SUMMARY
Model order reduction approaches, such as proper orthogonal decomposition (POD)‐Galerkin projection, provide a systematic manner to construct Reduced‐Order Models (ROM) from pregenerated high‐fidelity datasets. The current study focuses on the stabilization of ROMs built from high‐fidelity simulation data of a supersonic flow passing a circular cylinder, which features strong interactions between shockwaves and vortices. As shown in previous literatures and the current study, an implicit subspace correction (ISC) method is efficient in the stabilization of similar problems, but its accuracy is not consistent when applied on different ROMs; on the other hand, an eigenvalue reassignment (ER) method delivers superb accuracy when the mode number is small, but becomes too expensive and less robust as the number increases. A Hybrid method is proposed here to balance the computational cost while improving the overall robustness/accuracy in ROM stabilization. The Hybrid method first handles the majority of the modes using the ISC method and then applies the ER method to fine tune a smaller number of modes under a constraint for accuracy. Furthermore, when the typical L2 inner product is changed to a symmetry inner product in both POD computation and Galerkin projection, the performance of the stabilized ROMs is substantially improved for all methods.
With the increasing need for model-based numerical simulations with computational efficiency and mathematical rigorousness, reduced order models (ROM) derived from Galerkin projection have attracted much attention for the study of their performance in accuracy and stability. Typical POD-Galerkin projection approach computes basis functions from numerical snapshots simulated by high fidelity full order models (FOM). In the current study, snapshots were taken from direct numerical simulation using high-resolution Weighted Essentially Non-Oscillatory (WENO) scheme to solve nonlinear Euler equations. However, severe instabilities were noticed when a ROM derived by typical POD-Galerkin approach was directly applied to the simulation of a supersonic flow passing over a fixed cylinder, where strong shock-vortex interaction appears in downstream flow. To stabilize the POD-Galerkin ROM, we first compared typical stabilization treatments existed in literatures. Then, we proposed a hybrid approach for the improvement of accuracy, efficiency and robustness, and achieved better performance in the current study.
The systematic and physics‐infused construction of a projection‐based reduced‐order model (ROM) shows the capability to replicate the original high‐dimensional system's dynamical evolution but with a fractional computational cost. However, certain nonlinear features and high‐frequency contributions may be lost throughout the aggressive order reduction. Thus, ROMs in a broad category of fluid dynamics applications require stabilization and closure methods to compensate for the key contributions missed through the model order reduction. A new stabilization method for nonlinear ROMs is proposed in this study that learns a linear control law to drive the nonlinear ROM toward maximum agreement with the full‐order model, where a total power constraint guarantees the stability of the nonlinear ROM. This new method achieves both stability and accuracy for nonlinear proper orthogonal decomposition‐Galerkin ROMs in two chosen applications with strong shock‐wake interactions and unsteady oscillations, which trigger strong instabilities in the original ROMs before stabilization. A multistage layout is designed to further enhance the proposed stabilization method for more efficient and robust stabilization of nonlinear ROMs with a large number of unstable modes.
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