## An edge-based smoothed finite element method for primal-dual shakedown analysis of structures.(English)Zbl 1188.74073

Summary: An edge-based smoothed finite element method (ES-FEM) using three-node linear triangular elements was recently proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static, free and forced vibration analyses of solids. In this paper, ES-FEM is further extended for limit and shakedown analyses of structures. A primal-dual algorithm based upon the von Mises yield criterion and a non-linear optimization procedure is used to compute both the upper and lower bounds of the plastic collapse limit and the shakedown limit. In the ES-FEM, compatible strains are smoothed over the smoothing domains associated with edges of elements. Using constant smoothing function, only one Gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation. Three benchmark problems are presented to show the stability and accuracy of solutions obtained by the present method.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)

XFEM
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### References:

 [1] Bree, Elasticâplastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with applications to fast-nuclear-reactor fuel elements, Journal of Strain Analysis 2 pp 226– (1967) [2] Tran, Probabilistic limit and shakedown analysis of thin shells, Structural Safety 31 (1) pp 1– (2009) [3] Tran, Upper bound limit and shakedown analysis of shells using the exact Ilyushin yield surface, Computers and Structures 86 (17â18) pp 1683– (2008) [4] Heitzer, FEM-computation of load carrying capacity of highly loaded passive components by direct methods, Nuclear Engineering and Design 193 (3) pp 349– (1999) [5] Krabbenhoft, A general non-linear optimization algorithm for lower bound limit analysis, International Journal for Numerical Methods in Engineering 56 pp 165– (2003) · Zbl 1116.74404 [6] Andersen, An efficient primalâdual interior-point method for minimizing a sum of Euclidean norms, SIAM Journal on Scientific Computing 22 pp 243– (2000) [7] Vu, A primalâdual algorithm for shakedown analysis of structure, Computer Methods in Applied Mechanics and Engineering 193 pp 4663– (2004) [8] Vu, Analysis of pressure equipment by application of the primalâdual theory of shakedown, Communications in Numerical Methods in Engineering 23 (3) pp 213– (2007) [9] Garcea, Finite element shakedown analysis of two-dimensional structures, International Journal for Numerical Methods in Engineering 63 pp 1174– (2005) · Zbl 1084.74052 [10] Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081 [11] Rabczuk, Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering 193 (12â14) pp 1035– (2004) · Zbl 1060.74672 [12] Rabczuk, Cracking particles: a simplified meshfree method for arbitrary evolving cracks, International Journal for Numerical Methods in Engineering 61 (13) pp 2316– (2004) · Zbl 1075.74703 [13] Rabczuk, A meshfree thin shell for arbitrary evolving cracks based on an external enrichment, Computer Modeling in Engineering and Sciences 16 (2) pp 115– (2006) [14] Rabczuk, A meshfree thin shell method for non-linear dynamic fracture, International Journal for Numerical Methods in Engineering 72 (5) pp 524– (2007) · Zbl 1194.74537 [15] Liu, A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72 pp 1524– (2007) · Zbl 1194.74543 [16] Liu, A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72 pp 1524– (2007) · Zbl 1194.74543 [17] Liu, A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods 3 (4) pp 401– (2006) · Zbl 1198.74120 [18] Liu, On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM) (Letter to Editor), International Journal for Numerical Methods in Engineering 77 pp 1863– (2009) · Zbl 1181.74137 [19] Hung, Smooth finite element methods: convergence, accuracy and properties, International Journal for Numerical Methods in Engineering 74 pp 175– (2008) · Zbl 1159.74435 [20] Liu, A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems, Computers and Structures 87 pp 14– (2009) [21] Liu, A smoothed finite element for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047 [22] Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 pp 902– (2007) · Zbl 1194.74432 [23] Dai, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elements in Analysis and Design 43 pp 847– (2007) [24] Dai, Free sand forced vibration analysis using the smoothed finite element method (SFEM), Journal of Sound and Vibration 301 pp 803– (2007) [25] Nguyen-Thoi, Selective smoothed finite element method, Tsinghua Science and Technology 12 (5) pp 497– (2007) [26] Nguyen-Xuan, Addressing volumetric locking and instabilities by selective integration in smoothed finite elements, Communications in Numerical Methods and Engineering 25 pp 19– (2008) [27] Nguyen-Xuan, A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering 197 pp 1184– (2008) · Zbl 1159.74434 [28] Nguyen-Thanh, A smoothed finite element method for shell analysis, Computer Methods in Applied Mechanics and Engineering 198 pp 165– (2008) · Zbl 1194.74453 [29] Nguyen-Xuan, A stabilized smoothed finite element method for free vibration analysis of MindlinâReissner plates, Communications in Numerical Methods in Engineering 25 pp 882– (2009) · Zbl 1172.74047 [30] Cui, A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells, CMESâComputer Modeling in Engineering and Sciences 28 (2) pp 109– (2008) · Zbl 1232.74099 [31] Bordas, Strain smoothing in FEM and XFEM, Computers and Structures (2008) [32] Nguyen-Thoi, Adaptive analysis using the node-based smoothed finite element method (NS-FEM), Communications in Numerical Methods in Engineering (2009) · Zbl 1370.74144 [33] Liu, A novel alpha finite element method (Î{$$\pm$$} FEM) for exact solution to mechanics problems using triangular and tetrahedral elements, Computer Methods in Applied Mechanics and Engineering 197 pp 3883– (2008) · Zbl 1194.74433 [34] Liu, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, Journal of Sound and Vibration 320 pp 1100– (2009) [35] Nguyen-Xuan, An edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric structures, Smart Materials and Structures (2009) [36] Nguyen-Thoi, A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, International Journal for Numerical Methods in Engineering 78 (3) pp 324– (2009) · Zbl 1183.74299 [37] Nguyen-Thoi, A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh, Computer Methods in Applied Mechanics and Engineering (2009) · Zbl 1230.74193 [38] Arnold, Mixed finite element methods for elliptic problems, Computer Methods in Applied Mechanics and Engineering 82 pp 281– (1990) · Zbl 0729.73198 [39] Mijuca, On the efficiency of the primal-mixed finite element scheme, Advances in Computational Structured Mechanics; Civil-Comp Press pp 61– (1998) [40] Rong, Generalized mixed variational principles and solutions for ill-conditioned problems in computational mechanics: Part I. Volumetric locking, Computer Methods in Applied Mechanics and Engineering 191 pp 407– (2001) · Zbl 1054.74737 [41] Rong, Generalized mixed variational principles and solutions for ill-conditioned problems in computational mechanics: Part II. Shear locking, Computer Methods in Applied Mechanics and Engineering 192 pp 4981– (2003) · Zbl 1055.74563 [42] Vu, A dual form for discretized kinematic formulation in shakedown analysis, International Journal of Solids and Structures 41 pp 267– (2004) · Zbl 1069.74005 [43] Prager, Theory of Perfectly Plastic Solids (1951) · Zbl 0044.39803 [44] Casciaro, A mixed formulation and mixed finite elements for limit analysis, International Journal for Numerical Methods in Engineering 18 pp 211– (1982) · Zbl 0478.73048 [45] Yan AM. Contribution to the direct limit state analysis of plastified and cracked structures. Dissertation, UniversitÃ© de LiÃ”ge, Belgium, 1997. [46] Heitzer M. Traglast- und Einspielanalyse zur Bewertung der Sicherheit passiver Komponenten. Berichte des Forschungszentrums JÃ{$$\tfrac14$$}lich, JÃ{$$\tfrac14$$}l-3704, Dissertation, RWTH Aachen, Germany, 1999. [47] Vu DK. Dual limit and shakedown analysis of structures. Dissertation, UniversitÃ© de LiÃ”ge, Belgium, 2001. [48] Tran TN. Limit and shakedown analysis of plates and shells including uncertainties. Dissertation, Technische UniversitÃ{\currency}t Chemnitz, Germany, 2008. Available from: http://archiv.tu-chemnitz.de/pub/2008/0025. [49] Gaydon, A theoretical investigation of the yield point loading of a square plate with a central circular hole, Journal of the Mechanics and Physics of Solids 2 pp 156– (1954) [50] Belytschko, Plane stress shakedown analysis by finite element, International Journal of Mechanical Sciences 14 pp 619– (1972) [51] Corradi, A linear programming approach to shakedown analysis of structures, Computer Methods in Applied Mechanics and Engineering 3 pp 37– (1974) [52] Genna, A non-linear inequality, finite element approach to the direct computation of the shakedown load safety factor, International Journal of Mechanical Sciences 30 pp 769– (1988) · Zbl 0669.73027 [53] Stein, Shakedown with non-linear strain-hardening including structural computation using finite element method, International Journal of Plasticity 8 pp 1– (1992) · Zbl 0766.73026 [54] Zhang, Boundary element methods for lower bound limit and shakedown analysis, Engineering Analysis of Boundary Methods 28 pp 905– (2004) · Zbl 1130.74477 [55] Gross-Wedge, On the numerical assessment of the safety factor of elasticâplastic structures under variable loading, International Journal of Mechanical Sciences 39 (4) pp 417– (1997) [56] Zouanin, An algorithm for shakedown analysis non-linear yield function, Computer Methods in Applied Mechanics and Engineering 191 pp 2463– (2002)
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