Keywords: modular steel structures, optimization, mixed-discrete, topology, evolution strategy, adaptive penalty function.

Modular three-dimensional steel frames, like steel pallet racks or shelving assemblies and scaffoldings are prefabricated structures with slender elements, mostly under compressive forces. The analysis includes different nonlinearities, e.g. the geometric nonlinearities and global and local stability effects. Optimization variables are often of mixed-discrete or topology type.

The optimization of these systems requires the most advanced solution techniques, but also a systematic preprocessing, sophisticated nonlinear structural analysis and an automated handling of stress- and displacement-constraints and other side-constraints resulting from the underlying code of practice. The objective of the optimization is the minimization of cost.

The optimization of these structures involves continuous and discrete variables in mixed form and is solved by a -evolutionary strategy (ES). Special adjustments of the ES regarding the discontinuous behaviour of the discrete variables have to be done. The mutation step size of discrete and topological variables is controlled by the probabilistic value, which decides whether a variable is changed or not. In this way, the mutation instalments determine the probability of a change of the variable . With the implementation of the mutation probability and the coincidence distribution of the component a differentiated control of the mutation process becomes possible [1].

The robustness and efficiency of the optimization procedure is increased by the combination of ES with a penalty function [2]. An adaptive penalty function is developed for this purpose, in which the penalty factor is adapted by the percentage of permissible individuals in the current population. A special selection scheme fitted to the optimization task is used to minimize the number of computationally intensive finite-element-calculations.

The use of a penalty function provides good results for the constrained optimization. The developed adaptive penalty function proves as reliable and robust for the described optimization problems and leads to a speed-up in convergence. It can be shown that the -ES in connection with the described variants represents a reliable and efficient method for the cost optimization of modular steel structures.

1

C. Ebenau and G. Thierauf, "Combination of Evolutionary Strategies and Mathematical Programming for the Optimization of Modular Steel Frames", in Topping, B.H.V. (Ed.), "Developments in Computational Mechanics with High Performance Computing", Civil-Comp Press, Edinburgh, 229-234, 1999. doi:10.4203/ccp.57.11.3

2

Z. Michalewicz, "A Survey of Constraint Handling Techniques in Evolutionary Computation Methods", Proceedings of the 4th Annual Conference on Evolutionary Programming, Cambridge, 135-155, 1995.

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