x

## 留言板

 引用本文:
 Citation:

## Improved complex variable element-free Galerkin method for viscoelasticity problems

Peng Miao-Juan, Liu Qian
PDF
• #### 摘要

基于改进的复变量移动最小二乘法，提出了二维黏弹性问题的改进的复变量无单元Galerkin方法. 采用改进的复变量移动最小二乘法建立形函数，根据Galerkin积分弱形式建立求解方程，并用罚函数法施加本质边界条件，推导了二维黏弹性问题的改进的复变量无单元Galerkin方法的计算公式. 最后，通过实际算例，将计算结果与复变量无单元Galerkin方法及有限元法的结果进行了对比，说明了本文方法具有更高的计算精度和计算效率.

#### Abstract

In this paper, based on the improved complex variable least-square (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional viscoelasticity problems is proposed. The ICVMLS approximation is used to form the shape function, the Galerkin weak form is used to obtain the system equations, and the penalty method is used to impose the essential boundary conditions, then the corresponding formulae of the ICVEFG method for two-dimensional viscoelasticity problems are obtained. Finally, some numerical examples are given, and the numerical results from the ICVEFG method are compared with those from the CVEFG method and finite element method, and the results show that the ICVEFG method in this paper has the high computational precision and efficiency.

#### 作者及机构信息

###### 1. 上海大学土木工程系, 上海 200072
• 基金项目: 国家自然科学基金（批准号：11171208）资助的课题.

#### Authors and contacts

###### 1. Department of Civil Engineering, Shanghai University, Shanghai 200072, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11171208).

