基于分区径向基函数配点法的大变形分析

王莉华; 李溢铭; 褚福运

doi:10.6052/0459-1879-19-005

基于分区径向基函数配点法的大变形分析

doi: 10.6052/0459-1879-19-005

*同济大学航空航天与力学学院, 上海 200092
†上海宇航系统工程研究所, 上海 201109

基金项目: 1) 国家自然科学基金项目(11572229)和中央高校基本科研业务费项目(22120180063)资助.

详细信息

通讯作者:
王莉华

中图分类号: O34;
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- 被引次数: 0
出版历程
- 收稿日期: 2018-01-04
- 刊出日期: 2019-05-18

FINTE SUBDOMAIN RADIAL BASIS COLLOCATION METHOD FOR THE LARGE DEFORMATION ANALYSIS

*School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
†Aerospace System Engineering Shanghai, Shanghai 201109, China

摘要

摘要: 无网格法因为不需要划分网格, 可以避免网格畸变问题,使得其广泛应用于大变形和一些复杂问题. 径向基函数配点法是一种典型的强形式无网格法,这种方法具有完全不需要任何网格、求解过程简单、精度高、收敛性好以及易于扩展到高维空间等优点,但是由于其采用全域的形函数, 在求解高梯度问题时存在精度较低和无法很好地反应局部特性的缺点. 针对这个问题,本文引入分区径向基函数配点法来求解局部存在高梯度的大变形问题. 基于完全拉格朗日格式,采用牛顿迭代法建立了分区径向基函数配点法在大变形分析中的增量求解模式.这种方法将求解域根据其几何特点划分成若干个子域, 在子域内构建径向基函数插值, 在界面上施加所有的界面连续条件,构建分块稀疏矩阵统一求解. 该方法仍然保持超收敛性, 且将原来的满阵转化成了稀疏矩阵, 降低了存储空间,提高了计算效率. 相比较于传统的径向基函数配点法和有限元法, 这种方法能够更好地反应局部特性和求解高梯度问题.数值分析表明该方法能够有效求解局部存在高梯度的大变形问题.
- 无网格 /
- 分区径向基函数配点法 /
- 大变形 /
- 高梯度 /
- 牛顿迭代法
Abstract: The meshfree methods can avoid grid distortion problems because it does not need to be meshed, which make them widely used in large deformations and other complicated problems. Radial basis collocation method (RBCM) is a typical strong form meshfree method. This method has the advantages of no need for any mesh, simple solution process, high precision, good convergence and easy expansion to high-dimensional problems. Since the global shape function is used, this method has the disadvantages of low precision and poor representation to local characteristics when solving high gradient problems. To resolve this issue, this paper introduces finite subdomain radial basis collocation method to solve the large deformation problem with local high gradients. Based on the total Lagrangian formulation, the Newton iteration method is used to establish the incremental solution scheme of the FSRBCM in large deformation analysis. This method partitions the solution domain into several subdomains according to its geometric characteristics, then constructs radial basis function interpolation in the subdomains, and imposes all the interface continuous conditions on the interfaces, which results in a block sparse matrix for the numerical solution. The proposed method has super convergence and transforms the original full matrix into a sparse matrix, which reduces the storage space and improves the computational efficiency. Compared to the traditional RBCM and finite element method (FEM), this method can better reflect local characteristics and solve high gradient problems. Numerical simulations show that the method can effectively solve the large deformation problems with local high gradients.
- meshfree /
- finite subdomain radial basis collocation method /
- large deformation /
- high gradient /
- Newton iteration method

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参考文献(1)

