Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Interpolating element-free Galerkin method for viscoelasticity problems

Zhang Peng-Xuan Peng Miao-Juan

Interpolating element-free Galerkin method for viscoelasticity problems

Zhang Peng-Xuan, Peng Miao-Juan
PDF
HTML
Get Citation
  • 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.
      Corresponding author: Peng Miao-Juan, mjpeng@shu.edu.cn
    [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

  • 图 1  受均布荷载的悬臂梁

    Figure 1.  A cantilever beam subjected to a distributed loading

    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

  • [1] Peng Miao-Juan, Liu Qian. Improved complex variable element-free Galerkin method for viscoelasticity problems. Acta Physica Sinica, 2014, 63(18): 180203. doi: 10.7498/aps.63.180203
    [2] Cheng Yu-Min, Cheng Rong-Jun. The meshless method for solving the inverse heat conduction problem with a source parameter. Acta Physica Sinica, 2007, 56(10): 5569-5574. doi: 10.7498/aps.56.5569
    [3] Cheng Yu-Min, Li Jiu-Hong. A meshless method with complex variables for elasticity. Acta Physica Sinica, 2005, 54(10): 4463-4471. doi: 10.7498/aps.54.4463
    [4] Cheng Yu-Min, Cheng Rong-Jun. Error estimates of element-free Galerkin method for potential problems. Acta Physica Sinica, 2008, 57(10): 6037-6046. doi: 10.7498/aps.57.6037
    [5] Cheng Rong-Jun, Cheng Yu-Min. Error estimate of element-free Galerkin method for elasticity. Acta Physica Sinica, 2011, 60(7): 070206. doi: 10.7498/aps.60.070206
    [6] Li Zhong-Hua, Qin Yi-Xiao, Cui Xiao-Chao. Interpolating reproducing kernel particle method for elastic mechanics. Acta Physica Sinica, 2012, 61(8): 080205. doi: 10.7498/aps.61.080205
    [7] Cheng Yu-Min, Dai Bao-Dong. Local boundary integral equation method based on radial basis functions for potential problems. Acta Physica Sinica, 2007, 56(2): 597-603. doi: 10.7498/aps.56.597
    [8] Du Hong-Xiu, Wei Hong, Qin Yi-Xiao, Li Zhong-Hua, Wang Tong-Zun. Interpolating particle method for mechanical analysis of space axisymmetric components. Acta Physica Sinica, 2015, 64(10): 100204. doi: 10.7498/aps.64.100204
    [9] Cheng Yu-Min, Chen Li. Reproducing kernel particle method with complex variables for elasticity. Acta Physica Sinica, 2008, 57(1): 1-10. doi: 10.7498/aps.57.1
    [10] Zou Shi-Ying, Xi Wei-Cheng, Peng Miao-Juan, Cheng Yu-Min. Analysis of fracture problems of airport pavement by improved element-free Galerkin method. Acta Physica Sinica, 2017, 66(12): 120204. doi: 10.7498/aps.66.120204
    [11] Du Chao-Fan, Zhang Ding-Guo. A meshfree method based on point interpolation for dynamic analysis of rotating cantilever beams. Acta Physica Sinica, 2015, 64(3): 034501. doi: 10.7498/aps.64.034501
    [12] Zheng Bao-Jing, Dai Bao-Dong. Improved meshless local Petrov-Galerkin method for two-dimensional potential problems. Acta Physica Sinica, 2010, 59(8): 5182-5189. doi: 10.7498/aps.59.5182
    [13] Yang Xiu-Li, Dai Bao-Dong, Li Zhen-Feng. Meshless local Petrov-Galerkin method with complex variables for elasticity. Acta Physica Sinica, 2012, 61(5): 050204. doi: 10.7498/aps.61.050204
    [14] Li Shu-Chen, Li Shu-Cai, Cheng Yu-Min. Meshless manifold method for dynamic fracture mechanics. Acta Physica Sinica, 2006, 55(9): 4760-4766. doi: 10.7498/aps.55.4760
    [15] Tang Bo, Li Jun-Feng, Wang Tian-Shu. Numerical simulation of liquid drop phenomenon by least square particle finite element method. Acta Physica Sinica, 2008, 57(11): 6722-6729. doi: 10.7498/aps.57.6722
    [16] Li Jun, Dong Hai-Ying. Modelling of chaotic systems using wavelet kernel partial least squares regression method. Acta Physica Sinica, 2008, 57(8): 4756-4765. doi: 10.7498/aps.57.4756
    [17] Feng Zhao, Wang Xiao-Dong, Ouyang Jie. The element-free Galerkin method based on the shifted basis for solving the Kuramoto- Sivashinsky equation. Acta Physica Sinica, 2012, 61(23): 230204. doi: 10.7498/aps.61.230204
    [18] Fan Ji-Hua, Zhang Ding-Guo. Bezier interpolation method for the dynamics of rotating flexible cantilever beam. Acta Physica Sinica, 2014, 63(15): 154501. doi: 10.7498/aps.63.154501
    [19] Cheng Yu-Min, Qin Yi-Xiao. Reproducing kernel particle boundary element-free method for elasticity. Acta Physica Sinica, 2006, 55(7): 3215-3222. doi: 10.7498/aps.55.3215
    [20] Ren Jin-Lian, Jiang Rong-Rong, Lu Wei-Gang, Jiang Tao. Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method. Acta Physica Sinica, 2020, 69(8): 080202. doi: 10.7498/aps.69.20191829
  • Citation:
Metrics
  • Abstract views:  230
  • PDF Downloads:  3
  • Cited By: 0
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.

