Hopf bifurcation calculations for a symmetric airfoil in transonic flow

The application of a sparse matrix solver for the direct calculation of Hopf bifurcation points arising for an airfoil moving in pitch and plunge in a transonic flow is considered. The iteration scheme for solving the Hopf equations is based on a modified Newton's method. Direct solution of the linear system for the updates has previously been restrictive for application of the method, and the sparse solver overcomes this limitation. Results of experiments with the approximation to the Jacobian matrix driving the iteration to convergence are presented. Finally, it is shown that an entire flutter boundary for the NACA0012 airfoil can be traced out in a time comparable to that required for a small number of time-response calculations.