# zbMATH — the first resource for mathematics

Numerical solution of functionally graded materials based on radial basis reproducing kernel particle method. (English) Zbl 07153252
Summary: In this paper, the radial basis function (RBF) is used to construct the approximating function of the reproducing kernel particle method (RKPM), which can eliminate the negative effect of different kernel functions on the calculating accuracy. The radial basis reproducing kernel particle method (RRKPM) is proposed. Furthermore, the RRKPM is applied to solve the elastic mechanical problems of the functionally graded materials (FGM), and the RRKPM for FGM is established. The corresponding formulae of the RRKPM for FGM are derived. Compared with the traditional RKPM, the RRKPM has higher calculating accuracy and stability. Finally, the numerical examples illustrate that the RRKPM is correct and effective to solve the elastic mechanical problems of the FGM.

##### MSC:
 74 Mechanics of deformable solids 41 Approximations and expansions
Full Text:
##### References:
 [1] Li, P. W.; Fan, C. M., Generalized finite difference method for two-dimensional shallow water equations, Eng Anal Bound Elem, 80, 58-71 (2017) · Zbl 1403.76133 [2] Gupta, A.; Talha, M.; Chaudhari, V. K., Natural frequency of functionally graded plates resting on elastic foundation using finite element method, Proc Technol, 23, 163-170 (2016) [3] Li, C.; Man, H.; Song, C. M.; Gao, W., Fracture analysis of piezoelectric materials using the scaled boundary finite element method, Eng Fract Mech, 97, 52-71 (2013) [4] Peixoto, R. G.; Penna, S. S.; Pitangueira, R. L.S.; Ribeiro, G. O., A non-local damage approach for the boundary element method, Appl Math Model, 69, 63-76 (2019) [5] Liu, M. B.; Liu, G. R., Restoring particle consistency in smoothed particle hydrodynamics, Appl Numer Math, 56, 19-36 (2016) · Zbl 1329.76285 [6] Liu, F. B.; Cheng, Y. M., The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels, Int J Appl Mech, 10, 4, Article 1850047 pp. (2018) [7] Chen, L.; Cheng, Y. M., The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations, Comput Mech, 62, 67-80 (2018) · Zbl 06965870 [8] Chu, F. Y.; He, J. Z.; Wang, L. H.; Zhong, Z., Buckling analysis of functionally graded thin plate with in-plane material inhomogeneity, Eng Anal Bound Elem, 65, 112-125 (2016) · Zbl 1403.74031 [9] Ghadiri Rad, M. H.; Shahabian, F.; Hosseini, S. M., A meshless local Petrov-Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping, Acta Mech, 226, 5, 1497-1513 (2015) · Zbl 1329.74148 [10] Khosravifard, A.; Hematiyan, M. R.; Bui, T. Q., Accurate and efficient analysis of stationary and propagating crack problems by meshless methods, Theor Appl Fract Mech, 87, 21-34 (2017) [11] Zhu, H. H.; Sun, P.; Cai, Y. C., Independent cover meshless method for the simulation of multiple crack growth with arbitrary incremental steps and directions, Eng Anal Bound Elem, 83, 242-255 (2017) · Zbl 1403.74094 [12] Cai, Y. C.; Han, L.; Tian, L. G.; Zhang, L. Y., Meshless method based on Shepard function and partition of unity for two-dimensional crack problems, Eng Anal Bound Elem, 65, 126-135 (2016) · Zbl 1403.74082 [13] Ma, Z. H.; Wang, H.; Pu, S. H., A parallel meshless dynamic cloud method on graphic processing units for unsteady compressible flows past moving boundaries, Comput Methods Appl Mech Eng, 285, 146-165 (2015) · Zbl 1423.76347 [14] Sarler, B.; Vertnik, R.; Zaloznik, M., Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method, Eng Anal Bound Elem, 30, 10, 847-855 (2006) · Zbl 1195.76325 [15] Liu, F. B.; Cheng, Y. M., The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems, Int J Comput Mater Sci Eng, 7, 3, Article 1850023 pp. (2018) [16] Liu, F. B.; Wu, Q.; Cheng, Y. M., A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems, Int J Appl Mech, 11, 1, Article 1950006 pp. (2019) [17] Parand, K.; Hemami, M., Numerical study of astrophysics equations by meshless collocation method based on compactly supported radial basis function, Int J Appl Comput Math, 3, 2, 1053-1075 (2017) · Zbl 1397.85003 [18] Kennett, D. J.; Timme, S.; Angulo, J.; Badcock, K. J., An implicit meshless method for application in computational fluid dynamics, Int J Numer Methods Fluids, 71, 8, 1007-1028 (2013) [19] Pandey, A.; Hardt, S.; Klar, A.; Tiwari, S., Brownian dynamics of rigid particles in an incompressible fluctuating fluid by a meshfree method, Comput Fluids, 127, 174-181 (2016) · Zbl 1390.76894 [20] Hossein Musavi, S.; Ashrafizaadeh, M., Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries, Comput Fluids, 181, 236-247 (2019) · Zbl 1410.76371 [21] Liu, M. B.; Liu, G. R.; Lam, K. Y.; Zong, Z., Smoothed particle hydrodynamics for numerical simulation of underwater explosion, Comput Mech, 30, 2, 106-118 (2003) · Zbl 1128.76352 [22] Assari, P.; Dehghan, M., A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations, Appl