位势问题改进的无网格局部Petrov-Galerkin法

郑保敬; 戴保东

doi:10.7498/aps.59.5182

摘要

将滑动Kriging插值法与无网格局部Petrov-Galerkin法相结合,采用Heaviside分段函数作为局部弱形式的权函数,提出改进的无网格局部Petrov-Galerkin法,进一步将这种无网格法应用于位势问题,并推导相应的离散方程.因为滑动Kriging插值法构造的形函数满足Kronecker函数性质,所以本文建立的改进的无网格局部Petrov-Galerkin法可以像有限元法一样直接施加边界条件;由于采用Heaviside分段函数作为局部弱形式的权函数,因此在计算刚度矩阵时只涉及边界积分,而没有区域积分.此外,还对本方法中一些重要参数的选取进行了研究.数值算例表明,本文建立的改进的无网格局部Petrov-Galerkin法具有数值实现简单、计算量小以及方便施加边界条件等优点.

关键词:

Abstract

In this paper, combining the moving Kriging interpolation method and meshless local Petrov-Galerkin method, an improved meshless local Petrov-Galerkin method is presented, in which the Heaviside step function is used as test function over the local weak form. The present method is applied to two-dimensional potential problems and the corresponding discrete equations are derived. Because the shape functions so-obtained possess the Kronecker delta property, the essential boundary conditions can be enforced as the FEM; furthermore, the Heaviside step function is used as the test function, there is no domain integral, and only a regular boundary integral is involved. In this paper, the choice of the important parameters is studied. Numerical examples show that the present method has simpler numerical procedures and lower computation cost, in addition, the essential boundary conditions can be implemented directly.

Keywords:

作者及机构信息

1.
太原科技大学工程力学系,太原 030024

基金项目: 山西省自然科学基金(批准号:2007011009), 山西省高校科技研究开发计划(批准号:20091131),太原科技大学博士启动基金(批准号:200708)资助的课题.

Authors and contacts

1.
Department of Engineering Mechanics, Taiyuan University of Science and Technology, Taiyuan 030024, china

参考文献

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[2]	Zhang X, Liu Y, Ma S 2009 Advance in Mechanics 39 1(in Chinese) [张雄、刘岩、马上 2009 力学进展 39 1]
[3]	Lu Y Y, Belytschko T, Tabbara M 1995 Comput. Methods Appl.Mech. Engrg 126 131
[4]	Li S, Hao W, Liu W K 2000 Comput. Mech. 25 102
[5]	Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军、程玉民 2008 57 6037]
[6]	Cheng R J, ChengY M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 56 5569]
[7]	Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065
[8]	Peng M J, Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[9]	Cheng Y M, Peng M J 2005 Sci.Chin. G Phys.Mech. and Astron. 48 641
[10]	Belytscko T, Lu Y Y, Gu L 1994 Int.J.Numer.Meth.Engng. 37 229
[11]	Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059
[12]	Liu W K, Jun S, Zhang Y F 1995 Int.J.Numer.Meth.Fluids. 20 1081
[13]	Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1(in Chinese) [陈丽、程玉民 2008 57 1]
[14]	Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. for Num. Meth. in Engin. 39 3839
[15]	Babuska I, Melenk J M 1997 Int.J.Numer.Meth.Engng. 40 727
[16]	Liu G R, Gu Y T. 2001 Int.J.Numer.Meth.Engng. 50 937
[17]	Zhang X, Liu X H and Song K Z 2001 Int.J.Numer.Meth.Engng. 51 1089
[18]	Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese) [程玉民、李九红 2005 54 4463]
[19]	Atluri S N, Zhu T L 1998 Comput. Mech. 22 117
[20]	Mukherjee Y X, Mukherjee S 1997 Int.J.Numer.Meth.Engng. 40 797
[21]	Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375
[22]	Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 55 3215]
[23]	Zhang Z, Zhao P, Liew K M 2009 Eng. Anal. Bound. Elem. 33 547
[24]	Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California)
[25]	Lancaster P L, Salkauskas K 1981 Math. Comput. 37 141
[26]	Krige D G 1951 Journal of the Chemical Metallurgical and Mining Society of South Africa 52 119
[27]	Gu L 2003 Int.J.Numer.Meth.Engng. 56 1
[28]	Bui Q T, Nguyen N T, Nguyen-Dang H 2009 Int.J.Numer.Meth.Engng.77 1371
[29]	Dai K Y, Liu G R, Lim K M, Gu Y T 2003 Comput. Mech. 32 60
[30]	Sacks J, Susannah S B, Welch W J 1989 Technometrics 31 41

施引文献

[1]	Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl.Mech. Engrg. 139 3
[2]	Zhang X, Liu Y, Ma S 2009 Advance in Mechanics 39 1(in Chinese) [张雄、刘岩、马上 2009 力学进展 39 1]
[3]	Lu Y Y, Belytschko T, Tabbara M 1995 Comput. Methods Appl.Mech. Engrg 126 131
[4]	Li S, Hao W, Liu W K 2000 Comput. Mech. 25 102
[5]	Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军、程玉民 2008 57 6037]
[6]	Cheng R J, ChengY M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 56 5569]
[7]	Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065
[8]	Peng M J, Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[9]	Cheng Y M, Peng M J 2005 Sci.Chin. G Phys.Mech. and Astron. 48 641
[10]	Belytscko T, Lu Y Y, Gu L 1994 Int.J.Numer.Meth.Engng. 37 229
[11]	Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059
[12]	Liu W K, Jun S, Zhang Y F 1995 Int.J.Numer.Meth.Fluids. 20 1081
[13]	Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1(in Chinese) [陈丽、程玉民 2008 57 1]
[14]	Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. for Num. Meth. in Engin. 39 3839
[15]	Babuska I, Melenk J M 1997 Int.J.Numer.Meth.Engng. 40 727
[16]	Liu G R, Gu Y T. 2001 Int.J.Numer.Meth.Engng. 50 937
[17]	Zhang X, Liu X H and Song K Z 2001 Int.J.Numer.Meth.Engng. 51 1089
[18]	Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese) [程玉民、李九红 2005 54 4463]
[19]	Atluri S N, Zhu T L 1998 Comput. Mech. 22 117
[20]	Mukherjee Y X, Mukherjee S 1997 Int.J.Numer.Meth.Engng. 40 797
[21]	Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375
[22]	Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 55 3215]
[23]	Zhang Z, Zhao P, Liew K M 2009 Eng. Anal. Bound. Elem. 33 547
[24]	Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California)
[25]	Lancaster P L, Salkauskas K 1981 Math. Comput. 37 141
[26]	Krige D G 1951 Journal of the Chemical Metallurgical and Mining Society of South Africa 52 119
[27]	Gu L 2003 Int.J.Numer.Meth.Engng. 56 1
[28]	Bui Q T, Nguyen N T, Nguyen-Dang H 2009 Int.J.Numer.Meth.Engng.77 1371
[29]	Dai K Y, Liu G R, Lim K M, Gu Y T 2003 Comput. Mech. 32 60
[30]	Sacks J, Susannah S B, Welch W J 1989 Technometrics 31 41

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