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Gavete, L.; Gavete, M. L.; Alonso, B.; Martín, A. J.

A posteriori error approximation in EFG method. (English) Zbl 1047.74081

Int. J. Numer. Methods Eng. 58, No. 15, 2239-2263 (2003).
Summary: Recently, considerable effort has been devoted to the development of the so-called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the a posteriori error should be approximated.
An a posteriori error approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error.
The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors.

Cited in 13 Documents

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
65N15 Error bounds for boundary value problems involving PDEs

Keywords:

element free Galerkin method; error approximation; meshless; moving least squares
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\textit{L. Gavete} et al., Int. J. Numer. Methods Eng. 58, No. 15, 2239--2263 (2003; Zbl 1047.74081)
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References:

[1] Monaghan, Why particle methods work, SIAM Journal on Scientific and Statistical Computing 3 pp 422– (1982) · Zbl 0498.76010
[2] Monaghan, An introduction to SPH, Computer Physics Communications 48 pp 89– (1988) · Zbl 0673.76089
[3] Perrone, A general finite difference method for arbitrary meshes, Computers and Structures 5 pp 45– (1975)
[4] Liszka, The finite difference method at arbitrary irregular grids and its application in applied mechanics, Computers and Structures 11 pp 83– (1980) · Zbl 0427.73077
[5] Orkisz, Computational Mechanics (1998)
[6] Lancaster, Surfaces generated by moving least squares methods, Mathematics of Computation 37 pp 141– (1981) · Zbl 0469.41005
[7] Nayroles, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068
[8] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077
[9] Lu, A new implementation of the element-free Galerkin method, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994) · Zbl 0847.73064
[10] Liu, Reproducing kernel particle methods for structural dynamics, International Journal for Numerical Methods in Engineering 38 pp 1655– (1995) · Zbl 0840.73078
[11] Duarte, H-P Cloud - An h-p meshless method, Numerical Methods for Partial Differential Equations 12 pp 673– (1996) · Zbl 0869.65069
[12] Babuska, The partition of unity method, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) · Zbl 0949.65117
[13] Oñate, A finite point method in computational mechanics. Applications to convective transport and fluid flow, International Journal for Numerical Methods in Engineering 39 pp 3839– (1996) · Zbl 0884.76068
[14] Gavete, Implementation of essential boundary conditions in a meshless method, Communications in Numerical Methods in Engineering 16 pp 409– (2000) · Zbl 0956.65104
[15] Krongauz, A Petrov-Galerkin diffuse element method (PG DEM) and its comparison to EFG, Computational Mechanics 19 pp 327– (1997) · Zbl 1031.74527
[16] Mukherjee, On boundary conditions in the element-free Galerkin method, Computational Mechanics 19 pp 264– (1997) · Zbl 0884.65105
[17] Duarte, TICAM Report 96-07 (1996)
[18] Laouar, Computational Mechanics (1998)
[19] Gavete, II ECCOMAS Conference on Numerical Methods in Engineering, ECCOMAS96 pp 499– (1996)
[20] Gavete, An error indicator for the element free Galerkin method, European Journal of Mechanics - A/Solids 20 pp 327– (2001) · Zbl 1047.74080
[21] Gavete, A procedure for approximation of the error in EFG method, International Journal for Numerical Methods in Engineering 53 pp 677– (2002) · Zbl 1112.74564
[22] Chung, An error estimate in the EFG method, Computational Mechanics 21 pp 91– (1998) · Zbl 0910.73060
[23] Liu, Reproducing least square kernel Galerkin method. (i) Methodology and convergence, Computer Methods in Applied Mechanics and Engineering 143 pp 422– (1997) · Zbl 0883.65088
[24] Timoshenko, Theory of Elasticity (1987)
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