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G. R. Liu;  H. Nguyen-Xuan, T. Nguyen-Thoi (2010). A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates. International Journal for Numerical Methods in Engineering, Volume 84, Issue 10, pp. 1222–1256.

Link: http://onlinelibrary.wiley.com/doi/10.1002/nme.2941/abstract


Abstract.
Incorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S-FEM) for solid mechanics problems. In these S-FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models. First, an assumed strain field derived from the Hellinger–Reissner variational principle is shown to be identical to the smoothed strain field used in the S-FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S-FEM models. Some numerical examples are given to verify the theoretical properties established. Copyright © 2010 John Wiley & Sons, Ltd.