    • 无网格方法作为一种新的数值方法, 因其只需要节点信息, 从而可以避免网格的划分以及重构[1], 因此具有前处理简单、计算精度高等特点, 已经成为科学和工程计算发展的重要趋势之一. 当前已经发展的无网格方法有无单元Galerkin方法(element-free Galerkin method, EFG)[1-3]、重构核粒子方法(reproducing kernel particle method, RKPM)[4-6]、有限点方法(finite point method, FPM)[7]、无网格局部Petrov-Galerkin方法(meshless local Petrov-Galerkin method, MLPG)[8]、单位分解法[9,10]、复变量无网格方法[11-15]、维数分裂无网格方法[16-19]和无网格的边界积分方程方法[20-26]等.

      无单元Galerkin方法是目前应用最广泛的无网格方法之一, 该方法基于移动最小二乘法构造逼近函数, 所以具有较高的计算精度. 但该方法的不足之处在于形成方程组时, 有时是病态的或者是奇异的. 因此, 有可能难以求解或者得到不正确的解. 为了改进该方法的不足, 程玉民和陈美娟[20]以及Cheng和Peng[21]提出了改进的移动最小二乘法. 该方法选取正交函数作为基函数, 从而致使法方程既不病态也不会奇异, 同时也不需要求解矩阵的逆, 可以直接得到方程组的解, 降低了计算量. Zhang等[27-29]采用改进的移动最小二乘法建立形函数, 提出了瞬态热传导、波动方程和弹性动力学等问题改进的无单元Galerkin方法. Cheng和Liew[30]以及Cheng和Wei[31]利用改进的无单元Galerkin方法求解波动方程和广义Camassa-Holm方程. 文献[32-35]建立了三维黏弹性、三维弹塑性和弹塑性大变形等问题改进的无单元Galerkin方法, 并将改进的无单元Galerkin方法用于机场复合道面的断裂力学分析. Wu等[36]建立了弹性力学拓扑优化问题改进的无单元Galerkin方法. Meng等[37-39]将维数分裂法与改进的无单元Galerkin方法相结合, 提出了三维势问题、瞬态热传导问题和波动方程的杂交无单元Galerkin方法.

      由于改进的移动最小二乘法构造的逼近函数不满足Kronecker $\delta $函数性质, 使得基于其形成的无网格方法不能直接施加本质边界条件. Lancaster和Salkauskas[40]提出了移动最小二乘插值法, 该方法得到的逼近函数满足Kronecker $\delta $函数性质, 从而可以直接施加本质边界条件.

      Ren和Cheng[41,42]提出了改进的移动最小二乘插值法, 该方法证明了Lancaster的形函数公式中的一些内积可以为零, 可得到更为简单的形函数计算公式, 从而减少了方程个数以及未知量个数, 提高了计算效率. Ren和Cheng [41,42]基于改进的移动最小二乘插值法, 建立了势问题和弹性力学的插值型无单元Galerkin方法(interpolating element-free Galerkin method, IEFG). Cheng等[43,44]建立了弹塑性和非线性大变形等问题的插值型无单元Galerkin方法. Deng等[45]建立温度场问题的插值型复变量无单元Galerkin方法.

