Adaptive stochastic coupling in the Arlequin method

C. Zaccardi* - L. Chamoin – R. Cottereau – H. Ben Dhia

MASCOT NUM 2011 Workshop - In honor of Anestis Antoniadis

23-25 Mars 2011, Villard de Lans

Context

•  Structure with local behavior modification (cuts, holes, different processes…)

•  Multiscale: close to singularities (load, BC…), close to / far from the defect…

Global homogeneous behavior with: •  Strong variability of parameters •  Insufficient knowledge of material

parameters •  Specific and complex physics

(cracking…)

Local Stochastic model superposed on a Deterministic one

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?

Summary

1.  Definition of the reference model 2.  Reduced model with the Arlequin Method

•  The Arlequin method •  Specificity of the coupling •  Raw results on a simple case

3.  Goal-oriented error estimation •  Quantity of interest •  Definition of the adjoint problem •  Error estimation and adaptivity

4.  Example of adaptive model with stochastic coupling

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1. Definition of the reference model

Considering a complete probability space, •  Equilibrium equation

•  Boundary conditions

•  Stochastic material property

Ω

∀x ∈ Ω,

Chapter 1

ContinuousDeterministic-Stochastic Coupling

RE DO THE INTRODUCTION Classical deterministic models are widely usedand satisfactory for a large range of industrial applications. However, when one isinterested in multiscale phenomena, local specific quantities, or local behaviors, thesemodels are either too coarse or require too much information for the identificationof their parameters. Stochastic methods have therefore been proposed. Further, inmany cases in industrial applications, local defects sharply change the behavior of astructure only locally while the rest of the structure is only smoothly modified. Inthese cases, it is neither reasonable nor tractable to model the entire structure witha complete fine-scale method. Multiscale methods are appealing. In this chapter,the Arlequin Method is used to couple a deterministic model with a stochastic one.

1.1 Continuum Stochastic Monomodel Problem

We consider here a linearized static elasticity problem where only the Young modulusK(x) is stochastic:

K(x, !) ! L 2(!,C 0(")) (1.1)

where (!,F , P ) is a complete probability space with ! a set of outcomes, F a "-algebra of events, and P : F " [0, 1] a probability measure. The problem is definedin a bounded domain "s, clamped on #D, with a deterministic volume force fieldf(x) ! L 2("s) defined on a part of "s and a surface force field g(x) ! L 2(#N )defined on #N (see Fig. ?? with "s # "0. Following the idea proposed in [?], weassume that K(x, !) also verifies:

$Kmin,Kmax ! (0,+%), such that P (K(x, !) ! [Kmin,Kmax] ,&x ! "s) = 1(1.2)

On another way: 0 < Kmin ' K(x, !) ' Kmax < %,&x ! "s, almost surely.

Remark 1 This condition on the stochastic field K(x, !) may be too constraining.Nevertheless, other hypotheses are studied in [?] while still preserving the solvabilityof the stochastic value problem.

The weak formulation of the stochastic mono-model problem reads:Find us ! V such that:

a(us, v) = #(v) &v ! V (1.3)MASCOT NUM 2011 4

u = 0 on ΓD

K(x, θ)∇u = g(x) on ΓN

a.s.

Reference model – Approximated model – Error estimation – Example

−∇ · (K(x, θ)∇u(x, θ)) = f(x)

K(x, θ) ∈ L 2(Θ,C 0(Ω))

0 < Kmin ≤ K(x, θ) ≤ Kmax < ∞, ∀x ∈ Ω

a.s.

In practice, the solution is unavailable.

2. Reduced model using the Arlequin method

Key points   model superposition   volume coupling of the models   distribution of the mechanical energy

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Ω

Reference model – Approximated model – Error estimation – Example

(α1(x),α2(x))

[Ben Dhia 1998-2008]

Equilibrium equation [Cottereau 2010]

Internal works External works

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Equilibrium of model 2

Equilibrium of model 1

Coupling of the models

Reference model – Approximated model – Error estimation – Example

ad(ud, v) + C(λ, v) = d(v), ∀v ∈ Vd

as(us,v)− C(λ,v) = s(v), ∀v ∈ Ws

C(µ,Πud−us) = 0, ∀µ ∈ Wc

Find (ud,us,λ) ∈ Vd ×Ws ×Wc such that:

ad(u, v) =

Ω1

α1(x)Kd(x)∇u∇v dΩas(u,v) = E

Ω2

α2(x)Ks(x, θ)∇u∇v dΩ

s(v) = E

Ω2

α2(x)fv dΩ

d(v) =

Ω1

α1(x)fv dΩ

Coupling operator and space [Cottereau 2011]

