Topology optimization of nonlinear structural dynamics with invariant manifold-based reduced order models

We present a structural topology optimization method to tailor the hardening/softening dynamic response of nonlinear mechanical systems. The coefficient that controls this behavior is computed analytically using the third-order normal-form parametrization of the Lyapunov subcenter manifold, which eliminates the need for expensive full-order simulations and numerical continuation to approximate the so-called backbone curve of the system. The method further leverages the adjoint method for efficiently computing sensitivities of the objective function and constraints, while the explicit formulation of nonlinear internal elastic forces through tensor notation simplifies these evaluations. Notably, this tensorial approach is computationally efficient, especially when applied to a regular grid of elements. Consequently, the proposed approach offers a robust and efficient framework for optimizing the dynamic performance of nonlinear mechanical structures modeled with high-dimensional finite element models. The findings are corroborated through examples of two geometrically nonlinear systems, a Messerschmitt-Bölkow-Blohm (MBB) beam and a microelectro-mechanical system (MEMS) inertial resonator.