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.

Spectral submanifolds (SSMs) have emerged as accurate and predictive model reduction tools for dynamical systems defined either by equations or data sets. While finite-elements (FE) models belong to the equation-based class of problems, their implementations in commercial solvers do not generally provide information on the nonlinearities required for the analytical construction of SSMs. Here, we overcome this limitation by developing a data-driven construction of SSM-reduced models from a small number of unforced FE simulations. We then use these models to predict the forced response of the FE model without performing any costly forced simulation. This approach yields accurate forced response predictions even in the presence of internal resonances or quasi-periodic forcing, as we illustrate on several FE models. Our examples range from simple structures, such as beams and shells, to more complex geometries, such as a micro-resonator model containing more than a million degrees of freedom. In the latter case, our algorithm predicts accurate forced response curves in a small fraction of the time it takes to verify just a few points on those curves by simulating the full forced-response.

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system’s parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient.