The presence of uncertainty in engineering systems is ubiquitous. Such uncertainties are typically due to intrinsic variabilities in the system or manufacturing processes, as well as the lack of knowledge in precisely describing the governing physics in terms of mathematical/computational models. For example, due to the current limitations of the additive manufacturing (AM) process, micro-scale defects often appear when AM is used to produce structural components that are designed across multiple scales using topology optimization. The macroscale structural properties are affected adversely because of these random defects, which ultimately limits the efficacy of topology optimization for applications with sensitive structural components. Accounting for these uncertainties in the design optimization process requires, for example, computation of the statistical moments of the objective and its gradients using a Monte Carlo approach. This may lead to exorbitant computational costs as many forward and adjoint solves are needed to perform. To alleviate this computational burden, in this talk, a new design optimization paradigm based on stochastic gradients will be introduced. In this approach, stochastic approximations of the gradients, using only a handful of random samples of the uncertainty, are constructed at every optimization iteration. This reduces the per iteration computational cost of design under uncertainty by a significant amount. Popular variants of the stochastic gradient descent algorithm (e.g., AdaGrad and Adam) from machine learning literature are used within this framework to illustrate its efficacy. Design problems with microscale uncertainty at each integration point in the finite element mesh lead to an extremely high-dimensional problem in the space of uncertain parameters and consequently large computational cost. The stochastic gradient-based approach can be used here to resolve the high-dimensional microscale uncertainty and reduce the cost of design optimization significantly. This will be illustrated in this talk for topology optimization of structures made of chopped fiber composite and randomly distributed inclusions in a matrix with uncertain material properties as microstructures. The stochastic gradient-based approach will also be extended to topology optimization problems with reliability constraints. Further, a novel bi-fidelity version of the stochastic gradient descent algorithm to improve the convergence of design optimization under uncertainty will be discussed in this talk. Finally, the influence of machine learning techniques for modeling and design optimization under uncertainty will be briefly explored.

Dr. Subhayan De is an Assistant Professor in the Department of Mechanical Engineering and heads the Uncertainty Quantification, Learning, Inference, and Design (UQLID) lab at NAU. Prior to joining NAU, he was a postdoctoral research associate in Aerospace Engineering Sciences at the University of Colorado Boulder. Subhayan received his Ph.D. in Civil Engineering from the University of Southern California (USC) in 2018, where he was supported by a Viterbi Ph.D. Fellowship, a Gammel Scholarship, and several NSF grants. At USC, he worked on probabilistic model validation, machine learning, uncertainty quantification, and structural control design. Subhayan also holds an MS in Electrical Engineering from USC and an MEng in Structural Engineering from the Indian Institute of Science, Bangalore. He received his BEng in Civil Engineering from Jadavpur University, Kolkata.