mathematical morphology random sets and …mipomodim.math-info.univ-paris5.fr/talk-djeulin1.pdf1...

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Dominique JeulinCentre de Morphologie

Mathématique

Workshop on Models and Images for Porous Media, Paris, 12 January 2009

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Mathematical Morphology Random sets and Porous Media

Dominique [email protected]

Centre de Morphologie Mathématique, Mathématiques et Systèmes

Centre des Matériaux P.M. Fourt, Mines-ParisTechDominique Jeulin

Centre de Morphologie Mathématique


2Origin – MotivationsG. Matheron, 1967

• Characterization of themorphology of a heterogeneous medium?• Prediction of the macroscopicbehaviour of a porous medium (composition of permeabilities)?• Representation of a heterogeneousmedium by a model?

L = 1 mm


Mathématique


3Introduction

• Extraction of quantitative information on microstructures (3D images, measurements)

• Models of random sets and simulation of microstructures

• Theory of random sets and tools to solve homogenization problems


Mathématique


4

Characterization of a randomstructure

•3D Images•Morphological Criteria•Probabilistic Criteria


Mathématique


5

3D Images of Intermetallic particles in Al alloys

• X ray microtomography performed at the ESRF (ID 19 line)

• 3D measurements• High resolution (0.7μm)• 3D complex shapes of particles (morphological

parameters, local curvature) (E. Parra-Denis PhD)


Mathématique


6

ZnO needles afterZnO needles after sublimation in sublimation in thethe solairsolair oven oven PROMES PROMES LaboratoryLaboratory (CNRS (CNRS OdeilloOdeillo))

Application : Application : ionicionic conduction conduction

Acquisition : Acquisition : --4242°° àà +37+37°° withwith 11°° increments increments

ResolutionResolution of images : x12000 of images : x12000 2 nm 2 nm ↔↔ 1 1 voxel voxel

ZnO by NanotomographyArnaud GROSJEAN

Dominique JEULIN

Maxime MOREAUD

Alain THOREL

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Mathématique


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ZnO by Nanotomography

ResolutionResolution : 2 nm : 2 nm ↔↔ 1 1 voxelvoxel


Mathématique


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Mathématique


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Characterization of a random structure: Main Criteria

• Morphological criteria– Size– Shape– Distribution in space (Clustering, Scales, Anisotropy)– Connectivity

• Probabilistic criteria– Probability laws (n points, supK)– Moments


Mathématique


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Characterization of a random set

• Models derived from the theory of RandomSets by G. MATHERON

• For a random closed set A (RACS), characterization by the CHOQUET capacityT(K) defined on the compact sets K

• In the euclidean space Rn, CHOQUET capacityand dilation operation

}{ { } )(11)( KQAKPAKPKT c −=⊂−=Φ≠= I

}{ KAxPKT x ⊕∈=)(


Mathématique


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Binary Morphology (Fe-Ag)

L = 250 µm


Mathématique


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Basic Operations of MM

Erosion hexagonal (2)

Dilation hexagonal (2)

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Mathématique


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Morphological interpretation

• Experimental estimation of T(K) by image analysis, using realizations of A, and dilation operation.

• General case: several realizations and estimation for every point x

• For a stationary random set, T(Kx) = T(K);• For an ergodic random set, T(K) estimated from a

single realization by measurement of a volume fraction

• Every compact set K (points, ball...) brings its owninformation on the random set A


Mathématique


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Calculation of the CHOQUET capacity

For a given model, the functional T isobtained:

• by theoretical calculation• by estimation

– on simulations– on real structures (possible estimation of the

parameters from the "experimental" T , andtests of the validity of assumptions).


