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標題: 有限元素結構網格的自動重排研究
The study of Automatic Remesh of Finite Element Structure
作者: 鄒孟達
Tsou, Meng-Ta
關鍵字: 有限元素法,網格平滑化,大變形分析;finite element analysis, mesh smoothing, Laplacian method
出版社: 機械工程學系所
引用: [1] L. Freitag and P. Plassmann. “Local optimization-based simplicial mesh untangling and improvement,” International Journal for Numerical Methods in Engineering, 2000, vol.49, no.1, pp.109–125. [2] T. S. Li, Y. C. Wong, S. M. Hon, C. G. Armstrong and R. M. McKeag, “Smoothing by optimisation for a quadrilateral mesh with invalid elements,” Finite Elements in Analysis and Design, 2000, vol.34, pp.37–60. [3] L. Freitag and C. Ollivier-Gooch, “Tetrahedral Mesh Improvement Using Swapping and Smoothing,” International Journal for Numerical Methods in Engineering, 1997, vol.40, pp.3979–4002. [4] L. Chen. “Mesh smoothing schemes based on optimal delaunay triangulations,” in Proceedings of 13th International Meshing Roundtable, 2004, pp.109–120. [5] J. Shewchuk, “What Is a Good Linear Element? Interpolation, Conditioning, and Quality Measures,” Eleventh International Meshing Roundtable, 2002. [6] G. A. Hansen, R. W. Doglass and A. Zardecki, “Mesh Enhancement,” Imperial College Press, 2005. [7] A. M. Winslow, “Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh,” Journal of Computational Physics, 1967, vol.2, pp.149–172. [8] P. Knupp, “Winslow smoothing on two-dimensional unstructured meshes,” Engineering with Computers, 1999, vol.15, pp.263–268. [9] B. T. Helenbrook, “Mesh deformation using the biharmonic operator,” International Journal for Numerical Methods in Engineering, 2003, vol.56, pp.1007–1021. [10] J. Hermansson and P. Hansbo, “A variable diffusion method for mesh smoothing,” Communications in Numerical Methods in Engineering, 2003, vol.19, no.11, pp.897–908. [11] J. Park and S. Shontz, “Two derivative-free optimization algorithms for mesh quality improvement,” in Proceedings of the 2010 International Conference on Computational Science, 2010, vol.1, pp.387–396. [12] Z. Chen, J. Tristano, W. Kwok, “Combined Laplacian and optimization-based smoothing for quadratic mixed surface meshes,” in Proceedings of 12th International Meshing Roundtable. [13] E. Amezua, M. Hormaza, A. Hernadez and M. Ajuria, “A method for the improvement of 3D solid finite-element meshes,” Advances in Engineering Software, 1995, vol.22, no.1, pp.45–54. [14] N. Amenta, M. Bern and D. Eppstein, “Optimal point placement for mesh smoothing,” in Proceedings of 8th ACM-SIAM Symposium on Discrete Algorithms, 1997, pp.528–537. [15] L. A. Freitag and P. Knupp, “Tetrahedral mesh improvement via optimization of the element condition number,” International Journal for Numerical Methods in Engineering, 2002, vol.53, pp.1377–1391. [16] P. Knupp, “Hexahedral mesh untangling and algebraic mesh quality metrics,” in Proceedings of the 9th International Meshing Roundtable, New Orleans, LA, 2000, pp.173–183. [17] P. Knupp, “Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I: A framework for surface mesh optimization,” International Journal for Numerical Methods in Engineering, 2000, vol.48, pp.401–420. [18] P. Knupp, “Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. Part II: A framework for volume mesh optimization,” International Journal for Numerical Methods in Engineering, 2000, vol.48, pp.1165–1185. [19] P. Knupp, “Algebraic mesh quality metrics,” SIAM Journal on Scientific Computing, 2001, vol.23, no.1, pp.193–218. [20] P. Knupp, L. Margolin, and M. Shashkov, “Reference jacobian optimization-based rezone strategies for arbitrary Lagrangian Eulerian methods,” Journal of Computational Physics, 2002, vol.176, pp.93–128. [21] P. Knupp, “A method for hexahedral mesh shape