Reduced Order Models for Design Optimization Under Uncertainty.

For large models, design optimization quickly becomes infeasible due to the computational costs of the repeated analysis of the model and is only amplified when uncertainty in the design is incorporated. This work seeks to alleviate the considerable computational cost of optimization, stochastic analysis, and the integration of the two, by employing a surrogate model in place of the full model analysis. The class of surrogate models considered in this work is a Reduced Order Modeling (ROM) technique applying a Galerkin projection scheme to the spatially and temporally discretized governing equation. The key for utilizing the Galerkin projection scheme in optimization and stochastic analysis is choosing a basis able to reproduce the response not only at the design it was built from, but also for changes in the design and/or random variable. The techniques used in this research are studied on to two different applications: a linear static model and a linear dynamic model. The major contribution of this thesis is developing the ROM technique to the two models and integrating them into a design optimization problem under uncertainty. The choice of the basis is a key component in the integration of the ROM in the design under uncertainty process. The basis, for both models, which are able to account for the changes in the design/random variables, is a basis which includes derivatives with respect to the design/random variables as well as catenating basis information from various points in the design/random space. The first study, presents a comparative study of ROM approaches for the uncertainty analysis of linear static problems in structural mechanics by Polynomial Chaos Expansion (PCE) methods. The ROMs of interest are based on a Galerkin projection of the full-order model into subspaces, either spanned by Krylov vectors or derivatives of the system response with respect to the random parameters. The combination of reduced order models and PCE are applied to two different finite element models. The results show a savings up to 96% can be achieved in computational cost while the accuracy lies within 1% of the full model analysis. A further example is incorporated into a design under uncertainty problem. The results confirm observations of previous related studies that combining ROMs and PCE methods significantly reduces the computational costs with only a minor penalty on the accuracy. The next study presents a ROM framework for approximating the transient response of linear elastic structures over a range of design and random parameters. The full-order response is projected onto a lower-dimensional basis spanned by modes computed from a Proper Orthogonal Decomposition (POD) or a multi-linear tensor analysis of full-order model simulation results at multiple calibration points. The basis is further enriched by gradients with respect to the design/random parameters. A truncation strategy is proposed to compensate for the increase in basis vectors due to the proposed enrichment strategies. The accuracy, efficiency, and robustness of the proposed framework are studied with a two-dimensional model problem. The numerical results suggest the proposed ROM approach is well suited for large parameter changes. The number of basis vectors needs to be increased only linearly with the number of design and random parameters to maintain a particular ROM performance. The application of the proposed ROM approach, to robust shape optimization, demonstrates significant savings in computational cost over using full-order models. In conclusion, this research has succeeded in the integration of ROMs in the framework of design optimization under...