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About applying directly the alpha finite element method (αFEM) for solid mechanics using triangular and tetrahedral elements

This novel combination of the FEM and NS-FEM makes the best use of the upper bound property of the NS-FEM and the lower bound property of the standard FEM. This paper concentrates on applying directly the αFEM for solid mechanics to obtain the very accurate solutions with a suitable computational cost by using α = 0.6 for 2D problems and α = 0.7 for 3D problems. | Vietnam Journal of Mechanics, VAST, Vol. 32, No. 4 (2010), pp. 235 – 246 ABOUT APPLYING DIRECTLY THE ALPHA FINITE ELEMENT METHOD (αFEM) FOR SOLID MECHANICS USING TRIANGULAR AND TETRAHEDRAL ELEMENTS Nguyen Thoi Trung1,2 , Nguyen Xuan Hung1,2 1 University of Science, VNU, HCM 2 Ton Duc Thang University, HCM Abstract. An alpha finite element method (αFEM) has been recently proposed to compute nearly exact solution in strain energy for solid mechanics problems using three-node triangular (αFEM-T3) and four-node tetrahedral (αFEM-T4) elements. In the αFEM, a scale factor α ∈ [0, 1] is used to combine the standard fully compatible model of the FEM with a quasi-equilibrium model of the node-based smoothed FEM (NS-FEM). This novel combination of the FEM and NS-FEM makes the best use of the upper bound property of the NS-FEM and the lower bound property of the standard FEM. This paper concentrates on applying directly the αFEM for solid mechanics to obtain the very accurate solutions with a suitable computational cost by using α = 0.6 for 2D problems and α = 0.7 for 3D problems. 1. INTRODUCTION For many decades, the constant finite elements such as three-node triangle and fournode tetrahedron are popular and widely used in practice. The reason is that these elements can be easily formulated and implemented very effectively in the finite element programs using piecewise linear approximation. Further more, most FEM codes for adaptive analyses are based on triangular and tetrahedral elements, due to the simple fact that triangular and tetrahedral meshes can be automatically generated. However, these elements possess significant shortcomings, such as poor accuracy in stress solution, the overly stiff behavior and volumetric locking in the nearly incompressible cases. In the development of new finite element methods, the strain smoothing technique [1] has been applied to the FEM to formulate and develop four smoothed finite element methods (S-FEM) including a cell-based S-FEM