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G. R. Liu,  H. Nguyen-Xuan, T. Nguyen-Thoi (2010). A variationally consistent αFEM (VCαFEM) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements. International Journal for Numerical Methods in Engineering.

Link: http://onlinelibrary.wiley.com/doi/10.1002/nme.2977/abstract #2010

Abstract. A variationally consistent alpha finite element method (VC αFEM) for quadrilateral isoparametric elements is presented by constructing an assumed strain field in which the gradient of the compatible strain field is scaled with a free parameter α. The assumed strain field satisfies the orthogonal condition and the Hellinger–Reissner variational principle is used to establish the discretized system of equations. It is shown that the strain energy is a second-order continuous function of α, and the VC αFEM can produce both lower and upper bounds to the exact solution in the strain energy for all elasticity problems by choosing a proper α∈[0, αupper]. Based on this bound property, an exact-α approach has been devised to give an ultra-accurate solution that is very close to the exact one in the strain energy. Furthermore, the exact-α approach also works well for volumetric locking problems, by simply replacing the strain gradient matrix by a stabilization matrix. In addition, a regularization-α approach has also been suggested to overcome possible hourglass instability. Intensive numerical studies have been conducted to confirm the properties of the present VC αFEM, and a very good performance has been found in comparing to a large number of existing FEM models. Copyright © 2010 John Wiley & Sons, Ltd.