A Methodology for Projection-Based Model Reduction with Black-Box High-Fidelity Models

TL;DR

This paper introduces a projection-based model reduction methodology for black-box high-fidelity models, enabling efficient parametric simulations by approximating linear operators directly from discretized equations, demonstrated on CFD problems.

Contribution

The paper proposes a novel approach to perform model reduction on black-box models with known governing equations, using discretization to approximate operators for ROM construction.

Findings

Successfully applied to a nonlinear PDE with exponential non-linearity.

Demonstrated on compressible inviscid flow past an airfoil.

Generated online databases of ROMs for various parameters.

Abstract

This paper presents a methodology that enables projection-based model reduction for black-box high-fidelity models such as commercial CFD codes. The methodology specifically addresses the situation where the high-fidelity model may be a black-box but there is complete knowledge of the governing equations. The main idea is that the linear operator matrix, resulting from the discretization of the linear differential terms is approximated directly using a suitable discretization method such as the Finite Volume Method and requires only the computational grid as input. In this regard, the governing equations are first cast in terms of a set of scalar observables of the state variables, leading to a linear set of equations. By applying the snapshots of the observables to the discrete linear operator, a right hand side vector is obtained, providing the necessary system matrices for the Galerkin projection step. This way an online database of ROMs are generated for various parameter snapshots which are then interpolated online to predict the state for new parameter instances. Finally, the reduced order model is posed as a non-linear constrained optimization problem that can be solved at a significantly cheaper cost compared to the full order model. The method is successfully demonstrated on on a canonical non-linear parametric PDE with exponential non-linearity, followed by the compressible inviscid ow past the NACA0012 airfoil. As a first step, this paper focuses only on establishing feasibility of the method.