RADIATION WAVES IN STOCHASTIC BINARY MIXTURES · SUMMARY Modelling radiation transport in non-homogenous medium is an issue because direct simulations are questionable This problem

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RADIATION WAVES IN

STOCHASTIC BINARY

MIXTURES

SEPTEMBER, 26TH 2014

9/23/2014 | PAGE 1WCMF’6 Santa Barbara | september 27th

2014

Jean-François Clouet

SUMMARY

Modelling radiation transport in non-homogenous medium

is an issue because direct simulations are questionable

This problem has strong connection with particle transport

in stochastic medium for which rigorous results can be

exhibited

Nevertheless, there is no well established model for the

full radiative transfer equations and we will propose a new

one.

9/23/2014 | PAGE 2WCMF 6 Santa Barbara | september 27th 2014

ICF MOTIVATION

The principle of Inertial Confinment Fusion is to implode a pellet containing DT fuel.

Whatever the driver for implosion, it leads to non-laminar hydrodynamical motion and

mixture between fuel and pusher occurs

Modelling radiation transport in this highly non-homogeneous medium is necessary

9/23/2014 | PAGE 3Santa Barbara | september 27th 2014

Pusher

Deuterium/Tritium fuelPellet

EXAMPLE OF HYDRODYNAMIC SIMULATION


Snapshots of hydro-dynamical simulation (density plots)

Fuel

Pusher Rayleigh-Taylor Instability

Non-linear growth

Turbulent-like behavior

Need for models for describing radiation transport in the mixture

EXPERIMENTAL EVIDENCE OF THE INFLUENCE OF

HETEROGENEITIES

In a series of experiments at Omega-facility, P.Keiter has demonstrated

the role of heterogenities in the speed of radiation waves


7,5 kJ Laser Energy Temperature

~2.5 Millions Kelvin

Gold Wall

Foam loaded withcalibrated gold particles

• The laser energy is converted into radiative energy by the gold wall

• The high temperature reached in the gold cavity initiates a radiation wave

• The speed of the wave through the foam can be measured


(P.Keiter and al,

Phys;Plasmas, 15 (2008))

Position of the radiation wave as a function of time for two distribution of particles

Size effect is demonstrated

The speed of the radiation

wave is

t

ctV

effσ∝)(

σeff is the effective opacity of

the mixture : the experiment

proves that is depend on the

size of particles

POMRANING-LEVERMORE MODEL


The Pomraning-Levermore model is very popular among physicist to explain this

effect. The simplest one relies on:

1. Modeling of transport in purely absorbing medium*

0))()(()( =−+ sBIssIds

dσ

I(s): radiative intensity

σ(s): opacity

s: curvilinear coordinate

B: equilibrium Planck function2. Markovian statistics for the binary mixture:

• Chord length are i.i.d. random variables

• Exponential distributions with length l1 and l2 is assumed

3. « Diffusion approximation »

with effective opacity [vanderhaegen 1988] )(~

1)()( s

ds

dBsBsI

σ−≈

1

0

)(~−

∞−

∫= dsesσσ

* There exists an extension of the model for scattering medium but it is not correct in the diffusion limit [larsen 2005]


Using renewal equations, this effective opacity can be computed:

)(1

~

21

21

σσσ

σσσσ

−++

+=

h

h

l

l

21

111

lllh

+=

This model is very convenient and easy to use: the influence

of size of particles on effective mean free path is explicit

The goal of this talk is:

• to apply multiscale analysis in random media to radiative

transport

• Study the connexion with Pomraning’s model

• Evaluate the importance of Markovian assumption

• Extend the model to take into account particles ablation

MATHEMATICAL MODELLING


Neglecting hydrodynamical motion, the radiation field I(x,Ω,ν,t) is solution of

04

)(.1

=

Ω−+∇Ω+∂ ∫ π

σd

IIxIIc

t

Radiation transport

Energy balance

T=T(x,t) is the temperature

νΩis the radiative energy

Bν(T) is the equilibrium Planck function

Next, we assume that T(x,t) is constant and neglect frequency dependance

∫∫ =Ω

−+∂

=+∇Ω+∂

04

)(),(

4

)(),(),(.

