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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
MASCOT NUM 2011 2
?
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
MASCOT NUM 2011 3
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
MASCOT NUM 2011 5
Ω
Reference model – Approximated model – Error estimation – Example
(α1(x),α2(x))
[Ben Dhia 1998-2008]
Equilibrium equation [Cottereau 2010]
Internal works External works
MASCOT NUM 2011 6
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
MASCOT NUM 2011 7
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
MASCOT NUM 2011 8
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:
MASCOT NUM 2011 9
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
MASCOT NUM 2011 10
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
MASCOT NUM 2011 11
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
MASCOT NUM 2011 12
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
MASCOT NUM 2011 13
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 :
MASCOT NUM 2011 14
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.
MASCOT NUM 2011 15
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
MASCOT NUM 2011 16
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.
MASCOT NUM 2011 17
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:
MASCOT NUM 2011 18
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
MASCOT NUM 2011 19
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
MASCOT NUM 2011 20
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
MASCOT NUM 2011 21
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
MASCOT NUM 2011 22
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
MASCOT NUM 2011 23
Thank you for your attention
MASCOT NUM 2011 24
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