Universal Mechanical Polycomputation in Granular Matter
Atoosa Parsa

Sven Witthaus

Nidhi Pashine

University of Vermont
Burlington, Vermont, USA
atoosa.parsa@uvm.edu

Yale University
New Haven, Connecticut, USA
sven.witthaus@yale.edu

Yale University
New Haven, Connecticut, USA
nidhi.pashine@yale.edu

Corey S. O’Hern

Rebecca Kramer-Bottiglio

Josh Bongard

Yale University
New Haven, Connecticut, USA
corey.ohern@yale.edu

Yale University
New Haven, Connecticut, USArebecca.kramer@yale.edu

University of Vermont
Burlington, Vermont, USA
josh.bongard@uvm.edu

ABSTRACTACM Reference Format:
Atoosa Parsa, Sven Witthaus, Nidhi Pashine, Corey S. O’Hern, Rebecca
Kramer-Bottiglio, and Josh Bongard. 2023. Universal Mechanical Polycomputation in Granular Matter. In Genetic and Evolutionary Computation Conference (GECCO ’23), July 15–19, 2023, Lisbon, Portugal. ACM, New York, NY,
USA, 9 pages. https://doi.org/10.1145/3583131.3590520

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 withthe environment, are one such class of devices. This information
processing can be manifested in various physical substrates, one ofwhich is granular matter. In a granular assembly, vibration can betreated as the information-bearing mode. This can be exploited torealize “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 toquantum 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 thematerial’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.

1

INTRODUCTION

Traditionally, combinational logic operations are embedded in silicon based electronic devices. However, in light of the recent advances in chemical, physical and material sciences, unconventional
practices of information processing in engineered materials are
now being explored [24]. Quantum computing [6], DNA computing [1], neuromorphic computing [12], optical computing [21], andphysical reservoir computing [14] are all examples of the new formsof embedding computation. Moreover, attempts have been madeto overcome the limitations of the conventional Integrated Circuits via combining them with soft conductive materials to develop
kinematically reconfigurable electrical circuits [5]. Exploring these
new computational paradigms provides novel opportunities for thedevelopment of advanced forms of intelligent matter in which sensing, decision making and actuation are combined [9]. One recent
example is logic-enabled textiles that can be embedded into robotic
wearables [17].

CCS CONCEPTS• Computing methodologies → Artificial intelligence; • Theory of computation → Models of computation; • Hardware →Emerging technologies.

KEYWORDS
Mechanical Computing, Granular Metamaterials, Unconventional
ComputingFigure 1: The size, shape, and position of the grains in a granular metamaterial can be evolved such that it “polycomputes”:
different logical functions are computed at the same place,at the same time, in the material. This can be accomplishedif bits, encoded as vibrations, are combined by summing vibrations at different frequencies. The power of response ata different output grain (red), at those frequencies, can beinterpreted as the output of the computation performed on
those different bit pairs.

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ACM ISBN 979-8-4007-0119-1/23/07. . . $15.00
https://doi.org/10.1145/3583131.3590520

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2.1

Logical functions embedded in metamaterials have also been
explored to some extent. Non-intuitive exotic properties of theseengineered materials make them great candidates as physical substrates to perform computation [20]. In [8] authors have proposed
3D printed cells with embedded bistable springs that implement
simple logic functions using mechanical signals. In [22] origami
units have been designed that can be used to develop a framework
for embedding computation into the robotic structures. [19] has exploited a nonlinear mass-spring-damper model to design a universal
logic gate.
Recently, the granular metamaterials [10] are gaining special
interest as the physical substrates for computation. Here, the mechanical vibrations can be used as the information-bearing mode
to realize various logical functions in a granular matter (Fig. 1).
Because these materials are made of individual particles with diverse material properties, they can also be responsive to external
stimuli (like temperature or electromagnetic fields) and thus possess a higher potential for an increased computational density. In[11] OR and AND logical gates are designed in a granular chain ofchrome steel grains. In [2] the bistability of an array of spiral-spring
resonators is exploited to build transistor-like switches. In all of theaforementioned works (except [16]), the computational modules
are hand-designed by human experts. As we move to higher-orderparameter spaces and explore more complicated structures, theneed for an automated optimization pipeline to design computational metamaterials seems unavoidable. Moreover, in none of theprevious works the scalability of the designed units to realize theuniversal computational gates is explored.
In this paper, we first use Evolutionary Algorithms to design
granular assemblies that can compute a NAND function. Next, byleveraging the superposition principle in the frequency response of
the coupled granular system under external vibrations, we explore“polycomputation” in the material and provide the optimization
results showing NAND functionality computed at two different
frequencies at the same time and in the same place. We provide an
exploratory analysis of the distribution of the computation throughout the material and then discuss a possible hardware platform forthe physical realization of our designed structures.