1

ddITcB

xTTC

TcBxTIxTII

c

tv

t

νπ

σ

πσσ

νν

ννν

[ ]πϕµ

ϕµ

ϕµ

µ

2;0],1;1[,

)(sin1

)(cos12

2 ∈−∈

−

−=Ω

PROBABILISTIC INTERPRETATION


This equation has a simple probabilistic interpretation which is suitable for Monte-

Carlo simulation of radiation transport

In infinite medium, with initial condition I0(x,Ω,ν)

2

0

on process jump)(

),(),,(

S

cdt

dX

XItxI

t

tt

tt

=Ω

Ω=

Ω=Ω

π

σ

4

d

)(

Ω

tXIntensity of jumps:

Distribution of jumps:

In binary mixtures, a natural assumption is to model the scattering coefficient σ(x) by

a random field :

σ(x)= σ1=1/λ1 in medium 1, volume fraction f1, characteristic length l1,

σ(x)= σ2=1/λ2 in medium 2 volume fraction f2, characteristic length l2

We are mainly interested in the case :

• L>>l1, l2 : small heterogeneities

• L>> λ2 : diffusion approximation valid in medium 2 at the macroscopic scale

• L ~ λ1 : transport behavior in medium 1 at the macroscopic scale


Introduce two parameters:

1

1

2

1

1

<<=

<<=

σ

σ

ε

q

L

l Small heterogeneities homogenization

High contrast ratio between the components

Diffusion approximation

Equation rewrites as

( ) ( ) 04

),(4

),(1

. 12 =Ω−+Ω−+∇Ω+∂ ∫∫ πωεσπωεσ νννdIIxdIIx

qIIt

With 0, →qε

We use classical representation of ergodic random fields with stationnary

measure P(dω)

Methods of pediodic homogenization apply )(),( ωτσωσ xx ≡


If ε<<q:

• Apply first homogenization theory with q~1

II →ε

( )1

1

2

2

04

.

σσσ

πσ

ff

dIIIIt

+=

=Ω−+∇Ω+∂ ∫

• Then use diffusion approximation of transport process Θ→ νν BI

∆Θ=Θ∂σ3

ct

This is the « atomic » mix limit:

• the size of heterogeneities is the

smallest lenght scale of the problem

• The medium is overall diffusive


The case ε ~q is the most interesting case. After time renormalization:

( ) ( ) 04

),(1

4),(

1. 12

2=Ω−+Ω−+

∇Ω+∂ ∫∫ π

ωε

σεπ

ωε

σεε

νν

dIIxdIIxIIt

We must perform homogenization and diffusion approximation at the same time:

we introduce the asymptotic expansion

...),,,(),,,(),,,(

),,,(),,,(

2210 +Ω+Ω+Ω=

Ω=Ω≡=

ωτεωτεωτ

ωτωτ

ε

ε

ε

ε

εε

yyy

xy

yx

txItxItxII

txItxII

And plug this expansion into the transport equation


This enables to prove Fredholm alternative for operator:

( )

0

2

)()(

4)()(.

=∂

∂=∇

Ω−+∇Ω≡ ∫

yyfy

f

dfffLf

ωτω

πωσω ν

1),(0 ∝Ω⇔= ωfLf

hLf = has a solution if and only if

And the solution writes as 0)(),( =ΩΩ∫ ωων dPdh

( ) cdthf yXtXy t+Ω=Ω=Ω=Ω ∫

∞

0

0

,),(),(0

ωτωτωτωτ

Need for a technical assumption similar to « finite horizon » hypothesis for

Sinai’s billard

( ) ( )surelyalmost ),,(),,(,,0

medium diffusive];[0),(,];[),,(

000

2

LytLytL

ctyymesxxctyymesytL

>ΩΩ∀∞<∃>∃

∩Ω+=>∩Ω+=Ω ωσ

It implies that no particle can escape to infinity without crossing diffusive medium


Leading order ε-2 ),(),,,(0 00txtxILI y Φ=Ω⇔= ωτ

Order ε-1

( )

),(.,),,,(

)4

)((.

0

0

0

11

11

txtxILI

dBLI

yXty

y

Φ∇=Ω=ΩΩ=Ω⇔Φ∇Ω−=

ΩΦ−Φ+Φ∇Ω−=

∫

∫∞

ωτωτωτ

πωτσ ν

Order 1, integrated with respect to stationary measure

Φ∇Ω∇Ω=Ω

∇Ω=Φ∂ ∫∫∞

...)(4

.0

0

1dtdP

dI tt ω

π

Where means that expectation is taken with respect to stationnary

measure of Markov process and random medium


Summary ),(),,,( txBtxI Φ→Ω νε

ν ω

Solution of the tensorial diffusion equation

( ) dtji

ij

t

t∫∞

ΩΩ=

Φ∇∇=Φ∂

0

0D

)D.(

This is an extension of previous result of [Bensoussan-Lions-Papanicolaou]:

• Periodic case stochastic case

• Mixture of diffusive/non diffusive medium link with billiard model

How is this result, which comes from asymptotic analysis, related to

Pomraning model?

