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The paper presents a rigorous formulation of adjoint systems to be solved for a robust design optimization using the first-order second-moment method. This formulation allows to apply the method for any objective function, which is demonstrated by considering deformation at certain point and maximum stress as objectives subjected to random material stiffness and geometry, respectively. The presented approach requires the solution of at most three additional adjoint systems per uncertain system response, when compared to the deterministic case. Hence, the number of adjoint systems to be solved is independent of the number of random variables. This comes at the expense of accuracy, since the objective functions are assumed to be linear with respect to random parameters. However, the application to two standard cases and the validation with Monte Carlo simulations show that the approach is still able to find robust designs.

Explicitly considering fail-safety within design optimization is computationally very expensive, since every possible failure has to be considered. This requires solving one finite element model per failure and iteration. In topology optimization, one cannot identify potentially failing structural members at the beginning of the optimization. Hence, a generic failure shape is applied to every possible location inside the design domain. In the current paper, the maximum stress is considered as optimization objective to be minimized, since failure is typically driven by the occurring stresses and thus of more practical relevance than the compliance. Due to the local nature of stresses, it is presumed that the optimization is more sensitive to the choice of the failure shape than compliance-based optimization. Therefore, various failure shapes, sizes and different numbers of failure cases are investigated and compared on the basis of a general load-path-based evaluation scheme. Instead of explicitly considering fail-safety, redundant structures are obtained at much less computational cost by controlling the maximum length scale. A common and easy to implement maximum length scale approach is employed and fail-safe properties are determined and compared against the explicit fail-safe approach.

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