Virtual boundary meshless least square collocation method for calculation of 2D multi-domain elastic problems. (English) Zbl 1351.74165
Summary: A virtual boundary meshless least square collocation method is developed for calculation of two-dimensional multi-domain elastic problems in this paper. This method is different from the conventional virtual boundary element method (VBEM) since it incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. This method can be used to analyze multi-domain composite structures with each subdomain having different materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes. The results show that this method leads to faster convergence and higher accuracy in comparison with the other methods considered in this study.
MSC:
| 74S25 | Spectral and related methods applied to problems in solid mechanics |
| 74B05 | Classical linear elasticity |
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Software:
Mfree2DReferences:
| [1] | Brebbia, C. A.; Walker, S., Boundary element techniques in engineering (1980), Pentech Press: Pentech Press London · Zbl 0444.73065 |
| [2] | Xu, Q.; Sun, H. C., Virtual boundary element least square method for solving three dimensional problems of thick shell, J Dalian Univ Technol, 36, 413-418 (1996), in Chinese · Zbl 0929.74112 |
| [3] | Xu, Q.; Sun, H. C., Virtual boundary element least-square collocation method, Chin J Comput Mech, 14, 166-173 (1997), in Chinese |
| [4] | Sun, H. C.; Zhang, L. Z.; Xu, Q.; Zhang, Y. M., Nonsingularity boundary element methods (1999), Dalian University of Technology Press: Dalian University of Technology Press Dalian, in Chinese |
| [5] | Xu, Q., Three-dimension virtual boundary element method for solving the problems of thin shells, Chin J Appl Mech, 17, 111-115 (2000), in Chinese |
| [6] | Xu, Q.; Sun, H. C., Unified way for dealing with three-dimensional problems of solid elasticity, Appl Math Mech, 22, 1357-1367 (2001) · Zbl 1143.74385 |
| [7] | Mukherjee, S.; Mukherjee, Y. X., The boundary node method for potential problems, Int J Numer Meth Eng, 40, 797-815 (1997) · Zbl 0885.65124 |
| [8] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math Comput, 37, 141-158 (1981) · Zbl 0469.41005 |
| [9] | Gu, Y. T.; Liu, G. R., A boundary point interpolation method for stress analysis of solids, Comput Mech, 28, 47-54 (2002) · Zbl 1115.74380 |
| [10] | Gu, Y. T.; Liu, G. R., A boundary radial point interpolation method (BRPIM) for 2D meshless structure analysis, Struct Eng Mech, 15, 535-550 (2003) |
| [11] | Liu G.R., A point assembly method for stress analysis for solid. In: Shim VPV et al. (editors). Impact response of materials & structures. 1999; Oxford, p. 475-80.; Liu G.R., A point assembly method for stress analysis for solid. In: Shim VPV et al. (editors). Impact response of materials & structures. 1999; Oxford, p. 475-80. |
| [12] | Liu, G. R.; Gu, Y. T., A point interpolation method for two-dimensional solids, Int. J. Numer. Meth. Eng., 50, 937-951 (2001) · Zbl 1050.74057 |
| [13] | Wang J.G., Liu G.R.. Radial point interpolation method for elastoplastic problems. In: Proc. of the 1st int. conf. on structural stability and dynamics, Dec. 7-9, 2000, Taipei, Taiwan, p. 703-8.; Wang J.G., Liu G.R.. Radial point interpolation method for elastoplastic problems. In: Proc. of the 1st int. conf. on structural stability and dynamics, Dec. 7-9, 2000, Taipei, Taiwan, p. 703-8. |
| [14] | Liu, G. R., Mesh free methods: moving beyond the finite element method (2002), CRC press: CRC press Boca Raton, USA |
