Inverse problems and control optimal in non-linear mechanics C. - - PowerPoint PPT Presentation

Inverse problems and control optimal in non-linear mechanics

• C. Stolz

1

2

Introduction

• Optimal control
• Some Inverse Problems
• Typical approaches
• Other applications of optimal control

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Optimal Control

Optimal Control

Equations of a dynamical system

x (to) = x d ˙ x = F( x , v, t) ...

are controled by v such that

J( x , ...) = tf

to

f( x , t)dt + g( x (tf))

is minimum. Picof12 3

Some Inverse Problems

1 Some Inverse Problems

• Boundary Condition Determination
• Determination of loading history
• Methods of resolution

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Identification of Boundary Conditions.

Identification of Boundary Conditions.

Γi Γo

A body Ω with unknown Boundary Conditions along Γi Data : Load and displacements are known along Γo

σ.n = To, u = uo,

• ver Γo

Goal : Determine the unknown B.C. on Γi This problem is not well-posed in Hadamard sense Picof12 5

Methods of resolution

Methods of resolution

Three typical methods

• Integration of Cauchy Problem : problem of local instabilities
• Quasi-Reversibility method : problem of higher order in gradient to solve
• Optimal Control approach.

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Optimal Control

Optimal Control

             ∆u = 0 ∆u = 0 u = uo over Γo u = v overΓi q.n = To overΓo q.n = To

not well-posed well-posed (PB2) Find the control v defined on Γi such that

J(v) =

• Γo

1 2 u(v, To) − uo 2 dS + r

• Γi

1 2 v 2 dS

is minimum.

• J. L. Lions: existence of local minima depends on J..

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Resolution

Resolution

(PB2) L(u, u∗) =

• Ω

∇u.K.∇u∗ +

• Γo

u∗qo dS J(u) =

• Γo

1 2 u − uo 2 dS + r

• Γi

1 2 u 2 dS

Minimization of

˜ J = L(u, u∗) + J(u), u∗ = 0 sur Γi

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Resolution

                   q = −K.∇u, q∗ = −K∇u∗ div q = 0, div q∗ = 0 q.n = To, over Γo q∗.n = u − uo u = v, over Γi u∗ = 0

direct adjoint Two linear problems and a condition of optimality to solve:

∂J ∂v δv =

• Γi

(q∗.n + rv)δv dS = 0

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Resolution Picof12 10

Resolution Picof12 11

Extension in elastoplasticity

Extension in elastoplasticity

Initial and final shapes and the constitutive law Determine the history of loading and/or the internal state ! Picof12 12

Extension in elastoplasticity Differents problems

• Determination of the history of the loading
• Détermination of the internal state at tf
• Estimation of plastic zone
• Estimation of maximal load

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Main Idea

Main Idea

Assume the loading hitory T(x, t) is known. Solve elastoplastic evolution by a direct problem: εp(x, t) is then determined Choose the best history, such that the final shape is closed to the measured residual one

J(T, εp, tf) =

• Γo

1 2 u − uo 2 dS

Estimation of the final state for prescribed uo, To = T(tf) at tf is then performed. The control variables are εp. Picof12 14

Main Idea

• Constitutive Law :

σ = C : (ε − εp) = C : ε − p

• Compatibility

ε = 1 2(grad u + gradT u), [ u ]Γ = 0

• Boundary Conditions :

σ.n = (C : ε).n − p.n = To

• Equilibrium condition :

div C : ε − div p = 0, [ σ ]Γ.n = 0.

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Main Idea If p is given : u is unique. If To and uo are given over Γo

min p 1 2

• Γo

u(p) − uo 2 dS + 1 2r

• Ω

p 2 dΩ

can be solved by introducing an adjoint state. The solution p is not unique. We must add some constraints. Picof12 16

Main Idea Example : Problem on a beam (L. Bourgeois)

εxx = u′ − yv′′, εp = α(x, y)ex ⊗ ex σ = σxxex ⊗ ex F = |σxx| − k ≤

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Main Idea The direct is easy to solve

N = 0, 0 = ES(u′− < α >), M ′′ = T(x), M = EIv′′ − ES < yα > m(x, t) =

• 3(h2 + M/k),

α = k E (sgn(y) − y/m(x)) v′′ = g(M) = k Em + 3M 2Eh3 , M < −2kh2 3 v′′ = 0

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Main Idea Control optimal formulation

J(m) = 1 2(v − vo)2 + M ∗(T − M ′′) + v∗(g(M) − v′′)

