A coupling method for stochastic continuum models at different scales. A new approach to numerical stochastic homogenization. R. Cottereau 1 , Y. Le Guennec 1 , D. Clouteau 1 , C. Soize 2 1 Laboratoire MSSMat, ´ Ecole Centrale Paris - CNRS, France 2 Laboratoire MSME, Universit´ e Paris-Est Marne-la-Vall´ ee - CNRS, France Funding: ANR TYCHE (ANR-2010-BLAN-0904) / Digiteo - R´ egion Ile-de-France (2009-26D) R. Cottereau (ECP-CNRS) Coupling of stochastic models S´ eminaire Navier, Oct.’13 1 / 26
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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
How to construct two models of random parameters that are physically reasonable one withrespect to the other ?
How to couple two stochastic models ?
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 2 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 3 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 4 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 8 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 10 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 10 / 26
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)
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 11 / 26
Examples
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R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 12 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 13 / 26
Stochastic-deterministic acoustic couplingDefinition of mediator space and coupling operator [Cottereau et al., 2011]
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 16 / 26
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).
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 17 / 26
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).
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 18 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 19 / 26
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”.
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 21 / 26
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.
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 22 / 26
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.
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 23 / 26
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.
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 24 / 26
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)
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 25 / 26
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
R. Cottereau (ECP-CNRS) Coupling of stochastic models Seminaire Navier, Oct.’13 26 / 26