Math Comput, 350, 249-265 (2019) · Zbl 1429.65308 [23] Ma, J. C.; Wei, G. F.; Liu, D. D.; Liu, G. T., The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM, Appl Math Comput, 309, 170-182 (2017) · Zbl 1411.74062 [24] Memari, A.; Khoshravan Azar, M. R.; Vakili-Tahami, F., Meshless fracture analysis of 3D planar cracks with generalized thermo-mechanical stress intensity factors, Eng Anal Bound Elem, 99, 169-194 (2019) · Zbl 07006027 [25] Wei, D. D.; Zhang, W. W.; Wang, L. H.; Dai, B. D., The complex variable meshless local Petrov-Galerkin method for elasticity problems of functionally graded materials, Appl Math Comput, 268, 1140-1151 (2015) · Zbl 1410.74076 [26] Kumar, A.; Sharma, A.; Vaish, R.; Kumar, R.; Jain, S., A numerical study on anomalous behavior of piezoelectric response in functionally graded materials, J Mater Sci, 53, 4, 2413-2423 (2018) [27] Dai, B. D.; Wei, D. D.; Ren, H. P.; Zhang, Z., The complex variable meshless local Petrov-Galerkin method for elastodynamic analysis of functionally graded materials, Appl Math Comput, 309, 17-26 (2017) · Zbl 1411.74052 [28] Lin, J.; Li, J.; Guan, Y.; Zhao, G.; Naceur, H.; Coutellier, D., Geometrically nonlinear bending analysis of functionally graded beam with variable thickness by a meshless method, Compos Struct, 189, 239-246 (2018) [29] Chu, F. Y.; Wang, L. H.; Zhong, Z.; He, J. Z., Hermite radial basis collocation method for vibration of functionally graded plates with in-plane material inhomogeneity, Comput Struct, 142, 79-89 (2014) [30] Hidayat, M. I.P.; Ariwahjoedi, B.; Parman, S.; Irawan, S., A meshfree approach for transient heat conduction analysis of nonlinear functionally graded materials, Int J Comput Methods, 15, 2, Article 1850007 pp. (2018) · Zbl 1404.74202 [31] Li, Y.; Li, J.; Wen, P. H., Finite and infinite block Petrov-Galerkin method for cracks in functionally graded materials, Appl Math Model, 68, 306-326 (2019) [32] Hirai, T.; Chen, L., Recent and prospective development of functionally graded materials in Japan, Mater Sci Forum, 308-311, 3, 509-514 (1999) [33] Kugler, S.; Fotiu, P. A.; Murin, J., Thermo-elasticity in shell structures made of functionally graded materials, Acta Mech, 227, 5, 1307-1329 (2016) · Zbl 1341.74044 [34] Zhen, W.; Ma, Y. T.; Ren, X. H.; Lo, S. H., Analysis of functionally graded plates subjected to hygrothermomechanical loads, AIAA J, 54, 11, 3667-3673 (2016) [35] Sola, A.; Bellucci, D.; Cannillo, V., Functionally graded materials for orthopedic applications – an update on design and manufacturing, Biotechnol Adv, 34, 5, 504-531 (2016) [36] Zhang, H. H.; Han, S. Y.; Fan, L. F.; Huang, D., The numerical manifold method for 2D transient heat conduction problems in functionally graded materials, Eng Anal Bound Elem, 88, 145-155 (2018) · Zbl 1403.74135 [37] Phung-Van, P.; Tran, L. V.; Ferreira, A. J.M.; Nguyen-Xuan, H.; Abdel-Wahab, M., Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads, Nonlinear Dyn, 87, 2, 879-894 (2016) · Zbl 1372.74058 [38] Thang, P. T.; Lee, J., Free vibration characteristics of sigmoid-functionally graded plates reinforced by longitudinal and transversal stiffeners, Ocean Eng, 148, 53-61 (2018) [39] Yan, W. T.; Ge, W. J.; Smith, J.; Lin, S.; Kafka, O. L.; Lin, F.; Liu, W. K., Multi-scale modeling of electron beam melting of functionally graded materials, Acta Mater, 115, 403-412 (2016) [40] Heuer, S.; Lienig, T.; Mohr, A.; Weber, Th.; Pintsuk, G.; Coenen, J. W.; Gormann, F.; Theisen, W.; Linsmeier, Ch., Ultra-fast sintered functionally graded Fe/W composites for the first wall of future fusion reactors, Compos Part B: Eng, 164, 205-214 (2019) [41] Wang, L. H.; Wang, Z.; Qian, Z. H.; Gao, Y. K.; Zhou, Y. T., Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions, Inverse Probl Sci Eng, 26, 12, 1695-1727 (2018) · Zbl 1428.65036 [42] Wang, L. H.; Qian, Z. H.; Wang, Z.; Gao, Y. K.; Peng, Y. B., An efficient radial basis collocation method for the boundary condition identification of the inverse wave problem, Int J Appl Mech, 10, 1, Article 1850010 pp. (2018) [43] Zhong, Z.; Yu, T., Analytical solutions to the bending problems of functionally gradient cantilever beam, J Tongji Univ (Nat Sci), 34, 4, 443-447 (2006), (In Chinese) · Zbl 1222.74029 [44] Zhang, T.; Wei, G. F.; Ma, J. C.; Gao, H. F., Radial basis reproducing kernel particle method for piezoelectric materials, Eng Anal Bound Elem, 92, 171-179 (2018) · Zbl 1403.74331 [45] Liu, Z.; Wei, G. F.; Wang, Z. M., Numerical analysis of functionally graded materials using reproducing kernel particle method, Int J Appl Mech, 11, 6, Article 1950060 pp. (2019) [46] Yu, T.; Zhong, Z., Analytical solutions to the bending problems of functionally gradient cantilever beams under uniformly distributed load, Chin J Solid Mech, 01, 15-20 (2006), (In Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.