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Interpolating element-free Galerkin method for viscoelasticity problems

## Interpolating element-free Galerkin method for viscoelasticity problems

Zhang Peng-Xuan, Peng Miao-Juan
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• #### Abstract

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.

#### References

 [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 物理学报 57 6037 Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 [3] Cheng Y M, Wang J F, Li R X 2012 Int. J. Appl. Mech. 4 1250042 [4] Chen L, Cheng Y M, Ma H P 2015 Comput. Mech. 55 591 [5] Chen L, Cheng Y M 2018 Comput. Mech. 62 67 [6] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [7] Cheng R J, Cheng Y M 2008 Appl. Numer. Math. 58 884 [8] Chen L, Liu C, Ma H P, et al. 2014 Int. J. Appl. Mech. 6 1450009 [9] 李树忱, 程玉民 2004 力学学报 36 496 Li S C, Cheng Y M 2004 Acta Mech. Sin. 36 496 [10] Gao H F, Cheng Y M 2010 Int. J. Comput. Meth. 7 55 [11] 程玉民, 李九红 2005 物理学报 54 4463 Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 [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 Cheng Y M, Peng M J, Li J H 2005 Acta Mech. Sin. 37 719 [14] Bai F N, Li D M, Wang J F, Cheng Y M 2012 Chin. Phys. B 21 020204 [15] Cheng Y M, Wang J F, Bai F N 2012 Chin. Phys. B 21 090203 [16] Cheng H, Peng M J, Cheng Y M 2017 Eng. Anal. Boundary Elem. 84 52 [17] Cheng H, Peng M J, Cheng Y M 2017 Int. J. Appl. Mech. 9 1750090 [18] Cheng H, Peng M J, Cheng Y M 2018 Int. J. Numer. Methods Eng. 114 321 [19] Cheng H, Peng M J, Cheng Y M 2018 Eng. Anal. Boundary Elem. 97 39 [20] 程玉民, 陈美娟 2003 力学学报 35 181 Cheng Y M, Chen M J 2003 Acta Mech. Sin. 35 181 [21] Cheng Y M, Peng M J 2005 Sci. China Ser. G 48 641 [22] 秦义校, 程玉民 2006 物理学报 55 3215 Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 [23] Peng M J, Cheng Y M 2009 Eng. Anal. Boundary Elem. 33 77 [24] Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065 [25] Ren H P, Cheng Y M, Zhang W 2010 Sci. China Ser. G 53 758 [26] Wang J F, Wang J F, Sun F X, Cheng Y M 2013 Int. J. Comput. Methods 10 1350043 [27] Zhang Z, Li D M, Cheng Y M, et al. 2012 Acta Mech. Sin. 28 808 [28] Zhang Z, Hao S Y, Liew K M, et al. 2013 Eng. Anal. Boundary Elem. 37 1576 [29] Zhang Z, Wang J F, Cheng Y M, et al. 2013 Sci. China Ser. G 56 1568 [30] Cheng R J, Liew K M 2012 Eng. Anal. Boundary Elem. 36 1322 [31] Cheng R J, Wei Q 2013 Chin. Phys. B 22 060209 [32] Peng M J, Li R X, Cheng Y M 2014 Eng. Anal. Boundary Elem. 40 104 [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 Cai X J, Peng M J, Cheng Y M 2018 Sci. China: Phys. Mech. Astron. 48 024701 [34] Yu S Y, Peng M J, Cheng H, Cheng Y M 2019 Eng. Anal. Boundary Elem. 104 215 [35] 邹诗莹, 席伟成, 彭妙娟, 程玉民 2017 物理学报 66 120204 Zou S Y, Xi W C, Peng M J, Cheng Y M 2017 Acta Phys. Sin. 66 120204 [36] Wu Y, Ma Y Q, Feng W, Cheng Y M 2017 Chin. Phys. B 26 080203 [37] Meng Z J, Cheng H, Ma L D, Cheng Y M 2018 Acta Mech. Sin. 34 462 [38] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Sci. China Ser. G 62 040711 [39] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Int. J. Numer. Methods Eng. 117 15 [40] Lancaster P, Salkauskas K 1981 Math. Comput. 37 141 [41] Ren H P, Cheng Y M 2011 Int. J. Appl. Mech. 3 735 [42] Ren H P, Cheng Y M 2012 Eng. Anal. Boundary Elem. 36 873 [43] Cheng Y M, Bai F N, Peng M J 2014 Appl. Math. Model. 38 5187 [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 [46] Wang J F, Sun F X, Cheng Y M 2012 Chin. Phys. B 21 090204 [47] Sun F X, Wang J F, Cheng Y M 2013 Chin. Phys. B 22 120203 [48] Sun F X, Wang J F, Cheng Y M 2016 Int. J. Appl. Mech. 8 1650096 [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 [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 [54] Liu F B, Wu Q, Cheng Y M 2019 Int. J. Appl. Mech. 11 1950006 [55] Yang H T, Liu Y 2003 Int. J. Solids Struct. 40 701 [56] Canelas A, Sensale B 2010 Eng. Anal. Boundary Elem. 34 845 [57] Cheng Y M, Li R X, Peng M J 2012 Chin. Phys. B 21 090205 [58] 彭妙娟, 刘茜 2014 物理学报 63 180203 Peng M J, Liu Q 2014 Acta Phys. Sin. 63 180203