2

Computation in Granular MatterTo encode information as sequences of bits, we must first establish
a useful representation based on our physical substrate. In digitalelectronic systems, inputs and outputs are electrical voltages. In ouranalogous mechanical system, inputs and outputs are vibrations.
The material is a two-dimensional assembly of circular particles
placed on a triangular lattice (Fig. 2). The macroscopic behaviorof such a granular system is determined by the properties of individual particles such as their arrangement, mass, modulus, andshape. In the present work, the EA was only allowed to vary aparticle’s stiffness ratio: it could be set to any value in [1, 10]. Thesemicroscopic properties influence how vibrations propagate throughthe material; our goal here is to evolve particle stiffnesses such thatthe material’s vibrational response resembles the operation of adesired logic gate.

Figure 2: System setup. A logic gate with two inputs (green
and blue particles), one output (red particle) and one powersource (magenta particle) embedded in a 2D granular assembly with particles of various stiffness properties (indicatedas different shades of gray). 𝐴1 and 𝐴2 are the amplitudes ofoscillations applied in 𝑥 direction to the input ports which
will result in displacements from the initial positions (𝑥 10and 𝑥 20 ). 𝐴𝑠 is the amplitude of the source signal. 𝜔 is theoperational frequency. The truth table is shown on the right:
𝐷 represents the diameter of particles. 𝑂𝑖 𝑗 (𝑖, 𝑗 ∈ 0, 1) is themagnitude of the vibrations in the output.
In order to apply the input signal, two particles are arbitrarily
chosen as input ports (particles with green and blue markers in
Fig. 2). The red particle represents the output port used to observe
the system’s response. A fourth particle (magenta) is connected toan external power source, which is realized as constant vibration ofthat particle, regardless of the vibrations supplied to the inputs inthe material. All input and power vibrations are sinusoidal waves
with amplitude 𝐴𝑖 and frequency 𝜔 applied in the 𝑥 direction.
Other particles may vibrate more or less in 𝑥 and 𝑦 directions.
Applying this vibration to the particle causes a displacement fromits initial position (𝑥𝑖0 ). The amplitude of this displacement signal
over time (𝑥𝑖 (𝑡)) is interpreted as a bit: (considering the amplitudeof the power source as a baseline) small or no displacement isinterpreted as a ‘0‘; high magnitude of displacement is interpreted asa ‘1‘. The truth table in Fig. 2 shows this representation. Oscillation
amplitude arriving at the output port (𝑂𝑖 𝑗) at a pre-determined

GENERALIZATION

Boolean logic gates are at the root of any computational model.
Logical NAND gates are of particular interest because any Booleanfunction can be implemented using circuits constructed from only
NAND gates. Therefore, to take a step toward general-purpose
computation embedded in granular metamaterials, in this paper
we focus on evolving NAND gates. In this section, we start withintroducing the digital abstraction in our physical platform. We
then describe the details of the granular material simulator usedby the Evolutionary Algorithm. After discussing the EA itself, we
present the results of evolving a NAND gate in a material and thenshow common patterns that consistently arise across independent
solutions, suggesting deeper insight into how various materials canrealize new forms of computation.

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frequency is recorded in each of the four input cases (‘00‘, ‘01‘, ‘10‘,and ‘11‘). This quantity is then used to calculate the system’s gain
(𝐺𝑖 𝑗 ) defined as:
𝐺𝑖 𝑗 (𝜔) =𝑓ˆ(𝑂𝑖 𝑗 )𝑓ˆ(in𝑖 ) + 𝑓ˆ(in 𝑗 )𝑖, 𝑗 ∈ 0, 1

