A coupling method for stochastic continuum models … coupling method for stochastic continuum models at diﬀerent scales. A new approach to numerical stochastic homogenization. R

A coupling method for stochastic continuum models at different scales.A new approach to numerical stochastic homogenization.

R. Cottereau1, Y. Le Guennec1, D. Clouteau1, C. Soize2

1Laboratoire MSSMat, Ecole Centrale Paris - CNRS, France2Laboratoire MSME, Universite Paris-Est Marne-la-Vallee - CNRS, France

Funding: ANR TYCHE (ANR-2010-BLAN-0904) / Digiteo - Region Ile-de-France (2009-26D)

R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 1 / 26

IntroductionStochastic multi-scale analysis - shell/elastic coupling

[Rateau, 2003]

f

e

p

l l l

Coupling of models of different nature

Stochastic model of elastic constitutive tensor

Stochastic model of shell parameters


IntroductionStochastic multi-scale analysis - shell/elastic coupling

[Rateau, 2003]

f

e

p

l l l

Coupling of models of different nature

Stochastic model of elastic constitutive tensor

Stochastic model of shell parameters

Two issues to be discussed

How to construct two models of random parameters that are physically reasonable one withrespect to the other ?

How to couple two stochastic models ?


Outline

1 Introduction

2 Coupling Stochastic models in the Arlequin frameworkDescription of the stochastic mono-modelsThe Arlequin framework

3 Stochastic-stochastic acoustic couplingDescription of the mono-modelsRandom fields of parameters at different scalesCoupling formulation: stochastic-deterministic acoustic caseCoupling formulation: stochastic-stochastic acoustic caseExample: 1D bar in traction

4 New numerical homogenization method

5 Conclusions


Outline

1 Introduction




5 Conclusions


Outline

1 Introduction




5 Conclusions


Random fields of parameters at different scales

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Micro-scale mechanical parameter km(x)

Point-wise nonlinear transformation of a gaussian random field km(x) = h(gm(x))

Given PSD for gm(x): Rm(ζ)

Parameters: average km, variance σ2m and correlation length ℓm =

∫

R|Rm(y)|dy/σ2

m


Random fields of parameters at different scales

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Micro-scale mechanical parameter km(x)

Point-wise nonlinear transformation of a gaussian random field km(x) = h(gm(x))

Given PSD for gm(x): Rm(ζ)

Parameters: average km, variance σ2m and correlation length ℓm =

∫

R|Rm(y)|dy/σ2

m

Meso-scale mechanical parameter kM (x)

hierarchy of upscaled parameter fields by trimming the tails of Rm(ζ)

First-order marginal distribution assumed conserved through upscaling

Parameter evolution constrained by homogenization ℓM > ℓm, σ2M

< σ2m, k

M6= k

m


Relation between the models at different scales

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Natural segmentation of (Θ,F , P ) into (ΘM ,FM , PM ) and (Θm,Fm, Pm)

(Θ,F) = (ΘM ×Θm,FM ⊗Fm)

P is the unique product measure of Pm and PM , such that ∀Xm ∈ Fm and XM ∈ FM ,P (Xm ×XM ) = Pm(Xm)PM (XM ).

km(x) is defined on F ⊗ B(Ω)

kM (x) is defined on FM ⊗ B(Ω)


Examples

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Stochastic-deterministic acoustic couplingGeneral formulation [Cottereau et al., 2011]

Mixt BVP

Find (um, uM ,Φ) ∈ Vm(Ωm)× VM (ΩM )× Vc(Ωc) such that

am(um, vm) + C(Φ, vm) = fm(vm), ∀v ∈ Vm(Ωm)

aM (uM , vM )− C(Φ, vM ) = fM (vM ), ∀v ∈ VM (ΩM )

C(Ψ, um − uM ) = 0, ∀Ψ ∈ Vc(Ωc)

,

Micro-scale stochastic acoustic model

am(u, v) = E

[∫

Ωm

αm(x)km(x, θm)∇u · ∇v dx

]

Macro-scale deterministic acoustic model

aM (u, v) =

∫

ΩM

αM (x)K∗(x)∇u · ∇v dx

Natural embedding of functional spaces

Coupling operator and mediator space to be determined


Stochastic-deterministic acoustic couplingDefinition of mediator space and coupling operator [Cottereau et al., 2011]