#### Cited By

• 图 1  受均布荷载的悬臂梁

图 2  不同节点分布下有限元法解的方差

Figure 2.  The variances of the solutions of FEM under different node distributions.

图 3  不同节点分布下的相对误差

Figure 3.  The relative error under different node distributions.

图 4  节点布置

Figure 4.  Node distribution.

图 5  不同影响域比例参数下的相对误差

Figure 5.  The relative error for different scale parameters of influence domains.

图 6  $t = 20 \;{\rm{s}}$时悬臂梁中轴线上各点的挠度

Figure 6.  Vertical displacements of nodes on the neutral axis of the beam when $t = 20 \;{\rm{s}}$.

图 7  梁右端中点的挠度随时间的变化

Figure 7.  Time history of vertical displacement of midpoint in the right end of the beam.

图 8  纯弯曲的梁

Figure 8.  A beam subjected to simple bending

图 9  节点分布

Figure 9.  Node distribution.

图 10  $t = 30 \;{\rm{s}}$时梁中轴线上的节点挠度

Figure 10.  Vertical displacements of nodes on the neutral axis of the beam when $t = 30 \;{\rm{s}}$.

图 11  梁右端中点的挠度随时间$t$的变化

Figure 11.  Time history of vertical displacement of midpoint in the right end of the beam.

图 12  受均布内压的厚壁圆筒

Figure 12.  Circular ring under a distributed inner pressure

图 13  受均布内压1/4圆筒

Figure 13.  A quarter of the circular ring under a distributed inner pressure

图 14  1/4圆筒的节点分布

Figure 14.  Node distribution of a quarter of the circular ring.

图 15  $t = 30 \;{\rm{s}}$时沿${x_2} = {\rm{0}}$线上节点的位移

Figure 15.  Radial displacements at${x_2} = {\rm{0}}$ when $t = 30 \;{\rm{s}}$.

图 16  $(2,0)$的径向位移随时间$t$的变化

Figure 16.  Time history of radial displacement at point $(2,0)$.

图 17  受静水压力的混凝土水坝

Figure 17.  A concrete dam under hydrostatic pressure.

图 18  混凝土水坝的节点分布

Figure 18.  Node distribution of a concrete dam.

图 19  $t = 500$ d时沿${x_1} = 15$方向上节点的水平位移

Figure 19.  Horizontal displacements at ${x_1} = 15$ when $t = 500\; {\rm d}$.

图 20  混凝土坝上点$(15, 50)$的水平位移与时间的关系

Figure 20.  Time history of horizontal displacement of the point $(15, 50)$.

