Data-driven Modeling of Granular Chains with Modern Koopman Theory

Abstract

Externally driven dense packings of particles can exhibit nonlinear wave phenomena that are not described by effective medium theory or linearized approximate models. Such nontrivial wave responses can be exploited to design sound-focusing/scrambling devices, acoustic filters, and analog computational units. At high amplitude vibrations or low confinement pressures, the effect of nonlinear particle contacts becomes increasingly noticeable, and the interplay of nonlinearity, disorder, and discreteness in the system gives rise to remarkable properties, particularly useful in designing structures with exotic properties. In this paper, we build upon the data-driven methods in dynamical system analysis and show that the Koopman spectral theory can be applied to granular crystals, enabling their phase space analysis beyond the linearizable regime and without recourse to any approximations considered in the previous works. We show that a deep neural network can map the dynamics to a latent space where the essential nonlinearity of the granular system unfolds into a high-dimensional linear space. As a proof of concept, we use data from numerical simulations of a two-particle system and evaluate the accuracy of the trajectory predictions under various initial conditions. By incorporating data from experimental measurements, our proposed framework can directly capture the underlying dynamics without imposing any assumptions about the physics model. Spectral analysis of the trained surrogate system can help bridge the gap between the simulation results and the physical realization of granular crystals and facilitate the inverse design of materials with desired behaviors.

Figures (12)

• Figure 1: Overview of the proposed method. (a) A chain of elastic spherical particles is studied as a discrete-time autonomous system. Given the structural parameters (i.e. particles' size, mass, and stiffness) and initial conditions (positions and velocities of the particles), the system can be numerically simulated to obtain the trajectories of particle displacements in time. (b) The harmonic balance method predicts the particle trajectories as a superposition of the Linear Normal Modes (LNMs). Such prediction is a good approximation only in the linear regime. (c) Our proposed data-driven framework can predict particle trajectories in linear and nonlinear regimes by training a deep neural network using real or simulated system observations. Instead of LNMs, an unlimited number of complex-valued Koopman modes are obtained from the trained K Network to provide a more accurate prediction of particles' trajectories in the strongly nonlinear regime.
• Figure 2: A granular chain that is made of two types of elastic spherical particles. Hertz's lawHertz1882 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.
• Figure 3: Deep neural network architecture for approximating the Koopman operator. The encoder network learns the Koopman embedding ($\Vec{Y}_t$) from the measurements of the system's states ($\Vec{X}_t$). The K Network predicts the latent state in the next time step ($\Vec{Y}_{t+1}$). The loss function (\ref{['eq:koopmanloss']}) is designed to ensure that the physical constraints of the dynamical system (prediction loss over the input trajectories) and mathematical assumptions of the Koopman theory (reconstruction loss in the encoder/decoder networks and linearity of K network) are met.
• Figure 4: Value of the loss function \ref{['eq:koopmanloss']} over the training and validation datasets. The deep Koopman network is trained on the displacement and velocity trajectories obtained from a numerically simulated two-particle system. The particles' initial positions are selected randomly from a predefined range indicated in \ref{['tab:KP_sim']}.
• Figure 5: Koopman eigenvalues for the unforced two-particle system. The eigenvalues determine the frequency of the Koopman modes ($Im(\lambda)$) and their decay/growth rate ($Re(\lambda)$). The blue outline shows the unit circle, and the color of each mode indicates its dominance ($\parallel c_i \parallel$).