Abstract
This paper presents a simple and efficient approach for predicting the plastic limit loads in cracked planestrain structures.We use two levels of mesh repartitioning for the finite element limit analysis. The master level handles an adaptive primal-mesh process through a dissipation-based indicator. The slave level performs the subdivision of each triangle into three sub-triangles and constitutes a dual mesh from a pair of two adjacent sub-triangles shared by common edges of the primal mesh. Applying a strain smoothing projection to the strain rates on the dual mesh, the incompressibility constraint and the flow rule constraint are imposed over the edge-based smoothing domains and everywhere in the problem domain. The limit analysis problem is recast into the compact form of a second-order cone programming (SOCP) for the purpose of exploiting interior-point solvers. The present method retains a low number of optimization variables. It offers a convenient way for designing and solving the large-scale optimization problems effectively. Several benchmark examples are given to show the simplicity and effectiveness of the present method.
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References
Hill R. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids, 1952, 1(1): 19–30
Ewing D J F, Richards C E. The yield-point loading of singlynotched pin loaded tensile strips. Journal of the Mechanics and Physics of Solids, 1974, 22(1): 27–36
Miller A G. Review of limit loading of structures containing defects. International Journal of Pressure Vessels and Piping, 1988, 32(1–4): 197–327
Koiter W T. General theorems for elastic plastic solids. Progress in Solid Mechanics, Sneddon I N, Hill R, eds. Nord-Holland, Amsterdam, 1960, 1: 165–221
Melan E. Theorie statisch unbestimmter Systeme aus ideal plastischem Baustoff. Sitzber. Akad. Wiss. Wien IIa, 1936, 145: 195–2182
Prager W, Hodge PG. Theory of Perfectly Plastic Solids. New York: Wiley, 1951, 3
Chakrabarty J. Theory of Plasticity. 3rd ed. Elsevier Butterworth- Heinemann, 2006, 4
Yan A M, Nguyen-Dang H. Limit analysis of cracked structures by mathematical programming and finite element technique. Computational Mechanics, 1999, 24(5): 319–333
Vu D K. Dual Limit and Shakedown analysis of structures. Dissertation for the Doctoral Degree. Belgium: Université de Liège, 2001, 5
Khan I A, Ghosh A K. A modified upper bound approach to limit analysis for plane strain deeply cracked specimens. International Journal of Solids and Structures, 2007, 44(10): 3114–3135
Khan I A, Bhasin V, Chattopadhyay J, Singh R K, Vaze K K, Ghosh A K. An insight of the structure of stress fields for stationary crack in strength mismatch weld under plane strain mode–I loading–Part II: Compact tension and middle tension specimens. International Journal of Mechanical Sciences, 2014, 87: 281–296
Le C V, Askes H, Gilbert M. A locking-free stabilized kinematic EFG model for plane strain limit analysis. Computers & Structures, 2012, 106–107: 1–8
Tran T N, Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. An edgebased smoothed finite element method for primal-dual shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 2010, 82(7): 917–9386
Nguyen-Xuan H, Rabczuk T, Nguyen-Thoi T, Tran T N, Nguyen- Thanh N. Computation of limit and shakedown loads using a nodebased smoothed finite element method. International Journal for Numerical Methods in Engineering, 2012, 90(3): 287–310
Nguyen-Xuan H, Thai C H, Bleyer J, Nguyen P V. Upper bound limit analysis of plates using a rotation-free isogeometric approach. Asia Pacific Journal on Computational Engineering, 2014, 1(1): 12
Nguyen-Xuan H, Tran L V, Thai C H, Le C V. Plastic collapse analysis of cracked structures using extended isogeometric elements and second-order cone programming. Theoretical and Applied Fracture Mechanics, 2014, 72: 13–27
Nagtegaal J C, Parks D M, Rice J R. On numerically accurate finite element solutions in the fully plastic range. Computer Methods in Applied Mechanics and Engineering, 1974, 4(2): 153–177
Sloan S W, Kleeman P W. Upper bound limit analysis using discontinuous velocity fields. Computer Methods in Applied Mechanics and Engineering, 1995, 127(1–4): 293–314
Capsoni A, Corradi L. A finite element formulation of the rigidplastic limit analysis problem. International Journal for Numerical Methods in Engineering, 1997, 40(11): 2063–2086
Christiansen E, Andersen K D. Computation of collapse states with von Mises type yield condition. International Journal for Numerical Methods in Engineering, 1999, 46(8): 1185–1202
