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## Meshless local Petrov-Galerkin method with complex variables for elasticity

Yang Xiu-Li, Dai Bao-Dong, Li Zhen-Feng
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• #### 摘要

复变量移动最小二乘法构造形函数, 其优点是采用一维基函数建立二维问题的试函数, 使得试函数中所含的待定系数减少, 从而有效提高计算效率. 文章基于复变量移动最小二乘法和局部Petrov-Galerkin弱形式, 采用罚函数法施加边界条件, 推导相应的离散方程, 提出弹性力学的复变量无网格局部Petrov-Galerkin法. 数值算例验证了该方法的有效性.

#### Abstract

In this paper, the shape functions are obtained by the moving least-squares method with complex variable (MLSCV). The advantages of MLSCV are that the approximation function of a two-dimensional (2D) problem is formed with one-dimensional (1D) basis function, and the number of the undetermined coefficients is reduced, so it effectively improves the computational efficiency. Based on the MLSCV and meshless local Petrov-Galerkin method, the essential boundary conditions are imposed by the penalty method and the corresponding discrete equations are derived, then a meshless local Petrov-Galerkin method with complex variables is presented for 2D elasticity problems. Some examples given in this paper demonstrate the effictiveness of the present method.

#### 作者及机构信息

###### 1. 太原科技大学工程力学系, 太原 030024; 2. 太原科技大学交通与物流学院, 太原 030024
• 基金项目: 国家自然科学基金(批准号: 51078250)资助的课题.

#### Authors and contacts

###### 1. Department of Engineering Mechanics, Taiyuan University of Science and Technology, Taiyuan 030024, China; 2. College of Transportation and Logistics, Taiyuan University of Science and Technology, Taiyuan 030024, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51078250).

#### 参考文献

 [1] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl. Mech. Eng. 139 3 [2] Belytscko T, Lu Y Y, Gu L 1994 Int. J. Numer. Meth. Eng. 37 229 [3] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059 [4] Atluri S N, Zhu T L 1998 Comput. Mech. 22 117 [5] Zheng B J, Dai B D 2010 Acta Phys. Sin. 59 5182 (in Chinese) [郑保敬, 戴保东 2010 物理学报 59 5182] [6] Chen L, Liew K M 2010 Comput. Mech. 47 455 [7] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081 [8] Cheng R J, Liew K M 2009 Comput. Mech. 45 1 [9] Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. Num. Meth. Eng. 39 3839 [10] Cheng R J, ChengYM2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军, 程玉民 2007 物理学报 56 5569] [11] Liu G R, Gu Y T 2001 Int. J. Numer. Meth. Eng. 50 937 [12] 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] Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375 [14] 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] [15] 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] [16] Li X G, Dai B D, Wang L H 2010 Chin. Phys. B 19 120202 [17] Lancaster P, Salkauskas K 1981 Math. Comput. 37 141 [18] Cheng R J, ChengY M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军, 程玉民 2008 物理学报 57 6037] [19] Wang J F, Bai F N, Cheng Y M 2011 Chin. Phys. B 20 030206 [20] Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California) [21] Cheng Y M, Peng M J, Li J H 2005 Acta Mech. Sin. 37 719 (in Chinese) [程玉民, 彭妙娟, 李九红 2005 力学学报 37 719] [22] 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] [23] Cheng Y M, Li J H 2005 Science in China Ser. G 35 548 (in Chinese) [程玉民, 李九红 2005 中国科学 (G辑) 35 548] [24] 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] [25] 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] [26] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [27] Liew K M, Cheng Y M 2009 Comput. Meth. Appl. Mech. Eng. 198 3925 [28] Cheng Y M, Liew K M, Kitipornchai S 2009 Int. J. Numer. Meth. Eng. 78 1258

#### 施引文献

•  [1] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl. Mech. Eng. 139 3 [2] Belytscko T, Lu Y Y, Gu L 1994 Int. J. Numer. Meth. Eng. 37 229 [3] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059 [4] Atluri S N, Zhu T L 1998 Comput. Mech. 22 117 [5] Zheng B J, Dai B D 2010 Acta Phys. Sin. 59 5182 (in Chinese) [郑保敬, 戴保东 2010 物理学报 59 5182] [6] Chen L, Liew K M 2010 Comput. Mech. 47 455 [7] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081 [8] Cheng R J, Liew K M 2009 Comput. Mech. 45 1 [9] Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. Num. Meth. Eng. 39 3839 [10] Cheng R J, ChengYM2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军, 程玉民 2007 物理学报 56 5569] [11] Liu G R, Gu Y T 2001 Int. J. Numer. Meth. Eng. 50 937 [12] 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] Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375 [14] 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] [15] 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] [16] Li X G, Dai B D, Wang L H 2010 Chin. Phys. B 19 120202 [17] Lancaster P, Salkauskas K 1981 Math. Comput. 37 141 [18] Cheng R J, ChengY M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军, 程玉民 2008 物理学报 57 6037] [19] Wang J F, Bai F N, Cheng Y M 2011 Chin. Phys. B 20 030206 [20] Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California) [21] Cheng Y M, Peng M J, Li J H 2005 Acta Mech. Sin. 37 719 (in Chinese) [程玉民, 彭妙娟, 李九红 2005 力学学报 37 719] [22] 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] [23] Cheng Y M, Li J H 2005 Science in China Ser. G 35 548 (in Chinese) [程玉民, 李九红 2005 中国科学 (G辑) 35 548] [24] 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] [25] 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] [26] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [27] Liew K M, Cheng Y M 2009 Comput. Meth. Appl. Mech. Eng. 198 3925 [28] Cheng Y M, Liew K M, Kitipornchai S 2009 Int. J. Numer. Meth. Eng. 78 1258
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##### 出版历程
• 收稿日期:  2011-05-05
• 修回日期:  2012-06-25
• 刊出日期:  2012-03-05

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