| [15] | Gu Y.T., Liu G.R.. Using radial function basis in a boundary-type meshless method, boundary point interpolation method (BPIM)”. In: Int. conf. on sci. & engr. comput., March 19-23, 2001, Beijing. p .68.; Gu Y.T., Liu G.R.. Using radial function basis in a boundary-type meshless method, boundary point interpolation method (BPIM)”. In: Int. conf. on sci. & engr. comput., March 19-23, 2001, Beijing. p .68. |
| [16] | Gu Y.T., Liu G.R.. A boundary point interpolation method (BPIM) using radial function basis”. In: First MIT conference on computational fluid and solid mechanics, June 2001, MIT, p. 1590-2.; Gu Y.T., Liu G.R.. A boundary point interpolation method (BPIM) using radial function basis”. In: First MIT conference on computational fluid and solid mechanics, June 2001, MIT, p. 1590-2. |
| [17] | Liu, G. R.; Gu, Y. T., Boundary meshfree methods based on the boundary point interpolation methods, Eng Anal Bound Elem, 28, 475-487 (2004) · Zbl 1130.74466 |
| [18] | Liu, G. R.; Gu, Y. T., An introduction to meshfree methods and their programming (2005), Springer press: Springer press Dordrecht |
| [19] | Wu, Z. M., Compactly supported positive definite radial function, Adv Comput Math, 4, 283-292 (1995) · Zbl 0837.41016 |
| [20] | Kita, E.; Kamiya, N., Trefftz method: an overview, Adv Eng Software, 24, 3-12 (1995) · Zbl 0984.65502 |
| [21] | Wang, J.; Mogilevskaya, S. G.; Crouch, SL, Benchmarks results for the problem of interaction between a crack and a circular inclusion, J Appl Mech, 70, 619-621 (2003) · Zbl 1110.74740 |
| [22] | Mogilevskaya, S. G.; Crouch, S. L., A Galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers, Int J Solids Struct, 41, 1285-1311 (2004) · Zbl 1106.74420 |
| [23] | Wang, J.; Mogilevskaya, S. G.; Crouch, S. L., An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials, Int J Solids Struct, 42, 4588-4612 (2005) · Zbl 1119.74536 |
| [24] | Budiansky, Y., On the elastic moduli of heterogeneous materials, J Mech Phys Solids, 13, 223-227 (1965) |
| [25] | Hill, R., A self-consistent mechanics of composite materials, J Mech Phys Solids, 13, 213-222 (1965) |
| [26] | Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with mis-fitting inclusions, Acta Metall, 21, 571-583 (1973) |
| [27] | Weng, G. J., Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions, Int J Eng Sci, 22, 845-856 (1984) · Zbl 0556.73074 |
| [28] | Benveniste, Y., A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech Mater, 6, 147-157 (1987) |
| [29] | Huang, Y.; Hu, K. X.; Wei, X.; Chandra, A., A generalized self-consistent mechanics method for composite materials with multiphase inclusions, J Mech Phys Solids, 42, 491-504 (1994) · Zbl 0794.73039 |
| [30] | Isida, M.; Igawa, H., Analysis of zigzag array of circular holes in an infinite solid under uniaxial tension, Int J Solids Struct, 27, 849-864 (1991) |
| [31] | Day, A. R.; Snyder, K. A.; Garboczi, E. J.; Thorpe, M. F., The elastic moduli of a sheet containing circular holes, J Mech Phys Solids, 40, 1031-1051 (1992) |
| [32] | Yao Z.H., Kong F.Z., Wang P.B., Simulation of 2D elastic solids with randomly distributed inclusions by boundary element method. In: Proc WCCM V Vienna, 2002.; Yao Z.H., Kong F.Z., Wang P.B., Simulation of 2D elastic solids with randomly distributed inclusions by boundary element method. In: Proc WCCM V Vienna, 2002. |
| [33] | Yao, Z. H.; Kong, F. Z.; Wang, H. T.; Wang, P. B., 2D Simulation of composite materials using BEM, Eng Anal Bound Elem, 28, 927-935 (2004) · Zbl 1130.74476 |
| [34] | Yao, Z. H.; Wang, H. T., Boundary element methods (2010), Higher Education Press: Higher Education Press Beijing, in Chinese |
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