Sensitivity analysis of limit cycle oscillations

Many unsteady problems equilibrate to periodic behavior. For these problems the sensitivity of periodic outputs to system parameters are often desired, and must be estimated from a finite time span or frequency domain calculation. Sensitivities computed in the time domain over a finite time span can take excessive time to converge, or fail altogether to converge to the periodic value. Additionally, finite span outputs can exhibit local extrema in parameter space which the periodic outputs they approximate do not, hindering their use in optimization. We derive a theoretical basis for this error and demonstrate it using two examples, a van der Pol oscillator and vortex shedding from a low Reynolds number airfoil. We show that output windowing enables the accurate computation of periodic output sensitivities and may allow for decreased simulation time to compute both time-averaged outputs and sensitivities. We classify two distinct window types: long-time, over a large, not necessarily integer number of periods; and short-time, over a small, integer number of periods. Finally, from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation, respectively.