Dissipation Dilution-Driven Topology Optimization for Maximizing the $Q$ Factor of Nanomechanical Resonators

TL;DR

This work tackles maximizing the quality factor of nanomechanical resonators through dissipation-dilution optimization. It proposes a FE-based topology optimization framework that uses the ratio of geometrically nonlinear to linear modal stiffnesses, $D_Q ≈ (Φ^T Ktilde Φ)/(Φ^T Kbar Φ)$, as the objective and derives an adjoint sensitivity formulation. The method is demonstrated on square and hexagonal domains, achieving high-Q designs with four tethered connections in the hexagonal case, and reveals the interplay between resonance frequency and Q. The approach is predictive and scalable, avoids reliance on empirical models, and offers a path to geometry- and stress-aware high-Q resonator designs for sensing and quantum applications.

Abstract

The quality factor ($Q$ factor) of nanomechanical resonators is influenced by geometry and stress, a phenomenon called dissipation dilution. Studies have explored maximizing this effect, leading to softly-clamped resonator designs. This paper proposes a topology optimization methodology to design two-dimensional nanomechanical resonators with high $Q$ factors by maximizing dissipation dilution. A formulation based on the ratio of geometrically nonlinear to linear modal stiffnesses of a prestressed finite element model is used, with its corresponding adjoint sensitivity analysis formulation. Systematic design in square domains yields geometries with comparable $Q$ factors to literature. We analyze the trade-offs between resonance frequency and quality factor, and how these are reflected in the geometry of resonators. We further apply the methodology to optimize a resonator on a full hexagonal domain. By using the entire mesh -- i.e., without assuming any symmetries -- we find that the optimizer converges to a two-axis symmetric design comprised of four tethers.

Figures (15)

• Figure 1: Dissipation dilution computed by Eqs. \ref{['eq:comsolQ']} and \ref{['eq:DQ objective']} as a function of geometric parameters that define the geometry of the resonators. (left) H-beam resonators by Li et al. zichao23, for which the length of the central beam $L_b$ is varied; (right) Trampoline resonators by Norte et al. norte16, for which the length of the outer frame $L_f$ is varied.
• Figure 2: Schematic representation of a finite element model based on shell elements, used in combination with a generic density-based topology optimization model. The latter prescribes a (density) field of values between 0 and 1, to indicate material and void areas respectively. A shape represented by this density field is shown on the right, with a cutout showing the individual elements and the corresponding density field that defines the shape. In this work, the element formulation by Van Keulen keulen93 was used, for which the degrees of freedom corresponding to a single node can be found in the cutout on the left.
• Figure 3: Schematic of the problem considered in Section \ref{['sec:square simulations']}. The inset shows the finite element mesh used to model a quarter of the domain to reduce computational cost, where symmetry conditions are indicated by red dotted lines. Darker regions indicate the fixed non-design domain regions where the design density was fixed as $\rho_e = 1$ (these correspond to the black inner square and outermost boundary). Dashed lines indicate the clamped boundary where both the displacement and rotation fields are homogeneous.
• Figure 4: Optimization of a square resonator with a prescribed lower frequency $f_{\min}= 300kHz$. The top row shows the evolution of the intermediate projected density, while the second and third rows depict, respectively, the element-level contributions to the geometrically nonlinear and linear modal stiffnesses; these are computed, respectively, as $\widetilde{\kappa}_e = \bm{\mathit{\Phi}}_e^\intercal \tilde{\bm{\mathit{k}}}_e \bm{\mathit{\Phi}}_e/2000$, and $\overline{\kappa}_e = \bm{\mathit{\Phi}}_e^\intercal \bar{\bm{\mathit{k}}}_e \bm{\mathit{\Phi}}_e / 2$ (they are also normalized). The color bar has been capped for both quantities to give a clearer depiction of these fields (as this limit was exceeded locally). The bottom two graphs show the evolution of the dilution factor (left) and fundamental resonance frequency (right) throughout the optimization for all three fields of the robust projection.
• Figure 5: Optimization of a square resonator with a prescribed lower frequency $f_{\min}= 350kHz$. The top row shows the evolution of the intermediate projected density, while the second and third rows depict, respectively, the element-level contributions to the geometrically nonlinear and linear modal stiffnesses; these are computed, respectively, as $\widetilde{\kappa}_e = \bm{\mathit{\Phi}}_e^\intercal \tilde{\bm{\mathit{k}}}_e \bm{\mathit{\Phi}}_e/2000$, and $\overline{\kappa}_e = \bm{\mathit{\Phi}}_e^\intercal \bar{\bm{\mathit{k}}}_e \bm{\mathit{\Phi}}_e / 2$ (they are also normalized). The color bar has been capped for both quantities to give a clearer depiction of these fields (as this limit was exceeded locally). The bottom two graphs show the evolution of the dilution factor (left) and fundamental resonance frequency (right) throughout the optimization for all three fields of the robust projection.