Adjoint and Acceleration Methods for Projection-based Reduced Order Modeling

Abstract

Computational modeling is a pillar of modern aerospace research and is increasingly becoming more important as computer technology and numerical methods grow more powerful and sophisticated. However, computational modeling remains expensive for many aerospace engineering problems, including high-fidelity solutions to three-dimensional unsteady simulations, large-scale aeroservoelastic control problems, and multidisciplinary design optimization. Reduced-order models (ROMs) have therefore garnered interest as an alternative means of preserving high fidelity at a lower computational cost. Among these methods is the class of projection-based ROMs, which utilizes the original physics and equations of the high-fidelity system but resolves the state projected onto a lower-order trial manifold with the system projected onto a low-order test manifold. The lower-order trial manifold is typically chosen to be a linear basis, and this thesis focuses on the proper orthogonal decomposition method (POD). Hyper-reduction is also necessary to reduce the complexity of nonlinear problems, and this thesis is focused on the the discrete empirical interpolation method (DEIM), an interpolation method that uses a sparse sampling of the nonlinear function values. The benefit of the method of snapshots is that given a representative set of solution samples, a linear basis that projects the solution space with very low error can be constructed. However, accuracy in state projection is only one part requirement for a ROM to be useful for engineering applications.

Low errors in state projection do not necessarily mean that outputs are accurately predicted as the proportion of the domain that is used to calculate the output may be relatively very small. Quantification of the output error is thus important to assessing the quality of a ROM. Methods for estimating output error for POD-DEIM models exist; however, the application of these methods to fine-grain adaptation is limited. Furthermore, the commonly used Galerkin formulation of ROMs, where the test and trial spaces are the same, is known to be inaccurate for many problems. These inaccuracies arise from the inability of a state basis to appropriately project the physical system. However, the construction of a tailored Petrov-Galerkin test basis that yields the appropriate dynamics from system projection is not trivial. Finally, the implementation of ROMs for engineering applications may be difficult due to their intrusive nature. Projection-based models require the ability to call subroutines of the original high-fidelity model; however, code modification may be complex and hindered by intellectual property and export control protections.

The research presented investigates the use of adjoint-based methods to achieve those goals. For error estimation, this thesis applies adjoint-weighted residuals in order to assess output error, presents methods for localizing the contribution of output error to individual ROM degrees of freedom, and derives adaptation schemes using those error localizations. For constructing dynamically useful test bases, this thesis derives a novel test basis that yields dynamics from system projection that minimize the state error of the ROM. This test basis is composed of the reduced state adjoints and stability and convergence studies of this test basis are presented. Additionally, to overcome issues of portability and intrusive implementation, a novel hybridization of machine learning and hyper-reduced methods is presented that supplants the intrusive portions of projection-based model reduction with element-level neural networks. The implementation of the neural networks at a low level allows for ROM time-marching to be used, which appropriately propagates the influence of states on future times.