      为了克服移动最小二乘插值法因权函数奇异导致的计算困难并避免产生截断误差, Wang等[46]和Sun等[47,48]提出了基于非奇异权函数改进的移动最小二乘插值法, 建立了势问题、弹性问题和弹塑性问题改进的插值型无单元Galerkin方法. 该方法形函数的待定系数比传统的移动最小二乘法少一个, 求逆矩阵的阶数比移动最小二乘法少一阶, 提高了计算效率和计算精度. 基于非奇异权函数改进的移动最小二乘插值法, Wang等[49,50]研究了两点边值问题改进的插值型无单元Galerkin方法的误差估计和插值型移动最小二乘法的误差估计. Sun等[51]分析了在n维空间的插值型移动最小二乘法的误差估计. Liu等[52-54]建立了弹性大变形、弹塑性大变形和凝胶非均匀溶胀大变形等问题改进的插值型无单元Galerkin方法.

      黏弹性问题在力学和工程中具有十分重要的应用. 由于其具有非线性的特点, 目前国内外主要是运用有限元法或者边界元法等方法来求解黏弹性问题. 但是, 近年来无网格方法发展快速, 越来越多的学者运用此方法来解决黏弹性问题. Yang和Liu[55]把无单元Galerkin方法和时域精细算法结合, 用于求解黏弹性问题. Canelas和Sensale[56]运用无网格边界节点法分析了弹性和黏弹性材料中的简谐波问题. 彭妙娟和程玉民等建立了改进的无单元Galerkin方法[32]、黏弹性问题复变量无单元Galerkin方法[57]和改进的复变量无单元Galerkin方法[58].

      本文基于改进的移动最小二乘插值法构建形函数, 根据Galerkin积分弱形式建立方程, 提出了黏弹性问题的插值型无单元Galerkin方法. 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; 与无单元Galerkin方法以及有限元的解或解析解比较, 说明了该方法具有提高计算效率的优点.

    2.   改进的移动最小二乘插值法
    • 对于改进的移动最小二乘插值法[41,42], 定义奇异权函数

      其中参数$\alpha $是正整数, ${\rho _I}$是影响域${{{x}}_I}$的影响半径.

      定义任意函数$f({{x}})$$g({{x}})$的内积为

      其中参数$\bar \varOmega = \varOmega \cup \partial \varOmega $, ${C^0}(\bar \varOmega )$$\bar \varOmega $域中所有连续函数的集合, $n$是影响域覆盖场点$x$的节点数. 这样可得

      首先对基函数${p_i}({{x}})$做标准化处理. 在空间${\rm{span}} ({p_1},{p_2}, \ldots,{p_m})$中, 在${{x}}$点将基函数${p_1}({{x}}) \equiv 1$单位化,

      再将${p_2}({{x}}),{p_3}({{x}}), \ldots,{p_m}({{x}})$${q_1}({{x}})$正交化, 可得

      将新的基函数${q_i}({{x}})$应用于移动最小二乘法可得

      其中

      将(10)式代入(6)式可得

      (16)式可以写成

      其中

      为形函数.

      由于改进的移动最小二乘插值法建立的形函数满足Kronecker $\delta $函数性质, 因此在建立离散方程时, 可以直接施加本质边界条件, 从而减少了计算时间, 提高了计算效率.

    3.   黏弹性问题的基本方程
    • 黏弹性问题的应力-应变关系与时间相关, 其材料参数是时间的函数. 假定${{\sigma }}(t)$在整个加载过程中恒定, 然而${{\varepsilon}} ({{t}})$和位移${{u}} ({{t}})$随着时间变化. 如果给定求解域$\varOmega $内的体力${{b}}$, 应力边界${\varGamma _t}$上的面力${\bar {{t}}}$及位移边界${\varGamma _u}$上的位移${\bar {{u}}}$(边界条件与时间无关), 下面给出二维黏弹性问题的基本方程和边界条件.