•  Coupling operator

•  Coupling space

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Random field with a spatially varying mean

Deterministic

Random functions, perfectly spatially correlated

Reference model – Approximated model – Error estimation – Example

C : Wc ×Wc → R

Wc = v(x) + θIc(x)|v ∈ H1(Ωc),θ ∈ L

2(Θ,R),

Ωc

v(x) dΩ = 0

C(u,v) = E

Ωc

κ0uv + κ1∇u∇v dΩ

Meanings of the coupling

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C(u, v) =

Ωc

κ0uv + κ1∇u∇v dΩ

With

Equality between the mean of the stochastic field and the

deterministic one

Average cancelling of the stochastic field variability

Reference model – Approximated model – Error estimation – Example

C(µ,Πud − us) = 0, ∀µ ∈ Wc

= C(E [µ] , ud − E [us]) + E

θ

Ωc

(ud − us)dΩ

Simple application

•  Monodimensional application (reference model)

•  Stochastic distributed following a uniform law of bounds [0.2294 ; 1.7706], with parameters:

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Reference model – Approximated model – Error estimation – Example

E [K(x, θ)] = 1

Lcorrelation = 0.01

σ = 0.2

Arlequin approximation

•  Monodimensional application

•  Deterministic model described by:

•  distributed following a uniform law with parameters: Remark : To ensure the physical meaning of the coupling

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Ks(x, θ)

K−1d = E

K−1

s

Reference model – Approximated model – Error estimation – Example

E [Ks(x, θ)] = 1

Lcorrelation = 0.01

σ = 0.2

Kd(x) = 0.7537

Solution : gradient of the displacement

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Dashed black lines: mean and 90% confidence interval with a full stochastic monomodel

Reference model – Approximated model – Error estimation – Example

0 0.2 0.4 0.6 0.8 12

1

0

1

2

3

4

5

x [ ]

grad

ient

[]

0 0.2 0.4 0.6 0.8 12

1

0

1

2

3

4

5

x [ ]

grad

ient

[]

Solution : gradient of the displacement

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Continued black lines: deterministic solution, and mean of the stochastic one Yellow zone: 90% confidence interval (representation of the fluctuation) Dashed black lines: mean and 90% confidence interval with a full stochastic monomodel

Reference model – Approximated model – Error estimation – Example

3. Error estimation

Interest:   Control the quality of the solution   Drive an adaptive model

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Reference model – Approximated model – Error estimation – Example

Modeling, reducing

Numerical approximation

Modeling error

Numerical error

Goal-oriented error estimation [Prudhomme 1999] [Oden 2001]

•  Local quantity of interest: -  Mean of the displacement of a point -  Local average of the standard deviation of the stress field

•  Use of global error estimation with Extraction techniques: Example with the mathematical expectation of a displacement

•  Related error

•  Parameters :

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q(u)

q(u) = E [u(x = xm)]

Reference model – Approximated model – Error estimation – Example

q(u) = E

Ωδ(x− xm)u dΩ

η = q(uex)− q(u0)

Ls, hd, hs

Definition of the adjoint problem

•  Primal reference problem (1)

•  Adjoint problem if is linear:

-  The adjoint problem is still defined on the reference model.

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Find u ∈ V such that:

a(u, v) = (v), ∀v ∈ V

Reference model – Approximated model – Error estimation – Example

q

Find p ∈ V such that:

q(u) = E

Ωδ(x− xm)u dΩ

a(v, p) = q(v), ∀v ∈ V

Approximation of the adjoint problem

Using the Arlequin method Primal: Adjoint with linear: Quality control by estimation of the global error

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q

Reference model – Approximated model – Error estimation – Example

ad(ud, v) + C(λ, v) = d(v), ∀v ∈ Vd

as(us,v)− C(λ,v) = s(v), ∀v ∈ Ws

C(µ,Πud − us) = 0, ∀µ ∈ Wc

Find (ud,us,λ) ∈ Vd ×Ws ×Wc such that:

ad(v, pud) + C(v, pλ) = qd(v), ∀v ∈ Vd

as(v, pus)− C(v, pλ) = qs(v), ∀v ∈ Ws

C(Πpud − pus ,µ) = 0, ∀µ ∈ Wc

Find (pud , pus , pλ) ∈ Vd × Ws × Wc such that:

Error estimation

•  Estimation of the error for linear quantity of interest:

Where the residual is defined by: and where and are projections of and of in respectively.

