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 Picof12 2

Optimal Control Optimal Control Equations of a dynamical system x ( t o ) = x d x = F ( x , v , t ) ˙ ... are controled by v such that � t f J ( x , ... ) = f ( x , t )dt + g ( x ( t f )) t o is minimum. Picof12 3

Some Inverse Problems 1 Some Inverse Problems • Boundary Condition Determination • Determination of loading history • Methods of resolution Picof12 4

Identification of Boundary Conditions. Identification of Boundary Conditions. Γ i Γ o A body Ω with unknown Boundary Conditions along Γ i Data : Load and displacements are known along Γ o σ . n = T o , u = u o , over Γ o Goal : Determine the unknown B.C. on Γ i 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. Picof12 6

Optimal Control Optimal Control  ∆ u = 0 ∆ u = 0      u = u o over Γ o u = v over Γ i  q . n = T o over Γ o q . n = T o      not well-posed well-posed (PB2)  Find the control v defined on Γ i such that 1 1 � � 2 � u ( v, T o ) − u o � 2 d S + r 2 � v � 2 d S J ( v ) = Γ o Γ i is minimum. J. L. Lions: existence of local minima depends on J .. Picof12 7

Resolution Resolution � � ∇ u.K. ∇ u ∗ + ( PB 2) L ( u, u ∗ ) = u ∗ q o d S Ω Γ o � 1 � 1 2 � u − u o � 2 d S + r 2 � u � 2 d S J ( u ) = Γ o Γ i Minimization of J = L ( u, u ∗ ) + J ( u ) , u ∗ = 0 sur Γ i ˜ Picof12 8

Resolution  q ∗ = − K ∇ u ∗ q = − K. ∇ u,    div q ∗ = 0   div q = 0 ,     q . n = T o , over Γ o q ∗ . n = u − u o  u ∗ = 0  u = v, over Γ i       direct adjoint  Two linear problems and a condition of optimality to solve: ∂J � ( q ∗ . n + rv ) δv d S = 0 ∂v δv = Γ i Picof12 9

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 t f • Estimation of plastic zone • Estimation of maximal load Picof12 13

Main Idea Main Idea Assume the loading hitory T ( x, t ) is known. Solve elastoplastic evolution by a direct problem: ε p ( x, t ) is then determined Choose the best history, such that the final shape is closed to the measured residual one � 1 2 � u − u o � 2 d S J ( T, ε p , t f ) = Γ o Estimation of the final state for prescribed u o , T o = T ( t f ) at t f is then performed. The control variables are ε p . Picof12 14

Main Idea • Constitutive Law : σ = C : ( ε − ε p ) = C : ε − p • Compatibility ε = 1 2(grad u + grad T u ) , [ u ] Γ = 0 • Boundary Conditions : σ . n = ( C : ε ) . n − p . n = T o • Equilibrium condition : div C : ε − div p = 0 , [ σ ] Γ . n = 0 . Picof12 15

Main Idea If p is given : u is unique. If T o and u o are given over Γ o 1 � � u ( p ) − u o � 2 d S + 1 � � p � 2 dΩ min 2 r 2 p Γ o Ω 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) ε xx = u ′ − yv ′′ , ε p = α ( x, y ) e x ⊗ e x σ = σ xx e x ⊗ e x F = | σ xx | − k ≤ 0 Picof12 17

Main Idea The direct is easy to solve 0 = ES ( u ′ − < α > ) , N = 0 , M ′′ = T ( x ) , M = EIv ′′ − ES < yα > α = k 3( h 2 + M/k ) , � m ( x, t ) = E ( sgn ( y ) − y/m ( x )) M < − 2 kh 2 Em + 3 M k v ′′ = g ( M ) = 2 Eh 3 , 3 v ′′ = 0 Picof12 18

Main Idea Control optimal formulation � � 1 2( v − v o ) 2 + M ∗ ( T − M ′′ ) + v ∗ ( g ( M ) − v ′′ ) � J ( m ) = d x Adjoint Problem ( v ∗ ) ′′ = v − v o , v ∗ (0) = v ∗ ( L ) = 0 ( M ∗ ) ′′ = v ∗ dg/dM, M ∗ (0) = M ∗ ( L ) = 0 Optimality condition � ( M ∗ + rT ) dT d x = 0 dJ = Picof12 19