      平衡方程

      其中${{L}}$是微分算子矩阵

      ${{\sigma}} $是域内任意一点的应力

      ${{b}}$是域内任意一点的体力

      几何方程

      其中${{\varepsilon}} $${{u}}$分别为域内任意点的应变和位移,

      物理方程

      其中

      $J(t)$为不同流变模型的蠕变柔量, 对Maxwell模型,

      对Kelvin 模型,

      对三参数模型,

      其中$G$, ${G_{\rm{1}}}$${G_{\rm{2}}}$分别是各模型中弹簧的剪切弹性模量, $\eta $是各模型中黏壶的黏性系数, $K$为体积弹性模量.

      边界条件

      其中

      ${n_{\rm{1}}}$${n_{\rm{2}}}$是边界点的外法线方向余弦.

      由于可以直接施加本质边界条件, 因此黏弹性问题的Galerkin积分弱形式为

    4.   黏弹性问题的插值型无单元Galerkin方法
    • 设问题求解域$\varOmega $内配有$M$个节点, 节点的影响域${\varOmega _I}$($I = 1, 2, \cdots, M$)的并集覆盖了整个域$\varOmega $. 由改进的移动最小二乘插值法, 位移向量

      的插值函数为

      (36)式可以写为

      其中

      黏弹性平面问题的应变张量可表示为

      其中

      黏弹性平面问题的应力张量可表示为

      其中

      是正应变矩阵;

      是偏应变矩阵.

      对平面应力情况, 即${\sigma _{33}} = 0$, 可得

      其中${\beta _1}$${\beta _2}$是常数,

      则(53)式可写为

      对平面应变情况, 即${\varepsilon _{{\rm{33}}}} = {\rm{0}}$, 可得

      将(49)式和(50)式代入(48)式得到

      将(44)式和(61)式代入(35)式得到

      $\delta {{{U}}^{\rm{T}}}$的任意性, 得到最后的离散系统方程为

      (63)式中, 对于平面应力问题,

      对于平面应变问题,

      以上即为黏弹性问题的插值型无单元Galerkin方法.

    5.   数值算例
    • 为了验证上述理论的准确性, 以下通过该方法对4个算例进行了计算, 并将计算结果与无单元Galerkin方法以及有限元方法的计算结果或解析解进行对比. 在算例中, 构造移动最小二乘插值法的形函数采用线性基函数和奇异权函数. 在每个积分单元, 采用$4 \times 4$的Gauss积分.

      定义相对误差公式为

      其中${L^2}$范数定义为

      定义方差公式为

      其中$u_k^I$为第$I$种节点分布时变量$u$在节点${x_k}$处的数值解, $\bar u_k^{IJ}$为第$I$种和$J$种节点分布时变量$u$在节点${x_k}$处数值解的平均值.

      算例1 受均布荷载的悬臂梁

      图1所示的受均布荷载作用的悬臂梁, 梁长$L = {\rm{8\;}}{\rm{m}}$, 梁高$D = {\rm{2\;}}{\rm{m}}$, 取单位厚度, 按平面应力计算. 材料体积变化是弹性的, $E = 1.0 \times 1{0^8}\; {\rm{Pa}}$, $v = 0.25$, 剪切变形流变性质满足Kelvin黏弹性模型, 其参数为$G = 2.0 \times 1{0^8} \;{\rm{Pa}}$, $\eta = 6.0 \times 1{0^8} \;{\rm{Pa}} \cdot {\rm{s}}$, 均布荷载为$p = 3.0 \times 1{0^4} \;{\rm{Pa}}$, 不计自重.

      Figure 1.  A cantilever beam subjected to a distributed loading

      为保证有限元法(FEM)数值解的精确性, 进行不同节点布置以得到较为精确的数值解. 采用四边形4节点单元, 具体节点布置为: $17 \times 9$, $21 \times $11, $29 \times 11$, $33 \times 11$, $33 \times 13$, $37 \times 17$, 图2给出了不同节点分布下有限元法解的方差. 从图2可以看出, 节点分布越密方差越小, 说明有限元法得到的解是收敛的. 在此选用$33 \times 13$的节点布置.