algorithms have long been used effectively for many real-world
applications; more recently they have been applied for designing
granular matter [13] and computational granular matter [15]. In
this section, we used the Age-Fitness Pareto Optimization (AFPO)
algorithm [18] to find granular configurations that act as a single
NAND gate at a pre-determined frequency.
Each individual in the population is a granular assembly madeof particles with different stiffness values. There are many design
parameters in this system that can be included in the optimization. In initial experiments, it was found that the positions of theports can also affect the gate’s operation. In these experiments,
we examined 5 setups with different parameter sets including thestiffness vector, frequency of the input vibrations, phase offset between the input vibrations, and placement of the input ports (Fig.3).
We cannot conclude that one algorithm variant had statisticallysignificantly better performance than another, but the results doessuggest that input port position can be helpful in the optimization.
For this reason, we expanded the evolutionary algorithm to also
evolve the position of the input ports. To reduce the dimensionalityof the search space, we fixed the position of the source and output
ports. We used a direct encoding, with a genome that is made oftwo parts. The first part is a real vector of length 30 (the latticeis 5 × 6) that represents the stiffness of each particle (the values
range between 1 (soft) and 10 (stiff)). The second part of the genome
contains two integers that dictate the input port positions. Twomutation operators were employed: the first acts on the real vector
and is a Gaussian mutation operator (with standard deviation of
0.1). The second randomizes the position of one of the two input
ports, chosen randomly. We performed 5 independent trials, eachwith a different random initial population.

(1)

In this equation, 𝑓ˆ is the fast Fourier transform at the driving
frequency (𝜔), which is the frequency applied to the input andpower ports. For the material to resemble a NAND gate, we expect
to see high amplitude oscillation at the output in the ‘00‘, ‘01‘ and‘10‘ cases (which translates to high 𝐺 00 , 𝐺 01 and 𝐺 10 ), but a low
amplitude in the ‘11‘ case (low 𝐺 11 ). Using this formulation, we
can evaluate each candidate granular assembly of particles anddetermine its similarity to the desired logical function. To selectfor materials that are increasingly NAND-like, the fitness function(which was minimized during the optimization) was defined asfollows:
𝐹 = max (|1 − 𝐺 00 (𝜔)|, |1 − 𝐺 01 (𝜔)|, |1 − 𝐺 10 (𝜔)|, |0 − 𝐺 11 (𝜔)|)
(2)

2.2

Simulation

We simulated our granular system using the Discrete Element
Method (DEM) [23]. The particles are assumed to be frictionlesscircular disks with a fixed diameter (𝐷), placed on a 5 by 6 twodimensional hexagonal lattice (a total number of 30 particles) withperiodic boundary in the 𝑥 direction. Gravity is ignored and theparticle-particle interaction is modeled as a purely repulsive force
with a Lennard-Jones potential. The Fast Inertial Relaxation Engine (FIRE) [3] is used to adjust the initial positions of the particles
and ensure a statistically stable particle packing at the start ofthe simulation. Table 1 reports the main parameters used in our
experiments.
Table 1: Experimental Setup.
Simulator
Particle Diameter
0.1
Particle Mass
1
Stiffness Ratio∈ [1, 10]
Packing Fraction
0.91
Simulation Time
1𝑒4

Optimizer
Algorithm
AFPO
Population Size
50
Generations
500
Runs
5
Mutation
GaussianFigure 3: Examination of the parameter space. The evolutionary progress of 5 experiments, each with a different set of
optimized parameters (details in the text) is plotted. The plotsshow the average fitness of the population per generation,
averaged over 5 independent runs. The orange plot indicatesthe experiment with a 10 × 10 grid of particles. Because of thelimited computational resources, only 2 evolutionary runs
were performed in this case.

Source code for the experiments in this paper can be found at:https://github.com/AtoosaParsa/gecco-2023.

2.3

Optimization

There are many parameters in a granular system that affect its
macroscopic response to the applied vibrations: particle shapes,
sizes, placements and material properties. These affect the normal modes of the system, and consequently the propagation or
suppression of the input and power vibrations. Deciding on the
optimal parameters to achieve a desired behavior in the material isa challenging task for a human expert and thus optimization algorithms can be used to automate this design process. Evolutionary

2.4

Results

Fig. 4 shows the evolutionary progress. Panel (a) shows the fitness(Eqn. 1) decreasing over generations and reaching a value of 0.1

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and time space when supplying the four different input signals. We
see a low amplitude of oscillation when both of the input ports areactivated. This confirms the material functions as a logical NAND
gate at 𝜔 = 10𝐻𝑧.

at the end of the optimization. The best solutions from each of the
independent evolutionary runs are shown in Fig. 5.

Figure 4: (a) Evolutionary progress across the five trials dedicated to evolving a single NAND gate into the material. (b)reports evolutionary change within one trial. Each line reports the best individual within that AFPO clade at eachgeneration. Terminated lines denote clade extinction events.

Figure 6: One of the five best materials acting as a single
NAND gate at 𝜔 = 10𝐻𝑧. (a): particle configuration (left);frequency spectrum (right). (b-e): the response of the input(blue and green) and output (red) particles in the frequency(left) and time (right) domains.