[ ]πϕµ

ϕµ

ϕµ

µ

2;0],1;1[,

)(sin1

)(cos12

2 ∈−∈

−

−=Ω


Algorithm to compute the diffusion coefficient:

• Generate a random medium in a box of size L

• Solve the cell problem with periodic-boundary conditions

dxd

LD

dx

ji

ij

iiii

x

πχ

πχχσχ

4

1)(

4)(.

3

ΩΩ=

Ω=

Ω−+∇Ω

∫

∫

• Take the limit as ∞→L

This is as expensive as solving the full problem we need for simplified

formulas to evaluate the effective coefficients


Application to a stratified stochastic medium

Chord lengths (Lip) and (Lf

i) are independant random variables of given law.

In this case diffusion tensor take the form:

=

y

xD

σ

σ10

01

3

1

9/23/2014 | PAGE 19

Computation of σx

∫

∫

=

=

−+∇

−

2)(),()3(

2)()(

1 µωωµµχσ

µµχχωσωχµ

ddP

d

x

x

• Integrate with respect to dµ: ∫ =∇ 0µµχdx

This quantity is deterministic

• Multiply by µ and integrate with respect to dµ:

3

2)()()(2 =+∇ ∫∫ µωµχωσµωχµ ddx

• Integrate with respect to dP(ω):

∫

∫∫

=

=

)()(

3

1

2)(),()()(

ωωσσ

µωωµµχωωσ ν

dP

d

dPdP

x


Computation of σy µµχχωσωχµ =

−+∇ ∫ 2

)()(d

y

)()(

ωσ

µωχ =⇒

Because 0)( =∇ ωσy

1

2)(

)(

)(2)(

3

1−

== ∫

∫∫ωσ

ω

ωµ

ωσ

µσ

dP

dPd

y

This extends to transport equations classical results for homogenization of

diffusion equation is stratified medium:

• Effective opacity is the arithmetic average in x-direction

• Effective opacity is the harmonic average in y-direction

In one-dimensional medium, homogenization and diffusion approximation are

commuting:

• This is not true in general 3D medium

• This is not recovered by Pomraning model with scattering


For general 3D medium there is no longer a simple formula but we can prove

the following bound:

For isotropic random medium

33

1 0

)(

eff

∫∞

−

<=

dye

D

yσ

σ

In anisotropic medium : similar bounds for each component of the tensor D

This bound is exactly that of Pomraning model without scattering


The proof uses the probabilistic representation of diffusion coefficient:

∑

∫ ∑

>

+

∞

+

−+=

−==

0

100

2

0

100

)(

)(

0

i

iii

i

iiit

TTT

TTdtD

µµµ

µµµµTi: time of jumps for jump

process (µt, τXtω)

The first term is just

3

0

)(

∫∞

−dye

yσ

And the series writes as ∑∑≥>

+ −=−0

0

0

10 ()()(n

n

i

iii FFTT κκσµµ

(κn): sequence of places where process (µt) has a direction change

F(κ): centered random variable

∫∫∞

−

∫=

0

)(_1

1

0 2

)(

t

s ds

edtd

Fκτσ µµ

κ


0)(1

lim)()()(2

2

0

N

2

0

0

0 ≥+= ∑∑≥

∞→≥ n

n

n

n FN

FFF κκκκ

Using stationarity of Markov chain (κn) ijFFFF ijji ≥= − ,)()()()( 0 κκκκ

Hence:

And 0

2

0TD µ≥

This quantity can be computed given the chord-length distribution of the

random media.

For simplicity, we will assume isotropy of the stochastic binary mixture.

NUMERICAL SIMULATIONS


L2

L2

L2

L2 L1

L1

L1

A « natural » choice which correspond to P.Keiter’s experiment: overlapping

spheres of random radius (with distribution f(r)dr) whose centers constitute a

Poisson process

Each chord is a random variable: the sequence (L1)i(L2)i is a renewal process:

• chord length in background material is exponential

• chord length in overlapping spheres is not known analytically

If the spheres do not overlap the distribution in background is approximatively

exponential (Olson, jqsrt 2002) and inside spheres is

∫=max

2/

2)(

2)(

R

l

dssfr

lld


• A sample of the random medium is generated in a box of size L (large in front of

characteristic length scales)