optimization,” International Journal for Numerical Methods in Engineering, 2003, vol.58, pp.319–332. [22] D. Vartziotis, T. Athanasiadis, I. Goudas and J. Wipper, “Mesh smoothing using the geometric element transformation method,” Computer Methods in Applied Mechanics and Engineering, 2008, no.197, pp.3760–3767. [23] D. Vartziotis and J. Wipper, “The geometric element transformation method for mixed mesh smoothing,” Engineering with Computers, 2009, vol.25, no.3, 287–301. [24] E. J. Lopez, N. M. Nigro and M. A. Storti, “Simultaneous untangling and smoothing of moving grids,” International Journal for Numerical Methods in Engineering, 2008, vol.76, pp.994–1019. [25] P. Knupp, “Updating Meshes on Deforming Domains,” Communications in Numerical Methods in Engineering, 24, 2008, pp.467–476. [26] S. Shontz, S. Vavasis, “A mesh warping algorithm based on weighted Laplacian smoothing,” in Proceedings of the 12th International Meshing Roundtable, 2003, pp.147–158. [27] D. A. Knoll and D. E. Keyes. “Jacobian-free Newton–Krylov methods: a survey of approaches and applications,” Journal of Computational Physics, 2004, vol.193, pp.357–397. [28] C. Johnson, “Numerical solutions of partial differential equations by the finite element method,” Studentlitteratur, 1987. [29] S. Shontz, S. Vavasis, “An algorithm based on finite element weights for warping tetrahedral meshes,” arXiv:cs/0410045v3, http://arxiv. orq/abs/cs. NA/0410045, 2005. [30] S. Shontz, “Numerical methods for problems with moving meshes,” Ph. D. thesis, Cornell University, 2005. [31] T. Zhou, and K. Shimada, “An Angle-Based Approach to Two-dimensional Mesh Smoothing,” Proceedings of the 9th International Meshing Roundtable, 2000, pp.373–384. [32] P. Knupp, “Matrix norms and the condition number,” in Proceedings of the 8th International Meshing Roundtable, Lake Tahoe, CA, 1999, Sandia Report SAND 99-2288, Sandia National Laboratories, Albuquerque, NM, 1999, pp.13–22.
結構體在大變形有限元素分析過程中(包括非線性分析),迭代式運算是必須觸及的過程。這種過程經常會由於變形而使得有限元素網格在運算途中產生覆蓋或重疊的情形,而產生無效網格、自動中斷分析。過去處理這類問題最簡單的方法就是使用拉普拉斯法(Laplacian method)在產生無效網格的前一個步驟重排網格的形狀,以避開下一步驟的無效網格產生現象。但是這個方法並無法保證一定能讓網格排列品質變好。因而陸續又發表了許多的權重法與最佳化法來處理這個問題。可惜的是這些發表的許多方法中,不是公式複雜,就是計算時間相當長,或是程式繕寫不易,在實用上的價值較低。

During the large deformation of finite element analysis (including nonlinear analysis) of structures, the iteration process has to be involved. Because of the deformation, during the computing process, the finite element mesh may frequently encounter folding; an invalid mesh or an interruption of the computing will be encountered. In the past, the simplest solution is to use Laplacian method to remesh in updating the previous step so as to avoid the occurrence of the invalid mesh. However, this method does not guarantee to make a better mesh quality. Therefore, many methods have been introduced, such as the weighting method or optimization so as resolve this issue. Unfortunately, among those known methods, neither them provide a simplified formula, nor shortened costing time. Their program is costly to write rendering less practical value.
The paper introduces a novel re-meshing method which has been proved to be very useful in large deformation and nonlinear analysis. This method features primarily to put a weighting factor into the matrix of Laplacian method during the computing process. The weighting factor is derived directly from the determinant attained by computation of the matrix of the Laplacian method. In this paper, further elaboration and explanation of how the formula is derived, and many examples will be used to prove its practical utilization. Examples attest this method is very practical and meaningful.
其他識別: U0005-0406201310510900
Appears in Collections:機械工程學系所

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