3

DISTRIBUTED COMPUTATION

The dynamics of a granular system is mathematically equivalent toa system of coupled oscillators. In such a system, the steady-state
response to an arbitrary external oscillation is a superposition of thevibrational normal modes. The normal modes (or eigenfrequencies)of the granular assembly are determined by the material properties of the individual particles and the boundary conditions of theconfining structure. So fundamentally, our proposed optimization
platform is searching for a configuration that excites normal modes

Figure 5: The five best materials drawn from the five singleNAND evolutionary trials. Fitness (Eqn. 2) is reported below
each material.
We verify the performance of one of the evolved solutions by
testing the configuration’s response when different vibrations areapplied to the input ports. Fig. 6b-e shows the response in frequency

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compatible with our desired logical function (which is the NAND
operation at frequency 𝜔 = 10𝐻𝑧).
It can be informative to inspect the system’s behavior at otherfrequencies in the spectrum and to investigate if there is any amountof “NAND-ness” present in the system at those frequencies, without
any explicit selection pressure in the evolutionary optimization.
Because the fitness function (Eqn. 2) is designed to only consider
the worst case of the four gain values, We define the following
metric to combine the system’s gains in all four cases and obtain
one single number to measure the amount of “NAND-ness” in thematerial:

𝑀“NAND-ness” (𝜔) =𝐺 00 (𝜔)𝐺 01 (𝜔)𝐺 10 (𝜔)𝐺 11 (𝜔)(3)

Fig. 7 presents the result of sweeping through the frequency
spectrum and calculating the “NAND-ness” at each frequency for
one of the most fit materials with the evolved stiffness vector andport placements. As we expected, there is a high peak at 𝑓 = 10𝐻𝑧
which is the frequency used in the optimization. But we also noticethat the plot is not completely flat at frequencies other than 10𝐻𝑧:
there are noticeable peaks at 7𝐻𝑧, 18𝐻𝑧, and 35𝐻𝑧 (marked with
red stars in Fig. 7.a). Investigating the gains (Fig. 7.b) at these frequencies indicates some of the four cases comprising the NAND
operation are being performed better or worse at these other frequencies. Taken together, this suggests that, with a multiobjective
optimization method, we may be able to evolve NAND gates that
operate simultaneously at different frequencies. This would in turnincrease the computational density of the material. This idea isinvestigated in the next section.
To further how these materials evolved to compute the NAND
function, we looked at the amount of NAND-ness at every particle
in the five best materials (Fig. 8). Surprisingly, in all five cases, there
was another particle that was much better at NAND operation thanthe pre-designated output particle (red star). This suggests thatevolving output port position will, in future experiments, greatly
facilitate the evolution of computation in these kinds of materials.

4

Figure 7: For one of the most fit materials evolved to exhibit
NAND-ness at 𝜔 = 10𝐻𝑧, latent potential to act as a NAND
gate at other frequencies can be seen in (a). Red markers show
the potential frequencies (other than 10𝐻𝑧) for acting as a
NAND gate. (b) shows how the four NAND cases contributeor detract from that potential.

The simulator and experimental setup are the same as describedin the previous section. The only difference is the number of objectives in the evolutionary algorithm. Here, a tri-objective optimization method was employed: the AFPO age objective was minimized,
as was NAND-ness at 𝜔 = 10𝐻𝑧 (𝑓1 ) and NAND-ness at 𝜔 = 20 (𝑓2 ).
We used the same fitness function defined in Eqn. 2. Because of theincreased number of objectives, we also increase the population size
to 100 to minimize the likelihood of premature convergence (however this may have occurred in some of the trials). Fig. 9 presents
the result of this optimization.
The material at the knee of the Pareto optimal set after 500
generations was chosen for analysis (red marker in Fig. 9b). It wasevaluated at 𝜔 = 10𝐻𝑧 and 𝜔 = 20, and the material’s response isplotted in the frequency and time domains in Fig. 10.
Fig. 11 reports that the output particle has moderate amounts of
NAND-ness at both low (𝜔 = 10𝐻𝑧) and high (𝜔 = 20𝐻𝑧) frequencies. Moreover, it also shows that other particles throughout thematerial have specialized to exhibit varying amounts of NAND-nessat the two frequencies. As in previous experiments, one particularparticle exhibits as much, if not more NAND-ness, as the output
particle, for both frequencies.

