Abstract
The paper ‘A 99-line topology optimization code written in Matlab’ by Sigmund (Struct Multidisc Optim 21(2):120–127, 2001) demonstrated that SIMP-based topology optimization can be easily implemented in less than hundred lines of Matlab code. The published method and code has been used even since by numerous researchers to advance the field of topology optimization. Inspired by the above paper, we demonstrate here that, by exploiting the notion of topological-sensitivity (an alternate to SIMP), one can generate Pareto-optimal topologies in about twice the number of lines of Matlab code. In other words, optimal topologies for various volume fractions can be generated in a highly efficient manner, by directly tracing the Pareto-optimal curve.
References
Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
Amstutz S (2006) Sensitivity analysis with respect to a local perturbation of the material property. Asymptot Anal 49(1–2):87–108
Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57(8):1177–1196
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in optimal design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224
Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362
Céa J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimizations connection. Comput Methods Appl Mech Eng 188:713–726
Chen TY, Wu S-C (1998) Multiobjective optimal topology design of structures. Comput Mech 21:483–492
Cohon JL (1978) Multiobjective programming and planning, vol 140. Academic, London
Dambrine M, Vial G (2005) Influence of a boundary perforation on the Dirichlet energy. Control Cybern 34(1):117–136
Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–389
Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51
Feijóo RA, Novotny AA, Taroco E, Padra C (2003) The topological derivative for the Poisson’s problem. Math Models Methods Appl Sci 13(12):1825–1844
Feijóo RA, Novotny AA, Taroco E, Padra C (2005) The topological-shape sensitivity method in two-dimensional linear elasticity topology design. In: Idelsohn VSSR (ed) Applications of computational mechanics in structures and fluids. CIMNE, Barcelona
Gopalakrishnan SH, Suresh K (2008) Feature sensitivity: a generalization of topological sensitivity. Finite Elem Anal Des 44(11):696–704
Hamda H, Roudenko O, Schoenauer M (2002) Application of a multi-objective evolutionary algorithm to topology optimum design. In: Fifth international conference on adaptive computing in design and manufacture
Lin J, Luo Z, Tong L (2010) A new multi-objective programming scheme for topology optimization of compliant mechanisms. Struct Multidisc Optim 30:241–255
Luo Z, Chen L, Yang J, Zhang Y, Abdel-Malek K (2005) Compliant mechanism design using multi-objective topology optimization scheme of continuum structures. Struct Multidisc Optim 30:142–154
Madeira JFA, Rodrigues H, Pina H (2005) Multi-objective optimization of structures topology by genetic algorithms. Adv Eng Softw 36:21–28
Messac A, Ismail-Yahaya A (2001) Required relationship between objective function and Pareto frontier orders: practical implications. AIAA J 39(11):2168–2174
Messac A, Sundararaj GJ, Tappeta RV, Renaud JE (2000) Ability of objective functions to generate points on non-convex Pareto frontiers. AIAA J 38(6):1084–1091
Norato JA, Bendsøe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33:375–386
Novotny AA, Feijóo RA, Taroco E, Padra C (2003) Topological-shape sensitivity analysis. Comput Methods Appl Mech Eng 192(7):803–829
Novotny AA, Feijóo RA, Taroco E, Padra C (2005) Topological sensitivity analysis for three-dimensional linear elasticity problem. In: 6th world congress on structural and multidisciplinary optimization, Rio de Janeiro
Novotny AA, Feijóo RA, Taroco E, Padra C (2006) Topological-shape sensitivity method: theory and applications. Solid Mech Appl 137:469–478
Padhye N (2008) Topology optimization of compliant mechanism using multi-objective particle swarm optimization. In: GECCO’08, July 12–16. ACM, Atlanta
Papalambros PY (2002) The optimization paradigm in engineering design: promises and challenges. Comput Aided Des 34(12):939–951
Rozvany GIN (2001a) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidisc Optim 21(2):90–108
Rozvany GIN (2001b) Stress ratio and compliance based methods in topology optimization—a critical review. Struct Multidisc Optim 21(2):109–119
Samet B (2003) The topological asymptotic with respect to a singular boundary perturbation. Comptes Rendus Mathematique 336(12):1033–1038
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127
Sokolowski J, Zochowski A (1999) On topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272
Sokolowski J, Zochowski A (2003) Optimality conditions for simultaneous topology and shape optimization. SIAM J Control Optim 42(4):1198–1221
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246
Zhang WH, Yang HC (2002) Efficient gradient calculation of the Pareto optimal curve in multicriteria optimization. Struct Multidisc Optim 23:311–319
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89:197–224
Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier, Amsterdam
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Suresh, K. A 199-line Matlab code for Pareto-optimal tracing in topology optimization. Struct Multidisc Optim 42, 665–679 (2010). https://doi.org/10.1007/s00158-010-0534-6
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DOI: https://doi.org/10.1007/s00158-010-0534-6