Abstract
To date the design of structures via topology optimisation methods has mainly focused on single-objective problems. However, real-world design problems usually involve several different objectives, most of which counteract each other. Therefore, designers typically seek a set of Pareto optimal solutions, a solution for which no other solution is better in all objectives, which capture the trade-off between these objectives. This set is known as a smart Pareto set. Currently, only the weighted sums method has been used for generating Pareto fronts with topology optimisation methods. However, the weighted sums method is unable to produce evenly distributed smart Pareto sets. Furthermore, evenly distributed weights have been shown to not produce evenly spaced solutions. Therefore, the weighted sums method is not suitable for generating smart Pareto sets. Recently, the smart normal constraints method has been shown to be capable of directly generating smart Pareto sets. This work presents an updated smart normal constraint method, which is combined with a bi-directional evolutionary structural optimisation algorithm for multi-objective topology optimisation. The smart normal constraints method has been modified by further restricting the feasible design space for each optimisation run such that dominant and redundant points are not found. The algorithm is tested on several different structural optimisation problems. A number of different structural objectives are analysed, namely compliance, dynamic and buckling objectives. Therefore, the method is shown to be capable of solving various types of multi-objective structural optimisation problems. The goal of this work is to show that smart Pareto sets can be produced for complex topology optimisation problems. Furthermore, this research hopes to highlight the gap in the literature of topology optimisation for multi-objective problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Pareto, V.: Cour Deconomie Politique. Librarie Droz-Geneve, Geneva (1964)
Michell, A.G.M.: The limits of economy of material in frame structures. Philos. Mag. 8, 589–597 (1904)
Prager, W.: A note on discretized Michell structures. Comput. Methods Appl. Mech. Eng. 3(3), 349–355 (1974)
Svanberg, K.: Optimization of geometry in truss design. Comput. Methods Appl. Mech. Eng. 28(1), 63–80 (1981)
Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984)
Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Springer, Berlin (1992)
Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988)
Bendsøe, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1(4), 193–202 (1989)
Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49, 885–896 (1993)
Prager, W., Rozvany, G.I.N.: Optimization of the structural geometry. In: Bednarek, A., Cesari, L. (eds.) Dynnamical Systems, pp. 265–293. Academic Press, New York (1977)
Rozvany, G.I.N., Zhou, M., Birker, T.: Generalized shape optimization without homogenization. Struct. Optim. 4, 250–254 (1992)
Yang, X.Y., Xie, Y.M., Steven, G.P., Querin, O.M.: Bidirectional evolutionary method for stiffness optimization. AIAA J. 37, 1483–1488 (1999)
Munk, D.J., Vio, G.A., Steven, G.P.: Topology and shape optimization methods using evolutionary algorithms: a review. Struct. Multidiscip. Optim. 52(3), 613–631 (2015)
Rozvany, G.I.N.: A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37, 217–237 (2009)
Munk, D.J., Kipouros, T., Vio, G.A., Parks, G.T., Steven, G.P.: Multiobjective and multi-physics topology optimization using an updated smart normal constraints bi-directional evolutionary structural optimization method. Struct. Multidiscip. Optim. (2017, in press)
Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8(1), 59–60 (1963)
Haimes, Y.Y., Lasdon, L.S., Wismer, D.A.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man. Cybern. 1(3), 296–297 (1971)
Marler, R.T., Arora, J.S.: The weighted sum method for multiobjective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2010)
Das, I., Dennis, J.E.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 (1997)
Messac, A., Sundararaj, G.J., Tappeta, R.V., Renaud, J.E.: Ability of objective functions to generate points on non-convex Pareto frontiers. AIAA J. 38, 1084–1091 (2000)
Messac, A., Ismail-Yahaya, A., Mattson, C.A.: The normalized normal constraint method for generating the Pareto frontier. Struct. Multidiscip. Optim. 25, 86–98 (2003)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Zhao, S.Z., Suganthan, P.: Two-lbests based multi-objective particle swarm optimizer. Eng. Optim. 43(1), 1–17 (2011)
