A meshless numerical investigation based on the RBF-QR approach for elasticity problems

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Amirkabir University of Technology

2 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)

Abstract

In the current research work, we present an improvement of meshless boundary element method (MBEM) based on the shape functions of radial basis functions-QR (RBF-QR) for solving the two-dimensional elasticity problems. The MBEM has benefits of the boundary integral equations (BIEs) to reduce the dimension of problem and the meshless attributes of moving least squares (MLS) approximations. Since the MLS shape functions don’t have the delta function property, applying boundary conditions is not simple. Here, we propose the MBEM using RBF-QR to increase the accuracy and efficiency of MBEM. To show the performance of the new technique, the two-dimensional elasticity problems have been selected. We solve the mentioned model on several irregular domains and report simulation results.

Keywords

Main Subjects


[1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless method: an overview and recent developments, Comput. Methods Appl. Mech. Eng., 139 (1996) 3-47.
[2] J. S. Chen, C. Pan, C.T. Wu, W.K. Liu, Reproducing kernel particle methods for large deformation analysis of non-linear structures, Comput. Methods Appl. Mech. Eng., 139 (1996) 195-227.
[3] L. Chen, Y. M. Cheng, The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations, Computational Mechanics, 62 (2018) 67-80.
[4] Y. M. Cheng, F. Bai, C. Liu, M. Peng, Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method, International Journal of Computational Materials Science and Engineering, 5 (2016) 1650023.
[5] Y. M. Cheng, C. Liu, F. N. Bai, M. J. Peng, Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method, Chinese Physics B, 24 (2015).
[6] M. Dehghan, M. Abbaszadeh, A local meshless method for solving multi-dimensional Vlasov-Poisson and VlasovPoisson–Fokker–Planck systems arising in plasma physics, Engineering with Computers, 33 (2017) 961-981.
[7] M. Dehghan, M. Abbaszadeh, Proper orthogonal decomposition variational multiscale element free Galerkin (PODVMEFG) meshless method for solving incompressible Navier–Stokes equation, Computer Methods in Applied Mechanics and Engineering, 311 (2016) 856-888.
[8] M. Dehghan, M. Abbaszadeh, An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation, Engineering Analysis with Boundary Elements 92 (2018) 244-256.
[9] M. Dehghan, H. Hosseinzadeh, Calculation of 2D singular and near singular integrals of boundary elements method based on the complex space C, Applied Mathematical Modelling, 36 (2012) 545-560.
[10] M. Dehghan, H. Hosseinzadeh, Improvement of the accuracy in boundary element method based on high-order discretization, Computers & Mathematics with Applications, 62 (2011) 4461-4471.
[11] M. Dehghan, H. Hosseinzadeh, Obtaining the upper bound of discretization error and critical boundary integrals of circular arc boundary element method, Mathematical and Computer Modelling, 55 (2012) 517-529.
[12] W. Elleithy, Analysis of problems in elasto-plasticity via an adaptive FEM-BEM coupling method, Comput. Methods Appl. Mech. Eng., 197 (2008) 3687-3701.
[13] B. Fornberg, E. Larsson, N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011) 869-892.
[14] R. Gowrishankar, S. Mukherjee, A pure boundary node method for potential theory, Commun. Numer. Meth. Eng. 18 (2002) 411-427.
[15] S. Jun, W.K. Liu, T. Belytschko, Explicit reproducing kernel particle methods for large deformation problems, Int. J. Numer. Methods Eng. 41 (1998) 137- 166.
[16] S. J. Kim, J. T. Oden, Finite element analysis of a class of problems in finite elastoplasticity based on the thermodynamical theory of materials of type N, Comput. Methods Appl. Mech. Eng., 53 (1985) 277-302.
[17] E. Larsson, E. Lehto, A. Heryudono, B. Fornberg, Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions, SIAM J. Sci. Comput. 35(4) (2013) A2096-A2119.
[18] D. M. Li, K. M. Liew, Y. M. Cheng, An improved complex variable element-free Galerkin method for two-dimensional large deformation elastoplasticity problems, Comput. Methods Appl. Mech. Eng., 269 (2014) 72-86.
[19] X. Li, J. Zhu, On a Galerkin boundary node method for potential problems, Advances in Engineering Software, 42 (2011) 927-933.
[20] X. Li, J. Zhu, A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math., 230 (2009) 314-328.
