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Applications of Computational Homology on Granular Media

Julio Cesar Santiago B.S. in Mathematics and B.S. in Economics

Junior 16 Duke University

Abstract: We provide a robust review of computational homology tools and their application to granular media, as well as, recent developments in the field. Two key questions arose in the literature: (1) Can we characterizer force chains in some useful way in terms of topological measures? (2) Can these topological measures allow us to quantify the strength of the force for a given part of the network? We synthesize numerical, and experimental results, which suggest Betti numbers, and persistence diagrams are the answer.

 

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Acknowledgements I thank Dr. Robert Behringer for his patience and forward guidance throughout the development of this paper.

 

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Outline. This paper is organized as follows. Section 1.1 Motivations behind

Computational Homology Section 1.2. Computing the Topology of Granular Systems

Section 1.3 Persistent Homology of Force Networks Section 1.4 Recent Developments in

Computational Homology

1.1 Motivation behind Computational Homology

The behavior of granular media under load remains among the most challenging phenomena

to characterize, predict and control. It’s multiple phase behavior [1], rheology [4,9],

temporal transitions [23], self-organizing pattern formation under deformation [22], and

transition dynamics under jamming, compression, shear banding, and taping [2, 3, 5, 16] are

all hallmarks of complex behavior. There is a recent trend within the literature towards the

development of a universal method for describing and quantifying the relevant features of

interparticle forces, which have since been understood as fundamental in determining the

mechanical properties of complex granular systems [6]. In some respects, the recent

proliferation of computational homology in the literature [10-11, 15-17, 19-20] can be

attributed to both the insights gained and the scientific challenges that remain in

understanding the process of jamming. Or, in broader terms, an understanding of the many

distinctive mechanical responses dissipative media exhibits when applied by an external

force [2]. For readers who are not familiar with jamming thresholds, [18] provides a

comprehensive description of experiments that probe this behavior in dense granular media.

Direct observations of stresses in granular packings using photoelastic discs [23] showed

that stress propagates in the form of structures or chains that define a force network. The

findings in [23] suggested some measure of structural stability in dense granular materials.

In other words, the onset of a jammed state arises from the spontaneous development of

 

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internal structures as a response to external loads. Unraveling the details of these

mesoscopic structures and associated frictional rearrangements has been a major focus in

recent condensed particle physics research. Recent developments using readily available

tools from network graph analysis and statistical inference have shed light on a number of

features in granular media: (1) a qualitative change in the probability density distribution for

interparticle normal forces can be used to distinguished between flowing and jammed

systems (2) the shape of the PDF is related to the transition from fluid-like to solid-like

behavior in granular media, more specifically, the average force per contact and the

deviation of the average number of contacts from the isostatic value. Despite these, and

many other, important results, the mechanism by which a flowing media stops its dynamics

and develops a collective arrested state is not yet well understood. See [In order to develop

better (more complete) models, one requires a more rigorous characterization of 3-cycle

loops and force chains present in compressed granular media. See Figure [4] for a graphical

representation of these 3-cycle loops and force chains captured in compressed granular

media using standard statistical inference tools from the numerical experiments in [21].

From the standpoint of material characterization and modeling, these emergent structures

pose great scientific challenges as their analytical formulation embodies strong

nonlinearities and multiscale dynamics in space and time, and therefore cannot be fully

appreciated using the statistical tools employed to get Figure [4]. The root cause of this

complexity is inherently a geometric one: there are just too many internal degrees of

freedom and interaction that govern their evolution. An ongoing and exciting new trend in

the way in which complex granular networks are studied can be shown in [7, 8, 10, 11, 12,

 

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15-17, 19, 21, 24]. These studies have shifted from characterizing these networks as high

dimensional geometric structures with spatial distributions and pushing towards

characterizing these networks as algebraic structures with topological invariants.

Geometric Realization Theorem Every abstract simplicial complex of dimension d has a geometric realization in R2d+1

The proof for which is given in section III.1 of [8]. As a direct result of this theorem,

computational homology enables a complex to be constructed algebraically, and thus one

does not need to worry about placing it in a Euclidean space or how to relate it with rigid

and endless degrees of freedom.

