Optimality Conditions in Optimal Control of Elastoplasticity Roland - - PowerPoint PPT Presentation

Optimality Conditions in Optimal Control of Elastoplasticity

Roland Herzog Gerd Wachsmuth Christian Meyer

Numerical Mathematics TU Dortmund

Workshop on Control and Optimization of PDEs

Graz, October 10, 2011

DFG SPP 1253

Roland Herzog Optimality Conditions in Plasticity Control

Outline

1

The Elastoplastic Forward Problem Introduction The Plastic Multiplier Comparison to Obstacle Problem

2

An Elastoplastic Control Problem MPCCs C-Stationarity B-Stationarity

Roland Herzog Optimality Conditions in Plasticity Control

Outline

1

The Elastoplastic Forward Problem Introduction The Plastic Multiplier Comparison to Obstacle Problem

2

An Elastoplastic Control Problem MPCCs C-Stationarity B-Stationarity This talk: static (incremental) setting See talk by Gerd Wachsmuth (Tue, 9:30) for quasi-static setting and numerics

Roland Herzog Optimality Conditions in Plasticity Control

Typical Configuration in Linear Elasticity

g = 0 ΓD ΓN g f

Material laws and boundary conditions

C−1σ = ε(u) in Ω Hooke’s Law ∇ · σ = −f in Ω equilibrium condition u = 0

• n ΓD

displacement b/c σ · n = g

• n ΓN

stress b/c Cijkl = λ δij δkl + µ (δik δjl + δil δjk)

Variables

σ stress tensor u displacement vector ε(u) lin. strain tensor ε(u) = 1

2(∇u + ∇u⊤)

Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Linear elasticity

Minimize

1 2a(σ, σ)

s.t. b(σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Linear elasticity

Minimize

1 2a(σ, σ)

s.t. b(σ, v) = ℓ, v for all v ∈ V

Bilinear and linear forms

a(σ, τ ) =

• Ω

σ : C−1τ dx b(σ, v) = −

• Ω

σ : ε(v) dx, ε(u) = 1

2(∇u + ∇u⊤)

ℓ, v = −

• Ω

f · v dx −

• ΓN

g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Linear elasticity

Minimize

1 2a(σ, σ)

s.t. b(σ, v) = ℓ, v for all v ∈ V

Bilinear and linear forms

a(σ, τ ) =

• Ω

σ : C−1τ dx b(σ, v) = −

• Ω

σ : ε(v) dx, ε(u) = 1

2(∇u + ∇u⊤)

ℓ, v = −

• Ω

f · v dx −

• ΓN

g · v ds σ ∈ S = L2(Ω; Rd×d

sym ),

u ∈ V = H1

ΓD(Ω; Rd),

• u = 0 on ΓD
• Roland Herzog

Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Linear elasticity

Minimize

1 2a(σ, σ)

s.t. b(σ, v) = ℓ, v for all v ∈ V

Bilinear and linear forms

a(σ, τ ) =

• Ω

σ : C−1τ dx b(σ, v) = −

• Ω

σ : ε(v) dx, ε(u) = 1

2(∇u + ∇u⊤)

ℓ, v = −

• Ω

f · v dx −

• ΓN

g · v ds ε(u) σ σ ∈ S = L2(Ω; Rd×d

sym ),

u ∈ V = H1

ΓD(Ω; Rd),

• u = 0 on ΓD
• Roland Herzog

Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Linear elasticity

Minimize

1 2a(σ, σ)

s.t. b(σ, v) = ℓ, v for all v ∈ V

Bilinear and linear forms

a(σ, τ ) =

• Ω

σ : C−1τ dx b(σ, v) = −

• Ω

σ : ε(v) dx, ε(u) = 1

2(∇u + ∇u⊤)

ℓ, v = −

• Ω

f · v dx −

• ΓN

g · v ds ε(u) σ σ ∈ S = L2(Ω; Rd×d

sym ),

u ∈ V = H1

ΓD(Ω; Rd),

• u = 0 on ΓD
• Roland Herzog

Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization

Static plasticity (linear kinematic hardening)

Minimize

1 2a(Σ, Σ),

Σ = (σ, χ) s.t. b(Σ, v) = ℓ, v for all v ∈ V and Σ ∈ K (convex)

Bilinear and linear forms

a(Σ, T) =

• Ω

σ : C−1τ dx +

• Ω

χ : H−1µ dx b(Σ, v) = −

• Ω

σ : ε(v) dx, ε(u) = 1

2(∇u + ∇u⊤)

