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基于改进的移动最小二乘插值法, 提出了黏弹性问题的插值型无单元Galerkin方法. 采用改进的移动最小二乘插值法建立形函数, 根据黏弹性问题的Galerkin弱形式建立离散方程, 推导了相应的计算公式. 与无单元Galerkin方法相比, 本文提出的黏弹性问题的插值型无单元Galerkin方法具有直接施加本质边界条件的优点. 通过数值算例讨论了影响域、节点数对计算精确性的影响, 说明了该方法具有较好的收敛性; 将计算结果与无单元Galerkin方法和有限元方法或解析解比较, 说明了该方法具有提高计算效率的优点.
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关键词:
- 无网格方法 /
- 改进的移动最小二乘插值法 /
- 插值型无单元Galerkin方法 /
- 黏弹性问题
In this paper, based on the improved interpolating moving least-square (IMLS) approximation, the interpolating element-free Galerkin (IEFG) method for two-dimensional viscoelasticity problems is presented. The shape function constructed by the IMLS approximation can overcome the shortcomings that the shape function of the moving least-squares (MLS) can-not satisfy the property of Kronecker function, so the essential boundary conditions can be directly applied to the IEFG method. Under a similar computational precision, compared with the meshless method based on the MLS approximation, the meshless method using the IMLS approximation has a high computational efficiency. Using the IMLS approximation to form the shape function and adopting the Galerkin weak form of the two-dimensional viscoelasticity problem to obtain the final discretized equation, the formulae for two-dimensional viscoelasticity problem are derived by the IEFG method. The IEFG method has some advantages over the conventional element-free Galerkin (EFG) method, such as the concise formulae and direct application of the essential boundary conditions, For the IEFG method of two-dimensional viscoelasticity problems proposed in this paper, three numerical examples and one engineering example are given. The convergence of the method is analyzed by considering the effects of the scale parameters of influence domains and the node distribution on the computational precision of the solutions. It is shown that when dmax = 1.01−2.00, the method in this paper has a good convergence. The numerical results from the IEFG method are compared with those from the EFG method and from the finite element method or analytical solution. We can see that the IEFG method in this paper is effective. The results of the examples show that the IEFG method has the advantage in improving the computational efficiency of the EFG method under a similar computational accuracy. And the engineering example shows that the IEFG method can not only has higher computational precision, but also improve the computational efficiency.-
Keywords:
- meshless method /
- improved interpolating moving least-squares approxiamtion /
- interpolating element-free Galerkin method /
- viscoelasticity problem
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[1] 程玉民 2015 无网格方法 (北京: 科学出版社) 第1−13 页
Cheng Y M 2015 Meshless Methods (Beijing: Science Press) pp1−13 (in Chinese)
[2] 程荣军, 程玉民 2008 57 6037
Google Scholar
Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037
Google Scholar
[3] Cheng Y M, Wang J F, Li R X 2012 Int. J. Appl. Mech. 4 1250042
Google Scholar
[4] Chen L, Cheng Y M, Ma H P 2015 Comput. Mech. 55 591
Google Scholar
[5] Chen L, Cheng Y M 2018 Comput. Mech. 62 67
Google Scholar
[6] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204
Google Scholar
[7] Cheng R J, Cheng Y M 2008 Appl. Numer. Math. 58 884
Google Scholar
[8] Chen L, Liu C, Ma H P, et al. 2014 Int. J. Appl. Mech. 6 1450009
Google Scholar
[9] 李树忱, 程玉民 2004 力学学报 36 496
Google Scholar
Li S C, Cheng Y M 2004 Acta Mech. Sin. 36 496
Google Scholar
[10] Gao H F, Cheng Y M 2010 Int. J. Comput. Meth. 7 55
Google Scholar
[11] 程玉民, 李九红 2005 54 4463
Google Scholar
Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463
Google Scholar
[12] Cheng Y M, Li J H 2006 Sci. China Ser. G 49 46
[13] 程玉民, 彭妙娟, 李九红 2005 力学学报 37 719
Google Scholar
Cheng Y M, Peng M J, Li J H 2005 Acta Mech. Sin. 37 719
Google Scholar
