×

zbMATH — the first resource for mathematics

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)
Software:
XFEM
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.