We describe a systematic approach for the robust optimal design of linear elastic structures subjected to unknown loading using minmax and topology optimization methods. Assuming only the loading region and norm, we distribute a given amount of material in the design domain to minimize the principal compliance, i.e. the maximum compliance that is produced by the worst-case loading scenario. We evaluate the principal compliance directly by satisfying the optimality conditions which take the form of a Steklov eigenvalue problem and thus we eliminate the need of an iterative nested optimization. To generate a well-posed topology optimization problem we use relaxation which requires homogenization theory. Examples are provided to demonstrate our algorithm.
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