Abstract
The objective of this paper is to look for structural designs arising from topological optimization procedures that aim at maximizing the loading capacity regarding incipient plastic collapse. The mechanical problem is described by limit analysis (LA) formulation that allows a direct determination of the load that produces the plastic collapse phenomenon without information about the load history. In case of proportional loading processes, LA consists of computing a critical load factor such that the structure undergoes plastic collapse when the reference load is amplified by this factor. In this case, LA can be cast mathematically as a convex constrained optimization problem. The design optimization is formally stated as the maximization of the collapse load factor subject to a fixed quantity of available material. The design is controlled by solid isotropic microstructure with penalization (SIMP) technique. In the particular case of the chosen objective function, the solution of the adjoint problem in sensitivity analysis coincides with the Newton–Raphson update vector obtained at the convergence of the procedure developed to solve the LA optimization problem, fact that reduces the numerical cost of gradient calculations. In order to keep the implementation straightforward, the optimality conditions are solved by a classical heuristic element-by-element density updating algorithm, well known in the literature. The set of tested examples brings encouraging results with structures being stressed to ultimate bearing states. Implementation was kept as simple as possible, leaving the field open to further investigations. Numerical tests show that, despite having similar geometries, plastic collapse factor obtained with compliance optimal designs are lower than those obtained with present formulation.
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References
Alberdi R, Khandelwal K (2017) Topology optimization of pressure dependent elastoplastic energy absorbing structures with material damage constraints. Finite Elem Anal Des 133:42–61. https://doi.org/10.1016/j.finel.2017.05.004
Amir O (2017) Stress-constrained continuum topology optimization: a new approach based on elasto-plasticity. Struct Multidiscip Optim 55(5):1797–1818. https://doi.org/10.1007/s00158-016-1618-8
Andersen KD, Christiansen E, Overton ML (1998) Computing limit loads by minimizing a sum of norms. SIAM J Sci Comput 19(3):1046–1062. https://doi.org/10.1137/s1064827594275303
Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications, 2nd. Springer, Berlin. https://doi.org/10.1063/1.3278595
Borges LA, Zouain N, Huespe AE (1996) Nonlinear optimization procedure for limit analysis. Eur J Mech A/Solid 15(3):487–512
Cardoso EL, Fonseca JSO (2003) Complexity control in the topology optimization of continuum structures. J Braz Soc Mech Sci Eng 25(3):293–301. https://doi.org/10.1590/s1678-58782003000300012
Christiansen E (1980) Limit analysis in plasticity as a mathematical programming problem. CALCOLO 17(1):41–65. https://doi.org/10.1007/BF02575862
Christiansen E (1981) Computation of limit loads. Int J Numer Methods Eng 17(10):1547–1570. https://doi.org/10.1002/nme.1620171009
Christiansen E (1996) Limit analysis of collapse states. In: Handbook of Numerical Analysis, Elsevier, pp 193–312. https://doi.org/10.1016/s1570-8659(96)80004-4
Cohn M, Maier G (1977) Engineering plasticity by math programming. In: Proceedings of the NATO Advances Study Institute, Ontario
Emmendoerfer H Jr, Fancello EA (2015) Otimização topológica com restrições de tensão local usando uma equação de reação-difusão baseada em level sets. In: Proceedings of the XXXVI Iberian Latin American Congress on Computational Methods in Engineering, ABMEC Brazilian Association of Computational Methods in Engineering. https://doi.org/10.20906/cps/cilamce2015-0764
Fancello EA (2006) Topology optimization for minimum mass design considering local failure constraints and contact boundary conditions. Struct Multidiscip Optim 32(3):229–240. https://doi.org/10.1007/s00158-006-0019-9
Feijóo RA, Zouain N (1987) Variational formulations for rates and increments in plasticity. 1st Int Cong on Comput Plast I:33–57