Mathématique


15

Models of Random Structures


Mathématique


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Point Processes

Most simple kind of random structure: verysmall defects isolated in a matrix

• Particular RACS: Choquet capacity T(K)• Probability generating function GK(s) of

the random variable N(K) (number of points of the process contained in K)


Mathématique


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Poisson Point ProcessPrototype random process without any order


Mathématique


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Random sets and Random Functions Models

Starting from a point process, more generalmodels, called grain models:

• The Boolean model• The dead leaves model• Random function models

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Mathématique


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Boolean Model

• The Boolean model (G. Matheron) isobtained by implantation of random primarygrains A’ (with possible overlaps) on Poisson points xk with the intensity q:

• Any shape (convex or non convex, and evennon connected) can be used for the grain A’

xkk AxA 'U=


Mathématique


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Boolean Model (G. Matheron, 1967)

Fe-Ag Alloy Boolean Model of Spheres (0.5)


Mathématique


21Boolean Model of Spheres3D Simulation

D. Jeulin

M. Faessel

(CMM)


Mathématique


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Hard Spheres 3D Simulation

D. Jeulin

M. Faessel

(CMM)


Mathématique


232 scales Cox Boolean model of Spheres3D Simulation

D. Jeulin

M. Faessel

(CMM)


Mathématique


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Boolean Model

WC Co (J.L. Chermant, M. Coster, J.L. Quennec ’het D. Jeulin) L = 40 µm

Poisson Boolean Model(P. Delfiner)

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Mathématique


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Boolean Model

• Choquet capacity, with

• Ex: contact distribution (ball), covariance

• Percolation threshold obtained from simulations: 0:2895 +-0:0005 for spheres with a single diameter

• Percolation threshold related to the zeroes of NV – GV (p)

{ }cAxPq ∈=

( ) )'()'(

1)'(exp1)(1)( AKA

n nn

qKAKQKT μμ

μθ⊕

−=⊕−−=−=


Mathématique


26

Percolation threshold

• Materials made of components with a high contrast of properties: strong effect on the macroscopic properties when a given phase (e.g. the pores) percolates through the structure (connected paths in the samples of the medium)

• For a given model, estimation of the critical percolation threshold ρc (volume fraction above which a component percolates)


Mathématique


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Percolation of random structures

No percolation : no path connects

two opposite faces

Percolation: one aggregate connects two

opposite facesDominique Jeulin



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How to estimate a percolation threshold?

Labelisation of aggregates allows easy extraction of aggregates connecting two opposite faces

Percolation threshold ρc = volume fraction of objects when 50% of the realizations have aggregates connecting two

opposite faces


Mathématique


29

Carbon nanotubes composite materials

Outstanding mechanical, electrical or chemical properties, mainly due to

their low percolation thresholdDominique Jeulin



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Percolation of a Boolean model of spheres

Volume : 2000Volume : 200033

sphere radius : 10sphere radius : 10

0.28970.2897±±0.00040.000448 s.48 s.about 5 000 000about 5 000 000

Percolation Percolation thresholdthreshold

AverageAveragecomputation time computation time for one for one realizationrealization

AverageAverage NbNb. of . of simulatedsimulatedspheresspheres

(*) (*) 0.28950.2895±±0.0050.005M.D. Rintoul et S. Torquato, Precise determination of the criticM.D. Rintoul et S. Torquato, Precise determination of the critical threshold and al threshold and exponents in a threeexponents in a three--dimensional continuum percolation model, J. Phys. A: Math. dimensional continuum percolation model, J. Phys. A: Math. Gen. 30, L585Gen. 30, L585--L592, 1997L592, 1997

PC PIV 2.6GHz RAM 768 MoPC PIV 2.6GHz RAM 768 Mo

[*] Jeulin D., Moreaud M. Multi-scale simulation of random spheres aggregates – application to nanocomposites, Proc. 9th European Congress on Stereology and Image Analysis, Zakopane, Poland, May 10-13 2005, Vol. I p. 341-348.

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Mathématique


31Percolation of a Boolean model of spheres (complementary set)

Estimation of Estimation of thethe percolation percolation thresholdthreshold of of the the complementarycomplementary randomrandom set of a set of a booleanbooleanmodel of model of spheresspheres (constant radius)(constant radius)

ρρcc = = 0.05400.0540±±0.0050.005


Mathématique


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One scale simulationPeriodic simulations along axis x and y (50 realizations)

V=3003, l=50, r=2, Vv=0.02

Percolation of a Boolean model of sphero-cylinders with uniform orientations

AR=l / r

0.097800.011450.002320.00037ρc

101005003000Aspect ratio l/r

Jeulin D., Moreaud M. Percolation of multi-scale fiber aggregates, Proc. 6th International Conferenceon Stereology, Spatial Statistics and Stochastic Geometry, Czech Republic, June 26-26, 2006.


