Keywords: structural optimization, heuristic algorithms, non-linearity, non-convexity, threshold accepting algorithm, differential evolution.

Realistic problems of structural optimization are characterized by non-linearity, non-convexity and by continuous and/or discrete design variables. For their solution heuristic algorithms are well suited. The present contribution describes a combination of the Threshold Accepting Algorithm with Differential Evolution with particular emphasis on structural optimization. The Threshold Accepting Algorithm is similar to Simulated Annealing. Differential Evolution is based on Genetic Algorithms. With the presented combination of the Threshold Accepting Algorithm and Differential Evolution the speed of convergence of the optimization process is increased considerably. At the same time the risk of an early convergence to local optima is minimized. Both algorithms are simply structured and can be easily implemented into available programs for the analysis of structures.

Techniques of stochastic search are widely used for structural optimization, but combinations of the various techniques are rare. S. Botello et al. [1] combine the search operators selection, crossover, mutation of genetic algorithms (GA) with the acceptance operator of the Simulated Annealing (SA) and call this the General Stochastic Search Algorithm. The unmodified individuals of a population (before variation by recombination and mutation) are compared with the varied ones. The acceptance operator selects solutions to be carried over to the next generation. S. W. Mafoud and D.E. Goldberg [2] present the Parallel Recombinative Simulated Annealing. After the initialization of a population and choice of a system temperature T, parents are chosen. The offsprings are produced by recombination and mutation, followed by a comparison between parents and their offsprings. This is carried out e.g. by a comparison between solutions i and j, where i will be the winner with a chance of . Subsequently, the parents are replaced by the winners and T is reduced. This process is repeated for the complete population in every iteration.

In both publications the search mechanisms of GA are applied and the acceptance operator of SA is used for the selection of the individuals. In this paper, the combination with the Threshold Accepting Algorithm, which computes the functionals in every cycle of the iteration, is essential for the increased performance. The Differential Evolution helps to avoid local optima. By application of penalty functions even inadmissible solutions are allowed, whereby the approximation of the global optimum is possible either from the admissible as well as from the inadmissible direction. The standard variation is preset by a sensitivity analysis. Admissible results are stored and treated similar to the elite individual in the GA.

The combined Threshold Accepting - Differential Evolution Algorithm is a heuristic technique for the optimal design of civil engineering structures, which is characterized by non-linearity, non-convexity and by continuous and/or discrete design variables. Grouping of the cross-sections according to structural side-constraints, allows for the optimization of large structures with linear and nonlinear functional constraints. The proposed combination and modification of two known heuristic algorithms seems to be particularly suited for realistic problems in structural engineering. The analysis is based on finite element computations and because of the fairly simple structure of the algorithms any available finite element program can be used.

1

Botello, S., Marraquin, J.L., Oñate, E., Van Horebeek, J.: Solving Structural Optimization Problems with Genetic Algorithms and Simulated Annealing, International Journal for Numerical Methods in Engineering, 1999/45, S. 1069-1084. doi:10.1002/(SICI)1097-0207(19990720)45:8<1069::AID-NME620>3.0.CO;2-E

2

Mahfoud, W.S., Goldberg, D.E., Parallel Recombinative Simulated Annealing: A Genetic Algorithm, IlliGAL Report No. 93006, Department of Computer Science, University of Illinois, 1994

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