Book Description

In this study, we demonstrate the ability to perform large-scale PDE-constrained optimizations using Large Eddy Simulation (LES). We first outline the challenges associated with performing gradient-based optimization using LES, specifically chaotic divergence of the sensitivity functions. We then demonstrate that shape optimization using LES and Mesh Adaptive Direct Search Method (MADS) is feasible for aerodynamic design. Next, we introduce a Dynamic Polynomial Approximation (DPA) procedure, which allows the high-order solution polynomial representation used by the flow solver to be increased, or decreased, depending on the poll size being used by MADS. This allows rapid convergence towards the optimal design space using lower-fidelity simulations, followed by an automatic transition to higher-fidelity simulations when close to the optimal design point. Additionally, this study proposes a new physics-constrained data-driven approach for sensitivity analysis and uncertainty quantification of large-scale chaotic dynamical systems. Unlike conventional sensitivity analysis, the proposed approach can manipulate the unsteady sensitivity function (i.e., tangent) for PDE-constrained optimizations. In this new approach, high-dimensional governing equations from physical space are transformed into an unphysical space (i.e., Hilbert space) to develop a closure model in the form of a Reduced-Order Model (ROM). Afterward, a new data sampling approach is proposed to build a data-driven approach for this framework. To compute sensitivities, Least-Squares Shadowing (LSS) minimization is applied to the ROM. It is shown that the proposed approach can capture sensitivities for large-scale chaotic dynamical systems, where Finite Difference (FD) approximations fail. Therefore, we expect that implementing the proposed optimization approach can be applied to large-scale chaotic problems, such as turbulent flows, and this approach significantly reduces computational cost and data storage requirements.