A critical analysis of expected-compliance model in volume-constrained robust topology optimization with normally distributed loading directions, using a minimax-compliance approach alternatively

Uncertainty is an important consideration in topology optimization to produce robust and reliable solutions. There are several possibilities to take into account the uncertainty in the topology optimization of continuum structures. In this paper, we assume that the only source of uncertainty is the variability of the applied load directions. Most models in this area apply parametric statistical tools to describe the directional uncertainty of the applied loads to produce robust structures which are insensitive to the directional uncertainty as much as possible. In the most popular parametric statistical approach the expected-compliance, based on the directional normality assumption, is used as the preferred measure of robustness. We will proof indirectly that this approach is far from the engineering practice and may give hardly interpretable or totally misleading results, which will be demonstrated by two carefully selected counter-examples. The counter-examples validate the fact that the expected-compliance, as a statistical abstraction based on more or less theoretical assumption about normality, not a general applicable measure of robustness. It will be shown, that the non-parametric and really robust volume-constrained worst-load-direction-oriented minimax-compliance model, used in this paper only as a proofing tool in a very simple form, is a viable alternative of the parametric expected-compliance model and its results and its problem solving process as a whole are very close to the engineering thinking. The worst-load-direction-oriented minimax-compliance-model provides expressive, rigorous, generally applicable, and objective information about the robustness. The parametric expected-compliance in itself as the preferred measure of robustness is unable to characterize the compliance variability, in contrast of the minimax approach which can be describe the compliance variability by a robust range-like measure computed very easily as the difference of the maximal- and minimal-compliance on the set of the feasible loading directions.