# Selected Works

For the full list of my publications, please refer to my Google Scholar page.

Data-driven Modeling of Granular Crystals with Modern Koopman Theory

Parsa, A., Bagrow, J., O'Hern, C. S., Kramer-Bottiglio, R., & Bongard, J. ⚫ in preparation ⚫ code

Externally driven dense packings of particles can exhibit nonlinear wave phenomena that are not described by effective medium theory and linearized approximate models. Such nontrivial wave responses can be exploited to design sound-focusing/scrambling devices, filters, and acoustic 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 can give rise to remarkable properties, particularly useful in designing structures with exotic properties. However, assumptions such as linearity, periodicity, and homogeneity, break down in the strongly nonlinear regime and physical realizations of the simulated models. Periodicity, although a useful mathematical construct, does not hold in real-world systems that are spatially finite and prone to wave reflections at fixed boundaries. Discrepancies in the properties of individual particles will occur in the fabrication process and can give rise to unintentional disorder leading to wave scattering and attenuation. The discrete nature of the system leads to the opening and closing of interparticle gaps that are not captured with a continuum model. Nonlinear

contacts will distort the propagating waves by frequency-mixing and demodulation phenomena that don’t exist in a linearized model. In this paper, we build upon the data-driven methods in dynamical system analysis to devise an analysis framework and show that 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. Here, we use data from numerical simulations of granular crystals to prove the applicability of our framework, however, one can

envision incorporating the experimental data instead, to directly model the dynamics without any assumptions of the underlying physics model. We hope our proposed framework can bridge the gap between the simulation results and the physical realization of granular materials to facilitate the inverse design of increasingly multifunctional granular machines.

### Frequency Mixing

### Unraveling the Dynamical Complexity of Granular Materials with Koopman Spectral Analysis

Parsa, A. ⚫ ongoing project

### Interplay of Nonlinearity and Disorder

Unconventional computing devices leverage the intrinsic dynamics of a physical substrate to perform fast, energy-efficient, and special-purpose computations. Granular metamaterials have great potential for creating such computing devices. However, there is no general framework for the inverse design of large-scale granular materials. Here, we develop a gradient-based optimization framework for harmonically driven granular materials to obtain a target wave response. Using this framework, we design basic logic gates in which mechanical vibrations carry the information at predetermined frequencies. Our findings show that a gradient-based optimization method can greatly expand the design space of computational metamaterials and provide the opportunity to systematically traverse their parameter space to find materials with the desired functionalities.

Optimization Progress

Optimized AND Gate

Unconventional computing devices are increasingly of interest as they can operate in environments hostile to silicon-based electronics, or compute in ways that traditional electronics cannot. Mechanical computers, wherein information processing is a material property emerging from the interaction of components with the environment, are one such class of devices. This information processing can be manifested in various physical substrates, one of which is granular matter. In a granular assembly, vibration can be treated as the information-bearing mode. This can be exploited to realize polycomputing: materials can be evolved such that a single grain within them can report the result of multiple logical operations simultaneously at different frequencies, without recourse to quantum effects. Here, we demonstrate the evolution of a material in which one grain acts simultaneously as two different NAND gates at two different frequencies. NAND gates are of interest as any logical operations can be built from them. Moreover, they are nonlinear thus demonstrating a step toward general-purpose, computationally dense mechanical computers. Polycomputation was found to be distributed across each evolved material, suggesting the material’s robustness. With recent advances in material sciences, hardware realization of these materials may eventually provide devices that challenge the computational density of traditional computers.

Thin, planar sheets can be programmed to morph into complex shapes through stretching and out-of-plane bending, with applicability to shape-shifting soft robots. One way to make a morphing sheet is to use variable stiffness fibers that can modulate their tensile stiffness attached to the surface of a volumetrically expanding sheet. Adjusting local stiffnesses via tensile fiber jamming during sheet expansion allows control of the local shape tensor. However, finding the fiber placements and jamming policies to achieve a set of desired shapes is a non-trivial inverse design problem. We present an additive inverse design framework using an evolutionary algorithm to find optimal jamming fiber patterns to match multiple target shapes. We demonstrate the utility of our optimization pipeline with two input curvature pairs: 1) cylinder and sphere curvatures and 2) simple saddle and monkey saddle curvatures. Our method is able to find a diverse set of sufficient solutions in both cases. By incorporating hardware constraints into our optimization pipeline, we further explore the transfer of evolved solutions from simulation to reality.

Under an externally applied load, granular packings form force chain networks that depend on, among other things, the contact network and stiffness of the grains. In this work, we investigate packings of variable stiffness particles, where we can direct the force chains by changing the stiffness of individual particles within the packing on demand. Each variable stiffness particle is made of a silicone shell that encapsulates a core made of a low-melting-point metallic alloy (Field’s metal). By sending an electric current through a co-located copper heater, the Field’s metal internal to each particle can be melted via Joule heating, which causes softening of the particle. As the particle cools to room temperature, the alloy solidifies and the particle recovers its original stiffness. To optimize the mechanical response of granular packings containing both soft and stiff particles, we employ an evolutionary algorithm coupled with a discrete element method simulation to predict the stiffness patterns that will yield a particular force output on the assembly boundary. The predicted stiffness patterns were built in experiments using a 2D assembly of variable stiffness particles and the force outputs at different points of the assembly boundary were measured using photoelasticity. This result is a first step towards making robotic granular metamaterials that can dynamically adapt their mechanical properties such as force transmission, elastic moduli, and frequency response on demand.

Digital signal processors are widely used in today's computers to perform advanced computational tasks. But, the selection of digital electronics as the physical substrate for computation a hundred years ago was influenced more by technological limitations than substrate appropriateness. In recent decades, advances in chemical, physical and material sciences have provided new options. Granular metamaterials are one such promising target for realizing mechanical computing devices. However, their high-dimensional design space and the unintuitive relationship between microstructure and desired macroscale behavior makes the inverse design problem formidable. In this paper, we use multiobjective evolutionary optimization to solve this inverse problem: we demonstrate the design of basic logic gates embedded in a granular metamaterial, and that the designed material can be "reprogrammed" via frequency modulation. As metamaterial design advances, more computationally dense materials may be evolved, amenable to reprogramming by increasingly sophisticated programming languages written in the frequency domain.

Granular metamaterials are a promising choice for the realization of mechanical computing devices. As preliminary evidence of this, we demonstrate here how to embed Boolean logic gates (AND and XOR) into a granular metamaterial by evolving where particular grains are placed in the material. Our results confirm the existence of gradients of increasing “AND-ness” and “XOR-ness” within the space of possible materials that can be followed by evolutionary search. We measure the computational functionality of a material by probing how it transforms bits encoded as vibrations with zero or non-zero amplitude. We compared the evolution of materials built from mass-contrasting particles and materials built from stiffness-contrasting particles, and found that the latter were more evolvable. We believe this work may pave the way toward evolutionary design of increasingly sophisticated, programmable, and computationally dense metamaterials with certain advantages over more traditional computational substrates.

Please also check out my Masters and Bachelor research and selected course projects I worked on.