• Numerical simulations are performed on the periodized geometry

• Effective opacity depends on the following parameters:

limitmix atomic in theopacity :

spheres ofdepth optical :

ratiocontrast :

fraction volume:,

2211

1

2

1

21

σσσ

σ

σ

σ

ff

r

ff

+=

),,,( 1

2

121eff rff σ

σ

σσσ Ψ=Dimensional analysis

Mean chord length is related to mean radius by

S

Vl 4= =4r/3 for constant

radius


Numerical simulations evidence the influence of optical depth of spheres

Eff

ective o

pacity

Volume fraction of spheres

Next step is to compare with the bound

)(1

~

21

21

σσσ

σσσσ

−++

+=

h

h

l

l

1

0

)(~−

∞−

∫= dsesσσ


Using renewal theory, Vanderhaegen (jqsrt 1988) has computed

for various chord length distributions and concluded that Markov assumption

(each chord length has exponential distribution) is a good approximation

)(1

~

21

21

σσσ

σσσσ

−++

+=

h

h

l

l

We have to relate mean chord length to radius distributions:2

3

3

44

r

r

S

Vl ==

σ

σ

σ

σ eff and ~


• Except when optical depth of spheres is of order 1, the bound is a good

approximation of effective opacity

• Since statistics of the random medium is not perfectly known, Levermore-

Pomraning Model is well adapted to practical situations

• It is quite surprising that diffusion approximation of a 1D purely absorbing

model compares well with a model with scattering in 3D geometry!

MODEL FOR RADIATION WAVES


The assumption of constant temperature is no longer valid

( )

( )( ) 0)()(

4)()(.

4

4

=Ω−+∂

=+∇Ω+∂

∫ dITxTx

TxIxII

t

t

εεεεε

εεεεφε

σγ

πσσ

Asymptotic analysis can be performed on a linearized version of the system: we

obtain weak convergence to a 3-temperature system

( ) ( )

( )( ) 0

0

3

1

4

2222

4

1111

eff

4

222

4

111

=−+∂

=−+∂

∆=−+−+∂

EaTT

EaTT

EaTEfaTEfE

t

t

t

σγ

σγ

σσσ

4

raTE =

LEVERMORE-POMRANING MODEL WITH

SCATTERING


21

2

2

1

1222222

1

1

2

2111111

2

2

III

IIdIIII

IIdIIII

xt

xt

+=

−+=+∂+∂

−+=+∂+∂

∫

∫

λλµ

µσσµ

λλµ

µσσµ

• This model is exact without scattering (renewal equations with exponential

distribution)

• Does not have the correct asymptotic behavior in 1D-diffusive medium

RENEWAL EQUATIONS


)0(ˆ)0(ˆ)()(

00

2211

0

22

0

11

2

0

)(

21

0

)(

1

0

)(

000

GfGfdxxGfdxxGf

IdxefIdxefdxed

xxx

dyydyydyy

+=+=

∈∫

+∈∫

=∫

=

∫∫

∫∫∫

∞∞

∞ −∞ −∞ − σσσ

(Laplace transforms)

Using conditional expectation with

respect to first chord length, we obtain

∫

∫

++−=

++−=

+−−

−

+−−

−

xx

lly

y

xx

lly

y

l

edy

l

eeyxGxG

l

edy

l

eeyxGxG

0 22

)1

(

2

12

0 11

)1

(

1

21

1)()(

1)()(

22

2

2

11

1

1

σ

σ

σ

σ

σ

σ

+

+

++=

+

+

++=

2

2

1

2

222

1

1

2

1

111

1

1)(ˆ)

1()(ˆ

1

1)(ˆ)

1()(ˆ

l

pGl

ppGl

l

pGl

ppGl

σσ

σσ

1D ASYMPTOTICS


02

112

=

−+∂+∂ ∫

µϕϕσσ

εϕµ

εϕ εεεε d

xt

02

11 ,,

2

,, =

−

+∂+∂ ∫

µϕϕσ

εϕµ

εϕ εεεε d

qx qqq

x

q

t

q

x

q

t

qx

ϕσ

ϕ ∂

∂=∂3

1

q

xx

q

t ϕσ

ϕ ∂=∂3

1

0→ε

0→ε

0→q 0→q


Switzer (Ann.Math.Stat, 1965) has given a construction of a 2D binary mixture with

markovian chord length distribution:

• Given a disk of radius R, choose n=(πR)/λ couples (θi,ρi) uniformly in [0, π]X(-

R,R)

• The n lines (x cos(θi)+y sin(θi)= ρi ) defines polygonal cells whose chord length

is exp(λ)

• Choose randomly the kind of material in each cell according to volume fraction

note that it is not possible to find a distribution of radii such that the chord

length distribution is exponential

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Documents

RADIATION WAVES IN STOCHASTIC BINARY MIXTURES · SUMMARY Modelling radiation transport in non-homogenous medium is an issue because direct simulations are questionable This problem