It is not completely clear, if these new computational tools will provide more (or better)

information than traditional measures. Computational homology seems to provide

researches with two advantages: (1) Devising a method to study contact networks even in

the case where contacts cannot be exactly determined. Using computational homology, the

topology of contact networks is sufficient to distinguish granular media with the same

density (i.e. packing fraction) but in different state of equilibrium (i.e. packing intensity)

(2) Mathematical models based on persistent homology can be used to study the evolution

of the force network in compressed granular material and tracking the density for different

polydispersities and friction coefficients. Using computational homology, we are able to

uncover the distinctive behavior displayed by different system, and it is shown to be richer

in information than the pair correlation function, the bond orientational order parameter and

the distribution of forces. The first step towards using algebraic topology to characterize the

 

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geometric structure associated with dense granular media is to create complexes through

which the geometry of the particles can be expressed.

1.2 Computing Topology of Granular Systems Computational homology determines geometric invariants within a granular system, and

relates these homeomorphisms to key measures of the granular system’s underlying

statistical and dynamical properties.

Def. Homeomorphisms are continuous transformations between topological spaces (i.e.

mappings that preserve all the topological properties of a given space).

Homology then associates to each phase of a granular system, X, a series of homology

groups Hi (CN) i=0,1,2… where If X is isomorphic to Y, then Hi(X) ≅ Hi (Y) for every i.

Homology groups captured holes in topological spaces by focusing on what surrounds them.

Their main ingredients are group operations and maps that relate topologically meaningful

subsets of a space with each other. We refer the reader to [8] for comprehensive proofs

behind these definitions, as well as, several theorems and lemmas that are fundamental in

understanding the algebraic topology behind CH tools. We introduce a type of data

structures, complexes, to characterize the topology of granular media. Complexes are

computable algebraic structures that represent some topological space, simplices being a

decomposition of such a space into its lower dimensional (d-1) components. We refer the

reader to [11] for a rigorous development of simplicial, and chain complexes for both

contact and interaction force networks of granular media. To relate the homology groups of

 

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a complex, we define the boundary of a p-chain as the sum of its (p-1) dimensional faces.

For the sake of clarity, recall a p-chain is the formal sum of p-simplices in X. Let δp denote

the boundary homeomorphism, or boundary map, for both simplices and chain complexes.

The chain complex is the sequence of chain groups connected by boundary

homeomorphisms. Once we have well defined notions for simplicial and chain complex we

can compute Betti numbers, which are non-negative integers, one for each dimension.

Def. The i-th Betti number of X is defined as follows:

Dim (Hi(X))= Bi (X)

where B0 , B1 are considered topological invariants. B0 counts the number of connected

components and B1 counts the number of linearly independent, non-boundary loops in X.

Recent work [16] in slowly compressed model granular materials in two spatial dimensions,

showed that the zeroth Betti number, B0, can distinguish among force networks in systems

with varying density, friction and polydispersities of the grains. See Figure 1 for graphical

results of B0 normalized by the number of particles, and as a function of the force threshold

F and packing fraction ρ for different frictions. The packing fraction φ is proven not to be a

monotonous function of the tapping intensity Γ in [13,14]. Robust definitions of φ , Γ are

presented in [18]. See Figure 2 for a graphical result of the non-monotonicity between φ, Γ.

We table our discussion of Betti one, B1, until Section 1.3.

1.3 Persistent Homology of Force Networks Under experimental conditions, often contacts or forces between adjacent particles are not

well defined. Despite significant progress, the analysis of the force network is limited by the

complexity of the problem and the need to answer a particular set of questions of relevance

 

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to dense granular media. The results presented in [15] proved that an accurate determination

of the contacts or forces is not required, in order to, quantify topological differences in

granular media with the same density at a different mechanical equilibrium. It is also well

accepted that the geometry of force chains plays an important role in determining the

macroscopic properties of dense granular media.