Wang, S., Tai, K., Wang, M.Y.: An enhanced genetic algorithm for structural topology optimization. Int. J. Numer. Methods Eng. 65(1), 18–44 (2006)
Madeira, J.A., Rodrigues, H., Pina, H.: Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing. Struct. Multidiscip. Optim. 32(1), 31–39 (2006)
Wildman, R.A., Gazonas, G.A.: Multiobjective topology optimization of energy absorbing materials. Struct. Multidiscip. Optim. 51(1), 125–143 (2015)
Sigmund, O., Maute, K.: Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)
Zavala, G.R., Nebro, A.J., Luna, F., Coello, C.A.C.: A survey of multi-objective metaheuristics applied to structural optimization. Struct. Multidiscip. Optim. 49(4), 537–558 (2014)
Eschenauer, H.A., Olhoff, N.: Topology optimization of continuum structures: a review. Appl. Mech. Rev. 54(4), 331–390 (2001)
Tai, K., Prasad, J.: Target-matching test problem for multiobjective topology optimization using genetic algorithms. Struct. Multidiscip. Optim. 34(4), 333–345 (2007)
Cardillo, A., Cascini, G., Frillici, F.S., Rotini, F.: Multi-objective topology optimization through GA-based hybridization of partial solutions. Eng. Comput. 29(3), 287–306 (2013)
Sigmund, O.: Design of multiphysics actuators using topology optimization - part I: one-material structures. Comput. Method Appl. Mech. Eng. 190, 6577–6604 (2001)
Steven, G.P., Li, Q., Xie, Y.M.: Evolutionary topology and shape design for general physical field problems. Comput. Mech. 26(2), 129–139 (2000)
Deaton, J.D., Grandhi, R.V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49, 1–38 (2014)
Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Applications. Springer, Berlin (2004)
Huang, X., Xie, Y.M.: Evolutionary Topology Optimization of Continuum Structures. Wiley, Chichester (2010)
Rozvany, G.I.N., Lewinski, T.: Topology Optimization in Structural and Continuum Mechanics. Springer, Vienna (2013)
Proos, K.A., Steven, G.P., Querin, O.M., Xie, Y.M.: Multicriterion evolutionary structural optimization using the weighting and the global criterion methods. AIAA J. 39(10), 2006–2012 (2001)
Proos, K.A., Steven, G.P., Querin, O.M., Xie, Y.M.: Stiffness and inertia multicriteria evolutionary structural optimization. Eng. Comput. 18, 1031–1054 (2001)
Kim, W.Y., Grandhi, R.V., Haney, M.: Multiobjective evolutionary structural optimization using combined static/dynamic control parameters. AIAA J. 44, 794–802 (2006)
Izui, K., Yamada, T., Nishiwaki, S., Tanaka, K.: Multiobjective optimization using an aggregative gradient-based method. Struct. Multidiscip. Optim. 51(1), 173–182 (2015)
Sato, Y., Izui, K., Yamada, T., Nishiwaki, S.: Gradient-based multiobjective optimization using a discrete constraint technique and point replacement. Eng. Optim. 48(7), 1226–1250 (2015)
Sato, Y., Izui, K., Yamada, T., Nishiwaki, S.: Pareto frontier exploration in multiobjective topology optimization using adaptive weighting and point selection schemes. Struct. Multidisc. Optim. 55, 409–422 (2017)
Messac, A., Mattson, C.A.: Normal constraints method with guarantee of even representation of complete Pareto frontier. AIAA J. 42, 2101–2111 (2004)
Mattson, C.A., Mullur, A.A., Messac, A.: Smart Pareto filter: obtaining a minimal representation of multiobjective design space. Eng. Optim. 36, 721–740 (2004)
Rynne, B.: Linear Functional Analysis. Springer, New York (2007)
Hancock, B.J., Mattson, C.A.: The smart normal constraints method for directly generating a smart Pareto set. Struct. Multidiscip. Optim. 48, 763–775 (2013)
Zuo, Z., Xie, Y.M., Huang, X.: Evolutionary topology optimization of structures with multiple displacements and frequency constraints. Adv. Struct. Eng. 15, 385–398 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Munk, D.J., Vio, G.A., Steven, G.P., Kipouros, T. (2018). Producing Smart Pareto Sets for Multi-objective Topology Optimisation Problems. In: Schumacher, A., Vietor, T., Fiebig, S., Bletzinger, KU., Maute, K. (eds) Advances in Structural and Multidisciplinary Optimization. WCSMO 2017. Springer, Cham. https://doi.org/10.1007/978-3-319-67988-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-67988-4_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67987-7
Online ISBN: 978-3-319-67988-4
eBook Packages: EngineeringEngineering (R0)