•  [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 物理学报 57 6037 Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 [3] Cheng Y M, Wang J F, Li R X 2012 Int. J. Appl. Mech. 4 1250042 [4] Chen L, Cheng Y M, Ma H P 2015 Comput. Mech. 55 591 [5] Chen L, Cheng Y M 2018 Comput. Mech. 62 67 [6] Chen L, Cheng Y M 2010 Chin. Phys. B 19 090204 [7] Cheng R J, Cheng Y M 2008 Appl. Numer. Math. 58 884 [8] Chen L, Liu C, Ma H P, et al. 2014 Int. J. Appl. Mech. 6 1450009 [9] 李树忱, 程玉民 2004 力学学报 36 496 Li S C, Cheng Y M 2004 Acta Mech. Sin. 36 496 [10] Gao H F, Cheng Y M 2010 Int. J. Comput. Meth. 7 55 [11] 程玉民, 李九红 2005 物理学报 54 4463 Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 [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 Cheng Y M, Peng M J, Li J H 2005 Acta Mech. Sin. 37 719 [14] Bai F N, Li D M, Wang J F, Cheng Y M 2012 Chin. Phys. B 21 020204 [15] Cheng Y M, Wang J F, Bai F N 2012 Chin. Phys. B 21 090203 [16] Cheng H, Peng M J, Cheng Y M 2017 Eng. Anal. Boundary Elem. 84 52 [17] Cheng H, Peng M J, Cheng Y M 2017 Int. J. Appl. Mech. 9 1750090 [18] Cheng H, Peng M J, Cheng Y M 2018 Int. J. Numer. Methods Eng. 114 321 [19] Cheng H, Peng M J, Cheng Y M 2018 Eng. Anal. Boundary Elem. 97 39 [20] 程玉民, 陈美娟 2003 力学学报 35 181 Cheng Y M, Chen M J 2003 Acta Mech. Sin. 35 181 [21] Cheng Y M, Peng M J 2005 Sci. China Ser. G 48 641 [22] 秦义校, 程玉民 2006 物理学报 55 3215 Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 [23] Peng M J, Cheng Y M 2009 Eng. Anal. Boundary Elem. 33 77 [24] Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065 [25] Ren H P, Cheng Y M, Zhang W 2010 Sci. China Ser. G 53 758 [26] Wang J F, Wang J F, Sun F X, Cheng Y M 2013 Int. J. Comput. Methods 10 1350043 [27] Zhang Z, Li D M, Cheng Y M, et al. 2012 Acta Mech. Sin. 28 808 [28] Zhang Z, Hao S Y, Liew K M, et al. 2013 Eng. Anal. Boundary Elem. 37 1576 [29] Zhang Z, Wang J F, Cheng Y M, et al. 2013 Sci. China Ser. G 56 1568 [30] Cheng R J, Liew K M 2012 Eng. Anal. Boundary Elem. 36 1322 [31] Cheng R J, Wei Q 2013 Chin. Phys. B 22 060209 [32] Peng M J, Li R X, Cheng Y M 2014 Eng. Anal. Boundary Elem. 40 104 [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 Cai X J, Peng M J, Cheng Y M 2018 Sci. China: Phys. Mech. Astron. 48 024701 [34] Yu S Y, Peng M J, Cheng H, Cheng Y M 2019 Eng. Anal. Boundary Elem. 104 215 [35] 邹诗莹, 席伟成, 彭妙娟, 程玉民 2017 物理学报 66 120204 Zou S Y, Xi W C, Peng M J, Cheng Y M 2017 Acta Phys. Sin. 66 120204 [36] Wu Y, Ma Y Q, Feng W, Cheng Y M 2017 Chin. Phys. B 26 080203 [37] Meng Z J, Cheng H, Ma L D, Cheng Y M 2018 Acta Mech. Sin. 34 462 [38] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Sci. China Ser. G 62 040711 [39] Meng Z J, Cheng H, Ma L D, Cheng Y M 2019 Int. J. Numer. Methods Eng. 117 15 [40] Lancaster P, Salkauskas K 1981 Math. Comput. 37 141 [41] Ren H P, Cheng Y M 2011 Int. J. Appl. Mech. 3 735 [42] Ren H P, Cheng Y M 2012 Eng. Anal. Boundary Elem. 36 873 [43] Cheng Y M, Bai F N, Peng M J 2014 Appl. Math. Model. 38 5187 [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 [46] Wang J F, Sun F X, Cheng Y M 2012 Chin. Phys. B 21 090204 [47] Sun F X, Wang J F, Cheng Y M 2013 Chin. Phys. B 22 120203 [48] Sun F X, Wang J F, Cheng Y M 2016 Int. J. Appl. Mech. 8 1650096 [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 [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 [54] Liu F B, Wu Q, Cheng Y M 2019 Int. J. Appl. Mech. 11 1950006 [55] Yang H T, Liu Y 2003 Int. J. Solids Struct. 40 701 [56] Canelas A, Sensale B 2010 Eng. Anal. Boundary Elem. 34 845 [57] Cheng Y M, Li R X, Peng M J 2012 Chin. Phys. B 21 090205 [58] 彭妙娟, 刘茜 2014 物理学报 63 180203 Peng M J, Liu Q 2014 Acta Phys. Sin. 63 180203
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•  Citation:
##### Metrics
• Abstract views:  230
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##### Publishing process
• Received Date:  09 July 2019
• Accepted Date:  31 July 2019
• Available Online:  26 November 2019
• Published Online:  01 September 2019

## Interpolating element-free Galerkin method for viscoelasticity problems

###### Corresponding author: Peng Miao-Juan, mjpeng@shu.edu.cn
• Department of Civil Engineering, Shanghai University, Shanghai 200444, China

Abstract: 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.

Reference (58)

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