Robust topology optimization of continuous structures using the Bernstein approximation

We propose a robust formulation for the topology optimization of continuous structures. The objective is to determine the optimal distribution of a linear elastic material within a reference domain subjected to both stochastic and deterministic external loads. A key feature of this formulation is the incorporation of a failure probability constraint defined in terms of compliance. The Bernstein approximation is used to derive an upper bound on the failure probability, yielding a more tractable formulation. By using the Solid Isotropic Material with Penalization (SIMP) method, where the material density is the main design variable, we reformulate the original stochastic optimization problem into a standard nonlinear optimization problem. We develop a numerical algorithm to solve this reformulation by iteratively solving a sequence of linear conic subproblems, which can be efficiently handled in polynomial time via interior-point methods. Numerical experiments demonstrate the effectiveness of the proposed approach.