Keywords: shape optimisation, spreading solutions, multi-objective.

This paper presents an approch to determine the best shape of a structure under some design conditions. The two objective functions minimize the weight of the structure and node displacements. The constraints for the optimisation problem are the Von Misses stress, the number of pieces and small holes in the structure. Our algorithm estimates a probability distribution to sample the candidate solutions. Using the Pareto dominance criterion the best solutions are chosen. Normally the number of Pareto optimal solutions is greater than our fixed size archive, then we proposed a new mechanism to ensure a good distribution of the solutions over the Pareto Front, giving representative solutions through the solution space.

The Pareto dominance criterion is applied over the objectives and constraints segmenting our population in two sets, the overall non-dominated solutions(ONDS) and the overall dominated solutions (ODS). From ONDS the feasible ones are selected. If the feasible ONDS solutions are not enough to fill up the entire archive, Pareto dominance criterion is applied over the remaining ONDS set considering constrains only, selecting the non-dominated. For the cases when the selected set is larger than our archive size a procedure to discriminate solutions is proposed using the distance among the points in the Pareto Front. This selection improves solutions spreading in the Pareto front, giving a better coverage over the whole solution space. The archive size is the number of probability vectors; thus, each probability vector is updated by one individual from the fixed size archive. In the other hand, the estimation of distribution algorithm (EDA) used to approach the shape optimisation problem improves exploration using a regularization procedure, and solves connectivity and small-holes problems which are very common when the problem is tackled with other strategies (genetic algorithms). The Maxmin algorithm can be used with other evolutionary algorithm, and some other novel ideas are independent too; such as using a probability distribution (vector) to perform a local search in a subspace, in our case each vector is exploring a region chosen with information of the solution space, thus it is a new strategy to solve multiobjective optimisation problems. A variance measure to applied some mechanism to improve exploration is other novel proposal which can be adapted for any algorithm.

The finite element method (FEM) along with structure constraints are used to evaluate the candidate solutions, every element represents a binary variable in the optimisation problem, eventhough we work with hundreds of variables (elements) the algorithm found good solutions. Some experiments are presented, including a comparison with truth solutions for a small case using exhaustive search, and real design problems. Metrics and visual results are presented proving the algorithm performance and effectiveness.

Finally, note that the methodology proposed is independent of finite element type, and can be applied to two-dimensional or three-dimensional shape optimisation problems.

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