An Implicit Harmonic Balance Method in Graphics Processing Units for Oscillating Blades

An implicit harmonic balance (HB) method for modeling the unsteady nonlinear periodic flow about vibrating airfoils in turbomachinery is presented. An implicit edge-based three-dimensional Reynolds-averaged Navier–Stokes equations (RANS) solver for unstructured grids, which runs both on central processing units (CPUs) and graphics processing units (GPUs), is used. The HB method performs a spectral discretization of the time derivatives and marches in pseudotime, 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 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 tenth standard aeroelastic configuration and a transonic compressor.