Using Broyden's Method to Improve the Computational Performance of a Harmonic Balance Aeroelastic Solution Technique

Abstract

Presented is a technique that significantly improves the computational performance of an existing harmonic balance-based flutter onset and limit-cycle oscillation aeroelastic solution methodology. The existing harmonic balance aeroelastic solution methodology has a computational cost proportional to the number of degrees of freedom of the aeroelastic model because an aeroelastic Jacobian matrix must be evaluated at each iteration step of the aeroelastic solution process. The following paper demonstrates the use of Broyden's method, which provides, after the first iteration, an estimate of the aeroelastic Jacobian matrix at each subsequent iteration step. The use of Broyden's method enables the existing aeroelastic solution methodology to have a computational cost that is "almost" independent of the number of degrees of freedom of the aeroelastic model. Presented are flutter-onset results for an inviscid flow model of the AGARD 445.6 wing configuration and aeroelastic limit cycle oscillation results for a three-degree-of-freedom plunge, pitch, and flap with flap free-play transonic airfoil configuration computed with the use of Broyden's method. For the AGARD 445.6 wing configuration, it is demonstrated that using Broyden's method reduces the computational cost of the existing harmonic balance aeroelastic solution methodology by roughly a factor of five. The number of degrees of freedom for the aeroelastic model of the AGARD 445.6 wing is proportional to the number of structural modes, and five is the number of structural modes provided for the AGARD 445.6 wing. For the particular three-degree-of-freedom plunge, pitch, and flap with flap free-play transonic airfoil configuration examined in this paper, the aeroelastic model consists of 33 degrees of freedom, and limit cycle oscillation solutions take roughly two to three days to compute when using Broyden's method. When using the existing harmonic balance aeroelastic solution methodology, such computations are expected to take a factor 33 times longer than this. Thus the great benefit of using Broyden's method.