A new method of quality improvement for quadrilateral mesh based on small polygon reconnection

Transcription

1 Acta Mech. Sin. (2012) 28(1): DOI /s x RESEARCH PAPER A new method of quality improvement for quadrilateral mesh based on small polygon reconnection Jian-Fei Liu Shu-Li Sun Yong-Qiang Chen Received: 2 April 2011 / Revised: 22 July 2011 / Accepted: 17 August 2011 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012 Abstract In this paper, a new method of topological cleanup for quadrilateral mesh is presented. The method first selects a patch of mesh around an irregular node. It then seeks the best connection of the selected patch according to its irregular valence using a new topological operation: small polygon reconnection (SPR). By replacing the original patch with an optimal one that has less irregular valence, mesh quality can be improved. Three applications based on the proposed approach are enumerated: (1) improving the quality of a quadrilateral mesh, (2) converting a triangular mesh to a quadrilateral one, and (3) adapting a triangle generator to a quadrilateral one. The presented method is highly effective in all three applications. Keywords Mesh improvement Topological clean-up Quadrilateral mesh Small polygon reconnection cleanup changes the mesh topology, i.e. node-element connectivity relationship. The present paper focuses on quadrilateral meshes. The effects of smoothing and cleanup on a quadrilateral mesh are shown in Figs. 1 and 2, respectively. Fig. 1 Smoothing a quadrilateral mesh. a Before smoothing; b After smoothing 1 Introduction Geometrical optimization (likewise called smoothing or node repositioning) and topological optimization (likewise called topological cleanup or local reconnection) are two main categories of the mesh improvement procedure. The smoothing relocates mesh nodes to improve mesh quality without changing mesh topology, whereas the topological In Fig. 1, a mesh is improved by smoothing to an ideal structure, composed of perfect rectangles. The result of smoothing usually depends on the mesh topology. Figure 2a is a mesh with the same node positions as Fig. 1a, but with different mesh topologies. The consequence is that Fig. 2a can not be improved by smoothing. Here, topological cleanup is necessary and is also feasible, transforming the mesh from Fig. 2a to Fig. 2b by changing the connectivity relationships between nodes. The project was supported by the National Natural Science Foundation of China ( , ) and National Basic Research Program of China (2010CB832701). J.-F. Liu S.-L. Sun ( ) Y.-Q. Chen LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing, China sunsl@mech.pku.edu.cn Fig. 2 Topological cleanup for a quadrilateral mesh; a Before topological cleanup; b After topological cleanup

2 A new method of quality improvement for quadrilateral mesh based on small polygon reconnection 141 In practice, a mesh improvement scheme will combine the above two methods, as shown in Fig. 3. The present paper seeks to develop a new cleanup technique and combine it with smoothing to achieve quadrilateral mesh improvement. Moreover, it aims to develop two extended applications, which will be discussed in Sect. 3. To optimize the topology of a quadrilateral mesh, the valence of its nodes must be improved. Node valence refers to the number of elements adjacent to a node. For example, the valence of node A, B, C, and D in Fig. 3a is 5, 3, 3, and 5, respectively. Note that the perfect number of node valence is 4. The node valences of A, B, C, and D are a perfect 4 in Fig. 3b. A node is regular if its valence is 4; otherwise, it is an irregular node. Decreasing the number of irregular nodes may improve the mesh quality. The purpose of topological cleanup for a quadrilateral mesh is to make irregular nodes as sparse as possible. Fig. 3 A typical mesh improvement scheme combining smoothing with cleanup. a Initial mesh; b Mesh after cleanup; c Mesh after smoothing A number of terms and definitions are provided in the following subsection. Two of them, namely, node valence and irregular node are mentioned above; however, they are listed again for completeness and clarity. 1.1 Terms and definitions Mesh patch A set of neighboring elements bounded with one simple polygon Polygon Boundary of a mesh patch; a polygon can be subdivided into quadrilateral elements by different ways Node valence Number of elements incident on a node Irregular node Node with valence other than 4 Irregular valence Evaluation of a node; set as the of a node difference between its valence and 4 Irregular valence Summation of irregular valence of a mesh of nodes in a mesh 1.2 Related works and outline As previously mentioned, the topological cleanup aims at improving mesh quality by changing its connectivity [1-6]. It is usually carried out after a mesh has been generated by a certain meshing algorithm. The goal is to decrease the number of those nodes to which the elements attached are either too few or too many. The methods presented in literature on this subject are usually based on predefined templates [2,3,6]. A template links a pair of specific configurations of the mesh connectivity. To perform the cleanup, the first configuration in a template should be switched to the second one with better node valence. Because the number of possible configuration of a mesh increases dramatically with its size, a previously recognized template can not be too large. Typically, the templates are local and limited in a region of a few elements at most. By applying local cleanup templates successively on a mesh, many problematic configurations can be resolved. However, certain optimization options of mesh connectivity may be skipped by template-based methods. Recently, Bunin [7] presented a topological meshing method wherein a mesh patch around a problematic node is selected, and boundary of the mesh patch is represented by a polygon. The polygon is then topologically meshed in a specific way, and the resulting mesh replaces the original one if it possesses a better topological quality. This method can treat a considerably bigger area than template-based methods. With this method, however, geometrical information may be lost. Thus, it may create inverted elements and the element size is not in compliance with the predefined one. Moreover, it can only re-mesh a region with three, four, or five logical sides, without guarantee that the suggested connectivity is optimized. In the current paper, we present a more flexible method to re-mesh arbitrary regions and improve mesh connectivity. The size of regions considered in the presented method is considerably bigger than template-based methods, and is similar to that proposed by Bunin. However, geometrical and element size information will be considered. Optimal solution for a target region is guaranteed since the basic idea involves searching for the best connectivity of a region based on all meshing possibilities.

