Robust topology optimization design of structures with multiple load cases

In practical engineering, the structural performance always exhibit some degree of variations due to the fact that the applied loads fluctuate dramatically throughout its service life-cycle. Thus, the need is highlighted to account for uncertainties in topology optimization stage of the structural design. Conventional deterministic topology optimization searches for minimum compliance without considering the uncertainties in operating processes. Recently, the robust structural design has attracted intensive attentions because it can reduce the variability of structural performance. However, existing robust design methods are confined to the size and shape optimization problems. This paper aims to incorporate the robust design strategy into the continuum topology optimization problem under multiple uncertain load cases by minimizing variation of the objective performance. Following the SIMP approach, an artificial isotropic material model with penalization for elastic constants is assumed and elemental relative density variables are used for describing the structural layout. The considered robust topology optimization problem is thus formulated as to find the optimal structural topology that minimizes the standard deviation of structural total compliance under the constraint on material volume. To avoid the difficulties associated with directly evaluating the standard deviation of the structural compliance, a convenient computing formula of the objective function is presented based on the stochastic finite element method. In addition, an adjoint variable method is employed for the efficient sensitivity analysis of the objective function. Then, the gradient based optimization algorithm (Method of Moving Asymptotes, MMA) is used to update the design variables in the optimization loop. Finally, three numerical examples for topology optimization of 2D and 3D structures illustrate the applicability and the validity of the present model as well as the proposed numerical techniques. The computational results reveal that the robust topology optimization could yield a material layout with less variation of structural compliance than the conventional deterministic approach. The novelty of the proposed robust topology optimization approach lies in that it introduces the conception of robustness into earlier stage of the structural design, which may be considered as especial useful in some circumstances.