Adjoint sensitivities for the optimization of nonlinear structural dynamics via spectral submanifolds

This work presents an optimization framework for tailoring the nonlinear dynamic response of lightly damped mechanical systems using spectral submanifold (SSM) reduction. We derive the SSM-based backbone curve and its sensitivity with respect to parameters up to arbitrary polynomial orders, enabling efficient and accurate optimization of the nonlinear frequency-amplitude relation. Sensitivity expressions are obtained via the adjoint method, which significantly reduces computational cost compared to direct differentiation as the number of parameters increases. A key feature of the framework is the automatic adjustment of the expansion order of SSM-based reduced-order models using user-defined error tolerances during optimization. We demonstrate the effectiveness of the approach through several numerical examples, including the first application of topology optimization in nonlinear structural dynamics via arbitrary-order SSMs. Hence, the proposed framework extends the applicability of SSM-based optimization to practical engineering problems, providing a robust tool for designing and optimizing nonlinear mechanical structures.