Model order reduction for multidisciplinary design optimization in higher-dimensional parameter spaces
Keywords: Aeronautics and Astronautics.

This thesis introduces a new framework for solving optimization problems efficiently in higher-dimensional parameter spaces constrained by partial differential equations (PDEs). These problems, such as optimal design and optimal control, have a central and vital role in industrial, technological, economic, medical, and environmental applications. Even though the state of numerical simulation for solving PDEs has reached impressive levels of maturity, enabling the prediction of complicated dynamics, solving these problems, especially large scale industrial problems, is computationally intensive. The problem is aggravated in a many-query scenario, such as PDE-constrained optimization, where the PDE is repeatedly solved for various configurations and design parameters. This situation is further complicated when the optimization problem has many design parameters, typical in industrial settings, where the computational cost can increase by potentially many orders of magnitude. To make the process of solving repeatedly large scale PDEs tractable, this dissertation employs projection-based reduced-order models (PROMs) based on the method of snapshots and proper orthogonal decomposition (POD). In recent years, PROMs have shown potential for solving complex parametric PDEs but with a relatively small number of design parameters. However, with a large number of design parameters, as in industrial PDE-constrained optimization problems, the large increase in computational cost due to the training in higher-dimensional parameter spaces and the PROMs' size makes the use of these surrogate models impractical. This dissertation introduces the concept of piecewise-global reduced-order basis and hyperreduced projection-based reduced-order models (HPROMs) to address these issues. This concept leverages inexpensive approximation models to significantly reduce the cost of solving, for instance, nonlinear PDE-constrained optimization problems in higher-dimensional parameter spaces. A piecewise-global HPROM, for a fixed level of the desired accuracy, is more computationally efficient than its counterpart approach based on a global HPROM. In fact, the piecewise-global model needs to be accurate only in a local region of the design space allowing the model to be of a smaller dimension. The proposed method is shown to find the optimal aerodynamic shape of a full aircraft configuration with roughly a 30-fold reduction in computational time for the optimization process compared to using a global HPROM. This dissertation also introduces the method of active manifold (AM) to mitigate the curse of dimensionality associated with high-dimensional design spaces during the PROMs training. The AM is discovered using a deep convolutional autoencoder for dimensionality reduction. It is proposed as a superior alternative to the concept of active subspace (AS), whose capabilities are limited by the associated affine approximation. It has the advantage of allowing the discovery of an approximation manifold of a lower dimension than an optimal, linear active subspace. Hence, it can reduce the computational cost of the offline phase associated with the construction of projection-based reduced-order models, reduce the number of iterations performed by an optimization procedure, and improve the computed solution's optimality. The AM method is shown to find the optimal solution to an optimization problem, reducing computational time and the dimension of design space compared to solving the problem using the AS method. The new approach is intertwined with the concepts of projection-based model order reduction, demonstrating the feasibility of the overall AM- and PROM-based computational framework for accelerating the solution of complex PDE-constrained optimization problems. To construct the AM for reducing the dimensionality of the optimization problem, instead of performing an empirical, a priori sampling, as proposed in the original AS method, this dissertation presents an alternative approach where the information is sampled by following the solution trajectory of an economical version of the optimization problem of interest. This strategy has the advantage of reducing the number of sampling points used to construct the AM and using information related to the optimization problem of interest, such as the problem constraints. The new sampling strategy is also applied to the method of active subspace, giving origin to a new take on the method of AS. The new take is shown to compute the AS in about half the time required by the original AS method. Additionally, the AS obtained using the new take on the method of AS is shown to outperform the AS obtained by the original AS method when solving a multidisciplinary design optimization (MDO) problem