Petrov-Galerkin Projection-Based Model Reduction with an Optimized Test Space

Computational modeling is a pillar of modern aerospace research and is increasingly becoming more important in the engineering process as computer technology and numerical methods grow more powerful and sophisticated. However, computational modeling remains expensive for many aerospace engineering problems, such as large-scale aeroservoelastic control problems and multidisciplinary design optimization. Model reduction methods have therefore garnered interest as an alternative means of preserving high-fidelity in computational fluid dynamics (CFD) at a much lower computational cost. Specifically, projection-based model reduction methods seek to identify highly-important modes of a full-order model (FOM), for use in a projection of the FOM to a low-rank but highlyaccurate reduced model. Many projection-based model reduction techniques are based on proper-orthogonal decomposition (POD), in which the approximation/trial basis is obtained from a singular value decomposition of a set of solution samples of the full-order model. Many applications derived from POD use a Galerkin approach, in that the test space used to project the equations is the same as the trial space. Yet, research has shown accuracy and stability benefits of using a Petrov-Galerkin approach, in which the test and trial spaces are different. An active area of research concerns the determination of this test space. This paper adapts methods for defining test functions for discontinuous Petrov-Galkerin PDE systems to the case of reduced models. Following the derivation, a steady linear model reduction example is used to verify the mathematical framework presented. Following this, a steady nonlinear problem and an unsteady nonlinear problem are presented to test the robustness of the optimal test space and to compare the method’s performance to the performance of other commonly-used POD methods.