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

      在不同节点分布$9 \times 3$, $11 \times 4$, $13 \times 5$, $15 \times 6$, $17 \times 7$$19 \times 8$的情况下, 利用本文提出的插值型无单元Galerkin方法计算, 图3给出了不同节点分布下数值解的相对误差. 从图3可以看出, 节点分布越密相对误差越小, 即插值型无单元Galerkin方法具有较好的收敛性. 在考虑计算效率的情况下, 采用图4所示的$17 \times 7$节点布置, 背景积分网格选取$16 \times 6$.

      Figure 3.  The relative error under different node distributions.

      Figure 4.  Node distribution.

      不同影响域比例参数对本文方法数值解的精度具有一定影响, 如图5所示, 当${d_{{\rm{max}}}} = $1.7—1.9时, 该方法数值解的精度较高. 本算例选用${d_{{\rm{max}}}} = 1.7$.

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

      采用插值型无单元Galerkin方法求解时, 节点布置为$17 \times 7$, 背景积分网格选取$16 \times 6$, ${d_{{\rm{max}}}} = 1.7$, 此时中轴线各点挠度和右端中点挠度随时间变化的总相对误差为$1.29\% $, 计算时间为31.19 s.

      采用无单元Galerkin方法求解时, 节点布置为$17 \times 7$, 背景积分网格选取$16 \times 6$, ${d_{{\rm{max}}}} = 3.0$, $\alpha = 2.3 \times 1{0^{10}}$, 从而可以得到与插值型无单元Galerkin方法相近的计算精度, 计算时间为34.67 s.

      上述两种方法所得的数值解与有限元法的数值解的对比如图6图7所示, 可以看出, 两种方法得到的数值解与有限元法的数值解较为符合, 但插值型无单元Galerkin方法能够提高计算效率.

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

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

      算例2 受纯弯曲的梁

      图8为受纯弯曲的梁, 几何参数为$L = 5\;{\rm{m}}$, $H = 2\;{\rm{m}}$, 单位厚度. 材料体积变化是弹性的, $E = 1.0 \times 1{0^6}\; {\rm{Pa}}$, $\nu = 0.3$, 剪切变形的流变性质满足三参数模型, 其参数为${G_1} = 5.0 \times 1{0^5}\; {\rm{Pa}}$, G2 = $ 1.0 \times 1{0^6} \;{\rm{Pa}}$$\eta = 2.0 \times 1{0^6} \;{\rm{Pa}} \cdot {\rm{s}}$. 其所受三角形分布荷载大小如图8所示, 不计体力, 按平面应力问题计算.

      Figure 8.  A beam subjected to simple bending

      由于对称性和反对称性, 只取1/4梁进行计算. 在${x_{\rm{1}}}$轴和${x_{\rm{2}}}$轴上, 位移${u_{\rm{1}}}$均为0, 为了消除结构在${x_{\rm{2}}}$方向上刚体位移, 可令原点处${u_{\rm{2}}}$为0. 有限元法采用四边形4节点单元, 节点布置为$21 \times 13$. 无单元Galerkin方法和插值型无单元Galerkin方法均采用图9所示的$15 \times 7$的节点布置, 背景积分网格选取$14 \times 6$.

      Figure 9.  Node distribution.

      采用插值型无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 2.0$, 此时中轴线各点挠度和右端中点挠度随时间变化的总相对误差为$0.30\% $, 计算时间为21.88 s.

      时采用无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 3.0$, $\alpha = 1.5 \times 1{0^8}$, 从而可以得到与插值型无单元Galerkin方法相近的计算精度, 计算时间为28.76 s.

      上述两种方法所得数值解与有限元法数值解的对比如图10图11所示. 可以看出, 插值型无单元Galerkin方法和无单元Galerkin方法都可以达到较高的精度, 但插值型无单元Galerkin方法能够提高计算效率.

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

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

      算例3 受均布内压的圆环

      受均布内压的圆环如图12所示, 其内表面承受均匀分布的压力$p = 30 \;{\rm{kPa}}$. 几何参数为$a = 1\;{\rm{m}}$, $b = 5\;{\rm{m}}$. 材料体积变化是弹性的, $E = 1.0 \times 1{0^7}\; {\rm{Pa}}$, $\nu = 0.25$, 剪切变形的流变性质采用Kelvin模型. 其参数为$G = 5.0 \times 1{0^5}\; {\rm{Pa}}$, $\eta = 2.0 \times 1{0^6} \;{\rm{Pa}} \cdot {\rm{s}}$. 不计体力, 按平面应变问题计算.

