Based on the improved complex variable moving least-squares (ICVMLS)
approximation, the improved complex variable element-free Galerkin
(ICVEFG) method for the bending problem of Kirchhoff plate is presented.
Compared with the moving least-squares (MLS) approximation, in the
ICVMLS approximation, the approximation function of two-dimensional
problems can be obtained with one-dimensional basis function, then
the computational efficiency of the shape functions is higher. Compared
with the meshless methods based on the MLS approximation, under the
same node distributions, the ones using the ICVMLS approximation can
obtain the solutions with higher computational accuracy; and under
the similar computational accuracy, the ones using the ICVMLS approximation
have higher computational efficiency. The ICVMLS approximation is
used to form the approximation function of the deflection of a Kirchhoff
plate, the Galerkin weak form of the bending problem of Kirchhoff
plates is adopted to obtain the discretized system equations, and
the penalty method is employed to enforce the essential boundary conditions,
then the corresponding formulae of the ICVEFG method for the bending
problem of Kirchhoff plates are presented. By computing and analyzing
four typical examples, it is shown that the ICVEFG method of Kirchhoff
plates in this paper is efficient. And the computational precision
of the numerical solutions is analyzed to select the basis function,
weight function, scaling factor, node distribution and penalty factor
in the ICVEFG method. Numerical examples show that the method in this
paper has better convergence and higher accuracy. When the quadratic
polynomial basis function and the cubic or quartic spline weight function
are used, and
国家自然科学基金(11571223)
[1] Hein P. Diffuse Element Method Applied to Kirchhoff Plates. Technical Report, Department of Civil Engineering, Northwestern University, Evanston. 1993. Google Scholar
[2] Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Comput Mech, 1995, 17: 26-35 CrossRef ADS Google Scholar
[3] Li S, Hao W, Liu W K. Numerical simulations of large deformation of thin shell structures using meshfree methods. Comput Mech, 2000, 25: 102-116 CrossRef ADS Google Scholar
[4] Liu G R, Chen X L. A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J Sound Vib, 2001, 241: 839-855 CrossRef ADS Google Scholar
[5] Gu Y T, Liu G R, A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analyses of thin plates. Comp Model Eng Sci, 2001, 2: 463–476. Google Scholar
[6] Sladek J, Sladek V. A meshless method for large deflection of plates. Comput Mech, 2003, 30: 155-163 CrossRef ADS Google Scholar
[7] Sladek J, Sladek V, Zhang C, et al. Analysis of orthotropic thick plates by meshless local Petrov-Galerkin (MLPG) method. Int J Numer Meth Engng, 2006, 67: 1830-1850 CrossRef ADS Google Scholar
[8] Chen L, Cheng Y M. The complex variable reproducing kernel particle method for elasto-plasticity problems. Sci China-Phys Mech Astron, 2010, 53: 954-965 CrossRef ADS Google Scholar
[9] Chen L, Cheng Y M, Ma H P. The complex variable reproducing kernel particle method for the analysis of Kirchhoff plates. Comput Mech, 2015, 55: 591-602 CrossRef ADS Google Scholar
[10] Cheng Y M, Chen M J, A boundary element-free method for linear elasticity (in Chinese). Acta Mech Sin, 2000, 25: 181–186 [程玉民, 陈美娟. 弹性力学的一种边界无单元法. 力学学报, 2000, 25: 181–186]. Google Scholar
[11] Wang J F, Sun F X, Cheng Y M, et al. Error estimates for the interpolating moving least-squares method. Appl Math Computation, 2014, 245: 321-342 CrossRef Google Scholar
[12]
Sun
F X,
Wang
J F,
Cheng
Y M, et al.
Error estimates for the interpolating moving least-squares method in
[13] Cheng Y, Li J. A complex variable meshless method for fracture problems. Sci China Ser G-Phys Mech Astron, 2006, 49: 46-59 CrossRef ADS Google Scholar
[14] Cheng Y M, Peng M J, Li J H. The complex variable moving least-square approximation and its application (in Chinese). Chin J Theor Appl Mech, 2005, 37: 719–723 [程玉民, 彭妙娟, 李九红. 复变量移动最小二乘法及其应用. 力学学报, 2005, 37: 719–723]. Google Scholar
[15] Li D M, Peng M J, Cheng Y M. The complex variable element-free Galerkin (CVEFG) method for elastic large deformation problems (in Chinese). Sci Sin-Phys Mech Astron, 2011, 41: 1003-1014 CrossRef Google Scholar
[16] Ren H, Cheng J, Huang A. The complex variable interpolating moving least-squares method. Appl Math Comput, 2012, 219: 1724-1736 CrossRef Google Scholar
[17] Zhang Z, Wang J F, Cheng Y M, et al. The improved element-free Galerkin method for three-dimensional transient heat conduction problems. Sci China-Phys Mech Astron, 2013, 56: 1568-1580 CrossRef ADS Google Scholar
[18] Zhang Z, Li D M, Cheng Y M, et al. The improved element-free Galerkin method for three-dimensional wave equation. Acta Mech Sin, 2012, 28: 808-818 CrossRef ADS Google Scholar
[19] Zhang Z, Hao S Y, Liew K M, et al. The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng Anal Bound Elem, 2013, 37: 1576-1584 CrossRef Google Scholar
[20] Peng M J, Li R X, Cheng Y M. Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method. Eng Anal Bound Elem, 2014, 40: 104-113 CrossRef Google Scholar
[21] Cheng Y, Bai F, Liu C, et al. Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method. Int J Comp Mat Sci Eng, 2016, 05: 1650023 CrossRef ADS Google Scholar
[22] Sun F, Wang J, Cheng Y. An improved interpolating element-free Galerkin method for elastoplasticity via nonsingular weight functions. Int J Appl Mech, 2016, 08: 1650096 CrossRef Google Scholar
[23] Peng M, Liu P, Cheng Y. The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems. Int J Appl Mech, 2009, 01: 367-385 CrossRef ADS Google Scholar
[24] Peng M, Li D, Cheng Y. The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems. Eng Struct, 2011, 33: 127-135 CrossRef Google Scholar
[25] Cheng Y, Wang J, Li R. The complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems. Int J Appl Mech, 2012, 04: 1250042 CrossRef ADS Google Scholar
[26] Cheng Y M, Li R X, Peng M J. Complex variable element-free Galerkin method for viscoelasticity problems. Chin Phys B, 2012, 21: 090205 CrossRef ADS Google Scholar
[27] Weng Y J, Zhang Z, Cheng Y M. The complex variable reproducing kernel particle method for two-dimensional inverse heat conduction problems. Eng Anal Bound Elem, 2014, 44: 36-44 CrossRef Google Scholar
[28] Li D M, Liew K M, Cheng Y M. Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method. Comput Mech, 2014, 53: 1149-1162 CrossRef ADS Google Scholar
[29] Deng Y, Liu C, Peng M, et al. The interpolating complex variable element-free Galerkin method for temperature field problems. Int J Appl Mech, 2015, 07: 1550017 CrossRef ADS Google Scholar
[30] Cheng Y M, Bai F N, Peng M J. A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl Math Model, 2014, 38: 5187-5197 CrossRef Google Scholar
[31] Ren H, Cheng Y. A new element-free Galerkin method based on improved complex variable moving least-squares approximation for elasticity. Int J Comp Mat Sci Eng, 2012, 01: 1250011 CrossRef ADS Google Scholar
[32] Timoshenko S, Woinowsky-Krieger S. Theory of Plates and Shells. 2nd ed. New York: McGraw-Hill, 1959. Google Scholar
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