Keywords: robust optimization, topology optimization, multi objective evolutionary algorithms, NSGA-II.

Research in topology optimization has gained importance during recent years. However, a robust approach which seeks to find the optimal (or near-optimal) structural layout taking into account the variability in the structural response subject to uncertain input conditions is still an open problem. To address this issue, this paper presents a robust topology optimization methodology based on multi-objective evolutionary algorithms (MOEA). The MOEA is guided by two robustness metrics (objectives), namely, the expected value and the variance of the compliance subject to uncertain input conditions. We used a design of experiments to estimate the expected value and the variance of the compliance under uncertain loading conditions. The MOEA was implemented in MatlabR using an elitist variant of NSGA-II and a graph representation of the topology. The algorithm starts by generating a random initial population of solutions which are then ranked based on their fitness values. Those solutions with a higher rank are assigned a greater probability of becoming parents. The new generation is generated by performing crossover and mutation operations on the genes of the selected parents. This process is repeated until the termination conditions are met, namely, after a given number of generations has passed or if the cumulative change in the values of the fitness functions over a predefined number of stall generations is smaller than a predefined function tolerance. At the end, we provide the designer with a wide choice of topologies, all of them, being part of the (approximate) Pareto front of non-dominated solutions. Topologies obtained using the MOEA approach are stiffer than those found using the deterministic approach (over 0.24%) and their response varies significantly less when subjected to uncertain input conditions (over 3% less variance). Furthermore, since the volume of the structure is fixed, this methodology yields topologies which have the same cost as deterministic solutions, but have better performance measures. Additionally, the proposed multi-objective approach to robust topology optimization allows for naturally extending the algorithm to include other objectives such as volume minimization and reliability metrics, as well as exploring the possibility of adding more sources of variability to the problem, including uncertainty in load magnitudes, nodal locations and material properties.

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