Definition of coupling operator

C(Ψ, v) = E

[∫

Ωc

(Ψv + κ∇Ψ · ∇v) dx]

,

Definition of mediator space

Vc(Ωc) =

Ψ(x) = wM (x) + ξm(θm), wM ∈ H1(Ωc), ξm ∈ L2(Θm,R)


Stochastic-deterministic acoustic couplingDefinition of mediator space and coupling operator [Cottereau et al., 2011]


C(Ψ, v) = E

[∫

Ωc

(Ψv + κ∇Ψ · ∇v) dx]

,


Vc(Ωc) =

Ψ(x) = wM (x) + ξm(θm), wM ∈ H1(Ωc), ξm ∈ L2(Θm,R)

Property

C(Ψ, um − uM ) = 0, ∀Ψ ∈ Vc(Ωc)

= C (E[Ψ],E[um]− uM ) + E

[

ξm

∫

Ωc

um − E[um]dx

]


Stochastic-stochastic acoustic couplingGeneral formulation [Le Guennec et al., 2013]

Mixt BVP

Find (um, uM ,Φ) ∈ Vm(Ωm)× VM (ΩM )× Vc(Ωc) such that

am(um, vm) + C(Φ, vm) = fm(vm), ∀v ∈ Vm(Ωm)

aM (uM , vM )− C(Φ, vM ) = fM (vM ), ∀v ∈ VM (ΩM )

C(Ψ, um − uM ) = 0, ∀Ψ ∈ Vc(Ωc)

,

Micro-scale stochastic acoustic model

am(u, v) = E

[∫

Ωm

αm(x)km(x, θm, θM )∇u · ∇v dx

]

Meso-scale stochastic acoustic model

aM (u, v) = EM

[

∫

ΩM

αM (x)kM (x, θM )∇u · ∇v dx

]

Natural embedding of functional spaces

Coupling operator and mediator space to be determined


Stochastic-stochastic acoustic couplingDefinition of mediator space and coupling operator [Le Guennec et al., 2013]


C(Ψ, v) = E

[∫

Ωc

(Ψv + κ∇Ψ · ∇v) dx]

,


Vc(Ωc) =

Ψ(x) = wM (x, θM ) + ξm(θm), wM ∈ L20(ΘM ,H1(Ωc)), ξm ∈ L2(Θm,R)

Property

C(Ψ, um − uM ) = 0, ∀Ψ ∈ Vc(Ωc)

= C (Em[Ψ],Em[um]− uM ) + Em

[

ξm

∫

Ωc

EM [um − Em[um]]dx

]


Example: 1D bar in tractionMono-models solutions

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

position x[− ]

displacemen

tum[−

]

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

position x[− ]

gradient∇um[−

]

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

position x[− ]

gra

dient∇uM[−

]

Figure: Micro-scale mono-model solution displacement (left figure) and gradient (center figure) and meso-scalemono-model solution gradient (right figure): average (dashed lines), 90%-confidence interval (grey shades) andone realization (solid lines).


Example: 1D bar in tractionProposed approach

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

gradient∇w[−

]

p osition x[− ]

Figure: Gradients of the micro-scale and meso-scale solutions of the Arlequin coupled problem: average (dashedlines), 90%-confidence interval (grey shades) and one realization (solid lines).


Outline

1 Introduction




5 Conclusions


New numerical homogenization method [Cottereau, 2012]

Stochastic-Deterministic acoustic coupling

Rationale

Surround the stochastic sample by a ”homogenized” model, with unknown parameter(s);

Perform numerical homogenization (KUBC-SUBC) at the boundary of homogenized model

Iterate on the unknown parameter(s) until the stochastic model becomes ”invisible”.

