Efficient Adjoint Petrov-Galerkin Reduced Order Models for fluid flows governed by the incompressible Navier-Stokes equations

TL;DR

This paper introduces an efficient Adjoint Petrov-Galerkin reduced order model for fluid flows governed by the Navier-Stokes equations, improving accuracy and computational efficiency over standard methods.

Contribution

The paper develops a new efficient APG-ROM formulation that leverages polynomial structure and data-driven optimization to enhance stability and reduce computational costs.

Findings

Superior accuracy and stability of APG-ROM over standard Galerkin ROMs

Significant reduction in computational cost with the eAPG formulation

Effective modeling of 3D turbulent flow around a cylinder

Abstract

This research paper investigates the Adjoint Petrov-Galerkin (APG) method for reduced order models (ROM) and fluid dynamics governed by the incompressible Navier-Stokes equations. The Adjoint Petrov-Galerkin ROM, derived using the Mori-Zwanzig formalism, demonstrates superior accuracy and stability compared to standard Galerkin ROMs. However, challenges arise due to the time invariance of the test basis vectors, resulting in high computational requirements. To address this, we introduce a new efficient Adjoint Petrov-Galerkin (eAPG) ROM formulation, extending its application to the incompressible Navier-Stokes equations by exploiting the polynomial structure inherent in these equations. The offline and online phases partition eliminates the need for repeated test basis vector evaluations. This improves computational efficiency in comparison to the general Adjoint Petrov-Galerkin ROM formulation. A novel approach to augmenting the memory length, a critical factor influencing the stability and accuracy of the APG-ROM, is introduced, employing a data-driven optimization. Numerical results for the 3D turbulent flow around a circular cylinder demonstrate the efficacy of the proposed approach. Error measures and computational cost evaluations, considering metrics such as floating point operations and simulation time, provide a comprehensive analysis.