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A node-based smoothed finite element method (NS-Fem) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes. (English) Zbl 1231.74432
Summary: 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.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
XFEM
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##### References:
 [1] Chen, J.S.; Wu, C.T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin meshfree method, Int. J. numer. methods engrg., 50, 435-466, (2001) · Zbl 1011.74081 [2] Liu, G.R., Meshfree methods: moving beyond the finite element method, (2009), CRC Press Boca Raton, FL [3] Nguyen, V.P.; Rabczuk, T.; Stephane, Bordas; Duflot, M., Meshless methods: review and key computer implementation aspects, Math. comput. simul., 79, 763-813, (2008) · Zbl 1152.74055 [4] Liu, G.R.; Dai, K.Y.; Nguyen-Thoi, T., A smoothed finite element method for mechanics problems, Comput. mech., 39, 859-877, (2007) · Zbl 1169.74047 [5] Liu, G.R.; Nguyen-Thoi, T.; Dai, K.Y.; Lam, K.Y., Theoretical aspects of the smoothed finite element method (SFEM), Int. J. numer. methods engrg., 71, 902-930, (2007) · Zbl 1194.74432 [6] Liu, G.R.; Nguyen-Thoi, T.; Nguyen-Xuan, H.; Dai, K.Y.; Lam, K.Y., On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM) (letter to editor), Int. J. numer. methods engrg., 77, 1863-1869, (2009) · Zbl 1181.74137 [7] Zhang, H.H.; Liu, S.J.; Li, L.X., On the smoothed finite element method (short communication), Int. J. numer. methods engrg., 76, 1285-1295, (2008) · Zbl 1195.74210 [8] Stéphane, P.; Bordas, A.; Natarajan, Sundararajan, On the approximation in the smoothed finite element method (SFEM) (letter to editor), Int. J. numer. methods engrg., 81, 660-670, (2010) · Zbl 1183.74261 [9] Nguyen-Xuan, Hung; Bordas, Stéphane; Nguyen-Dang, Hung, Smooth finite element methods: convergence, accuracy and properties, Int. J. numer. methods engrg., 74, 175-208, (2008) · Zbl 1159.74435 [10] Dai, K.Y.; Liu, G.R.; Nguyen-Thoi, T., An n-sided polygonal smoothed finite element method (nsfem) for solid mechanics, Finite elem. anal. des., 43, 847-860, (2007) [11] Dai, K.Y.; Liu, G.R., Free and forced analysis using the smoothed finite element method (SFEM), J. sound vib., 301, 803-820, (2007) [12] Nguyen-Thoi, T.; Liu, G.R.; Dai, K.Y.; Lam, K.Y., Selective smoothed finite element method, Tsinghua sci. technol., 12, 5, 497-508, (2007) [13] Nguyen-Xuan, H.; Bordas, S.; Nguyen-Dang, H., Addressing volumetric locking and instabilities by selective integration in smoothed finite elements, Int. J. numer. methods biomed. engrg., 25, 19-34, (2008) · Zbl 1169.74044 [14] Nguyen-Xuan, H.; Nguyen-Thoi, T., A stabilized smoothed finite element method for free vibration analysis of mindlin – reissner plates, Int. J. numer. methods biomed. engrg., 25, 882-906, (2009) · Zbl 1172.74047 [15] Cui, X.Y.; Liu, G.R.; Li, G.Y.; Zhao, X.; Nguyen-Thoi, T.; Sun, G.Y., A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells, CMES - comput. model. engrg. sci., 28, 2, 109-125, (2008) · Zbl 1232.74099 [16] Nguyen-Thanh, N.; Rabczuk, T.; Nguyen-Xuan, i H.; Bordas, S., A smoothed finite element method for shell analysis, Comput. methods appl. mech. engrg., 198, 165-177, (2008) · Zbl 1194.74453 [17] Nguyen-Van, H.; Mai-Duy, N.; Tran-Cong, T., A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures, CMES - comput. model. engrg. sci., 23, 3, 209-222, (2008) · Zbl 1232.74108 [18] Nguyen-Xuan, H.; Rabczuk, T.; Bordas, S.; Debongnie, J.F., A smoothed finite element method for plate analysis, Comput. methods appl. mech. engrg., 197, 1184-1203, (2008) · Zbl 1159.74434 [19] Bordas, S.; Rabczuk, T.; Nguyen-Xuan, H.; Nguyen Vinh, P.; Natarajan, S.; Bog, T.; Do Minh, Q.; Nguyen Vinh, H., Strain smoothing in FEM and XFEM, Comput. struct., (2010) [20] Liu, G.R.; Nguyen-Thoi, T.; Nguyen-Xuan, H.; Lam, K.Y., A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM), Comput. struct., 87, 14-26, (2009) [21] Nguyen-Thoi, T.; Liu, G.R.; Nguyen-Xuan, H., Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems, Int. J. comput. methods, 6, 4, 633-666, (2009) · Zbl 1267.74115 [22] Nguyen-Thoi, T.; Liu, G.R.; Nguyen-Xuan, H.; Nguyen-Tran, C., Adaptive analysis using the node-based smoothed finite element method (NS-FEM), Int. J. numer. methods biomed. engrg., (2010) · Zbl 1370.74144 [23] Zhang, Z.Q.; Liu, G.R., Temporal stabilization of the node-based smoothed finite element method (NS-FEM) and solution bound of linear elastostatics and vibration problems, Comput. mech., 46, 229-246, (2010) · Zbl 1398.74431 [24] H. Nguyen-Xuan, T. Rabczuk, T. Nguyen-Thoi, T.N. Tran, N. Nguyen-Thanh, A node-based smoothed finite element method (NS-FEM) using linear triangular elements for primal – dual limit and shakedown analyses of 2D structures, Comput. Methods Appl. Mech. Engrg, submitted for publication. · Zbl 1242.74142 [25] Liu, G.R.; Nguyen-Thoi, T.; Lam, K.Y., An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, J. sound vib., 320, 1100-1130, (2009) [26] Nguyen-Xuan, H.; Liu, G.R.; Nguyen-Thoi, T.; Nguyen-Tran, C., An edge – based smoothed finite element method (ES-FEM) for analysis of two – dimensional piezoelectric structures, Smart mater. struct., 18, 065015, (2009), 12pp [27] Nguyen-Thoi, T.; Liu, G.R.; Vu-Do, H.C.; Nguyen-Xuan, H., An edge-based smoothed finite element method (ES-FEM) for visco-elastoplastic analyses in 2D solids using triangular mesh, Comput. mech., 45, 23-44, (2009) · Zbl 1398.74382 [28] Nguyen-Xuan, H.; Liu, G.R.; Thai-Hoang, C.; Nguyen-Thoi, T., An edge-based smoothed finite element method with stabilized discrete shear gap technique for analysis of reissner – mindlin plates, Comput. methods appl. mech. engrg., 199, 471-489, (2009) · Zbl 1227.74083 [29] Ngoc, Thanh Tran; Liu, G.R.; Nguyen-Xuan, H.; Nguyen-Thoi, T., An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, Int. J. numer. methods engrg., 82, 917-938, (2010) · Zbl 1188.74073 [30] Chen, L.; Liu, G.R.; Nourbakhsh-Nia, N.; Zeng, K., A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks, Comput. mech., (2009) · Zbl 1398.74316 [31] Nguyen-Thoi, T.; Liu, G.R.; Lam, K.Y.; Zhang, G.Y., A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, Int. J. numer. methods engrg., 78, 324-353, (2009) · Zbl 1183.74299 [32] Nguyen-Thoi, T.; Liu, G.R.; Vu-Do, H.C.; Nguyen-Xuan, H., A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh, Comput. methods appl. mech. engrg., 198, 3479-3498, (2009) · Zbl 1230.74193 [33] Liu, G.R.; Nguyen-Thoi, T.; Lam, K.Y., A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements, Comput. methods appl. mech. engrg., 197, 3883-3897, (2008) · Zbl 1194.74433 [34] Carstensen, C.; Klose, R., Elastoviscoplastic finite element analysis in 100 lines of Matlab, J. numer. math., 10, 157-192, (2002) · Zbl 1099.74544 [35] Han, W.; Reddy, B.D., Computational plasticity: the variational basis and numerical analysis, Comput. mech. adv., 2, 283-400, (1995) · Zbl 0847.73078 [36] Carstensen, C.; Funken, S.A., Averaging technique for FE-a posteriori error control in elasticity. part 1: conforming FEM, Comput. methods appl. mech. engrg., 190, 2483-2498, (2001) · Zbl 0981.74063 [37] Liu, G.L.; Trung, Nguyen Thoi, Smoothed finite element methods, (2010), CRC Press Boca Raton, Florida
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