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Elastodynamic problems by meshless local integral method: analytical formulation. (English) Zbl 1287.65117

Summary: Analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
74R10 Brittle fracture

Software:

BDEM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aliabadi, M. H., The boundary element method, Applications in solids and structures, vol. 2 (2002), Wiley: Wiley Chicester · Zbl 0994.74003
[2] Portela, A.; Aliabadi, M. H.; Rooke, D. P., The dual boundary element method: effective implementation for crack problems, Int J Numer Methods Eng, 33, 1269-1287 (1992) · Zbl 0825.73908
[3] Portela, A.; Aliabadi, M. H.; Rooke, D. P., Dual boundary element analysis of cracked plates: singularity subtraction technique, Int J Fract, 55, 17-28 (1992)
[4] Wen, P. H., Dynamic fracture mechanics: displacement discontinuity method, Topics in engineering, vol. 29 (1996), Computational Mechanics Publications: Computational Mechanics Publications Southampton
[5] Wrobel, L. C., The boundary element method, Application to thermo-fluids, vol. 1 (2002), Wiley: Wiley Chicesater
[6] Brancati, A.; Aliabadi, M. H.; Bendetti, I., Hierarchical adaptive cross approximation GMRES technique for solution of acoustic problems using the boundary element method, CMES Comput Model Eng Sci, 43, 149-172 (2009) · Zbl 1232.65158
[7] Brancati, A.; Aliabadi, M. H.; Mallardo, M. H.V., A BEM sensitivity formulation for three-dimensional active noise control, Int J Numer Methods Eng, 90, 9, 1183-1206 (2012) · Zbl 1242.76191
[8] Mallardo, V.; Aliabadi, M. H., An adaptive fast multipole approach to 2D wave propagation, CMES Comput Model Eng Sci, 87, 2, 77-96 (2012) · Zbl 1356.74235
[9] Telles, J. C.F.; Brebbia, C. A., On the application of the boundary element method to plasticity, Appl Math Model, 3, 466-470 (1979) · Zbl 0422.73074
[10] Leitao, V.; Aliabadi, M. H.; Rooke, D. P., The dual boundary element formulation for elastoplastic fracture mechanics, Int J Numer Methods Eng, 38, 315-333 (1995) · Zbl 0831.73072
[11] Mallardo, V., Integral equations and nonlocal damage theory: a numerical implementation using the BDEM, Int J Fract, 157, 1-2, 13-32 (2009) · Zbl 1308.74158
[12] Pindea, E.; Aliabadi, M. H., Dual boundary element analysis for time-dependent fracture problems in creeping materials, Key Eng Mater, 383, 109-121 (2008)
[13] 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
[14] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin method, Int J Numer Methods Eng, 37, 229-256 (1994) · Zbl 0796.73077
[15] Liu, W. K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int J Numer Methods Eng, 20, 1081-1106 (1995) · Zbl 0881.76072
[16] Sladek, V.; Sladek., J.; Zhang, Ch., Comparative study of meshless approximations in local integral equation method, CMC Comput Mater Continua, 4, 177-188 (2006) · Zbl 1180.74065
[17] Atluri, S. N., The meshless method (MLPG) for domain and BIE discretizations (2004), Tech Science Press: Tech Science Press Forsyth, GA, USA · Zbl 1105.65107
[18] Wen, P. H.; Aliabadi, M. H., An improved meshless collocation method for elastostatic and elastodynamic problems, Commun Numer Methods Eng, 24, 8, 635-651 (2008) · Zbl 1159.74461
[19] Zhang, J.; Yao, Zhenhan; Tanaka, M., The meshless regular hybrid boundary node method for 2D linear elasticity, Eng Anal Boundary Elem, 27, 259-268 (2003) · Zbl 1112.74556
[20] Zhang, Xi; Yao, Zhenhan; Zhang, Zhangfei, Application of MLPG in large deformation analysis, Acta Mech Sin (English Series), 22, 331-340 (2006) · Zbl 1202.74206
[21] Miers, L. S.; Telles, J. C.F., On the NGF procedure for LBIE elastostatic fracture mechanics, Comput Model Eng Sci, 14, 161-169 (2006) · Zbl 1357.74085
[22] Khosravifard, A.; Hematiyan, M. R.; Marin, L., Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method, Appl Math Model, 35, 4157-4174 (2011) · Zbl 1225.74033
[23] Oh, Hae-Soo; Davis, C.; Kim, J. G.; Kwon, Y.-H., Reproducing polynomial particle methods for boundary integral equations, Comput Mech, 1-19 (2011) · Zbl 1273.65189
[24] Li, Xiaolin; Li, Shuling, A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique, Appl Math Model, 35, 3698-3709 (2011) · Zbl 1221.65311
[25] Barbieri, E.; Meo, M. A., A meshless cohesive segments method for crack initiation and propagation in composites, Appl Compos Mater, 18, 45-63 (2011)