3 142 J.-F. Liu, et al. In a series of studies conducted by Liu et al. [8 11], a method called small polyhedron reconnection (SPR) was proposed to improve mesh quality and recover the mesh boundary by optimizing the connectivity of target regions. In the said papers, however, the method was applied merely to tetrahedral mesh. By contrast, the concept of SPR in the present paper is extended from tetrahedral to quadrilateral mesh. Polyhedrons will be degenerated to polygons. The current paper is organized as follows. In Sect. 2, the basic idea and algorithm of SPR for quadrilateral mesh are described. Section 3 presents a number of applications of the proposed method, including improving the quality of a quadrilateral mesh, converting a triangular mesh to a quadrilateral one, and adapting a triangle generator to a quadrilateral one. Discussions and conclusions are briefly drawn in Sect SPR or small polygon reconnection for quadrilateral mesh 2.1 Basic idea and procedure The SPR method finds the best connecting configuration of any given polygon (with or without internal nodes). To illustrate this, meshing of a polygon with two internal nodes (marked with circles, as shown in Fig. 4a), is performed. Meshing can be performed in nine different ways, four of which are shown in Fig. 4b; the rest can be obtained by geometric symmetries. Evidently, the best way to mesh the polygon is shown in the rightmost image in Fig. 4b, with optimized node valences. Fig. 4 A polygon with two internal nodes and different ways to mesh it. a A polygon with two internal nodes; b Four different connecting ways of the polygon The basic procedure to optimize node valences of a mesh based on SPR are described as follows. First, a polygon is constructed from the mesh by merging a patch of neighboring elements. Second, SPR operation is applied to the polygon. Finally, the original mesh of the patch is replaced with the best mesh of the polygon found by SPR. Figure 5 roughly illustrates the proposed procedure. A patch from a mesh shown as solid lines in Fig. 5a is first selected and taken out, then a polygon is constructed as shown in Fig. 5b. The optimized connecting configuration is searched according to node valences. Finally, the best connectivity shown in Fig. 5c is retained to replace the original mesh of the selected patch. Fig. 5 A patch of mesh is replaced by a new one covering the same polygon. a A patch of mesh to be taken out; b A polygon formed; c The best solution applied to the mesh Node valences of a patch can possibly be improved if a few or all of its internal nodes are ignored. This is allowed in our SPR method if the element size of the optimized mesh is close to the initial one, as illustrated by an example in the next section.

