Using Pseudotime Marching for the Solution of Harmonic Balance Problems

The aim of this paper is to study approaches to implement implicit pseudotime marching for the harmonic balance system in the frequency domain. We first give a motivation for using pseudotime marching as a solution technique. It turns out that, when the discretization errors of the pseudospectral time-derivative and the pseudotime derivative are neglected, the harmonic balance solution converges to a stable periodic flow, provided that the initial solution is sufficiently close to a stable periodic solution. This motivates the choice of a robust pseudotime marching approach, e.g., an implicit solver based on backward Euler integration. This approach requires the Jacobian of the harmonic balance residual. As for the steady problem, the Jacobian can be approximated without changing the final solution as long as the solver converges. Therefore, a central question is which simplifications are appropriate in terms of the overall efficiency and robustness of the solver. As has been shown in the literature, the spectral time-derivative operator should be taken into account in the implicit system. On the other hand, the linearization of the flow residual can be simplified to a certain extent, especially if the system is solved in the frequency domain. In this paper, we show that, up to terms which scale with the amplitude of the disturbances, the linear system matrix is the sum of a scalar diagonal and a block diagonal matrix with identical blocks for each harmonic. The deviation from this structure is due to to the nonlinearity of the unsteady flow problem. We show that when the unsteadiness is small, the nonlinear coupling terms can be neglected in the implicit solver and the resulting special matrix structure can be exploited to massively speed up the solver. In contrast, when a strong disturbance is simulated, this simplification can lead to significant losses in robustness. To illustrate our findings, we apply the implemented methods to predict the flow response to a disturbance prescribed at the inlet of a transonic compressor. When the disturbance amplitude is increased, a strong oscillation is induced, and the harmonic balance solver converges only when the nonlinear coupling between the harmonics is taken into account.