Topology optimization of nonlinear forced response curves via reduction on spectral submanifolds

Forced response curves (FRCs) of nonlinear systems can exhibit complex behaviors, including hardening/softening behavior and bifurcations. Although topology optimization holds great potential for tuning these nonlinear dynamic responses, its use in high-dimensional systems is limited by the high cost of repeated response and sensitivity analyses. To address this challenge, we employ the spectral submanifolds (SSMs) reduction theory, which reformulates the periodic response as the equilibria of an associated reduced-order model (ROM). This enables efficient and analytic evaluation of both response amplitudes and their sensitivities. Based on the SSM-based ROM, we formulate optimization problems that optimize the peak amplitude, the hardening/softening behavior, and the distance between two saddle-node bifurcations for an FRC. The proposed method is applied to the design of nonlinear MEMS devices, achieving targeted performance optimization. This framework provides a practical and efficient strategy for incorporating nonlinear dynamic effects into the topology optimization of structures.