[14] Bai F N, Li D M, Wang J F, Cheng Y M 2012 Chin. Phys. B 21 020204
Google Scholar
[15] Cheng Y M, Wang J F, Bai F N 2012 Chin. Phys. B 21 090203
Google Scholar
[16] Cheng H, Peng M J, Cheng Y M 2017 Eng. Anal. Boundary Elem. 84 52
Google Scholar
[17] Cheng H, Peng M J, Cheng Y M 2017 Int. J. Appl. Mech. 9 1750090
Google Scholar
[18] Cheng H, Peng M J, Cheng Y M 2018 Int. J. Numer. Methods Eng. 114 321
Google Scholar
[19] Cheng H, Peng M J, Cheng Y M 2018 Eng. Anal. Boundary Elem. 97 39
Google Scholar
[20] 程玉民, 陈美娟 2003 力学学报 35 181
Google Scholar
Cheng Y M, Chen M J 2003 Acta Mech. Sin. 35 181
Google Scholar
[21] Cheng Y M, Peng M J 2005 Sci. China Ser. G 48 641
[22] 秦义校, 程玉民 2006 55 3215
Google Scholar
Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215
Google Scholar
[23] Peng M J, Cheng Y M 2009 Eng. Anal. Boundary Elem. 33 77
Google Scholar
[24] Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065
Google Scholar
[25] Ren H P, Cheng Y M, Zhang W 2010 Sci. China Ser. G 53 758
Google Scholar
[26] Wang J F, Wang J F, Sun F X, Cheng Y M 2013 Int. J. Comput. Methods 10 1350043
Google Scholar
[27] Zhang Z, Li D M, Cheng Y M, et al. 2012 Acta Mech. Sin. 28 808
Google Scholar
[28] Zhang Z, Hao S Y, Liew K M, et al. 2013 Eng. Anal. Boundary Elem. 37 1576
Google Scholar
[29] Zhang Z, Wang J F, Cheng Y M, et al. 2013 Sci. China Ser. G 56 1568
Google Scholar
[30] Cheng R J, Liew K M 2012 Eng. Anal. Boundary Elem. 36 1322
Google Scholar
[31] Cheng R J, Wei Q 2013 Chin. Phys. B 22 060209
Google Scholar
[32] Peng M J, Li R X, Cheng Y M 2014 Eng. Anal. Boundary Elem. 40 104
Google Scholar
[33] 蔡小杰, 彭妙娟, 程玉民 2018 中国科学: 物理学 力学 天文学 48 024701
Google Scholar
Cai X J, Peng M J, Cheng Y M 2018 Sci. China: Phys. Mech. Astron. 48 024701
Google Scholar
[34] Yu S Y, Peng M J, Cheng H, Cheng Y M 2019 Eng. Anal. Boundary Elem. 104 215
Google Scholar
[35] 邹诗莹, 席伟成, 彭妙娟, 程玉民 2017 66 120204
Google Scholar
Zou S Y, Xi W C, Peng M J, Cheng Y M 2017 Acta Phys. Sin. 66 120204
Google Scholar
[36] Wu Y, Ma Y Q, Feng W, Cheng Y M 2017 Chin. Phys. B 26 080203
Google Scholar
[37] Meng Z J, Cheng H, Ma L D, Cheng Y M 2018 Acta Mech. Sin. 34 462
Google Scholar
[38] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Sci. China Ser. G 62 040711
Google Scholar
[39] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Int. J. Numer. Methods Eng. 117 15
Google Scholar
[40] Lancaster P, Salkauskas K 1981 Math. Comput. 37 141
Google Scholar
[41] Ren H P, Cheng Y M 2011 Int. J. Appl. Mech. 3 735
Google Scholar
[42] Ren H P, Cheng Y M 2012 Eng. Anal. Boundary Elem. 36 873
Google Scholar
[43] Cheng Y M, Bai F N, Peng M J 2014 Appl. Math. Model. 38 5187
Google Scholar
[44] Cheng Y M, Bai F N, Liu C, Peng M J 2016 Int. J. Comput. Mater. Sci. Eng. 5 1650023
[45] Deng Y J, Liu C, Peng M J, Cheng Y M 2015 Int. J. Appl. Mech. 7 1550017
Google Scholar
[46] Wang J F, Sun F X, Cheng Y M 2012 Chin. Phys. B 21 090204
Google Scholar
[47] Sun F X, Wang J F, Cheng Y M 2013 Chin. Phys. B 22 120203
Google Scholar
[48] Sun F X, Wang J F, Cheng Y M 2016 Int. J. Appl. Mech. 8 1650096
Google Scholar
[49] Wang J F, Hao S Y, Cheng Y M 2014 Math. Probl. Eng. 2014 641592
[50] Wang J F, Sun F X, Cheng Y M, Huang A X 2014 Appl. Math. Comput. 245 321
[51] Sun F X, Wang J F, Cheng Y M 2015 Appl. Numer. Math. 98 79
Google Scholar
[52] Liu F B, Cheng Y M 2018 Int. J. Comput. Mater. Sci. Eng. 7 1850023
[53] Liu F B, Cheng Y M 2018 Int. J. Appl. Mech. 10 1850047
Google Scholar
[54] Liu F B, Wu Q, Cheng Y M 2019 Int. J. Appl. Mech. 11 1950006
Google Scholar
[55] Yang H T, Liu Y 2003 Int. J. Solids Struct. 40 701
Google Scholar
[56] Canelas A, Sensale B 2010 Eng. Anal. Boundary Elem. 34 845
Google Scholar
[57] Cheng Y M, Li R X, Peng M J 2012 Chin. Phys. B 21 090205
Google Scholar
[58] 彭妙娟, 刘茜 2014 63 180203
Google Scholar
Peng M J, Liu Q 2014 Acta Phys. Sin. 63 180203
Google Scholar
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