POLYCOMPUTATION

Silicon-based digital electronic computing devices have long stoodas the state of the art for efficient, fast, and scalable computing[7]. However, logical components in Integrated Circuits are only
capable of serial operation. This means that in order to implement
more complex operations, many logic gates need to be connectedin parallel or series and thus the number of required transistors
will increase with the complexity of the computation. Moreover,wiring between transistors further limits the integration density.
Polycomputing is defined as the ability of the same physical substrate to perform multiple functions in the same place and at thesame time [4] and might eventually provide new paths to realize computationally dense materials. Granular metamaterials havebeen shown to possess this ability to simultaneously compute twological functions (AND and XOR) by carrying out the operations
at different frequencies of vibrations [16]. In this section, this idea
has been extended to NAND gates at multiple frequencies.

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Figure 8: Distributed computation was found in the five best materials evolved to act as a single NAND gate. Lighter colors are
better in the NAND-ness and 𝐺 11 ; darker is better in the 𝐺 00 , 𝐺 01 , and 𝐺 10 subplots

5

ROBUSTNESS

Noise is an inevitable component of all signals in real-world applications. So it is critical to understand its effect on a system’s
performance and analyze its robustness when subject to various
degrees and types of noise. In this section, we investigated therobustness of our evolved mechanical NAND gates. We used one of
the evolved particle configurations from Section 2 (Fig.5) as a casestudy. We added increasing amounts of Additive White Gaussian
Noise (AWGN) to the input signals and recorded the system’s response at each signal-to-noise ratio (SNR) tested. Fig. 12b,c report
the effects on the NAND-ness (Eqn. 3) and individual gain values
for that material.
Although the added noise contains all of the frequency components (see the frequency spectrum in Fig.12a), we observe that if the
SNR exceeds −20𝑑𝐵, the noise does not interfere with the NAND
operation in the material.

6

CONCLUSION AND FUTURE WORKIn this paper, we utilized evolutionary algorithms to design granularmaterials capable of polycomputation. Our granular materials here
were simulated with an overly simplified model which assumed frictionless circular disks. Additionally, the only parameters consideredin the optimization were the stiffness of the particles and the placement of the input ports. There are various other parameters suchas the particle shapes, sizes, masses, and nonlinear inter-particle
contacts that can be considered in building a more realistic systemwith possibly more computational power. We will pursue this possibility in our future research. We also plan to study the nonlinear
dynamics of the evolved solutions and develop a formal analysis toinvestigate the reasoning behind the stiffness patterns that enable
the embedding of the logical functions at various input frequencies.

Figure 9: Evolution of polycomputation. The progress of eachof the fitness functions during evolution is plotted in panel(a), these plots are averaged over 5 evolutionary trials. Panel(b) shows the Pareto front for one of the trials, at three different stages during the optimization. The selected solution
from the last generation is marked with a red star.

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Figure 10: Polycomputation. An optimized configuration of particles functions as a NAND gate at two different frequencies:𝜔 = 10𝐻𝑧 in panel (b) and 𝜔 = 20𝐻𝑧 in panel (c)into computational materials. Acoustic granular switches have beenshown in one-dimensional granular systems, but such systems haveprimarily used passive particles[11]. Recent advances in materialscience have led to variable stiffness composites made out of phasechanging materials, and we have used such materials to develop
variable stiffness particles.
Our particles encapsulate a low melting-point alloy (Field’s
metal) in a soft silicone elastomer with a heater embedded inside.
By locally heating individual particles, we can change the stiffnessof each particle. Fig. 13 shows a two-dimensional granular system
where each particle is made out of a soft silicone elastomer. By sinusoidally actuating the source particle, we can generate a sustained
normal mode in the system. Using this class of polymeric materials
can allow us to create variable stiffness granular materials that canbe used to perform complex computations.

Figure 11: Distribution of polycomputation. Heatmap shows
the amount of NAND-ness in each particle at 𝜔 = 10𝐻𝑧. Thickness of each particle’s perimeter represents NAND-ness at
𝜔 = 20𝐻𝑧.

ACKNOWLEDGMENTS
We would like to thank Dong Wang for his help in developing thesimulator for the granular system used in the experiments in thispaper. We would like to acknowledge financial support from the
National Science Foundation under the DMREF program (award
number: 2118810). Computations were performed, in part, on the
Vermont Advanced Computing Center.

In Section 3 of the paper, we computed the amount of NAND functionality at the particles other than the pre-determined output port
and realized that other particles are found that also possess NAND
functionality. This might suggest the distributed nature of computation in the material and provide opportunities to design morecomputationally dense materials. In this regard, the informationtheoretic measures can be used to provide more insight into theflow of information processing in the material and shed light onsome of these observed phenomena.
Realizing physical polycomputational materials remains a futurechallenge. In preparation for this, we are developing granular assemblies whose configurations may be evolved to transform them

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