#### 参考文献

 [1] Belytschko T, Krongauz Y, Organ D 1996 Comput. Methods Appl. Mech. Engineer. 139 3 [2] Cheng Y M, Ji X, He P F 2004 Acta Mech. Sin. 36 43 (in Chinese)[程玉民, 嵇醒, 贺鹏飞 2004 力学学报 36 43] [3] Belytschko T, Lu Y Y, Gu L 1994 Int. J. Numeri. Methods Engineer. 37 229 [4] Zhang Z, Li D M, Cheng Y M, Liew K M 2012 Acta Mech. Sin. 28 808 [5] Zhang Z, Wang J F, Cheng Y M, Liew K M 2013 Sci. China G: Phys. Mech. Astron. 56 1568 [6] Zhang Z, Hao S Y, Liew K M, Cheng Y M 2013 Engineering Analysis with Boundary Elements 37 1576 [7] 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] [8] Monaghan J J 1998 Comput. Phys. Commun. 48 89 [9] Cheng R J, Cheng Y M 2008 Appl. Numer. Math. 58 884 [10] 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] [11] Cheng R J, Cheng Y M 2007 Acta Mech. Sin. 39 843(in Chinese)[程荣军, 程玉民 2007 力学学报 39 843] [12] Dai B D, Cheng J, Zheng B J 2013 Int. J. Appl. Mech. 5 1350011 [13] Li S C, Cheng Y M 2004 Acta Mech. Sin. 36 496(in Chinese)[李树忱, 程玉民 2004 力学学报 36 496] [14] Li S C, Li S C, Cheng Y M 2005 Theoret. Appl. Fract. Mech. 44 234 [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] Chen L, Cheng Y M 2010 Sci. China G: Phys. Mech. Astron. 40 242(in Chinese)[陈丽, 程玉民 2010 中国科学 G辑 物理学 力学 天文学 40 242] [17] Cheng Y M, Chen M J 2003 Acta Mech. Sin. 35 181(in Chinese)[程玉民, 陈美娟 2003 力学学报 35 181] [18] Cheng Y M, Peng M J 2005 Sci. China G: Phys. Mech. Astron. 35 435(in Chinese)[程玉民, 彭妙娟 2005 中国科学 G辑 物理学 力学 天文学 35 435] [19] Cheng Y M, Peng M J 2005 Sci. China G: Phys. Mech. Astron. 48 641 [20] 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] [21] 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] [22] Peng M J, Cheng Y M 2009 Engineering Analysis with Boundary Elements 33 77 [23] Cheng Y M, Liew K M, Kitipornchai S 2009 Int. J. Numer. Methods Engineer. 78 1258 [24] Yang H T, Liu Y 2003 Int. J. Solids Struct. 40 701 [25] Sladek J, Sladek V, Zhang C H 2005 Engineering Analysis with Boundary Elements 29 597 [26] Sladek J, Sladek V, Zhang C H, Schanz M 2006 Computat. Mech. 37 279 [27] Cheng Y M, Li R X, Peng M J 2012 Chin. Phys. B 21 090205 [28] 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] [29] Cheng Y M, Li J H 2005 Sci. China G: Phys. Mech. Astron. 35 548(in Chinese)[程玉民, 李九红 2005 中国科学 G辑 物理学 力学 天文学 35 548] [30] 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] [31] Cheng Y M, Li J H 2006 Sci. China G: Phys. Mech. Astron. 49 46 [32] Liew K M, Feng C, Cheng Y M, Kitipornchai S 2007 Int. J. Numer. Methods Engineer 70 46 [33] Cheng Y M, Wang J F, Li R X 2012 Int. J. Appl. Mech. 4 1250042 [34] Liew K M, Cheng Y 2009 Comput. Methods Appl. Mech. Engineer. 198 3925 [35] Gao H, Cheng Y 2010 Int. J. Computat. Methods 7 55 [36] Gao H F, Cheng Y M 2009 Acta Mech. Sin. 41 480(in Chinese)[高洪芬, 程玉民 2009 力学学报 41 480] [37] Bai F N, Li D M, Wang J F, Cheng Y M 2012 Chin. Phys. B 21 020204 [38] Cheng Y M, Wang J F, Bai F N 2012 Chin. Phys. B 21 090203 [39] Wang J F, Cheng Y M 2012 Chin. Phys. B 21 120206 [40] Wang J F, Cheng Y M 2013 Chin. Phys. B 22 030208 [41] Li D M, Bai F N, Cheng Y M, Liew K M 2012 Comput. Methods Appl. Mech. Engineer. 233–236 1 [42] Li D M, Liew K M, Cheng Y M 2014 Comput. Methods Appl. Mech. Engineer. 269 72 [43] Peng M, Liu P, Cheng Y 2009 Int. J. Appl. Mech. 1 367 [44] Liu P, Peng M J, Cheng Y M 2009 Comput. Aided Engineer. 18 10(in Chinese)[刘沛, 彭妙娟, 程玉民 2009 计算机辅助工程 18 10] [45] Peng M J, Li D M, Cheng Y M 2011 Engineer. Struct. 33 127 [46] Li D M, Peng M J, Cheng Y M 2011 Sci. China G: Phys. Mech. Astron. 41 1003(in Chinese)[李冬明, 彭妙娟, 程玉民 2011 中国科学 G辑 物理学 力学 天文学 41 1003] [47] Peng M J, Li R X, Cheng Y M 2014 Engineering Analysis with Boundary Elements 40 104 [48] Li D M, Liew K M, Cheng Y M 2014 Computat. Mech. 53 1149 [49] Chen L, Cheng Y 2010 Sci. China G: Phys. Mech. Astron. 53 954 [50] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [51] Chen L, Ma H P, Cheng Y M 2013 Chin. Phys. B 22 050202 [52] Weng Y J, Cheng Y M 2013 Chin. Phys. B 22 090204 [53] Weng Y J, Zhang Z, Cheng Y M 2014 Engineering Analysis with Boundary Elements 44 36 [54] Ren H, Cheng J, Huang A 2012 Appl. Math. Computat. 219 1724 [55] Flgge W 1975 Viscoelasticity (2nd Ed) (New York: Springer-Verlag)

