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On nonlinear analysis by the multipoint meshless FDM. (English) Zbl 1403.65112
Summary: The main objective of this paper is to present an attempt of an application of the recently developed higher order multipoint meshless FDM in the analysis of nonlinear problems. The multipoint approach provides a higher order approximation and improves the precision of the solution. In addition to improved solution quality, the essential feature of the multipoint approach is its potentially wide ranging applicability. This is possible, because in both the multipoint and standard meshless FDM, the difference formulas are generated at once for the full set of derivatives. Using them, we may easily compose any required FD operator. It is worth mentioning that all derivative operators depend on the domain discretization rather than on the specific problem being analysed. Therefore, the solution of a wide class of problems including nonlinear ones, may be obtained with this method. The numerical algorithm of the multipoint method for nonlinear analysis is presented in this paper. Results of selected engineering benchmark problems – deflection of the ideal membrane and analysis of large deflection of plates using the von Karman theory – are considered.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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[1] Zienkiewicz, O. C.; Taylor, R. L., Finite element method its basis and fundamentals, (2005), Elsevier · Zbl 1084.74001
[2] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis: toward integration of CAD and FEA, (2009), Wiley Chichester, UK · Zbl 1378.65009
[3] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math Comput Simul, 79, 763-813, (2008) · Zbl 1152.74055
[4] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its applications in applied mechanics, Comput Struct, 11, 83-95, (1980) · Zbl 0427.73077
[5] Orkisz, J., Finite difference method (part III), (Kleiber, M., Handbook of computational solid mechanics, (1998), Springer-Verlag Berlin), 336-432
[6] Jaworska, I.; Orkisz, J., Higher order multi point method - from Collatz to meshless FDM, Eng Anal Bound Elem, 50, 341-351, (2015) · Zbl 1403.74279
[7] Jaworska, I., On the ill-conditioning in the new higher order multipoint method, Comput Math Appl, 66, 3, 238-249, (2013) · Zbl 1347.65127
[8] Collatz, L., Numerische behandlung von differential-gleichungen, (1955), Springer-Verlag Berlin-Heidelberg
[9] Atluri, S. N., The meshless method (MLPG) for domain & bie discretizations, (2004), Tech Science Press · Zbl 1105.65107
[10] Jaworska, I.; Orkisz, J., On the multipoint meshless FD method using the local Petrov-Galerkin formulation, (Proceedings of the European congress on computational methods in applied sciences and engineering, ECCOMAS, (2012)), 6582-6591, 2012
[11] Jaworska, I.; Orkisz, J., On some aspects of a posteriori error estimation in the multipoint meshless FDM, 2014, 2737-2743, (2014), WCCM Barcelona
[12] Liszka, T., Finite difference at arbitrary irregular meshes and advances of its use in problems of mechanics, (1977), Cracow University of Technology Cracow, Poland, Ph.D. thesis, in Polish
[13] Stanuszek, M., FEM analysis of large deformations of membrane shells with wrinkling, Int J Finite Elem Anal Des, 39, 599-618, (2003)
[14] Milewski, S.; Orkisz, J., In search of optimal acceleration approach to iterative solution methods of simultaneous algebraic equations, Comput Math Appl, 68, 3, 101-117, (2014) · Zbl 1369.65065
[15] Byklum, E.; Amdahl, J., A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling, Thin Walled Struct, 40, 925-953, (2002)
[16] Timoshenko, S. P.; Gere, J. M., Theory of plates and shells, (1959), McGraw-Hill New York
[17] Liszka, T.; Orkisz, J., Solution of nonlinear problems of mechanics by the finite difference method at arbitrary meshes (in Polish), Mech Komput, 5, 117-130, (1983)
[18] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, Comput Methods Appl Mech Eng, 142, 1-88, (1997) · Zbl 0895.76040
[19] Jaworska, I.; Orkisz, J., Estimation of a posteriori computational error by the higher order multipoint meshless, (2018), FDM, (forthcoming)
[20] Ugural, A. C.; Fenster, S. K., Advanced strength and applied elasticity, (2003), Prentice Hall · Zbl 0486.73002
[21] Tworzydlo, W., Analysis of large deformations of membrane shells by the generalized finite difference method, Comput Struct, 27, 1, 39-59, (1987) · Zbl 0624.73101
[22] Stanuszek, M., Analiza dużych deformacji wiotkich układów cięgnowo-powłokowych metodą elementów skończonych, (1988), Cracow University of Technology Cracow, Poland, Ph.D. thesis, in Polish
[23] Chen, J.-S.; Hillman, M.; Chi, S.-W., Meshfree methods: progress made after 20 years, J Eng Mech, 143, 4, (2017)
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