Mathématique


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Simulation of oriented sphero-cylinders: Φ uniformly distributed, φ limited between –x and +x

V=3003, l=50, r=2, Vv=0.02, x=± 15°

Percolation of oriented sphero-cylinders

Materials like papers or carbon nanotubes enhanced elastomers are composed of oriented fibers


Mathématique


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More oriented the fibers are, higher ρc is in the two directions (axis Z and plane XY)

Percolation of oriented sphero-cylinders

φ between–x and +x


Mathématique


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Combination of basic random sets

• Starting from the basic models, more complex structures, such as superposition of scales, or fluctuations of the local volume fraction p of one phase

• Union or intersection of independentrandom sets

21 AAA I=( ){ } { } { }2121)( AKPAKPAAKPKP ⊂⊂=⊂= I


Mathématique


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Intersection of random sets

P = 0.49 P = 0.49 (P1 = P2 =0.7)

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Mathématique


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Nanocomposite carbon black - polymer (TEM)

Transmission micrographs (L. Savary, D. Jeulin, A. Thorel)


Mathématique


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Simulation of a Carbon black Nanocomposite

Intersection of 3 scales Boolean models of spheres(identification from thick sections)


Mathématique


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Percolation of multiscale aggregates

• Cox Boolean model

Simulation Labeling of aggregatesDominique Jeulin



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Percolation multiscale spherical aggregates

• Two scales of spheres

• Critical percolation threshold lower than for the Boolean model of spheres (0.2895)

Volume : 2000Volume : 200033

sphsphèères : 10res : 10

aggregates : 300aggregates : 300

ρρcc = 0.085= 0.085±±0.0030.003ss2= 0.0838 (s: = 0.0838 (s: ρρc SB c SB spheresspheres))

[*] Jeulin D., Moreaud M. Multi-scale simulation of random spheres aggregates – application to nanocomposites, Proc. 9th European Congress on Stereology and Image Analysis, Zakopane, Poland, May 10-13 2005, Vol. I p. 341-348.


Mathématique


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•Two-scale simulation, with sphero-cylinders randomly located into inclusion spheres according to a Cox point process•Dichotomic research used to estimate ρc with 20 realizations for each volume fraction ρρcc = = 0.05%

V=3003, l=30,

r=2, Vv=0.05V=3003,

r=60, Vv=0.3

Percolation of multi-scale distributions of sphero-cylinders


Mathématique


42Percolation of multi-scale distributions of sphero-cylinders

• ρc lower than for a homogenous distribution• lowest ρc obtained for a large diameter of spheres and a low volume fraction of spheres

0.000490.000470.00051ρc

0.340.440.51Vv of spheres2 x ARDiameter of spheres D1000 Volume dimensions = (5 x D) 3Aspect ratio AR

0.0120.0120.0140.0190.0160.014ρc

0.350.430.50.350.420.51Vv of spheres5 x AR2 x ARDiameter of spheres D

50 Volume dimensions = (5 x D) 3Aspect ratio AR

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Mathématique


43Color dead leaves(D. Jeulin, 1979)


Mathématique


44Color dead leaves: non overlaping grains (D.

Jeulin, 1997)

Size distribution (initial – intact); single symmetric convex grain in Rn for homogeneous model: VV < = 1 / 2n


Mathématique


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Random Functions Models

Continuous version of random sets models• Boolean RF: random implantation of primary

random functions on points of a Poisson point process.

• The U operation for overlapping grains is replacedby the supremum or by the infimum

• Change of support by Sup or by Inf (ExtremeValues)


Mathématique


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Boolean random functions

Cone Primary grains Boolean Variety


Mathématique


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Powders and dead leaves

D. Jeulin, CALGON L = 15.5 µmDominique Jeulin



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ReactionReaction--Diffusion ModelsDiffusion Models

ReactionReaction--Diffusion Diffusion ThesisL. Decker (1999)

Simulation of connected media

Turing texture

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Mathématique


49Change of scale in random media(Physics-Texture)

From Nano to Macro


Mathématique


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Homogenization

• Heterogeneous Medium (composite, porousmedium, metalli polycristal, rocks, biological medium, rough surfaces …)

• Eqyuvalent Homogeneous Medium?• Prediction of the « effective» properties


Mathématique


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Examples of homogenization problems

