Gradient-based Design of Computational Granular Crystals

TL;DR

The paper addresses inverse design of granular crystals for wave-based mechanical computation by introducing a differentiable, end-to-end simulator and a gradient-based optimization framework. It demonstrates design of an acoustic waveguide and basic logic gates (AND and XOR) by tuning particle stiffness to produce desired frequency- and amplitude-dependent responses, outperforming gradient-free baselines in efficiency and solution quality. The results show that gradient-based methods can significantly expand the design space of granular metamaterials for computation, though real-world transfer requires addressing model-reality gaps and multiobjective trade-offs. The work lays a foundation for robust, multifunctional granular substrates and suggests directions for integrating gradient-based and gradient-free strategies in future designs.

Abstract

There is growing interest in engineering unconventional computing devices that leverage the intrinsic dynamics of physical substrates to perform fast and energy-efficient computations. Granular metamaterials are one such substrate that has emerged as a promising platform for building wave-based information processing devices with the potential to integrate sensing, actuation, and computation. Their high-dimensional and nonlinear dynamics result in nontrivial and sometimes counter-intuitive wave responses that can be shaped by the material properties, geometry, and configuration of individual grains. Such highly tunable rich dynamics can be utilized for mechanical computing in special-purpose applications. However, there are currently no general frameworks for the inverse design of large-scale granular materials. Here, we build upon the similarity between the spatiotemporal dynamics of wave propagation in material and the computational dynamics of Recurrent Neural Networks to develop a gradient-based optimization framework for harmonically driven granular crystals. We showcase how our framework can be utilized to design basic logic gates where mechanical vibrations carry the information at predetermined frequencies. We compare our design methodology with classic gradient-free methods and find that our approach discovers higher-performing configurations with less computational effort. Our findings show that a gradient-based optimization method can greatly expand the design space of metamaterials and provide the opportunity to systematically traverse the parameter space to find materials with the desired functionalities.

Figures (15)

• Figure 1: Inverse design of computational granular crystals. When the granular crystal is vibrated at its boundary, the elastic compression waves (indicated by the red shades in the panels) propagate in the material until they are scattered or attenuated by disorder, affected by dispersion, or distorted by (self-)demodulation and frequency mixing at nonlinear interparticle contacts (Forward Pass). The waves arriving at the output particle(s) are recorded and the difference between the desired ($Y(t)$) and recorded response ($\hat{Y}(t)$) is utilized in a loss function ($\mathcal{L}$) to adjust the trainable parameters ($\theta$). $f$ relates the input $X_t$, parameters $\theta$, and the hidden state of the system $h_{t-1}$ to the hidden state at the next time step $h_t$. An end-to-end differentiable physics simulator allows us to track the partial derivatives in the Backward Pass indicated by the pink arrows in the figure. The particles' material properties can be optimized with a gradient-based method to produce the desired nonlinear wave response.
• Figure 2: A granular crystal is made of spherical particles with identical size (diameter $\sigma$) and various stiffnesses (represented in different shades of grey) in a confined configuration with fixed boundaries. Hertz's law describes the relation between the particle's overlap ($\delta=\sigma_{ij}-r_{ij}$) and applied force ($F$) as $F=\alpha \delta ^ \beta$. Here, $\beta$ is a constant that depends on the particle geometry and determines the nonlinearity of the contact forces. A commonly used value for spherical contacts is $\beta = \frac{3}{2}$, which produces a cubic nonlinearity in the equations of motion. $r_{ij}=|r_i-r_j|$ is the interparticle distance and $\sigma_{ij}=\frac{\sigma_i+\sigma_j}{2}$ is the maximum distance, after which the particles lose contact. As it is shown in the plot on the right, the interparticle potential is one-sided, and unlike a Hookean spring, the force becomes zero when the two particles lose contact.
• Figure 3: Experimental setup for the acoustic waveguide. The input particle (blue marker) is harmonically vibrated with the amplitude $A$ and one of the two predefined frequencies, $f_1$ or $f_2$. The applied elastic vibrations propagate through the material toward the output ports (red and gold markers). The existence of the input frequency in the displacement signal of the output particles indicates the computational response. Each of the two output particles is expected to only respond to one of the two input frequencies.
• Figure 4: Inverse design of an acoustic waveguide. The training loss is plotted over $200$ epochs. The mean and standard deviation of $5$ independent runs are shown with solid and dashed lines, respectively. Snapshots of the granular crystal are shown at intermediate stages during the training for one example trial.
• Figure 5: The optimized acoustic waveguide. The stiffness pattern enables the material to direct the vibration toward one of the output ports according to its frequency. The plots on the left show the horizontal displacement of the input (blue) and output particles (red and gold) during the simulation time. The optimized material directs the input vibration toward the top particle when the frequency is $f_1=7[Hz]$ and the bottom particle when the frequency is $f_1=15[Hz]$.