Robust compliance-based topology optimization: A discussion on physical consistency

Abstract In probabilistic robust optimization problems we generally take as objective function the weighted sum of the expected value and the standard deviation of the performance. Unfortunately, it has been observed that if the weight given to the standard deviation is too high, then the designs obtained may have no physical meaning at all. If this occurs we say that the design obtained is not consistent from the physical point of view. In this work we present a probabilistic robust optimization approach aimed to preserve physical consistency of the problem being addressed. The objective function is written as the p -norm of a vector composed of the expected value and the weighted standard deviation. We then demonstrate that the standard approach is a particular case of this p -norm approach with p = 1 . In the proposed approach we take p = 2 and prove that physical consistency is ensured provided the standard deviation is not given more weight than the expected value. A similar proof is not available for the standard approach. This makes statement of the robust optimization problem much easier with the proposed approach. The proposed approach can be adapted into existing computational routines, since the same statistical moments and sensitivities are required. The theoretical results are illustrated in the context of compliance-based topology optimization. In the numerical examples we show that the concept of physical consistency may have a significant impact on the designs obtained.