• Thermal conductivity of a two-component medium (thermal insulation)

• Transport properties of porous media• Elastic Moduli of heterogeneous medium

• Wave Propagation in heterogeneous medium(electromagnetic, acoustic…)

• Optical Properties of a heterogeneous medium or of a rough surface


Mathématique


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Examples of Physical Properties

PermeabilityPressureGradient

VelocityFluid Flow in porous media

Elastic Moduli

SrainStressElasticity

DielectricPermittivity

Electric Field

DielectricDisplacement

Electrostatics

Thermal Conductivity

TemperatureGradient

Heat FluxThermal conduction

PropertyProblem


Mathématique


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Homogenization

• Bounds of macroscopic properties (order 3: multicomponent random sets and functions)

• Optimal random Microstructures

• Probabilistic Definition of the RVE for numerical simulations


Mathématique


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Change of scaleApplications of the models of random media to the prediction of the macroscopic behavior of a physical

system from its microscopic behavior.• Estimation of the effective properties (overall

properties of an equivalent homogeneous medium) of random heterogeneous media from theirmicrostructure (Homogenization)– From variational principles, bounds of the effective

properties for linear constitutive equations.– Estimation of the effective behaviour from numerical

simulations on random media.• Fracture statistics models

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Mathématique


55Thermal Conductivity of Ceramics (Use of Boundsfor the Boolean Model)

• Textures AlN (λ=100) with a Y rich binder (λ= 10) C. Pélissonnier, D. Jeulin, A. Thorel


Mathématique


56Thermal Conductivity of Ceramics: 3rd Order Bounds of the Booleanmodel


Mathématique


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Random Composite with optimal effective properties


Mathématique


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Digital Digital MaterialsMaterials


Mathématique


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Digital Digital MaterialsMaterials

• Input of 3D images – Real images (confocal, microtomography)– Simulations from a random model

• Use of a computational code (FiniteElements, Fast Fourier Transform, PDE numerical solver)

• Homogenization• Representative Volume Element?


Mathématique


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Homogenization and Simulation

• Prediction of the effective properties by 3D FFT

• Nano Composites CarbonBlack - polymer: permittivity,,fromfrom a a multi-scale randommodel (PhD A. Delarue, 2001 M. Moreaud, 2007, D. Jeulin, A. Thorel, DGA, EADS)

• Charged Elastomers: elasticbehaviour (PhD A. Jean, Michelin, 2006-2009 (19 February))

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Mathématique


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Homogenization and SimulationField D(x) in each point

Field E(x) in each point

Method:Properties of components

+ Iterative algorithmbased on Fourier tansform, to solve

the Gauss equationof electrostatics, derived from the

Maxwell equations

Periodic boundaryconditions

EDε*=


Mathématique


62Ice cream: mechanical behavior(Unilever)

Three scales of observation

d L<<<< l

RVE

L dl

(thesis T. Kanit, 2003, S. Forest)


Mathématique


63Morphology andMorphology and effectives effectives propertiesproperties**Heterogeneities Heterogeneities in 2Din 2D

100 100 µµmm

CoarseCoarse microstructuremicrostructure Fine microstructureFine microstructure


Mathématique


64Morphology and Young’s modulusMorphology and Young’s modulusExperimental measurementsExperimental measurements (4(4--point point bendingbending test)test)

How to How to predict the Young’s moduluspredict the Young’s modulus for for differentdifferent microstructures ?microstructures ?


Mathématique


65AnalyticalAnalytical models: models: bounds andbounds and estimationsestimations

* * BoundsBounds :: veryvery largelarge..

*Estimation :*Estimation : doesdoes not not really take the morphology really take the morphology into accountinto account..

