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Structures today may be equipped with passive structural control devices to achieve some performance criteria. The optimal design of these passive control devices, whether via a formal optimization algorithm or a response surface parameter study, requires multiple solutions of the dynamic response of that structure, incurring a significant computational cost for complex structures. These passive control elements are typically point-located, introducing a local change (possibly nonlinear, possibly uncertain) that affects the global behavior of the rest of the structure. When the structure, other than these localized devices, is linear and deterministic, conventional solvers (e.g., Runge-Kutta, MATLAB's ode45, etc.) ignore the localized nature of the passive control elements. The methodology applied in this paper exploits the locality of the uncertain and/or nonlinear passive control element(s) by exactly converting the form of the dynamics of the high-order structural model to a low-dimensional Volterra integral equation. Design optimization for parameters and placement of linear and nonlinear passive dampers, tuned mass dampers, and their combination, as well as their design-under-uncertainty for a benchmark cablestayed bridge, is performed using this approach. For the examples considered herein, the proposed method achieves a two-orders-of-magnitude gain in computational efficiency compared with a conventional method of comparable accuracy.

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Topology optimization under uncertainty (TOuU) often defines objectives and constraints by statistical moments of geometric and physical quantities of interest. Most traditional TOuU methods use gradient-based optimization algorithms and rely on accurate estimates of the statistical moments and their gradients, e.g., via adjoint calculations. When the number of uncertain inputs is large or the quantities of interest exhibit large variability, a large number of adjoint (and/or forward) solves may be required to ensure the accuracy of these gradients. The optimization procedure itself often requires a large number of iterations, which may render TOuU computationally expensive, if not infeasible. To tackle this difficulty, we here propose an optimization approach that generates a stochastic approximation of the objective, constraints, and their gradients via a small number of adjoint (and/or forward) solves, per optimization iteration. A statistically independent (stochastic) approximation of these quantities is generated at each optimization iteration. The total cost of this approach is only a small factor larger than that of the corresponding deterministic TO problem. We incorporate the stochastic approximation of objective, constraints and their design sensitivities into two classes of optimization algorithms. First, we investigate the stochastic gradient descent (SGD) method and a number of its variants, which have been successfully applied to large-scale optimization problems for machine learning. Second, we study the use of the proposed stochastic approximation approach within conventional nonlinear programming methods, focusing on the Globally Convergent Method of Moving Asymptotes (GCMMA). The performance of these algorithms is investigated with structural design optimization problems utilizing a Solid Isotropic