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On the conservation of physical properties in operator interpolation of parameterized hydrodynamic systems

Yuto Nakamura, Shintaro Sato, Naofumi Ohnishi

TL;DR

This work addresses predicting parameter-dependent fluid dynamics by constructing operator-interpolated ROMs that preserve key physical properties. It develops two ROMs: GapiROM, which interpolates POD–Galerkin bases on the Grassmann manifold and projects to obtain reduced operators, and DoiROM, which interpolates continuous-time operators derived from DMD with subspace alignment. Across circular-cylinder and elliptical-cylinder flows, DoiROM consistently captures both fundamental frequencies and harmonics, while GapiROM accurately tracks the base-flow–driven changes when subspaces are properly interpolated; however, it struggles with higher harmonics in periodic flows. The results demonstrate that operator fidelity under interpolation hinges on subspace quality, basis variation with parameters, and careful handling of numerical errors, offering a robust framework for fast, physically meaningful parametric ROMs in wake dynamics.

Abstract

Reduced-order models (ROMs) that capture changes in fluid systems due to variations in parameters, such as the Reynolds number or the shape of a stationary body placed in the flow, are attracting increasing attention in engineering applications. In this study, we identify linear operators that characterize the behavior of fluid systems across a wide parameter range by using flow field datasets at several representative parameter values. We then comprehensively assess the applicability of ROMs constructed through the interpolation of these operators. Specifically, we consider two intrusive operator-based ROMs: one derived from Galerkin projection and the other based on operator inference using dynamic mode decomposition (DMD). The performance of these ROMs is evaluated for flows around circular and elliptical cylinders over a range of Reynolds numbers and aspect ratios. The Galerkin-based ROM successfully predicts only the eigenvalue and eigenmode corresponding to the fundamental frequency of the Kármán vortex shedding, while other frequencies are not captured and show a decaying behavior. In contrast, the DMD-based ROM accurately predicts both the fundamental frequency and its higher harmonics. Furthermore, visualization of the linear operator matrix elements reveals that interpolation fails when the subspace includes bases contaminated by numerical errors. However, by carefully selecting the subspace dimension and reference conditions, it is possible to accurately predict eigenvalues and corresponding modes even under conditions where multiple low-frequency modes exist outside the harmonic structure of the fundamental frequency. These findings underscore the robustness of the DMD-based parametric operator ROM approach.

On the conservation of physical properties in operator interpolation of parameterized hydrodynamic systems

TL;DR

This work addresses predicting parameter-dependent fluid dynamics by constructing operator-interpolated ROMs that preserve key physical properties. It develops two ROMs: GapiROM, which interpolates POD–Galerkin bases on the Grassmann manifold and projects to obtain reduced operators, and DoiROM, which interpolates continuous-time operators derived from DMD with subspace alignment. Across circular-cylinder and elliptical-cylinder flows, DoiROM consistently captures both fundamental frequencies and harmonics, while GapiROM accurately tracks the base-flow–driven changes when subspaces are properly interpolated; however, it struggles with higher harmonics in periodic flows. The results demonstrate that operator fidelity under interpolation hinges on subspace quality, basis variation with parameters, and careful handling of numerical errors, offering a robust framework for fast, physically meaningful parametric ROMs in wake dynamics.

Abstract

Reduced-order models (ROMs) that capture changes in fluid systems due to variations in parameters, such as the Reynolds number or the shape of a stationary body placed in the flow, are attracting increasing attention in engineering applications. In this study, we identify linear operators that characterize the behavior of fluid systems across a wide parameter range by using flow field datasets at several representative parameter values. We then comprehensively assess the applicability of ROMs constructed through the interpolation of these operators. Specifically, we consider two intrusive operator-based ROMs: one derived from Galerkin projection and the other based on operator inference using dynamic mode decomposition (DMD). The performance of these ROMs is evaluated for flows around circular and elliptical cylinders over a range of Reynolds numbers and aspect ratios. The Galerkin-based ROM successfully predicts only the eigenvalue and eigenmode corresponding to the fundamental frequency of the Kármán vortex shedding, while other frequencies are not captured and show a decaying behavior. In contrast, the DMD-based ROM accurately predicts both the fundamental frequency and its higher harmonics. Furthermore, visualization of the linear operator matrix elements reveals that interpolation fails when the subspace includes bases contaminated by numerical errors. However, by carefully selecting the subspace dimension and reference conditions, it is possible to accurately predict eigenvalues and corresponding modes even under conditions where multiple low-frequency modes exist outside the harmonic structure of the fundamental frequency. These findings underscore the robustness of the DMD-based parametric operator ROM approach.

Paper Structure

This paper contains 20 sections, 108 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of fundamental method and its theory
  3. POD
  4. POD-Galerkin projection approach
  5. DMD
  6. Grassmann manifold interpolation
  7. Parameterized ROM with operator interpolation
  8. Galerkin-projection-based interpolation ROM (GapiROM)
  9. DMD-based operator interpolation ROM (DoiROM)
  10. Result and discussion
  11. Growth of oscillations in flow around a circular cylinder at varying Reynolds numbers
  12. Periodic flow around a circular cylinder
  13. Flow around elliptical cylinders with aspect ratio variation
  14. Conclusion
  15. Derivation of basis transformations using exponential and logarithmic maps on the Grassmann manifold
  16. ...and 5 more sections

Figures (21)

  • Figure 1: Conceptual diagram of interpolation on the Grassmann manifold.
  • Figure 2: Conceptual diagram of the GapiROM.
  • Figure 3: Conceptual diagram of the DoiROM.
  • Figure 4: Spatial distribution of (a) the steady flow field (base flow) and (b) a representative snapshot from the dataset $X$.
  • Figure 5: (a) Eigenvalue distributions of the linearized operator at $Re = 50$ and $100$. (b) Reynolds number dependence of the growth rate and frequency. (c) Spatial distributions of the eigenmodes at $Re = 50$ (left) and $Re = 100$ (right).
  • ...and 16 more figures