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T. Nguyen-Thoia, b, , H.C. Vu-Doa, T. Rabczukc and H. Nguyen-Xuana, b (2010). A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes, Computer Methods in Applied Mechanics and Engineering, Volume 199, Issues 45-48, Pages 3005-3027.

a Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, Vietnam National University – HCM, 227 Nguyen Van Cu, District 5, Hochiminh City, Viet Nam
b Division of Computational Mechanics, Faculty of Civil Engineering, Ton Duc Thang University, 98 Ngo Tat To, Binh Thanh District, Ho Chi Minh City, Vietnam
c Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, 1599423 Weimar, Germany

Abstract: A node-based smoothed finite element method (NS-FEM) was recently proposed for the solid mechanics problems. In the NS-FEM, the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh. In this paper, the NS-FEM is further extended to more complicated visco-elastoplastic analyses of 2D and 3D solids using triangular and tetrahedral meshes, respectively. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic hardening and linear kinematic hardening. A dual formulation for the NS-FEM with displacements and stresses as the main variables is performed. The von-Mises yield function and the Prandtl–Reuss flow rule are used. In the numerical procedure, however, the stress variables are eliminated and the problem becomes only displacement-dependent. The numerical results show that the NS-FEM has higher computational cost than the FEM. However the NS-FEM is much more accurate than the FEM, and hence the NS-FEM is more efficient than the FEM. It is also observed from the numerical results that the NS-FEM possesses the upper bound property which is very meaningful for the visco-elastoplastic analyses which almost have not got the analytical solutions. This suggests that we can use two models, NS-FEM and FEM, to bound the solution, and can even estimate the global relative error of numerical solutions.

Keywords: Numerical methods; Meshfree methods; Node-based smoothed finite element method (NS-FEM); Finite element method (FEM); Upper bound; Visco-elastoplastic analyses