In this paper, a non-probabilistic based topology optimization method under an external load uncertainty is presented. In traditional topology optimization problems, external loadings that apply to structures are always assumed as deterministic, but an external loading with uncertainties is very common in many practical engineering applications. In this paper, load uncertainty is described as an unknown-but-bounded model and the maximum possible strain energy based topology optimization formulation under an uncertain load is solved for the worst case condition. This optimization problem can be rewritten as a two-level optimization problem: the upper level optimization problem is a deterministic topology optimization under a critical loading of the worst structure response, and the lower level optimization problem is to determine the critical loading corresponding to the worst structure response.
The challenge of the lower level optimization problem is on its non-convexity which makes many gradient based search methods ineffective. To overcome this issue, the lower level optimization problem is reformulated based on the KKT optimality conditions as an inhomogeneous eigenvalue problem and is solved for the critical loading corresponding to the worst structure response. After the worst loading case is identified, the upper level problem can be solved through the existing gradient based optimization algorithms. The effectiveness of the proposed topology optimization under unknown-but-bounded external loading uncertainty is demonstrated through a few numerical examples.