#### 施引文献

•  [1] Belytschko T, Krongauz Y, Organ D 1996 Comput. Methods Appl. Mech. Engineer. 139 3 [2] Cheng Y M, Ji X, He P F 2004 Acta Mech. Sin. 36 43 (in Chinese)[程玉民, 嵇醒, 贺鹏飞 2004 力学学报 36 43] [3] Belytschko T, Lu Y Y, Gu L 1994 Int. J. Numeri. Methods Engineer. 37 229 [4] Zhang Z, Li D M, Cheng Y M, Liew K M 2012 Acta Mech. Sin. 28 808 [5] Zhang Z, Wang J F, Cheng Y M, Liew K M 2013 Sci. China G: Phys. Mech. Astron. 56 1568 [6] Zhang Z, Hao S Y, Liew K M, Cheng Y M 2013 Engineering Analysis with Boundary Elements 37 1576 [7] 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] [8] Monaghan J J 1998 Comput. Phys. Commun. 48 89 [9] Cheng R J, Cheng Y M 2008 Appl. Numer. Math. 58 884 [10] 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] [11] Cheng R J, Cheng Y M 2007 Acta Mech. Sin. 39 843(in Chinese)[程荣军, 程玉民 2007 力学学报 39 843] [12] Dai B D, Cheng J, Zheng B J 2013 Int. J. Appl. Mech. 5 1350011 [13] Li S C, Cheng Y M 2004 Acta Mech. Sin. 36 496(in Chinese)[李树忱, 程玉民 2004 力学学报 36 496] [14] Li S C, Li S C, Cheng Y M 2005 Theoret. Appl. Fract. Mech. 44 234 [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] Chen L, Cheng Y M 2010 Sci. China G: Phys. Mech. Astron. 40 242(in Chinese)[陈丽, 程玉民 2010 中国科学 G辑 物理学 力学 天文学 40 242] [17] Cheng Y M, Chen M J 2003 Acta Mech. Sin. 35 181(in Chinese)[程玉民, 陈美娟 2003 力学学报 35 181] [18] Cheng Y M, Peng M J 2005 Sci. China G: Phys. Mech. Astron. 35 435(in Chinese)[程玉民, 彭妙娟 2005 中国科学 G辑 物理学 力学 天文学 35 435] [19] Cheng Y M, Peng M J 2005 Sci. China G: Phys. Mech. Astron. 48 641 [20] 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] [21] 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] [22] Peng M J, Cheng Y M 2009 Engineering Analysis with Boundary Elements 33 77 [23] Cheng Y M, Liew K M, Kitipornchai S 2009 Int. J. Numer. Methods Engineer. 78 1258 [24] Yang H T, Liu Y 2003 Int. J. Solids Struct. 40 701 [25] Sladek J, Sladek V, Zhang C H 2005 Engineering Analysis with Boundary Elements 29 597 [26] Sladek J, Sladek V, Zhang C H, Schanz M 2006 Computat. Mech. 37 279 [27] Cheng Y M, Li R X, Peng M J 2012 Chin. Phys. B 21 090205 [28] 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] [29] Cheng Y M, Li J H 2005 Sci. China G: Phys. Mech. Astron. 35 548(in Chinese)[程玉民, 李九红 2005 中国科学 G辑 物理学 力学 天文学 35 548] [30] 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] [31] Cheng Y M, Li J H 2006 Sci. China G: Phys. Mech. Astron. 49 46 [32] Liew K M, Feng C, Cheng Y M, Kitipornchai S 2007 Int. J. Numer. Methods Engineer 70 46 [33] Cheng Y M, Wang J F, Li R X 2012 Int. J. Appl. Mech. 4 1250042 [34] Liew K M, Cheng Y 2009 Comput. Methods Appl. Mech. Engineer. 198 3925 [35] Gao H, Cheng Y 2010 Int. J. Computat. Methods 7 55 [36] Gao H F, Cheng Y M 2009 Acta Mech. Sin. 41 480(in Chinese)[高洪芬, 程玉民 2009 力学学报 41 480] [37] Bai F N, Li D M, Wang J F, Cheng Y M 2012 Chin. Phys. B 21 020204 [38] Cheng Y M, Wang J F, Bai F N 2012 Chin. Phys. B 21 090203 [39] Wang J F, Cheng Y M 2012 Chin. Phys. B 21 120206 [40] Wang J F, Cheng Y M 2013 Chin. Phys. B 22 030208 [41] Li D M, Bai F N, Cheng Y M, Liew K M 2012 Comput. Methods Appl. Mech. Engineer. 