© Imperial College LondonPage 1 Topological Analysis of Packings Gady Frenkel, R. Blumenfeld, M....

© Imperial College London

Topological Analysis of Packings

Gady Frenkel, R. Blumenfeld, M. Blunt, P. King

Application to flow propertiesin granular porous media


Overview

Separation: Topology - Geometry

Analyzing networks

Obtaining grain and

pore networks connectivity and proximity

Statistical description (Entropic formalism) Combining back with shape

Network flow simulationsCombine back with shape

Flow & Electrical properties as

expectation values over partition functionNetwork flow simulations


Overview


Analyzing networks

Obtaining grain and



+ Shape degrees of freedom

}{})({)/})({( qdqgXqWExpZ


expectation values over partition function


Topological Representation of Granular Packing

• Grains: (transformed)– Polygons (2D) or Polyhedrons (3D)– Corners are the contact points between grains– Assumption: transformed grain is convex

• Pores: – “Convex” “empty” volumes that are surrounded by

transformed grains.

• Throats: (3D)– Surfaces. the openings that connect two pores:


2D Packing Example:

• GRAINS: – Straight lines and planes that

connect contacts instead of real boundaries


2D Packing Example:



Contact Points


2D Packing Example:



• PORES: – empty” volumes that are

surrounded by transformed grains.

• CONTACT POINT:


Obtaining pores 2D

Grain

Pore

Directed Grain-edge vectors:Grain: Anti-ClockwisePore: Clockwise


3D

Pål-Eric Øren

Grains polyhedra

Pores bounded by facets

Throat missing faces


Finding Throats:

• Facets of the pore are known

• Use the 2D algorithm where the radial vector sets the positive edge direction


Finding Throats:


Finding Throats:

Need to know which faces of grains belong to the pore


– Growing a deformable object : Inflating a balloon inside the pore until it is filled.

– Controlling the inflation by curvature.– Convexity prevents the balloon from exiting

the pore.

Obtaining Pores 3D: Positively Curved Inflating Balloon

X


Example: Beads in 2D1. Grains → Polygons2. Balloons are

inflated from each facet


2D Large Sample with friction

• Transformed system – polygons• Balloon inflated in the pores – perfect recognition of

pores and pore grain relations


Large 3D Sphere Packing

Created By: Zdenek Grof


Obtaining The transformed Grains


Foam like structure


Grain 418 face 2 Grain 418 face 3

Grain 995 face 3 Grain 501 face 0

Sample cells of foam like structure


Overview


Analyzing networks

Obtaining grain and


Statistical description (Entropic formalism)

+ Shape degrees of freedom


expectation values over partition function


Entropic formalism

• Statistical Mechanics approach– Averages, Fluctuations, Scaling, …

• Configurational entropy S

• Edwards Conjecture of compactivity: – (energy) H W (volume)

– (temperature) T X (compactivity)

• Partition Function:

}{})({})({

qdqgeZ XqW

SWX


Entropic formalism

}{})({})({

qdqgeZ XqW

Degrees of freedom

-Correct number

-Independent

-Parameterize the volume

Density of states

Simplifying Assumptions:

- Degrees of freedom are uncorrelated

- All q’s are of the same kind

Partition function is obtained by a single quadrilateral distributionn

qqX

A

dAAgeZq

)(


2D Tessellation: quadrilaterals

• Quadrilaterals tessellate space

• Number of degrees of freedom for the isostatic case = number of quadrilaterals

• The areas of quadrilaterals serve as the degrees of freedom

loop i’

g’

rpg

Rpg

loop i

g


Distribution of quadron Area For 5000 spheres with Radii uniformly distributed in (rmin,3rmin) and friction. The distribution is fitted with a Gamma distribution to obtain:- Mean volume per quadron: <P>=1.44/(2.13+1/X).- n-th moment of the volume per quadron: <Pn>=Γ(1.44+n)/[Γ(1.44)*(2.13+1/X)n].- Volume fluctuation per quadron: <ΔP2>=1.44/(2.13+1/X)2.

2D quadrilaterals area distribution


Summary


Analyzing networks

Obtaining grain and



Network flow simulationsCombine back with shape


expectation values over partition functionNetwork flow simulations

© Imperial College LondonPage 1 Topological Analysis of Packings Gady Frenkel, R. Blumenfeld, M....

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