ℓ, v = −

• Ω

f · v dx −

• ΓN

g · v ds ε(u) σ χ, σ ∈ S = L2(Ω; Rd×d

sym ),

u ∈ V = H1

ΓD(Ω; Rd),

• u = 0 on ΓD
• Roland Herzog

Optimality Conditions in Plasticity Control

Kinematic Hardening Model: Yield Condition

Von Mises yield condition (linear kinematic hardening)

K =

• (σ, χ) ∈ Rd×d

sym : |σD + χD|Frob ≤

σ0 :=

• 2/3 σ0
• K = {Σ = (σ, χ) ∈ S × S : (σ(x), χ(x)) ∈ K a.e. in Ω}

AD = A − 1

d (trace A) I

Roland Herzog Optimality Conditions in Plasticity Control

Kinematic Hardening Model: Yield Condition

Von Mises yield condition (linear kinematic hardening)

K =

• (σ, χ) ∈ Rd×d

sym : |DΣ| ≤

σ :=

• 2/3 σ0
• K = {Σ = (σ, χ) ∈ S × S : (σ(x), χ(x)) ∈ K a.e. in Ω}

AD = A − 1

d (trace A) I

DΣ = σD + χD

Roland Herzog Optimality Conditions in Plasticity Control

Two Ways to Write the Forward Problem

The unique minimizer (σ, χ, u) ∈ S × S × V is characterized by Σ ∈ K, a(Σ, T − Σ) + b(T − Σ, u) ≥ 0 for all T = (τ, µ) ∈ K b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Two Ways to Write the Forward Problem

The unique minimizer (σ, χ, u) ∈ S × S × V is characterized by Σ ∈ K, a(Σ, T − Σ) + b(T − Σ, u) ≥ 0 for all T = (τ, µ) ∈ K b(Σ, v) = ℓ, v for all v ∈ V Note: The displacement field u acts as a Lagrange multiplier.

Roland Herzog Optimality Conditions in Plasticity Control

Two Ways to Write the Forward Problem

The unique minimizer (σ, χ, u) ∈ S × S × V is characterized by Σ ∈ K, a(Σ, T − Σ) + b(T − Σ, u) ≥ 0 for all T = (τ, µ) ∈ K b(Σ, v) = ℓ, v for all v ∈ V Equivalently, there exists λ ∈ L2(Ω) (plastic multiplier) such that a(Σ, T) + b(T, u) + c(λ, Σ, T) = 0 for all T = (τ, µ) ∈ S × S b(Σ, v) = ℓ, v for all v ∈ V 0 ≤ λ ⊥

1 2

• |DΣ|2 −

σ2

• ≤ 0

a.e. in Ω c(λ, Σ, T) =

• Ω

λ DΣ : DT dx

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (GAMM, 2011)]

Two Ways to Write the Forward Problem

The unique minimizer (σ, χ, u) ∈ S × S × V is characterized by Σ ∈ K, a(Σ, T − Σ) + b(T − Σ, u) ≥ 0 for all T = (τ, µ) ∈ K b(Σ, v) = ℓ, v for all v ∈ V Equivalently, there exists λ ∈ L2(Ω) (plastic multiplier) such that a(Σ, T) + b(T, u) + c(λ, Σ, T) = 0 for all T = (τ, µ) ∈ S × S b(Σ, v) = ℓ, v for all v ∈ V 0 ≤ λ ⊥

1 2

• |DΣ|2 −

σ2

• ≤ 0

a.e. in Ω c(λ, Σ, T) =

• Ω

λ DΣ : DT dx L2 L∞ L2

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (GAMM, 2011)]

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form elliptic VI

Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form a algebraic, b 1st order elliptic VI a 2nd order

Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form a algebraic, b 1st order no regularity gain: L2 ∋ f → Σ ∈ L2 elliptic VI a 2nd order substantial regularity gain: L2 ∋ f → y ∈ H1

0 or even H2

Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form a algebraic, b 1st order moderate regularity gain: L2 ∋ f → Σ ∈ Lp elliptic VI a 2nd order substantial regularity gain: L2 ∋ f → y ∈ H1

0 or even H2

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011)]

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form a algebraic, b 1st order moderate regularity gain: L2 ∋ f → Σ ∈ Lp admissible set K more involved elliptic VI a 2nd order substantial regularity gain: L2 ∋ f → y ∈ H1

0 or even H2

admissible set K simple

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011)]

Comparison to the Obstacle Problem

Static Plasticity Problem

a(Σ, T − Σ) + b(τ − σ, u) ≥ 0 ∀T ∈ K b(σ, v) = ℓ, v ∀v ∈ V with ℓ, v = − ` f, v ´