Vu D K, Yan A M, Nguyen-Dang H. A primal-dual algorithm for shakedown analysis of structure. Computer Methods in Applied Mechanics and Engineering, 2004, 193(42–44): 4663–4674
Krabbenhøft K, Lyamin A V, Hjiaj M, Sloan S W. A new discontinuous upper bound limit analysis formulation. International Journal for Numerical Methods in Engineering, 2005, 63(7): 1069–1088
Vicente da Silva M, Antao A N. A non-linear programming method approach for upper bound limit analysis. International Journal for Numerical Methods in Engineering, 2007, 72(10): 1192–1218
Lyamin A V, Sloan S W. Upper bound limit analysis using linear finite elements and nonlinear programming. International Journal for Numerical and Analytical Methods in Geomechanics, 2002, 26(2): 181–216
Makrodimopoulos A, Martin C M. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. International Journal for Numerical Methods in Engineering, 2006, 66(4): 604–634
Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplex strain elements and second-order cone programming. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31(6): 835–865
Borges L A, Zouain N, Costa C, Feijoo R. An adaptive approach to limit analysis. International Journal of Solids and Structures, 2001, 38(10–13): 1707–1720
Lyamin A V, Sloan S W, Krabbenhoft K, Hjiaj M. Lower bound limit analysis with adaptive remeshing. International Journal for Numerical Methods in Engineering, 2005, 63(14): 1961–1974
Ciria H, Peraire J, Bonet J. Mesh adaptive computation of upper and lower bounds in limit analysis. International Journal for Numerical Methods in Engineering, 2008, 75(8): 899–944
Munoz J, Bonet J, Huerta A, Peraire J. Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading. International Journal for Numerical Methods in Engineering, 2009, 77(4): 471–501
Martin C M. The use of adaptive finite-element limit analysis to reveal slip-line fields. Géotechnique Letters, 2011, 1(April-June): 23–29
Van-Phuc P, Nguyen-Thoi T, Nguyen C H, Le V C. An effective adaptive limit analysis of soil using FEM and second-order cone programming. The International Conference on Advances in Computational Mechanics (ACOME), 2012, 177–190
Le V C. A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin–Reissner plates. International Journal for Numerical Methods in Engineering, 2013, 96: 231–2467
Nguyen-Xuan H, Liu G R. An edge-based finite element method (ES-FEM) with adaptive scaled-bubble functions for plane strain limit analysis. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 877–905
Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582
Wu C T, Hu W. A two-level mesh repartitioning scheme for the displacement-based lower-order finite element methods in volumetric locking-free analyses. Computational Mechanics, 2012, 50(1): 1–18
Nguyen-Xuan H, Wu C T, Liu G R. An adaptive selective ES-FEM for plastic collapse analysis. European Journal of Mechanics- A/ Solid, submitted, 2014, 8
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. Journal of Sound and Vibration, 2009, 320(4–5): 1100–1130
Chapelle D, Bathe K J. The inf-sup test. Computers & Structures, 1993, 47(4–5): 537–545
Andersen K D, Christiansen E, Conn A R, Overton M L. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM Journal on Scientific Computing, 2001, 22(1): 243–262
Andersen E D, Roos C, Terlaky T. On implementing a primal-dual interior-point method for conic quadratic programming. Mathematical Programming, 2003, 95(2): 249–277
Dorfler W. A convergent adaptive algorithm for Poisson’s equation. SIAM Journal on Numerical Analysis, 1996, 33(3): 1106–1124
Rivara M C, Venere M. Cost analysis of the longest-side (triangle bisection) refinement algorithms for triangulations. Engineering with Computers, 1996, 12(3-4): 224–234
Funken S, Praetorius D, Wissgott P. Efficient implementation of adaptive p1-FEM in Matlab. Preprint, 2008 (www.asc.tuwien.ac.at/ preprint/2008/asc19x2008.pdf)
Mosek. The MOSEK optimization toolbox for MATLAB manual. Mosek ApS, Version 5.0 Edition.9, 2009 (http://www.mosek.com)
Le C V, Nguyen-Xuan H, Askes H, Rabczuk T, Nguyen-Thoi T. Computation of limit load using edge-based smoothed finite element method and second-order cone programming. International Journal of Computational Methods, 2013, 10(1): 1340004
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Nguyen-Xuan, H., Rabczuk, T. Adaptive selective ES-FEM limit analysis of cracked plane-strain structures. Front. Struct. Civ. Eng. 9, 478–490 (2015). https://doi.org/10.1007/s11709-015-0317-7
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DOI: https://doi.org/10.1007/s11709-015-0317-7