Topology Optimization in nonlinear dynamics: internal and parametric resonances

Topology optimization techniques for the design of MEMS devices in a linear dynamics framework have already been studied in several works in the literature. However, since there is no control of the degree of nonlinearity of the structure during the optimization process, the outcome of these routines often departs significantly from linearity. The consequence is that the device cannot be used.

Recently, we have implemented Topology Optimization (TO) routines to tune the hardening (softening) behavior of the system, by leveraging the 3rd order coefficient of the backbone curve¹ using the adjoint formulation². These results were later extended to any expansion order³, allowing even more control on the accuracy of the solution and, most importantly, giving us the freedom to choose any objective function we desire.

Yet, many other challenges remain to be explored, for instance:

• extension to multiple vibration modes, allowing for internal resonances⁴;
• extension to parametrically modulated systems (i.e., systems whose parameters are modulated in time), featuring so called parametric resonances⁵

In this project, the candidate will consider one of the aforementioned topics.

Phase 1 : review of the state of the art, with focus on Spectral Submanifold (SSM) theory and the chosen application [1-2 months]

Phase 2: application of the SSM to a low-dimensional lumped parameter model (e.g, nonlinear spring-mass chain). Definition of the optimization problem and first tests using (automatic) finite differences to compute the sensitivities [1-2 months]

Phase 3: develop the analytic expression for the sensitivities using the direct method and/or the adjoint method. Use analytic sensitivities in the optimization problem of phase 2 for comparison. [2 months]

Phase 4: extension to generic FE model using either SSMTool or YetAnotherFEcode (or both). Validation of the method on a relevant example of topology optimization. [2 months]

Phase 5: thesis writing [1 month]

Other tools

Optionally, OpenLSTO (a code for Topology Optimization in C++) can be used.

References and notes

1. In nonlinear systems, the resonance frequency of the system depends on the amplitude. The backbone curve describes this relation. Using SSM theory, it can be shown that the backbone can be written as a polynomial of the type: $\Omega=\omega_0 +\gamma_3 \rho ^2 +\gamma_5\rho^4 + \mathcal{O}(5)$, where $\rho$ is the vibration amplitude. ↩︎
2. Pozzi, M., Marconi, J., Jain, S. et al. Topology optimization of nonlinear structural dynamics with invariant manifold-based reduced order models. Struct Multidisc Optim 68, 72 (2025). https://doi.org/10.1007/s00158-025-04010-1 ↩︎
3. Matteo Pozzi, Jacopo Marconi, Shobhit Jain, Mingwu Li, Francesco Braghin; Adjoint sensitivities for the optimization of nonlinear structural dynamics via spectral submanifolds. Proc. A 1 December 2025; 481 (2328): 20250244. https://doi.org/10.1098/rspa.2025.0244 ↩︎
4. An internal resonance occurs when two eigenfrequencies of the system are in a integer ratio (e.g., 1:3). When an internal resonance occurs, energy is pumped from one mode to another. This mechanism sits at the core of TET (Targeted Energy Transfer) and NES (Nonlinear Energy Sinks) strategies to manipulate (vibrational) energy. ↩︎
5. Even a linear system with stiffness time modulation like $m \ddot x + k(1+\varepsilon \cos(\Omega t))=0$ may become unstable if the modulation frequency $\Omega$ is two times the linear eigenfrequency $\omega_0$. Parametric modulation is often overlooked, but it is extremely dangerous (e.g., a rotorcraft shaft periodically axially loaded) ↩︎