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

[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

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

[1]	廖敦微, 郑月军, 陈强, 丁亮, 高冕, 付云起. 基于裂纹模板法的金属网格透明导电薄膜制备及性能改进. , 2022, 71(15): 154201. doi: 10.7498/aps.71.20220101
[2]	高效伟, 丁金兴, 刘华雩. 有限线法及其在流固域间耦合传热中的应用. , 2022, 71(19): 190201. doi: 10.7498/aps.71.20220833
[3]	辛建建, 陈振雷, 石凡, 石伏龙. 基于直角网格法的单个和阵列布置下柔性水翼绕流数值模拟. , 2020, 69(4): 044702. doi: 10.7498/aps.69.20191711
[4]	任金莲, 蒋戎戎, 陆伟刚, 蒋涛. 基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟. , 2020, 69(8): 080202. doi: 10.7498/aps.69.20191829
[5]	张鹏轩, 彭妙娟. 黏弹性问题的插值型无单元Galerkin方法. , 2019, 68(17): 170203. doi: 10.7498/aps.68.20191047
[6]	辛建建, 石伏龙, 金秋. 一种径向基函数虚拟网格法数值模拟复杂边界流动. , 2017, 66(4): 044704. doi: 10.7498/aps.66.044704
[7]	焦宝宝. 用重正交化Lanczos法求解大型非正交归一基稀疏矩阵的特征值问题. , 2016, 65(19): 192101. doi: 10.7498/aps.65.192101
[8]	冯德山, 杨道学, 王珣. 插值小波尺度法探地雷达数值模拟及四阶Runge Kutta辅助微分方程吸收边界条件. , 2016, 65(23): 234102. doi: 10.7498/aps.65.234102
[9]	杜红秀, 魏宏, 秦义校, 李中华, 王同尊. 轴对称构件受力分析的插值粒子法. , 2015, 64(10): 100204. doi: 10.7498/aps.64.100204
[10]	杜超凡, 章定国. 基于无网格点插值法的旋转悬臂梁的动力学分析. , 2015, 64(3): 034501. doi: 10.7498/aps.64.034501
[11]	彭妙娟, 刘茜. 黏弹性问题的改进的复变量无单元Galerkin方法. , 2014, 63(18): 180203. doi: 10.7498/aps.63.180203
[12]	陈浩, 华灯鑫, 张毅坤, 朱承炫. 米散射激光雷达剖面数据三次样条垂直水平插值法. , 2014, 63(15): 154204. doi: 10.7498/aps.63.154204
[13]	李中华, 秦义校, 崔小朝. 弹性力学的插值型重构核粒子法. , 2012, 61(8): 080205. doi: 10.7498/aps.61.080205
[14]	杨秀丽, 戴保东, 栗振锋. 弹性力学的复变量无网格局部 Petrov-Galerkin 法. , 2012, 61(5): 050204. doi: 10.7498/aps.61.050204
[15]	王炜, 张琪昌, 王雪娇. 待定固有频率法在分析系统混沌临界值问题中的应用. , 2009, 58(8): 5162-5168. doi: 10.7498/aps.58.5162
[16]	程荣军, 程玉民. 势问题的无单元Galerkin方法的误差估计. , 2008, 57(10): 6037-6046. doi: 10.7498/aps.57.6037
[17]	程荣军, 程玉民. 带源参数的热传导反问题的无网格方法. , 2007, 56(10): 5569-5574. doi: 10.7498/aps.56.5569
[18]	李树忱, 程玉民, 李术才. 动态断裂力学的无网格流形方法. , 2006, 55(9): 4760-4766. doi: 10.7498/aps.55.4760
[19]	秦义校, 程玉民. 弹性力学的重构核粒子边界无单元法. , 2006, 55(7): 3215-3222. doi: 10.7498/aps.55.3215
[20]	程玉民, 李九红. 弹性力学的复变量无网格方法. , 2005, 54(10): 4463-4471. doi: 10.7498/aps.54.4463

计量

文章访问数: 9656
PDF下载量: 1462
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板