[1]	Belytschko T, Lu YY, Gu L.Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256
[2]	Belytschko T, Krongauz Y, Organ D, et al.Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4): 3-47
[3]	Liu WK, Jun S, Zhang YF.Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8-9): 1081-1106
[4]	张雄, 刘岩, 马上. 无网格法的理论及应用.力学进展, 2009, 39(1): 1-36
[4]	(Zhang Xiong, Liu Yan, Ma Shang.Meshfree method and their applications. Advance in Mechanics, 2009, 39(1): 1-36 (in Chinese))
[5]	Chen JS, Hillman M, Chi SW.Meshfree methods: Progress made after 20 years. Journal of Engineering Mechanics, 2017, 143(4): 04017001
[6]	Chen JS, Pan C, Wu CT, et al.Reproducing kernel particle methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1-4): 195-227
[7]	程玉民, 李九红. 弹性力学的复变量无网格方法. 物理学报, 2005, 54(10): 4463-4471
[7]	(Chen Yuming, Li Jiuhong.A meshless method with complex variables for elasticity. Acta Physica Sinica, 2005, 54(10): 4463-4471 (in Chinese))
[8]	Liu MB, Liu GR.Smoothed particle hydrodynamics (SPH): An overview and recent developments. Archives of Computational Methods in Engineering, 2010, 17(1): 25-76
[9]	邓立克, 王东东, 王家睿等. 薄板分析的线性基梯度光滑伽辽金无网格法. 力学学报, 2019, 51(3)
[9]	(Deng Like, Wang Dongdong, Wang Jiarui, et al.A gradient smoothing Galerkin meshfree method for thin plate analysis with linear basis function. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3) (in Chinese))
[10]	Chen J, Wu C, Yoon S.A stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 2001, 50(2): 435-466
[11]	Wang D, Chen JS.Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12-14): 1065-1083
[12]	宋彦琦, 周涛. 基于S-R和分解定理的三维几何非线性无网格法. 力学学报, 2018, 50(4): 853-862
[12]	(Song Yanqi, Zhou Tao.Three-dimensional geometric nonlinearity element-free method based on S-R decomposition theorem. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(4): 853-862 (in Chinese))
[13]	马文涛. 二维弹性力学问题的光滑无网格伽辽金法. 力学学报, 2018, 50(5): 147-156
[13]	(Ma Wentao.A smoothed meshfree galerkin method for 2d elasticity problem. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(5): 1115-1124(in Chinese))
[14]	Zhang X, Song KZ, Lu MW, et al.Meshless methods based on collocation with radial basis functions. Computational Mechanics, 2000, 26(4): 333-343
[15]	Aluru NR.A point collocation method based on reproducing kernel approximations. International Journal for Numerical Methods in Engineering, 2015, 47(6): 1083-1121
[16]	Hu HY, Chen JS, Hu W.Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 2010, 69(13): 2736-2757
[17]	Wang L, Wang Z, Qian Z.A meshfree method for inverse wave propagation using collocation and radial basis functions. Computer Methods in Applied Mechanics & Engineering, 2017, 322: 311-350
[18]	王莉华, 仲政. 基于径向基函数配点法的梁板弯曲问题分析. 固体力学学报, 2012, 33(4): 349-357
[18]	(Wang Lihua, Zhong Zheng.Radial basis collocation method for bending problems of beam and plate. Chinese Journal of Solid Mechanics, 2012, 33(4): 349-357 (in Chinese))
[19]	Li J, Chen CS, Cheng AHD.A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundry Elements, 2003, 27(3): 251-257
[20]	Wang D, Wang J, Wu J.Superconvergent gradient smoothing meshfree collocation method. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 728-766
[21]	Gao XW, Gao LF, Zhang Y, et al.Free element collocation method: A new method combining advantages of finite element and mesh free methods. Computers & Structures, 2019, 215: 10-26
[22]	Chen JS, Hillman M, Chi SW.Meshfree methods: Progress made after 20 years. Journal of Engineering Mechanics, 2017, 143(4): 04017001
[23]	Zheng H, Zhang C, Wang Y, et al.A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals. Journal of Computational Physics, 2016, 305: 997-1014
[24]	Thomas A, Majumdar P, Eldho TI, et al.Simulation optimization model for aquifer parameter estimation using coupled meshfree point collocation method and cat swarm optimization. Engineering Analysis with Boundary Elements, 2018, 91: 60-72
[25]	Qi D, Wang D, Deng L, et al. Reproducing kernel mesh-free collocation analysis of structural vibrations. Engineering Computations, 2019,
[26]	Singh LG, Eldho TI, Kumar AV.Coupled groundwater flow and contaminant transport simulation in a confined aquifer using meshfree radial point collocation method (RPCM). Engineering Analysis with Boundary Elements, 2016, 66: 20-33
[27]	Kansa EJ.Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 1990, 19(8-9): 147-161
[28]	Wang L.Radial basis functions methods for boundary value problems: Performance comparison. Engineering Analysis with Boundary Elements, 2017, 84: 191-205
[29]	Chen JS, Wang L, Hu HY, et al.Subdomain radial basis collocation method for heterogeneous media. International Journal for Numerical Methods in Engineering, 2010, 80(2): 163-190
[30]	Wang L, Chen JS, Hu HY.Subdomain radial basis collocation method for fracture mechanics. International Journal for Numerical Methods in Engineering, 2010, 83(7): 851-876
[31]	Chu F, Wang L, Zhong Z.Finite subdomain radial basis collocation method. Computational Mechanics, 2014, 54(2): 235-254
[32]	Wong ASM, Hon YC, Li TS, et al.Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme. Computers & Mathematics with Applications, 1999, 37(8): 23-43
[33]	吴宗敏. 函数的径向基表示. 数学进展, 1998(3): 202-208
[33]	(Wu Zongmin.Radial basis functions-a survey. Advance in Mathematics, 1998(3): 202-208 (in Chinese))
[34]	Holden JT.On the finite deflections of thin beams. International Journal of Solids & Structures, 1972, 8(8): 1051-1055