Robust topology optimization for the worst load case based on aggregation of linear system

This paper proposes a robust topology optimization method for a linear elasticity de- sign problem subjected to an uncertain load. The robust design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control research is introduced to evaluate the value of the robust compliance. The aggregation is applied to provide the direct relationship between the uncertain input load and output displacement using a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method. According to the Rayleigh-Ritz theorem for symmetric matrices, the robust compliance minimization problem is formu- lated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix subject to the constraint that the Euclidean norm of the uncertain load set is fixed. More- over, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed robust structural optimization method is implemented as topology optimization using SIMP method, sensitivity analysis and the method of moving asymptotes (MMA). The numerical examples provided illustrate me- chanically reasonable structures and establish the worst load cases corresponding to these optimal structures.