[26] Skouras, E. D.; Bourantas, G. C.; Loukopoulos, V. C.; Nikiforidis, G. C., Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials, Eng Anal Boundary Elem, 35, 452-464 (2011) · Zbl 1259.80029
[27] M. B.; Shariati, M.; Eslami, M. R.; Hassani, B., Meshless analysis of cracked functionally graded materials under thermal shock, Mechanika, 4, 20-27 (2010)
[28] Ferronato, M.; Mazzia, A.; Pini, G. A., Finite Element enrichment technique by the Meshless Local Petrov-Galerkin method, Comput Model Eng Sci, 62, 205-223 (2010) · Zbl 1231.65210
[29] Wen, P. H.; Aliabadi, M. H., Elastic moduli of woven fabric composite by meshless local Petrov-Galerkin (MLPG) method, Comput Model Eng Sci, 61, 133-154 (2010) · Zbl 1231.74448
[30] Wen, P. H.; Aliabadi, M. H.; Liu, Y. W., Meshless method for crack analysis in functionally graded materials with enriched radial base functions, Comput Model Eng Sci, 30, 133-147 (2008) · Zbl 1232.74085
[31] Wen, P. H.; Aliabadi, M. H., An improved meshless collocation method for elastostatic and elastodynamic problems, Commun Numer Methods Eng, 24, 635-651 (2008) · Zbl 1159.74461
[32] Sladek, J.; Sladek, V.; Wen, P. H.; Aliabadi, M. H., Meshless local Petrov-Galerkin (MLPG) method for shear deformable shells analysis, Comput Model Eng Sci, 13, 103-117 (2006) · Zbl 1232.74073
[33] Wen, P. H.; Aliabadi, M. H., A variational approach for evaluation of stress intensity factors using the element free Galerkin method, Int J Solids Struct, 48, 1171-1179 (2011) · Zbl 1236.74093
[34] Wen, P. H.; Aliabadi, M. H., Evaluation of mixed-mode stress intensity factors by the mesh-free Galerkin method: static and dynamic, J Strain Anal Eng Des, 44, 273-286 (2009)
[35] Wen, P. H.; Aliabadi, M. H., Meshless method with enriched radial basis functions for fracture mechanics, SDHM Struct Durability Health Monit, 3, 107-119 (2007)
[36] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least square methods, Math Comp, 37, 141-158 (1981) · Zbl 0469.41005
[37] Golgerg, M. A.; Chen, C. S.; Karur, S. R., Improved multiquadric approximation for partial differential equations, Eng Anal Boundary Elem, 18, 9-17 (1996)
[38] Li, L. Y.; Wen, P. H.; Aliabadi, M. H., eshfree modeling and homogenization of 3D orthogonal woven composites, Compos Sci and Technol, 71, 15, 1777-1788 (2011)
[39] Wen, P. H.; P. H.; Aliabadi, M. H., Damage mechanics analysis of plain woven fabric composite micromechanical model for mesh-free simulations, J Compos Mater, 46, 18, 2239-2253 (2012)
[40] Sladek, V.; Sladek, J., Local integral equations implemented by MLS—approximation and analytical integrations, Eng Anal Boundary Elem, 34, 904-913 (2010) · Zbl 1244.74231
[41] Sladek, V.; Sladek, J.; Zhang, Ch., On increasing computational efficiency of local integral equation method combined with meshless implementations, CMES Comput Model Eng Sci, 63, 243-263 (2010) · Zbl 1231.65221
[42] Wen, P. H.; Aliabadi, M. H., Analysis of functionally graded plates by meshless method: a purely analytical formulation, Eng Anal Boundary Elem, 36, 639-650 (2012) · Zbl 1351.74060
[43] Soares, D.; Sladek, V.; Sladek, J., Modified meshless local Petrov-Galerkin formulations for elastodynamics, Int J Numer Methods Eng, 90, 1508-1528 (2012) · Zbl 1246.74070
[44] Soares, D.; Sladek, V.; Sladek, J.; Zmindak, M.; Medvecky, S., Porous media analysis by modified MLPG formulations, CMC Comput Mater Continua, 27, 101-127 (2012)
[45] Fleming, M.; Chu, Y. A.; Moran, B.; Belytschko, T.; Lu, Y. Y.; Gu, L., Enriched element-free Galerkin methods for crack-tip fields, Int J Numer Methods Eng, 40, 1483-1504 (1997)
[46] Rao, B. N.; Rahman, S. A., Coupled meshless-finite element method for fracture analysis of cracks, Int J Pressure Vessels Piping, 78, 647-657 (2001)
[47] Wen, P. H.; Aliabadi, M. H.; Rooke, D., The influence of elastic waves on dynamic stress intensity factors (two dimensional problem), Arch Appl Mech, 66, 5, 326-335 (1996) · Zbl 0844.73065
[48] Hon, Y. C.; Mao, X. Z., A multiquadric interpolation method for solving initial value problems, J. Sci Comput, 12, 51-55 (1997) · Zbl 0907.65062
[49] Durbin, F., Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, Comput J., 17, 371-376 (1975) · Zbl 0288.65072
[50] Hardy, R. L., Multiquadric equations of topography and other irregular surface, J Geophys Res, 76, 1905-1915 (1971)
[51] Rooke, D. P.; Cartwright, D. J., Compendium of stress intensity factors (1976), Her Majesty’s Stationery Office: Her Majesty’s Stationery Office London
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