4 A new method of quality improvement for quadrilateral mesh based on small polygon reconnection Searching technique in SPR operation The best way to mesh a polygon can be achieved through systematically searching, constructing, and comparing all feasible solutions. To construct a mesh, we follow the famous advancing front technique by digging away quadrilateral elements from the polygon in an individual manner (Fig. 6). Once the remains are empty, one meshing solution is found; it will be recorded if found to be better than the previous ones. Fig. 6 Using advancing front technique to obtain a feasible solution for meshing a polygon. a A polygon to be meshed; b An element is dug out; c Two elements created Theoretically, there are O(n!) possible meshing ways to construct and compare, where n is the number of nodes of the polygon. The efficiency of searching algorithm is closely related to the success of the SPR method. There are two simple checks that can prevent most meshing solutions from being constructed and compared: Valid check. An element can not be concave. It can not cross the polygon boundary, and it should meet the element size requirement implied by the initial mesh. Valence check. The procedure will maintain a variable named allowed irregular valence (AIV), which yields the allowed amount of irregular valence for the polygon. The initial value of AIV is counted from the initial mesh, and AIV continues to decrease whenever a better meshing method is determined. When constructing a new meshing solution, we consistently calculate its irregular valence. If, in a certain step, the amount of irregular valence is larger than AIV, the current constructing process will be aborted. Similar to its tetrahedral predecessor [11], the above searching process is organized in a recursive algorithm. It is sufficiently efficient to treat a two-layer polygon in reasonable time and a two-layer polygon is good enough for optimizing the connectivity of a quadrilateral mesh. and create a polygon around it to determine the best way to mesh the polygon with the least irregular valence. Figure 7 illustrates the procedure. Procedure 1: Improve mesh connectivity based on SPR operation. For each irregular node nd; Get a polygon poly by merging two layers of elements around nd; denote the original mesh as m0; do SPR operation for poly, i.e., from all possible meshes of poly, get the mesh mb with best connecting way. If mb has less irregular valence than m0, then replace m0 with mb and update local mesh bounded by poly. End of Procedure 1. 3 Applications of SPR The node valences of the whole mesh can be improved patch by patch using the above procedure. Topological connectivity and quality of mesh can be improved as well. In addition to improving the quality of a quadrilateral mesh, the proposed SPR method can be used to convert a triangular mesh to a quadrilateral mesh and to adapt a triangle generator to a quadrilateral generator. The details are shown below. 3.1 Quality improvement for quadrilateral mesh The idea is to perform SPR operation for each irregular node Fig. 7 One SPR operation to improve valence of an irregular node. a Creating a patch around the doted node by merging two layers of elements; b A patch replaced by a new one occupying the same polygon After applying the SPR operation to all irregular nodes, valences of the whole mesh can be improved. Smoothing operation is then undertaken to improve further the quality of the elements by relocating the nodes, as shown in Fig. 8. Mesquite [12] is used for this purpose. After the node positions are adjusted, a sweep of SPR is attempted again to optimize connectivity and subsequently perform smoothing

5 144 J.-F. Liu, et al. (see Procedure 2). After two or three iterations, the proce- dure results in a converged quality mesh (see Fig. 8). Fig. 8 Quality improvement by SPR coupled with smoothing. a Mesh after a sweep of SPR; b Mesh after smoothing; c Converged mesh Procedure 2: Couple SPR with smoothing to improve a quadrilateral mesh. Do{SPR for the whole mesh; Smoothing the mesh;} while (SPR has changed the connectivity); End of procedure. A practical example is presented in Fig. 9, where the original mesh is generated using the latest version of AN- SYS. Although it has very good quality, the mesh can be further improved using the proposed method. Fig. 10 Template of one-to-three conversion for converting a triangular mesh to a quadrilateral mesh Fig. 9 Quality improvement by SPR coupled with smoothing. a Original mesh; b Mesh optimized by SPR coupled with smoothing An example of a quadrilateral mesh converted from a triangular mesh is shown in Fig. 11. The original triangular mesh is generated by a famous tool, Triangle [13]; it is then converted to quadrilateral mesh, as shown in Fig. 11a, through one-to-three conversion. Evidently, the resulting mesh after SPR coupled with smoothing (Fig. 11b) has considerably better quality than the mesh prior to optimization (Fig. 11a). 3.2 Converting a triangular mesh to a quadrilateral mesh A triangular mesh is considerably easier to generate than a quadrilateral mesh. Using SPR, we can convert a triangular mesh to a quadrilateral mesh in a direct manner, following a template of one-to-three conversion shown in Fig. 10. In particular, this involves subdividing each triangle into three quadrilaterals. The resulting mesh is obviously not good for FEA since many of its nodes are naturally irregular. SPR operation, however, can optimize mesh node valences and improve mesh quality dramatically when coupled with smoothing. Fig. 11 Example of converting a triangular mesh to a quadrilateral mesh. a One-to-three conversion result of a triangular mesh; b Final quadrilateral mesh

