Keywords: structural design, multiobjective optimization, genetic algorithm, Pareto optimal, truss, rigid frame.

Multiple and conflicting objectives often exist in practical design problems. For example, the cost of products is to be minimized, while maximum performance is desired. Because all objective functions in such cases cannot be optimized simultaneously, some compromise solutions should be found alternatively. An optimization problem with some objective functions is called the multiobjective or multicriteria optimization problem (MOP). Many multiobjective optimization methods have been proposed since introduction of a concept of the Pareto optimality. These methods can search for Pareto optimal solutions, which constitute a trade-off surface called a Pareto optimal front in a coordinate system of objective functions, i.e. the objective space.

Traditional approaches, such as the weighting method and the constraint method, use single-objective optimization techniques, in which objective functions of the original MOP are transformed into a single objective function. Such scalarization methods have been adopted in various engineering problems. Min et al. [1] propose a unified topology design method to generate structures satisfying both static and vibration performance measures using the weighting method. Steven et al. [2] also apply the weighting method to the shape and topology design of continuum structures subjected to multiple load cases.

Many researchers, however, point out a disadvantage that the scalarization methods cannot search for rigorous Pareto optimal solutions if a MOP has a non-convex trade-off surface. On the other hand, various multiobjective optimization techniques based on the genetic algorithm are recently proposed [3,4,5,6], which are known generally as the multiobjective genetic algorithm (MOGA) or the multiobjective evolutionary algorithm. The MOGAs are also applied to various engineering problems, such as shape design of a wing for supersonic transport [7] and structural optimum design of trusses [8,9].

In this study, the strength Pareto evolutionary algorithm (SPEA), proposed by Zitzler and Thiele [10], is applied to the elastic design of frame structures, such as rigid frames and truss structures, since the SPEA is more efficient and accurate than other MOGAs. The present design problem can be formulated as a multiobjective optimization problem, which is to find cross-sectional size of members, such that the total volume, the maximum stress of members and the maximum nodal displacement are minimized. The MOGA can provide various design candidates and a trade-off relationship between objective functions, which are useful for final decision making in practical design. The application of the MOGA is illustrated in numerical examples with discussion on properties of both design solutions and the trade-off relationship between the objective functions.

1

S. Min, S. Nishiwaki and N. Kikuchi, "Unified Topology Design of Static and Vibrating Structures using multiobjective optimization", Computers and Structures, Vol. 75, pp.93-116, 2000. doi:10.1016/S0045-7949(99)00055-3

2

G. P. Steven, Q. Li and Y. M. Xie, "Multicriteria Optimization that Minimizes Maximum Sstress and Maximizes Stiffness", Computers and Structures, Vol. 80, pp.2433-2448, 2002. doi:10.1016/S0045-7949(02)00235-3

3

J. D. Schaffer, "Multiple Objective Optimization with Vector Evaluated Genetic Algorithms", Proc. of the 1st Int. Conf. on Genetic Algorithms, pp.93-100, 1985.

4

C. M. Fonseca and P. J. Fleming, "Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization", Proc. of the 5th Int. Conf. on Genetic Algorithms, pp.416-423, 1993.

5

J. Horn, N. Nafpliotis and D. E. Goldberg, "A Niched Pareto Genetic Algorithm for Multiobjective Optimization", Proc. of the 1st IEEE Conference on Evolutionary Computation, Vol. 1, pp.82-87, 1994. doi:10.1109/ICEC.1994.350037

6

K. Deb, "Multi-Objective Optimization using Evolutionary Algorithms", John Wiley & Sons, Chichester, UK, 2001.

7

S. Obayashi, D. Sasaki and Y. Takeguchi, "Multiobjecitve Aerodynamic Optimization Considering Supersonic and Transonic Cruises", Proc. of Dynamics and Design Conference '99 (The Japan Society of Mechanical Engineers), D548, 1999 (in Japanese).

8

F. Y. Cheng and D. Li, "Multiobjective Optimization Design with Pareto Genetic Algorithm", J. of Structural Engineering, Vol. 123, No. 9, pp.1252-1261, 1997. doi:10.1061/(ASCE)0733-9445(1997)123:9(1252)

9

C. A. Coello and A. D. Christiansen, "Multiobjective Optimization of Trusses using Genetic Algorithms", Computers and Structures, Vol. 75, pp.647-660, 2000. doi:10.1016/S0045-7949(99)00110-8

10

E. Zitzler and L. Thiele, "Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach", IEEE Transactions on Evolutionary Computation, 3(4), pp.257-271, 1999. doi:10.1109/4235.797969

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