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p (pud , pus , pλ)V

u (ud, us,λ)

3

K = 1.0418Lcorrelation = 0.01σ = 0.2

(5)

K(x, θ) ∈ L 2(Θ,C 0(Ω))η = R(u, p) ≈ R(u, p)R : V × V → RR(u, v) = (v)− a(u, v)

3

K = 1.0418Lcorrelation = 0.01σ = 0.2

(5)

K(x, θ) ∈ L 2(Θ,C 0(Ω))η = R(u, p) ≈ R(u, p)R : V × V → RR(u, v) = (v)− a(u, v)

u =

ud in Ωd \ Ωs

us in Ωs

Reference model – Approximated model – Error estimation – Example

For instance: [Prudhomme 2008]

η = q(uex)− q(u0) = R(u, p) ≈ R(u, p)

Error sources

•  Several error sources:   Modeling error (Arlequin and Stochastic homogenization)   Spatial discretization error (FEM)   Stochastic discretization (Truncation of the Monte Carlo method)

•  Introducing intermediate models   Continuum deterministic-stochastic Arlequin model (arl)   Arlequin model only discretized in space (arlh)   Arlequin model discretized in space and using Monte Carlo (arlhθ)

•  Decomposition of the error:

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ηm ηh ηθ

Reference model – Approximated model – Error estimation – Example

modeling error stochastic error discretization error

η = q(uex)− q(u0)= (q(uex)− q(uarl)) + (q(uarl)− q(uh)) + (q(uh)− q(u0))

4. Example of adaptive coupling

•  Approximated model (primal)

•  Adjoint problem

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Numerical model : discretization by FEM and Monte Carlo techniques Lc Ls

nMChd hs

Reference model – Approximated model – Error estimation – Example

f = 1

Evolution of the modeling error for different

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Reference model – Approximated model – Error estimation – Example

Ls

0.2 0.25 0.3 0.35 0.40.05

0.04

0.03

0.02

0.01

0

0.01

half length of the stochastic part

rela

tive

erro

r

etaetametahetaMC

Evolution of the discretization error for different

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hd, hs

0 0.01 0.02 0.03 0.04 0.057

6

5

4

3

2

1

0x 10 3

element size of the stochastic part hd [ ]

rela

tive

disc

retiz

atio

n er

ror

hd = 0.05hd = 0.01

Remark about the example

Errors on the example are relatively small

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0 0.2 0.4 0.6 0.8 10.2

0

0.2

0.4

0.6

0.8

1

1.2

x [ ]

solu

tion

u

90% confidence intervaldeterministic solutionmean of the stochastic solutionsolution exact

Conclusion and extensions

Outcomes   Efficient deterministic - stochastic coupling   Goal oriented error estimation with a linear quantity of interest   Error sources

Outlines   Other intermediate problems (sto-sto,semi-discretized…)   Importance of the projection into the reference admissible space   Extension to the coupling of particular stochastic model with a

deterministic continuum one

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Thank you for your attention

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H. Ben Dhia, Multiscale mechanical problems: the Arlequin method. Comptes-Rendus de l’Académie des Sciences – séries IIB (1998) 326(12):899-904 H. Ben Dhia, Further insights by theorical investigations of the multiscale Arlequin method. Int. J. Mult. Comp. Engrg. (2008) 6(3):215-232 R. Cottereau, H. Ben Dhia, D. Clouteau, Localized modeling of uncertainty in the Arlequin framework. Proceedings of IUTAM Symposium on the Vibration Analysis of structures with Uncertainties, R. Langley and A. Belyaev (eds), IUTAM Bookseries (2010) 27:477-488 R. Cottereau, D. Clouteau, H. Ben Dhia, C. Zaccardi, A stochastic-deterministic coupling method for continuum mechanics. Submitted to Comput. Meth. Appl. Mech. Engrg. (2010) S. Prudhomme, J.T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Meth. Appl. Mech. Engrg. (1999) 176:313-331 J.T. Oden, S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. (2001) 41:735-756 S. Prudhomme, L. Chamoin, H. Ben Dhia, Paul T. Bauman, An adaptive strategy for the control of modeling error in two-dimensional atomic-to-continuum coupling simulations. Comput. Meth. Appl. Engrg. (2009) 198:1887-1901