Def. A force network filtration

Given a complex CN and a monotone function f: CN -> R, the associated force network

filtration is the collection of all force network complexes

{FN (f, θ) | θ ∈ R} See Figure 7 for a demonstration of a network force filtration [10]. We can use homology,

in particular the Betti numbers, to characterize the geometry of each of the distinct force

networks in the force network filtration. It is important to note that only the normal force

is accounted for, that is, the component of the force projected on the line connecting the

centers of interacting particles.

Given a particular force network filtration, FN (f, θ): = {θ ∈ CN | f(θ) ≥ θ } ,generated

by a finite number of particles, there exist a finite number of values 0= θ0 <…< θk =

max f(θ). In other words, force networks are super-level sets focused on the geometry of a

part of the complex on which the forces exceed a specified level, which corresponds to the

part of the particle network experiencing force larger than θ. Though the Betti numbers

characterize the topology of a given force network, the vector space structure of its

homology plays an essential role, in that, it allows us to compare the topology of FN (f, θk)

with that of any other force network FN (f, θi). A key feature of persistent homology is

 

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that it reduces a scalar function to a persistence diagram, a collection of points in the plane

where each point encodes well-defined geometric information about the function, but does

not rely on a particular choice of threshold to do this. In other words, we no longer require

a precise definition for force chains. Force chains are self-organized gran columns that

rely on contacts from weakly stressed grain for lateral support to maintain and find new

stable states. The application of persistent homology can be viewed as a continuous non-

linear projection of contact and force networks to the space of persistence diagrams. This

feature suggests persistent homology is a good tool to study both the static and the

dynamical aspects of experimental and computational realizations of particulate systems.

The mechanical response of a dense granular media submitted to indentation by a rigid flat

punch is used when discussing the dynamics of compressed granular media. The resultant

deformation is viewed as a process of self-organization, which can be parameterized

through a force network filtration. Four aspects of the mechanical response can be observed:

(1) indentation resistance (2) Reynolds’s dilatancy (3) the undeforming dead zone are

characterized under the linear and cyclical structural building blocks of granular media self

organization; force chains and 3D cycle loops. The introduction of persistence allows us to

present positive results on all (4) aspects of granular media’s mechanical response. Force

network filtrations and their associated persistent homology provide more appropriate

metrics for understanding force networks than measurements performed at single thresholds.

To obtain stability results, we need to be able to compare across persistence diagrams.

 

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Def. Bottleneck distance.

Let PDk and PDi be the two n-th persistence diagrams, then [bd] distance is defined to be

dB (PDk and PDi) = inf sup || p- y(p) ||∞

where y: PDk -> PDi and p ∈PDk.

We now discuss some of the insights gained using computational homology to answer the

following set of research inquiries: (1) How does the load-indentation curve (i.e. principal

measurement in tests of the macroscopic response of materials under this type of boundary

condition- relate to the evolution of structural building blocks inside the material? (2) What

aspects of the evolution govern the four key aspects of its mechanical response? (3) How

can we characterize the irreversible deformation when shear is applied leading to long force

chains?

A materials deformation can thus be entirely tracked through its evolving network of

contacts, since all forces are transmitted through these contacts. Using persistent homology,

we are able to model the force chains and n-cycles as algebraic structures, known as

complexes, which have an inherent homology respectively. We are then able to model the

evolution of force chains and n-cycles in a dense granular media by studying how their

homologies persist or change over a filtration parameter. Figures 5-6 presents the evolution

of both force chains and n-cycles classified according to the number of particles (m, or n)

that form the structure. We observe a general decline in the population of force chains in

each class with increasing indentation. By contrast, the evolution of the minimal contact

cycles differs according to size n: the smallest members (i.e. 3-cycles) decrease in

 

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population while larger (n>6) increase in number. As a whole, an overall increase in the

porosity of the material (i.e. Reynolds’s dilatancy) as loading proceeds, the small 3-cycles

die to form larger components. The brunt of the indentation resistance is bared by the force

chains- the principal load bearing structures in a deforming granular material. The

degradation of the lateral supporting structures of 3-cycles, seen in Figures 5-6, precipitates

the collapse of force chains by buckling.