Main Idea Picof12 20

Main Idea Evolution Consider the final state at t f We search an history of loading T ( t ) , on Γ T such that u ( t f ) = u m on Γ T is given. The functional to minimize is now : � t f � 1 � 1 2 k � u ( t f ) − u m � 2 d S + T.H. ˙ ˙ J = T d Sdt (1) 2 Γ T 0 Γ T 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 A = − ∂w Free Energy , w ( ε , α ) , ∂α A = ∂ D Dissipation , D( ˙ α ) , ∂ ˙ α Normality rule determines the evolution of internal state Picof12 22

For Elasto viscoplasticity The direct problem corresponds to the set of equations Compatibility 2 ε ( u ) = ∇ u + ∇ t u , over Ω , u = 0 along Γ u , Equilibrium σ = ˙ div σ = 0 , n . ˙ T along Γ T , Constitutive law σ = ∂w A = − ∂w α = ∂ Φ ∂ ε , ∂α , ˙ ∂A Picof12 23

For Elasto viscoplasticity Equations of the problem (CS,2008) for direct and adjoint satisfies � t f L + 1 1 � � k � u ( t f ) − u o � 2 d S + T.H. ˙ ˙ J = T d Sdt 2 2 o Γ T Γ T � t f � t f � � T. u ∗ d Sdt ˙ α ) t . W . ( ε ∗ , α ∗ ) dΩ dt + L = − (˙ ε , ˙ o Ω o Γ T � t f α + ∂ Φ � ∂A ) − α ∗ ˙ A ∗ . ( − ˙ + A dΩ dt o Ω among a set of admissible fields Picof12 24

For Elasto viscoplasticity Optimisation B ) t = W : (˙ α ) , ( σ ∗ , B ∗ ) t = W : ( ε ∗ , α ∗ ) σ , ˙ ( ˙ ε , ˙ Equilibrium σ ∗ = 0 , div σ ∗ = 0 , σ = ˙ div ˙ σ = 0 , n . ˙ T over Γ T n . ˙ Constitutive laws A ∗ + B ∗ = 0 , ˙ A + B = 0 , α ∗ = − ∂ 2 φ α = ∂φ ∂A∂AA ∗ , α ∗ ( t f ) = 0 ˙ ∂A, ˙ Optimality � ( n . σ ∗ ( t f ) + k ( u ( t f ) − u o )) .δu t f d S = 0 Γ T Picof12 25

Cyclic loading Cyclic loading For periodic loading, the stress respons is periodic  ε p ( t ) = 0 ˙ Elastic Shakedown    � T ˙ ε p ( t ) dt = 0 Plastic Shakedown   � T ˙ ε p ( t ) dt � = 0 . Ratcheting  Criterion on ˙ ε p ( t ) : ⇒ determine the limit respons ! Picof12 26

Optimal Control Optimal Control A = − ∂w Free Energy , w ( ε , α ) , ∂α A = ∂ D Dissipation , D( ˙ α ) , ∂ ˙ α Control variables: α o The state : ( u, σ , A, α )( x, t ) is solution of direct problem Cost functional J disance to periodicity J ( α o ) = 1 � (∆ σ : S : ∆ σ + ∆ A : Z : ∆ A ) dΩ 2 Ω ( S, Z ) = ( W ′′ ) − 1 ∆ f = f ( x, T ) − f ( x, 0) , Picof12 27

Asymptotic Behaviour Asymptotic Behaviour The cyclic answer satisfies : ε p ∞ , α ∞ is solution of ε p ,α J ( ε p , α ) min M. Peigney, C Stolz (2002-2003). Picof12 28

Application 2 Application • Rigid punch on an half elastoplastic space • Vertical Displacement is imposed 40 élastique élastoplastique 35 30 25 20 p/k 15 10 5 0 −5 −3 −2 −1 0 1 2 3 x/a Picof12 29

Elasto - Plasticity Elasto - Plasticity −2 18 −1 16 0 14 12 1 y/a 10 2 8 6 3 4 4 2 5 −6 −4 −2 0 2 4 6 x/a Picof12 30

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... Picof12 31