Ω −

` g, v ´

ΓN

K = ˘ Σ = (σ, χ) : |DΣ| ≤ e σ0 ¯

Obstacle Problem

a(y, z − y) ≥ (f , z − y)Ω ∀z ∈ K with a(y, z) = ` ∇y, ∇z ´

Ω

K = ˘ y ∈ H1

0(Ω) : y ≥ 0

¯

VI in mixed form a algebraic, b 1st order moderate regularity gain: L2 ∋ f → Σ ∈ Lp admissible set K more involved

• Lagr. multiplier λ non-trivial

elliptic VI a 2nd order substantial regularity gain: L2 ∋ f → y ∈ H1

0 or even H2

admissible set K simple

• Lagr. multiplier λ := f + △y

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011)]

Outline

1

The Elastoplastic Forward Problem Introduction The Plastic Multiplier Comparison to Obstacle Problem

2

An Elastoplastic Control Problem MPCCs C-Stationarity B-Stationarity

Roland Herzog Optimality Conditions in Plasticity Control

Application: Control of Springback

Deep drawing

car body parts plane body parts packings

Roland Herzog Optimality Conditions in Plasticity Control

Application: Control of Springback

Deep drawing

car body parts plane body parts packings

Springback

release of stored elastic energy once the loads are withdrawn partial restoration away from the desired shape

Roland Herzog Optimality Conditions in Plasticity Control

A Control Problem in Plasticity

Upper-level problem

Minimize

1 2 u − ud2 L2(Ω;Rd) + ν1 2 f2 L2(Ω;Rd) + ν2 2 g2 L2(ΓN;Rd)

s.t. ℓ, v = −

• Ω f · v dx −
• ΓN g · v ds

and · · ·

Lower-level problem

Minimize

1 2a(Σ, Σ),

Σ ∈ K s.t. b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

A Control Problem in Plasticity

Upper-level problem

Minimize

1 2 u − ud2 L2(Ω;Rd) + ν1 2 f2 L2(Ω;Rd) + ν2 2 g2 L2(ΓN;Rd)

s.t. ℓ, v = −

• Ω f · v dx −
• ΓN g · v ds

and · · ·

Variational inequality

a(Σ, T − Σ) + b(T − Σ, u) ≥ 0 for all T = (τ, µ) ∈ K b(Σ, v) = ℓ, v for all v ∈ V MPEC K = {Σ : φ(Σ) ≤ 0}

Complementarity system

a(Σ, T) + b(T, u) + (λ φ′(Σ), T) = 0 for all T = (τ, µ) ∈ S × S b(Σ, v) = ℓ, v for all v ∈ V 0 ≤ λ ⊥ φ(Σ) ≤ 0 MPCC MPEC: Non-smooth control-to-state map. MPCC: Classical Lagrange multiplier approach for upper-level problem unsuitable, several stationarity concepts exist.

Roland Herzog Optimality Conditions in Plasticity Control

MPECs & MPCCs in Function Space

Contributors: Mignot, Puel Barbu Berm´ udez, Saguez Bonnans, Casas Bergounioux Mordukhovich Ito, Kunisch Hinterm¨ uller, Kopacka, Rautenberg, Surowiec, Tber, Wegner

• D. Wachsmuth

Farshbaf-Shaker

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

singly active point

Tlin( ) = TX( ) x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

bi-active point

Tlin( ) TX( ) x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 L(x, µ, λ) = f (x) − µ⊤x + λ x1x2

singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 ∇xL(x, µ, λ) = ∇f (x) − µ1 µ2

• + λ

x2 x1

• singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 ∇xL(x, µ, λ) = ∇f (x) − µ1 µ2

• + λ

x2 x1

• singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ2

• + λ

x∗

1

• = 0

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 ∇xL(x, µ, λ) = ∇f (x) − µ1 µ2

• + λ

x2 x1

• singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ2

• + λ

x∗

1

• = 0

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ1 µ2

• + λ
• = 0

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 ∇xL(x, µ, λ) = ∇f (x) − µ1 µ2

• + λ

x2 x1

• singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ2

• + λ

x∗

1

• = 0

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ1 µ2

• + λ
• = 0

redundant multiplier λ, MFCQ does not hold

Roland Herzog Optimality Conditions in Plasticity Control

Example (MPCC)

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0 ∇xL(x, µ, λ) = ∇f (x) − µ1 µ2