1.4 Recent developments in Computational Homology

To unravel further details on the mechanisms giving rise to dilatancy, as well as, failure and

dead zone, we need to examine the spatial distribution of force chains and 3-cycles. In [1-5],

all the m-force chains and n-cycles have been shown to exhibit cooperative behavior

invariant to loading conditions, material properties and dimensionality.

1.4.1 Local Stratified Homology

Using local stratified homology, we could gain new insights into the emergence and

cooperative interplay between force chains and 3-cycles in different localized regions across

a granular media sample. The local homology of frictional particles in a deforming dense

granular material reveals that favored cluster confirmation reside in distinct stability states.

To support axial loads, force chains typically reside in more stable states of stability,

preferring truss-like, and three cycle contact triangular topologies with neighboring grains.

Recent research has yielded insight into how the local structure behavior of column force

chains leads to inelastic deformation and failure at the macroscopic scale. Local stratified

homology would allow us to unravel the details of these mesoscopic structures and

 

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associated frictional re-arrangements, and their impact on bulk mechanical response-

particularly on the development of failure.

1.4.2 Sensor Networks and Time-Varying Persistent Homology

Another key mechanical response, which is of recent interest, is a granular media’s inability

to support load is the failure of force chains by buckling. Force chain buckling involves loss

of potential energy, loss of quasilinearity (i.e. alignment) and loss of contacts among its

constituent particles. Sensor network techniques have been used in [4] to construct a

network from the relationships between the particles’ time series, and to use the properties

of this network to detect emergent and persistent structures in the material. See Figures

[9,10] for graphical representation of a symbolic time-series for a force chain in [4].

Computational topology enters the study of sensor networks when we consider questions

like coverage. When does a set of sensors effectively monitor a region and when are there

gaps? Consider each particle in a granular network to be its own internal sensor which

observes its force chain status for each equilibrium state (record a 1 if the particle is part of a

force chain and record a 0 otherwise), then we obtain a symbolic time series of the force

chain status history of each particle in the assembly. [4] We refer the reader to [12] for a

more robust mathematical discussion on sensor networks and useful computational

algorithms.

 

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1.4.3 Sliding Windows and Time-Delayed Embedding Algorithm

The SW1PerS algorithm has been recently proposed in [25, 26] as a robust method for

studying time series data. It has never been used in the study of force chain. We invite the

reader to contemplate further applications of SW1PerS algorithm in their ongoing research

of granular media. Implementation guidelines of SW1PerS can be found in

http://cms.math.duke.edu/harer/?q=downloads. The SW1PerS algorithm is designed to

quantify periodicity in time series data. It measures the existence of a distinctive pattern in

the graph of a signal, and quantifies the extent to which it repeats. Using computational

homology, it measures the circularity of a point cloud derived from time-series data. We

refer the reader to [26] for a detailed mathematical derivation of sliding windows and

time-delayed embedding. The only empirical treatment of this technique using real data

can be found in [25]. It uses the SW1PerS algorithm to identify periodically expressed

genes across different processes.

From [5,9], we know the granular system overall evolves into a limit cycle under certain

conditions. See Figure [8] obtained from [5]. Understanding how failure propagates

spatially and temporally in granular media is key to our ability to control a broad range of

natural and human-made processes. We propose using sliding windows and time-delayed

embedding to track energy storage and dissipation at the particulate level and relate these

local processes to macroscopic behavior. The findings in [5] called for more accurate

methods to quantify the dynamics governing the spatial and temporal propagation of energy

that incorporate directionality (i.e. the capability to distinguish loading-up verses off-loading

of stored energy contacts). An attractive feature of the SW1PerS algorithm is that, unlike

 

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most standard signal processing measures, it does not use a set of patterns to identify signals

of interest, but instead attempts to discover patterns that exist organically within the data.

SW1PerS algorithm proved to be the best method for data with high shape variation,

damping and noise level in [25].

 

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Figure 1

 

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Figure 2

Figure 3

 

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Figure 4

Figure 5 Figure 6

 

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Figure 7

Figure 8

 

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Figure 9

Figure 10

 

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