Study Of Volumetric Locking And Shear Locking Based On Smoothed Finite Element Method

Smoothed finite element method (S-FEM) was recently proposed by incorporation of the strain smoothing operation of meshless method with the existing finite element techniques. S-FEM can give better results than that of FEM in both displacement and energy because the stiffness of S-FEM is softer than that of FEM. Due to the property of stiffness matrix is softer than original FEM, both the displacement result and stress result are more accuracy. Due to the introduced strain projection technique only shape function itself is involved in the calculation of field gradients and no coordinate transformation is necessary, domain discretization of S-FEM is more flexible. Thus, S-FEM is widely used in piezoelectric structures, viscoelastoplasticity, heat transfer problems, acoustics problems and results show more accuracy than the original FEM. Similarly to FEM, S-FEM also exists two serious drawbacks—volumetric locking and shear locking that are largely limit the applicability of S-FEM. This paper is focus on how to solve volumetric locking and shear locking in S-FEM.S-FEM family mainly includes cell based smoothed finite element method(CS-FEM), node-based finite element method(NS-FEM), edge based finite element method(ES-FEM) and face based finite element method(FS-FEM). This paper first introduces the domain discretization and formulations of these S-FEM models. We catch the characteristics of each model in essence and comparative study the differences of these models. Furtherly, the paper summarizes the general procedure for S-FEM models and then emphases on the two methods of how to construction of shape function.Based on study of volumetric locking and shear locking, the author proposes two schemes to solve volumetric locking and two schemes to solve shear locking in S-FEM. Scheme1that is based on CS-FEM to solve volumetric locking refers to the use of SC=1only on the λ-part that causes the volumetric locking and refers to the use of SC>1on the μ-part of material matrix. Scheme2is based on NS-FEM and ES-FEM, refers to the use of NS-FEM that is free of volumetric locking on the λ-part and refers to the use of ES-FEM on the μ-part of material matrix. Similarly Scheme1to solve volumetric locking, scheme1to solve shear locking is also based on CS-FEM and refers to the use of SC=1only on the shearing-part that causes the shear locking and refers to the use of SC>1on the bending-part of material matrix. Scheme4is combined ES-FEM and discrete shear gap technique in order to solve shear locking and improve the computation accuracy.For checking the validity and effectiveness of above theory and algorithm, the paper implements typical numerical examples research. Choose cantilever beam and infinite plate with a circular hole as the numerical examples of two schemes in solving volumetric locking. Choose square plate subjected to a concentrated load at the center and barrel vault shell subjected to its own weight as the numerical examples of two schemes in solving shear locking. By means of MATLAB coding, we got the numerical results and then compare the numerical results with the analytical results. The research demonstrates that all the schemes are with good performance in convergence, accuracy, stability and adaptability and can solve volumetric locking and shear locking effectively.