• + λ

x2 x1

• singly active point

Tlin( ) = TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ2

• + λ

x∗

1

• = 0

bi-active point

Tlin( ) TX( ) x∗ x∗ Tlin( )◦ = TX( )◦ x∗ x∗ ∇f (x∗) − µ1 µ2

• + λ
• = 0
• algorithmic difficulties

Roland Herzog Optimality Conditions in Plasticity Control

[Fletcher, Leyffer (2004)]

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

MPCC: stationarity

• L = f (x) −

µ⊤x ∇x L = ∇f (x) −

• µ1
• µ2
• µ1 ∈ R,

x1 ≥ 0,

• µ1 x1 = 0
• µ2 ∈ R,

x2 ≥ 0,

• µ2 x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

MPCC: weak stationarity

• L = f (x) −

µ⊤x ∇x L = ∇f (x) −

• µ1
• µ2
• µ1 ∈ R,

x1 ≥ 0,

• µ1 x1 = 0
• µ2 ∈ R,

x2 ≥ 0,

• µ2 x2 = 0

MPCC stationarity concepts

weak stat.

Roland Herzog Optimality Conditions in Plasticity Control

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

MPCC: weak stationarity

• L = f (x) −

µ⊤x ∇x L = ∇f (x) −

• µ1
• µ2
• µ1 ∈ R,

x1 ≥ 0,

• µ1 x1 = 0
• µ2 ∈ R,

x2 ≥ 0,

• µ2 x2 = 0

MPCC stationarity concepts

strong stat. ⇒ M-stationarity ⇒ C-stationarity ⇒ weak stat. differ only in conditions for µ1, µ2, if x1 = x2 = 0

Roland Herzog Optimality Conditions in Plasticity Control

[Scheel, Scholtes (2000) ]

Some Stationarity Concepts for MPCCs

Minimize f (x) s.t. x1 ≥ 0, x2 ≥ 0, x1 x2 = 0

KKT conditions

L = f (x) − µ⊤x + λ x1x2 ∇xL = ∇f (x) − µ1 µ2

• + λ

x2 x1

• µ1 ≥ 0,

x1 ≥ 0, µ1 x1 = 0 µ2 ≥ 0, x2 ≥ 0, µ2 x2 = 0

MPCC: weak stationarity

• L = f (x) −

µ⊤x ∇x L = ∇f (x) −

• µ1
• µ2
• µ1 ∈ R,

x1 ≥ 0,

• µ1 x1 = 0
• µ2 ∈ R,

x2 ≥ 0,

• µ2 x2 = 0

MPCC stationarity concepts

strong stat. ⇒ M-stationarity ⇒ C-stationarity ⇒ weak stat. differ only in conditions for µ1, µ2, if x1 = x2 = 0 ”Limits of regularized MPCCs satisfy C-(or M-)stationarity”

Roland Herzog Optimality Conditions in Plasticity Control

[Scheel, Scholtes (2000); Kanzow, Schwartz (2010)]

Regularization by Penalization

Lower level (forward) problem

Minimize

1 2a(Σ, Σ),

Σ ∈ K s.t. b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Regularization by Penalization

Lower level (forward) problem

Minimize

1 2a(Σ, Σ) + Iγ(Σ)

(penalize constraint violation) s.t. b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Regularization by Penalization

Lower level (forward) problem

Minimize

1 2a(Σ, Σ) + Iγ(Σ)

(penalize constraint violation) s.t. b(Σ, v) = ℓ, v for all v ∈ V

Regularized optimality conditions

a(Σ, T) + b(T, u) + I ′

γ(Σ) T = 0

for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Regularization by Penalization

Lower level (forward) problem

Minimize

1 2a(Σ, Σ) + Iγ(Σ)

(penalize constraint violation) s.t. b(Σ, v) = ℓ, v for all v ∈ V

Regularized optimality conditions

a(Σ, T) + b(T, u) + I ′

γ(Σ) T = 0

for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V Iγ(Σ) = γ

2Σ − PK(Σ)2

— Moreau-Yosida regularization I ′

γ(Σ) = γ (Σ − PK(Σ))

Roland Herzog Optimality Conditions in Plasticity Control

Regularization by Penalization

Lower level (forward) problem

Minimize

1 2a(Σ, Σ) + Iγ(Σ)

(penalize constraint violation) s.t. b(Σ, v) = ℓ, v for all v ∈ V

Regularized optimality conditions

a(Σ, T) + b(T, u) + I ′

γ(Σ) T = 0

for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V Iγ(Σ) = γ

2Σ − PK(Σ)2

— Moreau-Yosida regularization I ′

γ(Σ) = γ (Σ − PK(Σ)) = γ max

• 0, 1 − ˜

σ0 |σD + χD|−1 σD + χD σD + χD

• Roland Herzog

Optimality Conditions in Plasticity Control

Regularization by Penalization

This regularization corresponds to a visco-plastic model!

