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Topological Analysis of Packings
Gady Frenkel, R. Blumenfeld, M. Blunt, P. King
Application to flow propertiesin granular porous media
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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
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Overview
Separation: Topology - Geometry
Analyzing networks
Obtaining grain and
pore networks connectivity and proximity
Statistical description (Entropic formalism) Combining back with shape
+ Shape degrees of freedom
}{})({)/})({( qdqgXqWExpZ
Flow & Electrical properties as
expectation values over partition function
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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:
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2D Packing Example:
• GRAINS: – Straight lines and planes that
connect contacts instead of real boundaries
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2D Packing Example:
• GRAINS: – Straight lines and planes that
connect contacts instead of real boundaries
Contact Points
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2D Packing Example:
• GRAINS: – Straight lines and planes that
connect contacts instead of real boundaries
Contact Points
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2D Packing Example:
• GRAINS: – Straight lines and planes that
connect contacts instead of real boundaries
• PORES: – empty” volumes that are
surrounded by transformed grains.
• CONTACT POINT:
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Obtaining pores 2D
Grain
Pore
Directed Grain-edge vectors:Grain: Anti-ClockwisePore: Clockwise
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3D
Pål-Eric Øren
Grains polyhedra
Pores bounded by facets
Throat missing faces
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Finding Throats:
• Facets of the pore are known
• Use the 2D algorithm where the radial vector sets the positive edge direction
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Finding Throats:
Need to know which faces of grains belong to the pore
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– 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
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Example: Beads in 2D1. Grains → Polygons2. Balloons are
inflated from each facet
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2D Large Sample with friction
• Transformed system – polygons• Balloon inflated in the pores – perfect recognition of
pores and pore grain relations
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Grain 418 face 2 Grain 418 face 3
Grain 995 face 3 Grain 501 face 0
Sample cells of foam like structure
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Overview
Separation: Topology - Geometry
Analyzing networks
Obtaining grain and
pore networks connectivity and proximity
Statistical description (Entropic formalism)
+ Shape degrees of freedom
Flow & Electrical properties as
expectation values over partition function
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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
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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
)(
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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
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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
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Summary
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
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