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## Error estimate of element-free Galerkin method for elasticity

Cheng Yu-Min, Cheng Rong-Jun
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• #### 摘要

基于移动最小二乘法在Sobolev空间Wk,p(Ω)中的误差估计以及弹性力学问题的变分弱形式中出现的双线性形式的连续性和强制性,研究了弹性力学问题的无单元Galerkin方法的误差分析以及数值解的误差和影响域半径之间的关系,给出了弹性力学问题的无单元Galerkin方法在Sobolev空间中的误差估计定理,并证明了当节点和形函数满足一定条件时该误差估计是最优阶的.从误差分析中可以看出,数值解的误差与权函数的影响域半径密切相关.最后,通过算例验证了结论的正确性.

#### Abstract

Based on both the error estimates of moving least-square approximation in the Sobolev space Wk,p(Ω) and the continuity and coercion of the bi-linearity in the weak form of the elasticity, the error analysis of element-free Galerkin method for elasticity is discussed in this paper, the relationship between the error and the radius of the weight function is given, and the theorem of the error estimate presented. The error estimate proves to be of optimal order when nodes and shape functions satisfy some conditions. From the error analysis, it is shown that the error bound of the elasticity is directly related to the radius of the weight function. And a numerical example is given to verify the correctness of the given results.

#### 作者及机构信息

###### (1)上海大学上海市应用数学和力学研究所,上海 200072; (2)浙江大学宁波理工学院,宁波 315100;上海大学上海市应用数学和力学研究所,上海 200072
• 基金项目: 国家自然科学基金(批准号: 10871124,11026223)和上海市重点科学建设项目(批准号: S30106)资助的课题.

#### 参考文献

 [1] Belytschko T, Krongauz Y 1996 Computer Methods in Applied Mechanics and Engineering 139 3 [2] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059 [3] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 1 [4] Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese)[程荣军、程玉民 2008 物理学报 57 6037] [5] Cheng R J, Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)[程荣军、程玉民 2007 物理学报 56 5569] [6] Liu W K, Jun S, Zhang Y F 1995 International Journal for Numerical Methods in Engineering 20 1081 Cheng R J, Ge H X 2010 Chin. Phys. B 19 090201 Duarte C A, Oden J T 1996 Computer Methods in Applied Mechanics and Engineering 139 237 [7] Onate E, Idelsohn S, Zienkiewicz O C 1996 International Journal for Numerical Methods in Engineering 39 3839 [8] Atluri S N, Zhu T 1998 Computational Mechanics 22 117 [9] Aluru N R 2000 International Journal for Numerical Methods in Engineering 47 1083 [10] Liu W K, Chen Y J 1995 International Journal for Numerical Methods in Fluids 21 901 [11] Idelsohn S R, Onate E, Calvo N 2003 International Journal for Numerical Methods in Engineering 58 893 [12] Cheng Y M, Peng M J, Li J H 2005 Acta Mechanica Sinica 37 719 (in Chinese)[程玉民、彭妙娟、李九红2005 力学学报 37 719] Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)[程玉民、李九红 2005物理学报54 4463] [13] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090202 [14] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1 (in Chinese) [陈 丽、程玉民 2008 物理学报 57 1] [15] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 6047(in Chinese) [陈 丽、程玉民 2008 物理学报 57 6047] [16] Peng M J, Cheng Y M 2009 Engineering Analysis with Boundary Elements 23 77 [17] Zhu T, Zhang J D, Atluri S N 1998 Computational Mechanics 21 223 [18] Kothnur V S, Mukherjee S, Mukherjee Y X 1999 International Journal of Solids and Structures 36 1129 [19] Cheng Y M, Cheng M J 2003 Acta Mechanica Sinica 35 181 (in Chinese)) [程玉民、陈美娟 2003 力学学报 35 181] Cheng Y M, Peng M J 2005 Science in China 35 435 (in Chinese) [程玉民、彭妙娟 2005 中国科学 35 435] Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 物理学报 55 3215] [20] Dai B D, Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese) [戴保东、程玉民 2007 物理学报 56 597] [21] Ren H P, Zhang W 2009 Chin. Phys. B 18 4065 [22] Wendland H 2001 IMA Journal of Numerical analysis 21 285 [23] Armentano M G, Duran R G 2001 Applied Numerical Mathematics 37 397 [24] Armentano M G 2002 SIAM J. Numer. Anal. 39 38 [25] Zuppa C 2003 Bulletin of the Brazilian Mathematical Society 34 231 [26] Krysl P, Belytschko T 1997 Computer Methods in Applied Mechanics and Engineering 148 257 [27] Chung H J, Belytschko T 1998 Computational Mechanics 21 91 [28] Gavete L, Cuesta J L, Ruiz A 2002 International Journal for Numerical Methods in Engineering 53 677 [29] Gavete L, Gavete M L 2003 International Journal for Numerical Methods in Engineering 58 2239 [30] Gavete L, Faclon S 2001 Eur. J. Mech. A/Solids 20 327 [31] Rynne B P 2000 Advances in Computational Mathematics 12 251 [32] Rossi R, Alves M K 2004 Computational Mechanics 33 194 [33] Han W M, Meng X P 2001 Computer Methods in Applied Mechanics and Engineering 190 6157 [34] Cheng R J, Liew K M 2009 Computational Mechanics 45 1 [35] [王烈衡、许学军. 有限元方法的数学基础. (北京: 科学出版社) 2004] [36] [李开泰、黄艾香、黄庆怀. 有限元方法及其应用(修订本). (西安: 西安交通大学出版社) 1992] [37] Cheng R J, Cheng Y M 2008 Applied Numerical Mathematics 58 884

