Robust Optimization Applied to Uncertain Limit Analysis

Limit analysis (LA) is an efficient tool for computing in a direct manner the ultimate load of a structure made of a perfectly plastic material. The lower bound static approach amounts to maximize the load factor such that one can find an optimal stress field in equilibrium with such loading and satisfying strength conditions at each point in the domain. In the deterministic case, the ultimate load is obtained via the resolution of a convex optimization problem. When loading or strength properties are random, the data of such an optimization problem become uncertain. Robust optimization theory is a branch of mathematical optimization which aim at finding an optimal solution of uncertain problems among all possible realizations of the uncertainty within a known uncertainty set. Applying the concepts of robust optimization to uncertain limit analysis, one may compute a worst-case ultimate load estimate associated with a given uncertainty set, for instance in the case of uncertain strength properties or uncertain load cases. This paper discusses how robust limit analysis problems can be reformulated, either exactly or approximately, into deterministic problems. In particular, the distinction between static and adjustable robust counterparts is introduced. In the former case, uncertain LA problems are replaced with a deterministic problem with reduced strength properties. In the latter case, additional optimization variables must be introduced in order to obtain an extended LA problem in much higher dimension.