Not really useful for media with a high contrast in propertiesDominique Jeulin



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PrinciplePrinciple of of Homogenization TheoryHomogenization Theory•• TwoTwo--phase phase heterogeneous material with elastic tensorsheterogeneous material with elastic tensors CC11

andand CC22

•• Homogeneous equivalent material with the macroscopic Homogeneous equivalent material with the macroscopic elastic tensor elastic tensor CCeffeff

•• WeWe have: have: •• With theWith the spatial spatial averagesaverages

•• WhereWhere

~~ECeff=Σ

>=<Σ~~

σ >=<~~εE

~σ andand

~ε are are thethe local stress local stress and strain tensorsand strain tensors

∫>=<V

dxxPV

P )(1

12


Mathématique


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PrinciplePrinciple of of Homogenization TheoryHomogenization Theory

•• TwoTwo--phase phase heterogeneous material with elastic tensorsheterogeneous material with elastic tensors CC11andand CC22

•• AverageAverage elasticelastic energyenergy (Hill(Hill--Mandel Mandel lemmalemma) of a ) of a specimenspecimen V submittedsubmitted to one of to one of the followingthe following boundaryboundaryconditions:conditions:

-- KUBC : KUBC : kinematickinematic ((strainstrain) ) uniformuniform boundaryboundary conditionsconditions-- SUBC : SUBC : staticstatic (stress) (stress) uniformuniform boundaryboundary conditionsconditions-- PERIODIC : PERIODIC : periodicperiodic boundaryboundary conditionsconditions

><>=<Σ>=>=<<~~~~~~

:::: εεεεεσ CeffEC

~σ andand

~ε are are uncorrelateduncorrelated

RVE RVE and Integraland Integral RangeRange•• P(x)P(x): local : local random propertyrandom property ((indicator functionindicator function, , Young’s Young’s

modulusmodulus, …) in , …) in the domainthe domain VV..•• Local variance of Local variance of P: DP: D22[[P(x)P(x)]]•• MeanMean value of value of P P inin VV: :

•• Variance of Variance of the meanthe mean::

•• AA33 is the Integralis the Integral Range of Range of PP ((giving thegiving the size of a RVE for size of a RVE for PP))

•• If If AA33 << V, << V, V V may be subdivided intomay be subdivided into N = N = V /AV /A3 3 subdomains with uncorrelated propertiessubdomains with uncorrelated properties PPii

VAxPDPD 322 )]([][ =><

∫>=<V

dxxPV1P )(


Mathématique


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Integral Range and Covariance

• Fluctuations of the average values <P> of P(x) (stationary RF) in the domain V, as afunction of the centred covariance

•W2h)}()())(()({()(2 PExPPEhxPEhW −−+=

dxdyyxWV

VDVVP )(1)( 22

2 −= ∫∫


Mathématique


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Integral Range and Covariance

• For large specimens (V >> A3), where A3 isthe integral range, asymptotic formula for the variance:

dhhWD

A

VADVD

RP

PP

)(1with

)(

223

322

3∫=

=

where D2P is the point variance of P(x)


Mathématique


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Set Covariance

• Covariance C(h) of a random set A

• For a stationary random set, C(x,x+h) = C(h)

• For an ergodic random set, C(h) isestimated by the volume fraction of

Cx,x h Px ∈ A,x h ∈ A

A ∩ A−hDominique Jeulin



72Microstructure and Covariance

withoutAFP

withAFP

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Mathématique


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Set Covariance and field Covariance

• Connexion between the two?

• Generally, no direct link!!

• The Covariance of fields depends on all « sets » moments with n points…


Mathématique


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Determination of the integral range and of the RVE

• Start from realisations of the microstructure (images or simulations)

• Use appropriate Boundary Conditions (Periodic,…) to estimate the effectives propertiesof every realisation

• Estimate the average and the variance of effectives properties as a function of the volume of specimens

• Estimate A3 , the RVE and the number of fields to simulate as a function of the wanted precision


Mathématique


75Elastic Moduli and Thermal Conductivity of the 3D

Voronoï Mosaïc

• 3D Voronoï space tessellation: zones of inluence of random Poisson points

• Independent random coloration of each Voronoï polyhedron: Poisson mosaïc(approximation of some real two-phase textures)

• Finite element calculations on finite volume realizations of Voronoï mosaïc in V


Mathématique


76Thermal computation on Thermal computation on VoronoïVoronoï mosaïcmosaïc70% 70% iceice, , contrastcontrast = 100 in thermal = 100 in thermal conductivityconductivity

Computation of thermal Computation of thermal conductivityconductivity in 3 directionsin 3 directions

map of the flux in direction (z)with periodic b. c. (8281 d.o.f.)

map of the temperature with UGT b. c. (312481 d.o.f. on 20 pc)