233–236 1 [42] Li D M, Liew K M, Cheng Y M 2014 Comput. Methods Appl. Mech. Engineer. 269 72 [43] Peng M, Liu P, Cheng Y 2009 Int. J. Appl. Mech. 1 367 [44] Liu P, Peng M J, Cheng Y M 2009 Comput. Aided Engineer. 18 10(in Chinese)[刘沛, 彭妙娟, 程玉民 2009 计算机辅助工程 18 10] [45] Peng M J, Li D M, Cheng Y M 2011 Engineer. Struct. 33 127 [46] Li D M, Peng M J, Cheng Y M 2011 Sci. China G: Phys. Mech. Astron. 41 1003(in Chinese)[李冬明, 彭妙娟, 程玉民 2011 中国科学 G辑 物理学 力学 天文学 41 1003] [47] Peng M J, Li R X, Cheng Y M 2014 Engineering Analysis with Boundary Elements 40 104 [48] Li D M, Liew K M, Cheng Y M 2014 Computat. Mech. 53 1149 [49] Chen L, Cheng Y 2010 Sci. China G: Phys. Mech. Astron. 53 954 [50] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [51] Chen L, Ma H P, Cheng Y M 2013 Chin. Phys. B 22 050202 [52] Weng Y J, Cheng Y M 2013 Chin. Phys. B 22 090204 [53] Weng Y J, Zhang Z, Cheng Y M 2014 Engineering Analysis with Boundary Elements 44 36 [54] Ren H, Cheng J, Huang A 2012 Appl. Math. Computat. 219 1724 [55] Flgge W 1975 Viscoelasticity (2nd Ed) (New York: Springer-Verlag)
•  [1] 周浩, 李毅, 刘海, 陈鸿, 任磊生. 最优输运无网格方法及其在液滴表面张力效应模拟中的应用. 物理学报, 2021, 70(24): 240203. doi: 10.7498/aps.70.20211078 [2] 张鹏轩, 彭妙娟. 黏弹性问题的插值型无单元Galerkin方法. 物理学报, 2019, 68(17): 170203. doi: 10.7498/aps.68.20191047 [3] 邹诗莹, 席伟成, 彭妙娟, 程玉民. 运用改进的无单元Galerkin方法分析机场道面断裂力学问题. 物理学报, 2017, 66(12): 120204. doi: 10.7498/aps.66.120204 [4] 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. 物理学报, 2015, 64(10): 100204. doi: 10.7498/aps.64.100204 [5] 冯昭, 王晓东, 欧阳洁. 求解Kuramoto-Sivashinsky方程的平移基无单元Galerkin方法. 物理学报, 2012, 61(23): 230204. doi: 10.7498/aps.61.230204 [6] 李中华, 秦义校, 崔小朝. 弹性力学的插值型重构核粒子法. 物理学报, 2012, 61(8): 080205. doi: 10.7498/aps.61.080205 [7] 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. 物理学报, 2012, 61(5): 050204. doi: 10.7498/aps.61.050204 [8] 程荣军, 程玉民. 弹性力学的无单元Galerkin方法的误差估计. 物理学报, 2011, 60(7): 070206. doi: 10.7498/aps.60.070206 [9] 郑保敬, 戴保东. 位势问题改进的无网格局部Petrov-Galerkin法. 物理学报, 2010, 59(8): 5182-5189. doi: 10.7498/aps.59.5182 [10] 李 军, 董海鹰. 基于小波核偏最小二乘回归方法的混沌系统建模研究. 物理学报, 2008, 57(8): 4756-4765. doi: 10.7498/aps.57.4756 [11] 汤 波, 李俊峰, 王天舒. 水珠滴落的最小二乘粒子有限元方法模拟. 物理学报, 2008, 57(11): 6722-6729. doi: 10.7498/aps.57.6722 [12] 陈 丽, 程玉民. 瞬态热传导问题的复变量重构核粒子法. 物理学报, 2008, 57(10): 6047-6055. doi: 10.7498/aps.57.6047 [13] 陈 丽, 程玉民. 弹性力学的复变量重构核粒子法. 物理学报, 2008, 57(1): 1-10. doi: 10.7498/aps.57.1 [14] 程荣军, 程玉民. 势问题的无单元Galerkin方法的误差估计. 物理学报, 2008, 57(10): 6037-6046. doi: 10.7498/aps.57.6037 [15] 戴保东, 程玉民. 势问题的径向基函数——局部边界积分方程方法. 物理学报, 2007, 56(2): 597-603. doi: 10.7498/aps.56.597 [16] 程荣军, 程玉民. 带源参数的热传导反问题的无网格方法. 物理学报, 2007, 56(10): 5569-5574. doi: 10.7498/aps.56.5569 [17] 李树忱, 程玉民, 李术才. 动态断裂力学的无网格流形方法. 物理学报, 2006, 55(9): 4760-4766. doi: 10.7498/aps.55.4760 [18] 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. doi: 10.7498/aps.54.4463 [19] 郭常霖, 黄月鸿. 无标样X射线定量分析最小二乘法方程的稳定解. 物理学报, 1993, 42(7): 1106-1111. doi: 10.7498/aps.42.1106 [20] 张宏图. 具有二重对称轴的弹性各向异性晶体的范性问题. 物理学报, 1959, 15(1): 1-12. doi: 10.7498/aps.15.1
• 文章访问数:  5085
• PDF下载量:  409
• 被引次数: 0
##### 出版历程
• 收稿日期:  2014-04-29
• 修回日期:  2014-05-20
• 刊出日期:  2014-09-05

/