[21] X. Li, Meshless analysis of two-dimensional Stokes flows with the Galerkin boundary node method, Eng. Anal. Bound. Elem., 34 (2010) 79-91.
[22] X. Li, S. Li, Meshless boundary node methods for Stokes problems, Appl. Math. Model., 39 (2015) 1769-1783.
[23] X. Li , A meshless interpolating Galerkin boundary node method for Stokes flows, Eng. Anal. Bound. Elem., 51 (2015) 112-122.
[24] K. M. Liew, Y. Cheng, S. Kitipornchai, Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform, Int. J. Numer. Meth. Eng., 64 (2005) 1610-1627.
[25] G. R. Liu, Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer Science & Business Media, 2005.
[26] G. R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method, Taylor & Francis, 2009.
[27] G. R. Liu, G. Y. Zhang, Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM), Int. J. Numer. Meth. Eng., 74 (2008) 1128-1161.
[28] W. K. Liu, S. Jun, Multiple-scale reproducing kernel particle methods for large deformation problem, Int. J. Numer. Methods Eng., 41 (1998) 1339-1362.
[29] J. H. Lv, Y. Miao, H. P. Zhu, Boundary node method based on parametric space for 2D elasticity, Eng. Anal. Bound. Elem., 37 (2013) 659-665.
[30] H. P. Ren, Y. M. Cheng, W. Zhang, An interpolating boundary element-free method (IBEFM) for elasticity problems, Physics, Mechanics & Astronomy, 53 (2010) 758-766.
[31] Z. J. Meng, H. Cheng, L. D. Ma, Y. M. Cheng, The dimension split element-free Galerkin method for three-dimensional potential problems, Acta Mechanica Sinica/Lixue Xuebao, 34(3) (2018) 462-474.
[32] Miao Yu, W. Yuan-han, Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method, Appl. Math. Mech., 27 (2006) 673–681.
[33] D. Mirzaei, A new low-cost meshfree method for two and three dimensional problems in elasticity, Appl. Math. Model., 39 (2015) 7181-7196.
[34] Y. X. Mukherjee, S. Mukherjee, The boundary node method for potential problems, Int. J. Numer. Meth. Eng., 40 (1997) 797-815.
[35] M. Peng, D. Li, Y. M. Cheng, The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems, Engineering Structures, 33 (2011) 127-135.
[36] M. Tezer-Sezgin, Boundary element method solution of MHD flow in a rectangular duct, Internat. J. Numer. Methods Fluids, 18 (1994) 937-952.
[37] M. Tezer-Sezgin, S. Han Aydin, Solution of magnetohydrodynamic flow problems using the boundary element method, Eng. Anal. Bound. Elem., 30 (2006) 411-418.
[38] M. Tezer-Sezgin, C. Bozkaya, Boundary element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field, Comput. Mech., 41 (2008) 769-775.
[39] M. Tezer-Sezgin, C. Bozkaya, The boundary element solution of magnetohydrodynamic flow in an infinite region, J. Comput. Appl. Math., 225 (2009) 510-521.
[40] J. Sladek, V. A. Sladek, A meshless method for large deflection of plates, Comput. Mech. 30 (2) (2003) 155-163.
[41] F. Sun, J. Wang, Y. M. Cheng, An improved interpolating element-free Galerkin method for elasticity, Chinese Physics B, 22(12) (2013) 120203.
[42] F. Sun, J. Wang, Y.M.Cheng, A. Huang, Error estimates for the interpolating moving least-squares method in ndimensional space, Applied Numerical Mathematics, 98 (2015) 79-105.
[43] F. Tan, Y. Zhang, Y. Li, Development of a meshless hybrid boundary node method for Stokes flows, Eng. Anal. Bound. Elem. 37 (2013) 899-908.
[44] F. Tan, Y. Zhang, Y. Li, An improved hybrid boundary node method for 2D crack problems, Archive of Applied Mechanics, 85 (2015) 101-116.
[45] M. Tatari, F. Ghasemi, The Galerkin boundary node method for magneto-hydrodynamic (MHD) equation, J. Comput. Phys. 258 (2014) 634-649.
[46] S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, third ed., McGraw-Hill, New York, 1970.
[47] J. Wang, J. Wang, F. Sun, Y. M. Cheng, An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems, International Journal of Computational Methods, 10 (2013) 1350043.
[48] H. Xie, T. Nogami, J. Wang, A radial boundary node method for two-dimensional elastic analysis, Eng. Anal. Bound. Elem., 27 (2003) 853-862.
[49] F. Yan, X. T. Feng, H. Zhou, Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates, Appl. Math. Model. 35 (2011) 5691-5706.
[50] L. W. Zhang, K. M. Liew, An improved moving least-squares Ritz method for two-dimensional elasticity problems, Appl. Math. Comput., 246 (2014) 268-282.
[51] J. Zhang, Z. Yao, H. Li, A hybrid boundary node method, Int. J. Numer. Meth. Eng., 53 (2002) 751-763.
[52] Y. M. Zhang, F. L. Sun, D. L. Young, W. Chen, Y. Gud, Average source boundary node method for potential problems, Eng. Anal. Bound. Elem. 70 (2016) 114-125.