Mathématique


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* * Fluctuations of Fluctuations of thethe thermal thermal conductivityconductivity--volume fraction = 70% of hard phase (volume fraction = 70% of hard phase (iceice))--contrastcontrast in thermal in thermal conductivityconductivity = 100= 100--UGT : UGT : uniformuniform temperaturetemperature gradient gradient atat thethe boundaryboundary--UHF : UHF : uniformuniform heatheat flux flux atat thethe boundaryboundary--PERIODIC : PERIODIC : periodicperiodic boundaryboundary conditionsconditions

Results for the Voronoï mosaïc


Mathématique


78Results for the Voronoi mosaïc

30765thermal conductivity

number of RVE

number of RVE

= 5%= 1%ε relative ε relative

Minimal Minimal numbernumber of of realizationsrealizations for for periodicperiodic boundaryboundary conditionsconditionsExampleExample: RVE for : RVE for unbiasedunbiased volume = 125 grains volume = 125 grains

14


Mathématique


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Integral range (volume fraction P)

1.1770.9

1.1110.7P =

1.1780.5

Gilbert (1962): 1.179


Mathématique


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Integral range (elastic moduli)

1.637Périodicµ (P = 0.5)

1.863KUBCµ (P = 0.7)

2.088KUBCK (P = 0.7)

E1 / E2 = 100


Mathématique


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Integral range (thermal conductivity)

2.619Périodic

2.036UHFλ

2.335UGT

P = 0.7; λ1 / λ2 = 100


Mathématique


82

Integral Range and Representative Volume Element for the estimation of

the Elastic Moduli of the Boolean Model of Spheres

(joint work with F. Willot)(joint work with F. Willot)


Mathématique


83Elastic Fields solved by the Fourier transform (FFT)

Algorithm based on the Lippmann-Shwinger equations(Moulinec, Suquet, 1994). Introduction of a homogeneous

elastic tensor L(0) (reference medium) and of the corresponding Green function G(0)

• Iterations in the Fourier space (Green function) and in the real space. For G(0) with zeo mean:

• Periodic boundary conditions. Operation on images, without any mesh

• Convergence for an infinite contrast with the improved algorithm “increased Lagrangien (Moulinec, Suquet + Moulinec, 2001)

For an isotropic medium, in the Fourier space:


Mathématique


84

Effective Effective BulkBulk Modulus of the Boolean model of spheresModulus of the Boolean model of spheres

pc

ĸ µmatrice 1/3 ½inclusions 1000 1000

Bulk Modulus as a function of the concentration of the rigid phase and HS, Beran bounds

f=0,2f=0,2

15


Mathématique


85

Int. J. of Eng. Sc. 2008, in press

Effective Effective BulkBulk Modulus of the Boolean model of spheresModulus of the Boolean model of spheresporespores

Non compressible matrix (Poisson coeff. ~ 0,4999)

Sphere VV

Bulk

Modulus


Mathématique


86Integral Range for different properties

Sphere VV

Integral range A3


Mathématique


87

Representative Volume Element

• The RVE depends on:

– The morphology– The physical property (elastic moduli, thermal

conductivity, dielectric permittivity, …)– The contrast between properties of components– The type of boundary conditions (uniform,

periodic,;..)


Mathématique


88

Conclusion• Random models of structures, to simulate the

complex morphology of microstructures• Approach, based on measurements obtained by

image analysis: test and select appropriatemodels, estimate their parameters

• Physical and Morphological Modelling « Whichtextures for which properties? »

• Possible use in the synthesis of textures• Many domains of application: Materials, Nano

composites, Porous media, Biology, Food, Surfaces, Vision...


Mathématique


89

To learn more• Courses ENSMP, 60 Bd Saint-Michel,

Paris–– PhysicsPhysics and Mechanicsand Mechanics of of RandomRandom Media (30 Media (30

March March -- 3 3 AprilApril 2009)2009)–– Models of Models of RandomRandom Structures (Structures (NovemberNovember

2009)2009)• Groupe de Travail MECAMAT « Approches

probabilistes en Mécanique des Milieux Hétérogènes » (Bordeaux, 14-15 May 2009)

http://cmm.ensmp.fr/

mathematical morphology random sets and …mipomodim.math-info.univ-paris5.fr/talk-djeulin1.pdf1...

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