Incorporating improved refinement techniques for a grid-based geometrically-adaptive hexahedral mesh generation algorithm

Highlights

• A set of improved 27-refinement templates is proposed to improve the mesh quality.

• 8-Refinement based mesh generation algorithm is put forward.

• The date structure and procedures for realization of the algorithm are presented.

• A buffer layer is inserted on the concave domains to resolve propagation problems.

• Several examples demonstrate the effectiveness and robustness of the algorithm.

Abstract

This paper presents a density control based adaptive hexahedral mesh generation algorithm for three dimensional models. To improve the mesh quality of hexahedral elements, a set of improved 27-refinement templates is proposed and the refinement modes of these templates are given. A set of effective refinement templates for 8-refinement based mesh generation algorithm is employed. The corresponding date structure and the procedures for realization of the algorithm are also presented. A buffer layer is inserted on the concave domains to resolve the propagation problems. Finally, the effectiveness and robustness of the algorithm are demonstrated by using several examples.

Introduction

With the development of computer technology and numerical method, numerical simulation methods such as finite element method, finite volume method, and finite difference method play more and more important roles in the fields of science research and engineering applications. Mesh generation is the key technique in the preprocessing part of numerical analysis software, and its task is to discretize the solid model into a ‘mesh’ composed of a number of elements. The efficiency and accuracy of numerical analysis and the reliability of software computation are strongly dependent on the density and quality of mesh model. In three-dimensional numerical analysis, tetrahedron, hexahedron and a combination of them are usually used. Hexahedral element meshes have been proved to be superior to tetrahedral element meshes in terms of analysis accuracy, amount of meshes, distortion resistance and regeneration times [1]. This turns hexahedra an attractive choice for the numerical analysis of three-dimensional problems.

Due to the characteristics of finite element mesh, the quality of deformed mesh has a great effect on the accuracy of numerical analysis. It is crucial to improve the mesh quality. The quality of finite element mesh not only includes the quality of a single element mesh but also the quality of the whole element mesh. A main index of presenting the whole element mesh quality is the reasonable distribution of mesh density. In general, mesh density distribution depends on many factors, such as the geometry of solid models, the topology requirement from an adjacent mesh, field variables distribution, and user’s specification. Although user could manually guide the mesh generation according to experience, this manual density control way is tedious and very approximate, especially when the geometry of a solid model is very complex. Thus adaptive mesh generation techniques are very important. Generally, an effective adaptive mesh generation can give a reasonable mesh density distribution corresponding to the geometric features of engineering problems to be analyzed, enhance the analysis result accuracy and minimize the computational cost at the same time.

In particular, in bulk metal plastic forming analysis, as severe local stress and deformation concentration frequently occurs due to the gradual contact of deforming material with dies, the mesh elements are often distorted and the regeneration process is needed frequently. In addition, in the areas with non-uniform deformation and stress concentration, mesh elements should be meshed adaptively and with reasonable local refinement. Therefore, a sound mesh generation is necessary and it can significantly improve the accuracy and efficiency of the analysis. Up to now, many researches have been done in developing the automatic hexahedral mesh generation algorithms [2], [3], [4], [5], [6]. There are mainly four typical approaches proposed for all-hexahedral mesh generation, including mapping/sweeping method [7], [8], plastering method [9], [10], whisker-weaving method [11], [12], [13] and the grid-based method [14], [15]. The grid-based method is relatively simple to implement and easy to realize the local refinement. Recently, with the development of the adaptive techniques of mesh generation, grid-based method is modified by many researchers and used widely in the mesh [16], [17], [18], [19]. Unfortunately, there are no demonstrated methods for creating grid-based, good-quality, reasonably density distributed hexahedral meshes, especially when the local mesh refinement is required.

One basic requirement of mesh generation algorithm is mesh conformity. At present, the conformal refinement templates are widely used to realize the conformity of hexahedral meshes. The representative is the octree-based refinement templates proposed by Schneiders [20] as shown in Fig. 1, where the black points represent the refinement nodes. This set of refinement templates are named as all refinement, face refinement, edge refinement and node refinement, respectively. It can realize the local conformal refinement of hexahedral mesh elements. Many researchers such as Su et al. [21], Zhang and Bajaj [22], Ito et al. [23], et al. put forward the similar or improved refinement templates on the basis of the Schneiders’ templates. The all refinement template in Fig. 1a subdivides a hexahedral element into 27 similar sub-elements. In this paper, this kind of template is called as 27-refinement and the corresponding set of templates is called as 27-refinement templates. The 27-refinement templates based mesh generation algorithm is referred to as 27-refinement method hereafter. At present, the 27-refinement templates which many researchers adopted either produce much more new elements and nodes after once refinement, or simplify the templates and at the same time reduce the quality of sub-elements. Although the 27-refinement templates can realize the mesh transition, the transition is not smooth from dense meshes to sparse meshes. When the mesh elements within the transition areas are used for finite interpolation calculation, this unsmooth transition of the mesh density will cause a large difference among elements of stiffness matrix and mass matrix, and thereby increase the errors of numerical calculation. Moreover, the 27-refinement templates only can realize the refinement of the mesh whose element size ratio is multiples of 3, but cannot perform the mesh refinement flexibly according to various refinement requirements. Schneiders and Aachen [24] proposed another set of quadtree based templates which include all refinement, face refinement and edge refinement, as shown in Fig. 2. The all refinement template in Fig. 2a subdivides a hexahedral element into 8 similar sub-elements. In this paper, this kind of template is called as 8-refinement and the corresponding set of templates is called as 8-refinement templates. The 8-refinement templates based mesh generation algorithm is referred to as 8-refinement method hereafter. Fig. 2a and b shows the all refinement template and face refinement template, respectively. The face refinement can only be applied when four elements on the same plane and adjacent to one node. The edge refinement can be realized according to the template shown in Fig. 2c. Since the 8-refinement templates can realize gradual and conformal transition from dense meshes to sparse meshes, it is more favorable for the finite element calculation.

Aiming at solving the above problems of local refinement and mesh quality, this paper principally studies the density control based adaptive generation algorithm of three-dimensional hexahedral meshes. In order to improve the mesh quality of hexahedral element, this paper proposes a set of improved 27-refinement templates to perform local mesh refinement efficiently without creating any hanging nodes. At the same time, a set of effective 8-refinement templates is employed to ensure the hexahedral mesh transition gradually and conformably. A buffer layer is inserted on the transition refinement domains that contain concave corners to resolve the propagation problem. By using the 8-refinement method and the improved 27-refinement method, the local refinement of all-hexahedral meshes can be easily realized according to the different density requirement of users.