#### 施引文献

•  [1] Belytschko T, Krongauz Y 1996 Computer Methods in Applied Mechanics and Engineering 139 3 [2] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059 [3] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 1 [4] Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese)[程荣军、程玉民 2008 物理学报 57 6037] [5] Cheng R J, Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)[程荣军、程玉民 2007 物理学报 56 5569] [6] Liu W K, Jun S, Zhang Y F 1995 International Journal for Numerical Methods in Engineering 20 1081 Cheng R J, Ge H X 2010 Chin. Phys. B 19 090201 Duarte C A, Oden J T 1996 Computer Methods in Applied Mechanics and Engineering 139 237 [7] Onate E, Idelsohn S, Zienkiewicz O C 1996 International Journal for Numerical Methods in Engineering 39 3839 [8] Atluri S N, Zhu T 1998 Computational Mechanics 22 117 [9] Aluru N R 2000 International Journal for Numerical Methods in Engineering 47 1083 [10] Liu W K, Chen Y J 1995 International Journal for Numerical Methods in Fluids 21 901 [11] Idelsohn S R, Onate E, Calvo N 2003 International Journal for Numerical Methods in Engineering 58 893 [12] Cheng Y M, Peng M J, Li J H 2005 Acta Mechanica Sinica 37 719 (in Chinese)[程玉民、彭妙娟、李九红2005 力学学报 37 719] Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)[程玉民、李九红 2005物理学报54 4463] [13] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090202 [14] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1 (in Chinese) [陈 丽、程玉民 2008 物理学报 57 1] [15] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 6047(in Chinese) [陈 丽、程玉民 2008 物理学报 57 6047] [16] Peng M J, Cheng Y M 2009 Engineering Analysis with Boundary Elements 23 77 [17] Zhu T, Zhang J D, Atluri S N 1998 Computational Mechanics 21 223 [18] Kothnur V S, Mukherjee S, Mukherjee Y X 1999 International Journal of Solids and Structures 36 1129 [19] Cheng Y M, Cheng M J 2003 Acta Mechanica Sinica 35 181 (in Chinese)) [程玉民、陈美娟 2003 力学学报 35 181] Cheng Y M, Peng M J 2005 Science in China 35 435 (in Chinese) [程玉民、彭妙娟 2005 中国科学 35 435] Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 物理学报 55 3215] [20] Dai B D, Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese) [戴保东、程玉民 2007 物理学报 56 597] [21] Ren H P, Zhang W 2009 Chin. Phys. B 18 4065 [22] Wendland H 2001 IMA Journal of Numerical analysis 21 285 [23] Armentano M G, Duran R G 2001 Applied Numerical Mathematics 37 397 [24] Armentano M G 2002 SIAM J. Numer. Anal. 39 38 [25] Zuppa C 2003 Bulletin of the Brazilian Mathematical Society 34 231 [26] Krysl P, Belytschko T 1997 Computer Methods in Applied Mechanics and Engineering 148 257 [27] Chung H J, Belytschko T 1998 Computational Mechanics 21 91 [28] Gavete L, Cuesta J L, Ruiz A 2002 International Journal for Numerical Methods in Engineering 53 677 [29] Gavete L, Gavete M L 2003 International Journal for Numerical Methods in Engineering 58 2239 [30] Gavete L, Faclon S 2001 Eur. J. Mech. A/Solids 20 327 [31] Rynne B P 2000 Advances in Computational Mathematics 12 251 [32] Rossi R, Alves M K 2004 Computational Mechanics 33 194 [33] Han W M, Meng X P 2001 Computer Methods in Applied Mechanics and Engineering 190 6157 [34] Cheng R J, Liew K M 2009 Computational Mechanics 45 1 [35] [王烈衡、许学军. 有限元方法的数学基础. (北京: 科学出版社) 2004] [36] [李开泰、黄艾香、黄庆怀. 有限元方法及其应用(修订本). (西安: 西安交通大学出版社) 1992] [37] Cheng R J, Cheng Y M 2008 Applied Numerical Mathematics 58 884
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##### 出版历程
• 收稿日期:  2010-07-08
• 修回日期:  2010-10-10
• 刊出日期:  2011-07-15

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