Robust topology optimization for dynamic compliance minimization under uncertain harmonic excitations with inhomogeneous eigenvalue analysis

Variability of load magnitude/direction is a most significant source of uncertainties in practical engineering. This paper investigates robust topology optimization of structures subjected to uncertain dynamic excitations. The unknown-but-bounded dynamic loads/accelerations are described with the non-probabilistic ellipsoid convex model. The aim of the optimization problem is to minimize the absolute dynamic compliance for the worst-case loading condition. For this purpose, a generalized compliance matrix is defined to construct the objective function. To find the optimal structural layout under uncertain dynamic excitations, we first formulate the robust topology optimization problem into a nested double-loop one. Here, the inner-loop aims to seek the worst-case combination of the excitations (which depends on the current design, and is usually to be found by a global optimization algorithm), and the outer-loop optimizes the structural topology under the found worst-case excitation. To tackle the inherent difficulties associated with such an originally nested formulation, we convert the inner-loop into an inhomogeneous eigenvalue problem using the optimality condition. Thus the double-loop problem is reformulated into an equivalent single-loop one. This formulation ensures that the strict-sense worst-case combination of the uncertain excitations for each intermediate design be located without resorting to a time-consuming global search algorithm. The sensitivity analysis of the worst-case objective function value is derived with the adjoint variable method, and then the optimization problem is solved by a gradient-based mathematical programming method. Numerical examples are presented to illustrate the effectiveness and efficiency of the proposed framework.