×

zbMATH — the first resource for mathematics

A weighted nodal-radial point interpolation meshless method for 2D solid problems. (English) Zbl 1297.74138
Summary: A novel weighted nodal-radial point interpolation meshless (WN-RPIM) method is proposed for 2D solid problems. In the new approach, the moment matrices are performed only at the nodes to get nodal coefficients. At each computational point (node or integration point), the shape functions are obtained by weighting the nodal coefficients whose nodes are located in its support domain. The shape functions obtained by the new scheme preserve the Kronecker delta function property under certain conditions. This conclusion can be extended for the weighted nodal-interpolating moving least squares approximation studied in [T. Most and C. Bucher, Eng. Anal. Bound. Elem. 32, No. 6, 461–470 (2008; Zbl 1244.74228)]. Besides, the new method is much less time consuming than the RPIM method, since the number of nodes is generally much smaller than that of the integration points. Some numerical examples are illustrated to show the effectiveness of the proposed method. Some parameters that influence the performance of the proposed method are also investigated.

MSC:
74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
PDF BibTeX Cite
Full Text: DOI
References:
[1] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method diffuse approximation and diffuse elements, Comput Mech, 10, 307-318, (1992) · Zbl 0764.65068
[2] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int J Numer Methods Eng, 37, 229-256, (1994) · Zbl 0796.73077
[3] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[4] Chen, W., Meshfree boundary particle method applied to Helmholtz problems, Eng Anal Bound Elem, 16, 577-581, (2002) · Zbl 1013.65128
[5] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 117-127, (1998) · Zbl 0932.76067
[6] Zhang, X.; Song, K. Z.; Lu, M. W.; Liu, X., Meshless methods based on collocation with radial basis functions, Comput Mech, 26, 333-343, (2000) · Zbl 0986.74079
[7] Zhang, X.; Liu, Y., Meshless method, (2004), Tsinghua University Press Beijing
[8] Liu, G. R.; Gu, Y. T., An introduction to meshfree methods and their programming, (2005), Springer Netherland
[9] Cao, Y.; Yao, L. Q.; Yin, Y., New treatment of essential boundary conditions in EFG method by coupling with RPIM, Acta Mech Solida Sinica, 26, 302-316, (2013)
[10] Liu, G. R.; Gu, Y. T., A point interpolation method for two-dimensional solids, Int J Numer Methods Eng, 50, 937-951, (2001) · Zbl 1050.74057
[11] Wang, J. G.; Liu, G. R., A point interpolation method based on radial basis functions, Int J Numer Methods Eng, 54, 1623-1648, (2002) · Zbl 1098.74741
[12] Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J Approx Theory, 93, 258-272, (1998) · Zbl 0904.41013
[13] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J Geophys Res, 76, 1905-1915, (1971)
[14] Franke, R., Scattered data interpolationtest of some methods, Math Comput, 38, 181-200, (1982) · Zbl 0476.65005
[15] Kansa, E. J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics—I. surface approximations and partial derivative estimate, Comput Math Appl, 19, 127-146, (1990) · Zbl 0692.76003
[16] Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—II. solutions to hyperbolic, parabolic and elliptic partial differential equations, Comput Math Appl, 19, 149-161, (1990) · Zbl 0850.76048
[17] Franke, C.; Schaback, R., Solving partial differential equations by collocation using radial basis functions, Appl Math Comput, 93, 73-82, (1998) · Zbl 0943.65133
[18] Hon, Y. C.; Schaback, R., On unsymmetric collocation by radial basis functions, Appl Math Comput, 119, 177-186, (2001) · Zbl 1026.65107
[19] Wang, J. G.; Liu, G. R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Comput Methods Appl Mech Eng, 191, 2611-2630, (2002) · Zbl 1065.74074
[20] Wang, J. G.; Liu, G. R.; Lin, P., Numerical analysis of Biot’s consolidation process by radial point interpolation method, Int J Solids Struct, 39, 1557-1573, (2002) · Zbl 1061.74014
[21] Liu, G. R.; Dai, K. Y.; Lim, K. M.; Gu, Y. T., A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart Mater Struct, 12, 171-180, (2003)
[22] Liu, G. R.; Zhang, G. Y.; Gu, Y. T.; Wang, Y. Y., A meshfree radial point interpolation method (RPIM) for three-dimensional solids, Comput Mech, 36, 421-430, (2005) · Zbl 1138.74420
[23] Liu, G. R.; Gu, Y. T., Point interpolation method based on local residual formulation using radial basis functions, Struct Eng Mech, 14, 713-732, (2002)
[24] Liu, G. R.; Gu, Y. T., A local radial point interpolation method (LRPIM) for free vibration analysis of 2-D solids, J Sound Vib, 246, 29-46, (2001)
[25] Xia, P.; Long, S. R.; Cui, H. X., Elastic dynamic analysis of moderately thick plate using meshless LRPIM, Acta Mech Solida Sinica, 22, 116-124, (2009)
[26] Saeedpanah, I.; Jabbari, E., Local heaviside-weighted LRPIM meshless method and its application to two-dimensional potential flows, Int J Numer Methods Fluids, 59, 475-493, (2009) · Zbl 1394.76090
[27] Lu, Y. Y.; Belytschko, T.; Gu, L., A new implementation of the element-free Galerkin method, Comput Methods Appl Mech Eng, 113, 397-414, (1994) · Zbl 0847.73064
[28] Zhang, Z.; Liew, K. M.; Cheng, Y. M.; Lee, Y. Y., Analyzing 2D fracture problems with the improved element-free Galerkin method, Eng Anal Bound Elem, 32, 241-250, (2008) · Zbl 1244.74240
[29] Cao Y, Yao LQ, Yin Y. The meshless local Petrov-Galerkin method based on the improved moving least-squares approximation. Technical Report; 2011 [Proceedings of the International Conference on Computational & Experimental Engineering and Sciences, Nanjing].
[30] Most, T.; Bucher, C., New concepts for moving least squaresan interpolating non-singular weighting function and weighted nodal least squares, Eng Anal Bound Elem, 32, 461-470, (2008) · Zbl 1244.74228
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.