6 A new method of quality improvement for quadrilateral mesh based on small polygon reconnection Adapting a triangle generator to a quadrilateral generator Adapting a triangle generator to a quadrilateral generator is rather straightforward. A number of extra steps of element type conversion and quality optimization are required. For the purpose of integrality, the following main steps to generate a quadrilateral mesh using our adapted generator are illustrated in Fig. 12. (1) Nodes are created by way of sphere packing. Figures 12a and 12b show a region to be meshed and its sphere packing, respectively. The centers of spheres will be taken as nodes and the radius of spheres will be taken as the element size. (2) Nodes are connected to create triangles using advancing front techniques or Delaunay based methods, as shown in Fig. 12c. (3) Triangles are converted to quadrilaterals through one-to-three conversion, as shown in Fig. 12d. (4) The mesh is optimized using SPR coupled with smoothing, and the final mesh is obtained, as shown in Fig. 12e. As illustrated in Fig. 12, the quadrilateral mesh generated by the adapted generator is fairly good. Only a tiny part of nodes are irregular, with valences of either three or five that are close to a regular value of four. Fig. 12 Meshing steps of an adapted generator. a Region to be meshed; b Nodes created by way of sphere packing; c Creating triangles by connecting nodes; d Result after one-to-three conversion; e Final quadrilateral mesh 4 Discussion and conclusions We have proposed a new method to reduce irregular valence for a quadrilateral mesh. As an extension of Liu et al. s earlier work on tetrahedral mesh, this new method can improve mesh quality dramatically when coupled with node repositioning techniques. There are virtually no difficulties in applying the SPR method to quadrilateral mesh. However, there are indeed certain differences compared with the previously proposed tetrahedral SPR, where small polyhedrons with internal nodes are not considered and all the nodes of a target mesh will be retained after optimization. In the presented quadrilateral mesh version, there are rings of internal nodes in a small polygon. The optimized mesh may have all, several, or even none of these internal nodes in case it satisfies the element size condition. Further research must be performed in the future with priority given to extending SPR to hexahedral mesh. SPR is virtually a local operation regardless of the size of the polygon or polyhedron used, whereas the well-known concept of chords and sheets is taken from a global viewpoint. We believe that these two ideas can merge perfectly, using the global viewpoint induced from chords and sheets to guide the local SPR operation. References 1 Blacker, T.D., Stephenson, M.B.: Paving: A new approach to automated quadrilateral mesh generation. Int. Journal for Numerical Methods in Eng. 32(4), (1991) 2 Canann, S.A., Muthukrishnan, S.N., Phillips, R.K.: Topological improvement procedures for quadrilateral finite element meshes. Engineering with Computers 14(2), (1998) 3 Kinney, P.: Clean Up: Improving quadrilateral finite element meshes. In: Proc. of the 6th International Meshing Roundtable, (1997) 4 Lee, C.K., Lo, S.H.: A new scheme for the generation of graded quadrilateral meshes. Computers and Structures 52(5), (1994) 5 Owen, S.J., Staten, M.L., Canann, S.A., et al.: Advancing front quadrilateral meshing using triangle transformations. In: Proc. of the 7th International Meshing Roundtable, (1998) 6 Staten, M.L., Canann, S.A.: Post refinement element shape improvement for quadrilateral meshes. AMD-220, Trends in Unstructured Mesh Generation, ASME, 9 16 (1997) 7 Bunin G.: Non-local topological clean-up. In: Proc. of 15th International Meshing Roundtable, 3 20 (2006) 8 Liu, J.F., Sun, S.L.: Small polyhedron reconnection: a new way to eliminate poorly-shaped tetrahedra. In: Proc. of 15th International Meshing Roundtable, (2006) 9 Liu, J.F., Sun, S.L., Wang, D.C.: Optimal tetrahedralization for small polyhedron: A new local transformation strategy for 3- D mesh generation and mesh improvement. CMES: Computer Modeling in Engineering & Sciences 14(1), (2006) 10 Liu, J.F., Chen, B.: Boundary recovery after 3D Delaunay tetrahedralization without adding extra nodes. Inter. J. Numer. Meth. Eng. 72(6), (2007) 11 Liu, J.F., Chen, Y.Q., Sun, S.L.: Small polyhedron reconnection for mesh improvement and its implementation based on advancing front technique. Inter. J. Numer. Meth. Eng. 79(8), (2009) quake/triangle.html