⇒

c©J. Schwartz


Proposed approach [Cottereau, 2012]

Stochastic-Deterministic acoustic coupling

Algorithm

Initialization: K0 ←− E[k]I;while ‖Ki −Ki−1‖ > criterion do

choice of modulus: K ←− Ki;resolution of Arlequin stochastic-deterministic coupled system and estimation of (u, u,Φ);update of Ki+1 so as to minimize

∫

Ω1‖∇u− I‖dx

⇒

c©J. Schwartz


2D ExampleDefinition of model

homogenization of operator ∇ · k(x)∇u over domain Ω2 = [0, 1]× [0, 1]

k(x) a random field with log-normal first-order marginal density, triangular spectrum,E[k] =

√2, σ =

√2

correlation length ǫ = ℓc/|Ω2| = 10, 1, 0.1.Exact value of the homogenized coefficient:

K∗ =

[

1 00 1

]

.

(a) ǫ = 10 (b) ǫ = 1 (c) ǫ = 0.1

Figure: Map of realizations of k(x) (log scale) for (a) ǫ = ℓc/|Ω2| = 10, (b) ǫ = 1, (c) ǫ = 0.1.


2D ExampleKUBC/SUBC estimates

(a) ǫ = 10 (b) ǫ = 1 (c) ǫ = 0.1

Figure: Convergence of the homogenized coefficients Kǫ

N(dark grey crosses) and Kǫ

N(light grey circles) for

different correlation lengths ((a) ǫ = ℓc/|Ω2| = 10, (b) ǫ = 1, and (c) ǫ = 0.1) as a function of the numbersof Monte Carlo trials N . The dashed lines indicate the values of the arithmetic average E[k] and of the

harmonic average E[k−1]−1 and the solid lines indicate the value of the exact homogenized coefficientK∗ = 1.


2D ExampleArlequin estimate

(a) ǫ = 10 (b) ǫ = 1 (c) ǫ = 0.1

Figure: Convergence of the Arlequin estimate Kǫ

N(black pluses) for different correlation lengths ((a)

ǫ = ℓc/|Ω2| = 10, (b) ǫ = 1, and (c) ǫ = 0.1) as a function of the numbers of Monte Carlo trials N , and

comparison with the coefficients Kǫ

N(light grey crosses) and Kǫ

N(light grey circles). The dashed lines indicate

the values of the arithmetic average E[k] and of the harmonic average E[k−1]−1 and the solid lines indicatethe value of the exact homogenized coefficient K∗ = 1.


Conclusions

Generic coupling approach

Available for a wide range of models (potentially of different nature)

Available for deterministic-deterministic, deterministic-stochastic, stochastic-stochasticcouplings [Le Guennec et al., 2013]

(Some) mathematical analysis available [Cottereau et al., 2011]

Error estimation tools available [Zaccardi et al., 2013]

New stochastic homogenization method

Seems to work better than existing KUBC/SUBC methods for classical randomhomogenization [Cottereau, 2013]

Extendable to wider range of homogenization problems - upscaling (in particularλ-homogenization, and models of different nature)


Bibliography

[Ben Dhia, 1998]. H. Ben Dhia (1998). Multiscale mechanical problems: the Arlequin method. Comptes Rendus de l’Acadmie des Sciences de Paris

Srie IIb 326, pp. 899-904.

[Cottereau, 2013]. R. Cottereau (2013) Numerical strategy for the unbiased homogenization of random materials. Int. J. Numer. Meth. Engr. 95,pp. 71–90

[Cottereau et al., 2011]. R. Cottereau, D. Clouteau, H. Ben Dhia, C. Zaccardi (2011) A stochastic–deterministic coupling method for continuummechanics. Comp. Meth. Appl. Mech. Engr. 200, pp. 3280–3288.

[Le Guennec et al., 2013]. Y. Le Guennec, R. Cottereau, D. Clouteau, C. Soize (2013) A coupling method for stochastic continuum models atdifferent scales. Accepted for publication in Prob. Engr. Mech.

[Rateau, 2003] G. Rateau (2003) Methode Arlequin pour les problemes mecaniques multi-echelles. PhD. Thesis, Ecole Centrale Paris, France.

[Zaccardi et al., 2013]. C. Zaccardi, L. Chamoin, R. Cottereau, H. Ben Dhia (2013) Error estimation and model adaptation forstochastic-deterministic coupling in the Arlequin framework. Accepted for publication at Int. J. Numer. Meth. Engr.

https://github.com/cottereau/CArl


Documents

A coupling method for stochastic continuum models … coupling method for stochastic continuum models at diﬀerent scales. A new approach to numerical stochastic homogenization. R