An implicit harmonic balance method for modeling the unsteady non-linear periodic flow about vibrating airfoils in turbomachinery is presented. As departing point, an implicit edge-based three-dimensional Reynolds Averaged Navier-Stokes equations solver for unstructured grids that runs both on central processing units (CPUs) and graphics processing units (GPUs) is used. The harmonic balance method performs a spectral discretization of the time derivatives and marches in pseudo-time a new system of equations where the unknowns are the variables at different time samples.

The application of the method to vibrating airfoils is discussed. It is shown that a time spectral scheme may achieve the same temporal accuracy at a much lower computational cost than a Backward Finite Difference method at the expense of using more memory.

The performance of the implicit solver has been assessed with several application examples. A speed-up factor of 10 is obtained between the spectral and finite difference version of the code whereas and an additional speed-up factor of 10 is obtained when the code is ported to GPUs, totalizing a speed factor of 100. The performance of the solver in GPUs has been assessed using the 10th standard aeroelastic configuration and a transonic compressor.

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