Fourier Series for Accurate, Stable, Reduced-Order Models for Linear CFD Applications

Abstract

A new method, Fourier model reduction (FMR), for obtaining stable, accurate, low-order models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discrete-time Fourier coefficients of the large system transfer function are calculated and used to construct the Hankel matrix of an intermediate system with guaranteed stability. Explicit balanced truncation formulae are then applied to obtain the final reduced-order model, whose size is determined by the Hankel singular values of the intermediate system. In this paper, the method is applied to two computational fluid dynamic systems, which model unsteady motion of a twodimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, guarantees the stability of the reduced-order model, uses both input and output information, and is valid over a wide range of frequencies.