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Topology optimization under uncertainty using a stochastic gradient-based approach

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Abstract

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 topology optimization 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 material with penalization (SIMP), as well as an explicit level set method. These investigations, conducted on both two- and three-dimensional structures, illustrate the efficacy of the proposed stochastic gradient approach for TOuU applications.

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Acknowledgments

The authors acknowledge the support of the Defense Advanced Research Projects Agency (DARPA) TRADES project under agreement HR0011-17-2-0022. Computations for the results presented in Section 3 are performed on Lonestar5, a high-performance computing resource operated by the Texas Advanced Computing Center (TACC) at University of Texas, Austin.

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  1. Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, 80309, USA

    Subhayan De, Jerrad Hampton, Kurt Maute & Alireza Doostan

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  1. Subhayan De
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Correspondence to Alireza Doostan.

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Replication of results

The optimization algorithms of Sections 2.4 and 2.5 have been implemented in MATLAB and are accessible via the GitHub page https://github.com/CU-UQ/TOuU. The link also includes the necessary input files for the elastic bedding example of Section 3.2. The TO and XFEM calculations were performed using an in-house solver that is not at the stage of being publicly available. However, the MATLAB codes used in this study can interface with other TO and XFEM codes.

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De, S., Hampton, J., Maute, K. et al. Topology optimization under uncertainty using a stochastic gradient-based approach. Struct Multidisc Optim 62, 2255–2278 (2020). https://doi.org/10.1007/s00158-020-02599-z

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