      Figure 12.  Circular ring under a distributed inner pressure

      在极坐标系下, 采用Kelvin模型, 圆筒径向位移随时间变化的解析解为

      其中

      (70)式表明Kelvin模型加载瞬时位移为零, 当时间足够长, 位移将趋于稳定.

      图13, 由于对称性, 只取1/4区域为研究对象. 无单元Galerkin方法和插值型无单元Galerkin方法均选用如图14所示的${\rm{1}}7 \times 12$的节点布置, 背景积分网格选取$16 \times 11$.

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

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

      采用插值型无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 1.01$, 此时$t = 30 \;{\rm{s}}$时沿${x_2} = {\rm{0}}$线上节点的位移和点$(2,0)$的径向位移随时间变化的总相对误差为$1.10\% $, 计算时间为23.38 s.

      采用无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 1.5$, $\alpha = 8.0 \times 1{0^7}$, 此时可得到与上述插值型无单元Galerkin方法相近的相对误差, 计算时间为28.66 s.

      两种方法得到的数值解与解析解的对比如图15图16所示. 从以上分析和图15图16的对比可看出, 插值型无单元Galerkin方法和无单元Galerkin方法的数值解均与解析解符合得较好, 但插值型无单元Galerkin方法能够提高计算效率.

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

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

      算例4 工程算例: 受静水压力的混凝土水坝

      为了证明本方法在较复杂几何求解域的可行性, 本算例计算一个受到静水压力的混凝土水坝, 水坝的几何参数如图17所示. 坝高100 m, 上游水位90 m, 下游无水. 坝底部完全固定, 仅考虑坝体受到的上游水荷载. 坝体材料为混凝土, 所以需要考虑建造完工后期混凝土的流变效应. 假定混凝土材料体积变化是弹性的, $E = 2.0 \times 1{0^{10}}\; {\rm{Pa}}$, $\nu = 0.2$, 混凝土的早期流变性质满足三参数模型, 其参数为${G_1} = 8.33 \times 1{0^9}\; {\rm{Pa}}$, ${G_2} = 4.0 \times 1{0^{10}}\; {\rm{Pa}}$$\eta = 8.0 \times $$ 10^{16} \;{\rm{Pa}} \cdot {\rm{s}}$. 不计体力, 按平面应变问题计算.

      Figure 17.  A concrete dam under hydrostatic pressure.

      有限元法采用四边形4节点单元, 共布置234个节点. 无单元Galerkin方法和插值型无单元Galerkin方法均在求解域内布置如图18所示的255个节点, 背景积分网格选取231个.

      Figure 18.  Node distribution of a concrete dam.

      采用插值型无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 1.2$, $t = 500$ d时沿${x_1} = 15$方向上节点的水平位移和点$(15, 50)$的水平位移与时间关系的总相对误差为$0.27\% $, 计算时间为50.42 s.

      采用无单元Galerkin方法求解时, ${d_{{\rm{max}}}} = 3.0$, $\alpha = 2.0 \times 1{0^{12}}$, 其总相对误差为$1.36\% $, 计算时间为94.44 s.

      两种方法的数值解与有限元法的数值解的对比如图19图20所示. 从以上分析和图19图20的对比可以看出, 与无单元Galerkin方法相比, 插值型无单元Galerkin方法具有更高的精度和效率, 说明该方法在解决复杂工程问题时能够提高计算精度和计算效率.

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

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

    6.   结 论
    • 本文基于改进的移动最小二乘插值法构造形函数, 建立了黏弹性问题的插值型无单元Galerkin方法. 由于改进的移动最小二乘插值法的形函数满足Kronecker$\delta $函数性质, 避免了传统无单元Galerkin方法要利用Lagrange乘子法或者罚函数法来施加本质边界条件, 大大减少了计算量, 从而有效地提高了在求解黏弹性问题时的计算效率. 通过数值算例讨论了影响域大小、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; 将计算结果和有限元解或解析解进行对比, 说明了插值型无单元Galerkin方法较无单元Galerkin方法在求解黏弹性问题上具有提高计算效率的优点.

Reference (58)

Catalog

    /

    返回文章
    返回