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

Abstract

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.

Figures (12)

• Figure 1: Initial layout of the half MBB beam considered in the example of Sec. \ref{['ssec: example1']}. The total domain is 800 $\mathrm{\mu m}$ long and 100 $\mathrm{\mu m}$ high, and includes a non-design region of 100 $\mathrm{\mu m}$ in length. The gray area represents the designable region, while the black area indicates the fixed non-design region.
• Figure 2: Iteration history of the objective $\rho_{\max}$ (left panel) and the constraint value $\mathrm{Im}(\gamma)$ (right panel) for the case of $\gamma_{\mathrm{target}} = 5 \times 10^{-5}$.
• Figure 3: Optimal layouts obtained from the nonlinear optimization formulation \ref{['eq:opt-rho_max']} and the linear optimization formulation \ref{['eq:opt-linear']}. The first four panels correspond to nonlinear optimal layouts with different values of $\gamma_{\mathrm{target}}$. The last panel shows the layout obtained from the linear optimization formulation.
• Figure 4: FRCs of the optimal layouts shown in Fig. \ref{['fig:layouts_linear_nonlinear']} under two levels of forcing amplitude: $f^{\mathrm{ext}} = 5 \times 10^{9}$ (left panel) and $2 \times 10^{10}$ (right panel) $\mathrm{ng \cdot \mu m/ms^2}$.
• Figure 5: Initial layout of the half MBB beam considered in the example of Sec. \ref{['ssec:example2']}. The total domain is 500 $\mathrm{\mu m}$ long and 100 $\mathrm{\mu m}$ high, and includes a non-design region of 100 $\mathrm{\mu m}$ in length. The gray area represents the designable region, while the black area indicates the fixed non-design region.

Theorems & Definitions (2)

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