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Consistent pseudo-derivatives in meshless methods. (English) Zbl 0894.73156
Summary: A meshless Petrov-Galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of Shepard functions. A key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo-derivatives for Petrov-Galerkin method which pass the patch test. Numerical results show that the resulting method is substantially more accurate than the Galerkin method with Shepard approximants and exceeds the rate of convergence of linear finite elements.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Melenk, J.M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099
[2] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075
[3] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[4] Duarte, C.; Oden, J., Hp—clouds—an hp meshless method, Numerical methods for partial differential equations, 12, 673-705, (1996) · Zbl 0869.65069
[5] Hughes, T., The finite element method, (1987), Prentice Hall Englewood Cliffs, NJ
[6] Krongauz, Y.; Belytschko, T., Enforcement of essential boundary conditions in meshless approximations using finite elements, Comput. methods appl. mech. engrg., 131, 133-145, (1966) · Zbl 0881.65098
[7] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[8] Liu, W.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1680, (1995) · Zbl 0840.73078
[9] Liu, W.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[10] Liu, W.; Li, S.; Belytschko, T., Moving least square reproducing kernel methods: (i) methodology and convergence, Comput. methods appl. mech. engrg., 139, 159-193, (1996)
[11] Monaghan, J.J., Why particle methods work, SIAM J. scient. stat. comput., 3, 4, 422, (1982) · Zbl 0498.76010
[12] 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
[13] Shepard, D., A two dimensional function for irregularly space data, ()
[14] Strang, W.G.; Fix, G.J., An analysis of finite element method, (1973), Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116
[15] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1987), McGraw-Hill New York · Zbl 0266.73008
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