Methodology for numerically stabilizing a harmonic balance based aeroelastic solution approach
Presented is a methodology for numerically stabilizing harmonic balance based aeroelastic solution approaches. The governing harmonic balance based aeroelastic equations are a stationary/steady system of equations. In order to solve these equations, many recent solution method techniques in development add a pseudo time marching term to these equations, and the resulting pseudo time unsteady harmonic balance based aeroelastic equations are then pseudo time marched to the final stationary/steady harmonic balance aeroelastic solution. We demonstrate that in certain cases, pseudo time marching techniques can be unstable when pseudo time-marching the pseudo unsteady harmonic balance based aeroelastic equations. The reason for this is that the pseudo time-step may have the incorrect sign. In this paper, we present a methodology which uses gradient information provided by the unsteady aerodynamic Jacobian matrix to determine the correct sign, that is positive or negative, for the pseudo time-step, along with its maximal absolute value, which insures numerical stability of the pseudo time marching approach. Since terms of the unsteady aerodynamic Jacobian matrix are typically not highly sensitive to frequency, it is possible for the Jacobian matrix to only need to be computed once up front for a single frequency. We demonstrate the numerical stabilizing methodology for both a specified gust forced aeroelastic response problem, as well as a self-excited aeroelastic limit cycle oscillation problem, for a transonic Reynolds Averaged Navier-Stokes computational model of the F-16 wing. We show that the new methodology is able to produce converged aeroelastic solutions in cases where other approaches diverge.
