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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 177

An Equilibrium Model in the Element Free Galerkin Method

B.Q. Tinh1 and H. Nguyen-Dang2

1Department of Computational Mechanics, University of Natural Science, Ho Chi Minh City, Vietnam
2LTAS, Department of Fracture Mechanics, Institute of Mechanics and Civil Engineering, Liège, Belgium

Full Bibliographic Reference for this paper
B.Q. Tinh, H. Nguyen-Dang, "An Equilibrium Model in the Element Free Galerkin Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 177, 2006. doi:10.4203/ccp.83.177
Keywords: meshless method, EFG, equilibrium model.

The equilibrium approach with using the element free Galerkin method [1] is presented in this paper. Together, a scaling parameter study is carried out so as to better understand the effects of the interpolation domain size on the quality of the results.

We used the Lagrange multipliers to impose the essential boundary conditions and used the moving least squares method to approximate the stress fields from an Airy stress function. These comparison show that the accuracy and agreement of the two models. Nevertheless, our aim was not to replace the classical displacement model with the equilibrium model but rather to obtain a global error estimator based upon the combination of both models.

Furthermore, in the equilibrium model, for most of the values of took from 2.5 to 4.0 gave good results. The size of the support should be sufficiently large so that the moment matrix is regular and well conditioned and that the spatial distribution of neighbours is fairly even. On the other hand, selecting domains of influence that are too large leads to a great deal of computational expense in forming the approximations as well as assembling the stiffness matrix. Support sizes that are too large also detract from the local character of the approximation, for problems involving sharp gradients; some loss of accuracy is typically noted as the effect of the gradient is smeared. This model seems to need more nodes inside the support domain than the displacement model. However, as in Duflot et al. [2], the advantage is that there is only one degree of freedom associated with each node while in the traditional method; the number of degree of freedom for each node is equal to the number of space dimensions.

The method was illustrated in two-dimensional elasticity but it can be extended to other complex problems, to see Duflot et al. [2,3] in details. As the first step we will apply for three-dimension problem, thin plate problems and fracture mechanics problems, where the meshless method has proved to be particularly efficient.

T. Belytschko, Y.Y. Lu and L. Gu. "Element-free Galerkin method", International Journal for Numerical Methods in Engineering; 37:229-256, 1994. doi:10.1002/nme.1620370205
M. Duflot and N.D. Hung. "Dual analysis by a meshless method", International Journal for Numerical Methods in Engineering, 2002.
M. Duflot and N.D. Hung. "Global error estimation with displacement and equilibrium meshless method", Computational mechanics WCCM VI in conjunction with APCOM'04, Sept. 5-10, Beijing, China, 2004.

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