  We extend linear input/output (resolvent) analysis to take into account
nonlinear triadic interactions by considering a finite number of harmonics in
the frequency domain using the harmonic balance method. Forcing mechanisms that
maximize the drag are calculated using a gradient-based ascent algorithm. By
including nonlinearity in the analysis, the proposed frequency-domain framework
identifies the worst-case disturbances for laminar-turbulent transition. We
demonstrate the framework on a flat-plate boundary layer by considering
three-dimensional spanwise-periodic perturbations triggered by a few optimal
forcing modes of finite amplitude. Two types of volumetric forcing are
considered, one corresponding to a single frequency/spanwise-wavenumber pair,
and a multi-harmonic where a harmonic frequency and wavenumber are also added.
Depending on the forcing strategy, we recover a range of transition scenarios
associated with K-type and H-type mechanisms, including oblique and
Tollmien-Schlichting waves, streaks and their breakdown. We show that
nonlinearity plays a critical role in optimizing growth by combining and
redistributing energy between the linear mechanisms and the higher perturbation
harmonics. With a very limited range of frequencies and wavenumbers, the
calculations appear to reach the early stages of the turbulent regime through
the generation and breakdown of hairpin and quasi-streamwise staggered
vortices.
