polytechnic University.