  For mechanical systems subject to periodic excitation, forced response curves
(FRCs) depict the relationship between the amplitude of the periodic response
and the forcing frequency. For nonlinear systems, this functional relationship
is different for different forcing amplitudes. Forced response surfaces (FRSs),
which relate the response amplitude to both forcing frequency and forcing
amplitude, are then required in such settings. Yet, FRSs have been rarely
computed in the literature due to the higher numerical effort they require.
Here, we use spectral submanifolds (SSMs) to construct reduced-order models
(ROMs) for high-dimensional mechanical systems and then use multidimensional
manifold continuation of fixed points of the SSM-based ROMs to efficiently
extract the FRSs. Ridges and trenches in an FRS characterize the main features
of the forced response. We show how to extract these ridges and trenches
directly without computing the FRS via reduced optimization problems on the
ROMs. We demonstrate the effectiveness and efficiency of the proposed approach
by calculating the FRSs and their ridges and trenches for a plate with a 1:1
internal resonance and for a shallow shell with a 1:2 internal resonance.