Frémond M, FRIAA A (1982) Les Métodes Statique et Cinématique en Calcul à la Rupture et an Analyse Limite. Eur J App Comp Mech 1(Nro 5):881–905
Fusch P, Pisano AA, Weichert D (eds) (2015) Direct methods for limit and shakedown analysis advanced computational algorithms and material modelling. Springer, Berlin. https://doi.org/10.1007/978-3-319-12928-0
Kamenjarzh J (1996) Limit analysis of solids and structures. CRC Press, Boca Raton
Kammoun Z, Smaoui H (2014) A direct approach for continuous topology optimization subject to admissible loading. Comptes Rendus Mécanique 342(9):520–531. https://doi.org/10.1016/j.crme.2014.06.003
Kammoun Z, Smaoui H (2015) A direct method formulation for topology plastic design of continua. In: Fuschi P, Pisano A A, Weichert D (eds) Direct methods for limit and shakedown analysis of structures: Advanced computational algorithms and material modelling. Springer International Publishing, Cham, pp 47–63, DOI https://doi.org/10.1007/978-3-319-12928-0-3
Komkov V, Choi KK, Haug EJ (1986) Design sensitivity analysis of structural systems, vol 177. Academic Press, Cambridge
Krabbenhoft K, Damkilde L (2002) A general non-linear optimization algorithm for lower bound limit analysis. Int J Numer Methods Eng 56(2):165–184. https://doi.org/10.1002/nme.551
Li L, Zhang G, Khandelwal K (2017a) Design of energy dissipating elastoplastic structures under cyclic loads using topology optimization. Struct Multidiscip Optim 56(2):391–412. https://doi.org/10.1007/s00158-017-1671-y
Li L, Zhang G, Khandelwal K (2017b) Topology optimization of energy absorbing structures with maximum damage constraint. Int J Numer Methods Eng 112:737–775. https://doi.org/10.1002/nme.5531
Lubliner J (1990) Plasticity theory. Maxwell Macmillan international editions in engineering. Macmillan, London
Makrodimopoulos A, Martin CM (2007) Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Methods Geomech 31(6):835–865. https://doi.org/10.1002/nag.567
Pastor F, Loute E (2005) Solving limit analysis problems: an interior-point method. Commun Numer Methods Eng 21(11):631–642. https://doi.org/10.1002/cnm.779
Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidiscip Optim 26(1):50–66
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
de Saxcé G, Bousshine L (1998) Limit analysis theorems for implicit standard materials: Application to the unilateral contact with dry friction and the non-associated flow rules in soils and rocks. Int J Mech Sci 40(4):387–398. https://doi.org/10.1016/S0020-7403(97)00058-1, http://www.sciencedirect.com/science/article/pii/S0020740397000581
Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21(2):120–127. https://doi.org/10.1007/s001580050176
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75. https://doi.org/10.1007/BF01214002
Wallin M, Jönsson V, Wingren E (2016) Topology optimization based on finite strain plasticity. Struct Multidiscip Optim 54(4):783–793. https://doi.org/10.1007/s00158-016-1435-0
Yu Mh, Ma GW, Li JC (2009) Structural plasticity: limit, shakedown and dynamic plastic analyses of structures, 1st. https://doi.org/10.1007/978-3-540-88152-0
Zhang G, Li L, Khandelwal K (2017) Topology optimization of structures with anisotropic plastic materials using enhanced assumed strain elements. Struct Multidiscip Optim 55(6):1965–1988. https://doi.org/10.1007/s00158-016-1612-1
Zouain N, Herskovits J, Borges LA, Feijóo RA (1993) An iterative algorithm for limit analysis with nonlinear yield functions. Int J Solids Struct 30(10):1397–1417. https://doi.org/10.1016/0020-7683(93)90220-2
Zouain N, Borges L, Silveira LJ (2002) An algorithm for shakedown analysis with nonlinear yield functions. Comput Methods Appl Mech Eng 191(23-24):2463–2481. https://doi.org/10.1016/S0045-7825(01)00374-7
Zouain N, Borges L, Silveira JL (2014) Quadratic velocity-linear stress interpolations in limit analysis. Int J Numer Methods in Eng 98(7):469–491. https://doi.org/10.1002/nme.4636
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Fin, J., Borges, L.A. & Fancello, E.A. Structural topology optimization under limit analysis. Struct Multidisc Optim 59, 1355–1370 (2019). https://doi.org/10.1007/s00158-018-2132-y
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DOI: https://doi.org/10.1007/s00158-018-2132-y