• dx

Adjoint Problem

(v∗)′′ = v − vo, v∗(0) = v∗(L) = 0 (M ∗)′′ = v∗dg/dM, M ∗(0) = M ∗(L) = 0

Optimality condition

dJ =

• (M ∗ + rT)dT dx = 0

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Main Idea Picof12 20

Main Idea Evolution Consider the final state at tf We search an history of loading T(t), onΓT such that u(tf) = umonΓT is given. The functional to minimize is now :

J =

• ΓT

1 2k u(tf) − um 2 dS + tf

• ΓT

1 2 ˙ T.H. ˙ T dSdt

(1) To solve this, we consider a well-posed problem, controled by T(t) Picof12 21

For Elasto viscoplasticity

For Elasto viscoplasticity

Consider the constitutive behaviour Free Energy, w(ε, α),

A = −∂w ∂α

Dissipation, D( ˙

α), A = ∂D ∂ ˙ α

Normality rule determines the evolution of internal state Picof12 22

For Elasto viscoplasticity The direct problem corresponds to the set of equations Compatibility

2ε(u) = ∇u + ∇tu, over Ω, u = 0 along Γu,

Equilibrium

divσ = 0,

• n. ˙

σ = ˙ T along ΓT ,

Constitutive law

σ = ∂w ∂ε , A = −∂w ∂α , ˙ α = ∂Φ ∂A

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For Elasto viscoplasticity Equations of the problem (CS,2008) for direct and adjoint satisfies

J = L + 1 2

• ΓT

k u(tf) − uo 2 dS + tf

• ΓT

1 2 ˙ T.H. ˙ T dSdt L = − tf

• Ω

(˙ ε, ˙ α)t.W.(ε∗, α∗) dΩdt + tf

• ΓT

˙ T.u∗ dSdt + tf

• Ω

A∗.(− ˙ α + ∂Φ ∂A) − α∗ ˙ A dΩdt

among a set of admissible fields Picof12 24

For Elasto viscoplasticity Optimisation

( ˙ σ, ˙ B)t = W : (˙ ε, ˙ α), (σ∗, B∗)t = W : (ε∗, α∗)

Equilibrium

div ˙ σ = 0, divσ∗ = 0,

• n. ˙

σ = ˙ T over ΓT

• n. ˙

σ∗ = 0,

Constitutive laws

A∗ + B∗ = 0, ˙ A + B = 0, ˙ α = ∂φ ∂A, ˙ α∗ = − ∂2φ ∂A∂AA∗, α∗(tf) = 0

Optimality

• ΓT

(n.σ∗(tf) + k(u(tf) − uo)).δutf dS = 0

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Cyclic loading

Cyclic loading

For periodic loading, the stress respons is periodic

      

Elastic Shakedown

˙ εp(t) = 0

Plastic Shakedown

• T ˙

εp(t)dt = 0

Ratcheting

• T ˙

εp(t)dt = 0.

Criterion on ˙

εp(t) : ⇒ determine the limit respons !

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Optimal Control

Optimal Control

Free Energy, w(ε, α),

A = −∂w ∂α

Dissipation, D( ˙

α), A = ∂D ∂ ˙ α

Control variables: αo The state : (u, σ, A, α)(x, t) is solution of direct problem Cost functional J disance to periodicity

J(αo) = 1 2

• Ω

(∆σ : S : ∆σ + ∆A : Z : ∆A) dΩ ∆f = f(x, T) − f(x, 0), (S, Z) = (W ′′)−1

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Asymptotic Behaviour

Asymptotic Behaviour

The cyclic answer satisfies :

εp

∞, α∞ is solution of

min εp,α J(εp, α)

• M. Peigney, C Stolz (2002-2003).

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Application

2 Application

• Rigid punch on an half elastoplastic space
• Vertical Displacement is imposed

−3 −2 −1 1 2 3 −5 5 10 15 20 25 30 35 40

x/a p/k

élastique élastoplastique

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Elasto - Plasticity

Elasto - Plasticity

2 4 6 8 10 12 14 16 18 −6 −4 −2 2 4 6 −2 −1 1 2 3 4 5

x/a y/a

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CONCLUSION

CONCLUSION

• Some Inverse problems
• Comparaison with classical methods
• The main difficulty is to chooze a good cost function
• extension to other cases: Damage, cracks, ...
• Cyclic loading : elastic and plastic skakedown, accommodation...

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