2D Mesh smoothing based on Markov chain method
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ORIGINAL ARTICLE
2D Mesh smoothing based on Markov chain method Fan Yang1,2,3 · Dujiang Zhang1,2 · Hu Ren4 · JinXiu Xu4 Received: 24 March 2019 / Accepted: 21 May 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract The mesh quality is of vital importance to obtain the numerical results precisely. Poorly shaped or distorted elements can be produced by automatic mesh generation tools. In this article, the mesh smoothing algorithm based on the Markov chain Monte Carlo method is proposed to improve the quality of the mesh. The movement of nodes position is converted to a stochastic process to seek the best position for the element quality. Compared with the widely known Laplacian smoothing and optimization-based smoothing techniques, the mesh quality by the proposed method is found better than these methods. Examples are performed to illustrate the applicability of the approach. The numerical results show that the proposed algorithm is effective and valuable. Keywords Mesh quality · Laplacian smoothing · Optimization-based smoothing · Markov chain · Stochastic process
1 Introduction Numerical method is a powerful technique for solving partial differential equations by partitioning the domain with meshes. The quality of the mesh has a significant effect on the accuracy of the analysis results. For example, errors of gradient can be increased by large element angles and the condition of the stiffness matrix can be increased by smallangle elements during finite element analysis. However, automatic mesh generation process may not produce a very good mesh to meet the requirement of the analysis accuracy [1, 2]. In particular, it is difficult to generate a high-quality mesh automatically for a complex geometry. Therefore, the
* Fan Yang [email protected] 1
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
3
State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
4
Wuxi Hengding Supercomputing Center Ltd, Wuxi 214135, China
mesh quality needs to be improved after automatically mesh generation. There are two basic methods to improve the quality of unstructured mesh: mesh modification methods [3, 4] and mesh smoothing methods [5–7]. Mesh modification methods change the mesh topology which includes edge swapping, face swapping, node insertion and deletion, and local mesh refinement [8]. Meanwhile, the topology connection of node is not changed in mesh smoothing method [9, 10]. The Laplacian smoothing is a very famous method for mesh quality improvement [11]. This method is simple and effective with an iteration process by moving each node to the average of its neighboring nodes in most cases. However, the mesh quality is not always improved, especially applying on the case of the nonconvex el
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