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. We use the adjoint method to derive sensitivity expressions, which drastically reduces the computational cost compared to direct differentiation as the number of parameters increases. An important feature of this framework is the automatic adjustment of the expansion order of SSM-based ROMs using user-defined error tolerances during the optimization process. We demonstrate the effectiveness of the approach in optimizing the nonlinear response over several numerical examples of mechanical systems. Hence, the proposed framework extends the applicability of SSM-based optimization methods to practical engineering problems, offering a robust tool for the design and optimization of nonlinear mechanical structures.