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V Iγ(Σ) = γ

2Σ − PK(Σ)2

— Moreau-Yosida regularization I ′

γ(Σ) = γ (Σ − PK(Σ)) = γ max

• 0, 1 − ˜

σ0 |σD + χD|−1 σD + χD σD + χD

• Jγ(Σ) = maxε
• 0, γ (1 − ˜

σ0 |σD + χD|−1) σD + χD σD + χD

• Roland Herzog

Optimality Conditions in Plasticity Control

Regularization by Penalization

This regularization corresponds to a visco-plastic model!

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V Iγ(Σ) = γ

2Σ − PK(Σ)2

— Moreau-Yosida regularization I ′

γ(Σ) = γ (Σ − PK(Σ)) = γ max

• 0, 1 − ˜

σ0 |σD + χD|−1 σD + χD σD + χD

• Jγ(Σ) = maxε
• 0, γ (1 − ˜

σ0 |σD + χD|−1) σD + χD σD + χD

• DΣ = σD + χD

D⋆σ = σD σD

• Roland Herzog

Optimality Conditions in Plasticity Control

Regularization by Penalization

This regularization corresponds to a visco-plastic model!

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V Iγ(Σ) = γ

2Σ − PK(Σ)2

— Moreau-Yosida regularization I ′

γ(Σ) = γ (Σ − PK(Σ)) = γ max

• 0, 1 − ˜

σ0 |σD + χD|−1 σD + χD σD + χD

• Jγ(Σ) = maxε
• 0, γ (1 − ˜

σ0 |DΣ|−1)

• D⋆DΣ

DΣ = σD + χD D⋆σ = σD σD

• Roland Herzog

Optimality Conditions in Plasticity Control

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Facts

Quasi-linear elasticity system

Roland Herzog Optimality Conditions in Plasticity Control

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Facts

Quasi-linear elasticity system The control to state map G : ℓ → (Σ, u) is Lipschitz to Lp × W 1,p

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011)]

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Facts

Quasi-linear elasticity system Jγ : S2 → S2 is a Nemytskii operator, differentiable Lp → L2, p > 2 The control to state map G : ℓ → (Σ, u) is Lipschitz to Lp × W 1,p

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011)]

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Facts

Quasi-linear elasticity system Jγ : S2 → S2 is a Nemytskii operator, differentiable Lp → L2, p > 2 The control to state map G : ℓ → (Σ, u) is differentiable to L2 × W 1,2

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011); Herzog, Meyer, Wachsmuth (in revision)]

Differentiability of the Control-to-State Map

Regularized optimality conditions

a(Σ, T) + b(T, u) + Jγ(Σ) T = 0 for all T = (τ, µ) ∈ S2 b(Σ, v) = ℓ, v for all v ∈ V

Facts

Quasi-linear elasticity system Jγ : S2 → S2 is a Nemytskii operator, differentiable Lp → L2, p > 2 The control to state map G : ℓ → (Σ, u) is differentiable to L2 × W 1,2

Derivative

The derivative (δΣ, δu) of (Σ, u) in the direction δℓ solves the system a(δΣ, T) + b(T, δu) + J′

γ(Σ)(δΣ, T) = 0

for all T = (τ, µ) ∈ S2 b(δΣ, v) = δℓ, v for all v ∈ V

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (JMAA, 2011); Herzog, Meyer, Wachsmuth (in revision)]

Optimality Cond. for Regularized Problem

Regularized optimal control problem

Minimize

1 2 u − ud2 L2(Ω;Rd) + ν 2g2 L2(ΓN;Rd)

s.t. the regularized static plasticity problem with ℓ, v = −

• ΓN

g · v ds =: R g (Pγ)

Optimality Conditions

AΣγ + Jγ(Σγ) + B⋆uγ = 0 BΣγ = R gγ (A + J′

γ(Σγ))Υγ + B⋆wγ = 0

BΥγ = uγ − ud R⋆wγ + ν gγ = 0

Roland Herzog Optimality Conditions in Plasticity Control

Approximability of Solutions

Global Minimizers

If {gk} are global solutions to regularized problems with γk → ∞ and εk → 0 as k → ∞ then every weak accumulation point g is a global minimizer of the unregularized problem (and in fact a strong accumulation point).

Roland Herzog Optimality Conditions in Plasticity Control

Approximability of Solutions

Global Minimizers

If {gk} are global solutions to regularized problems with γk → ∞ and εk → 0 as k → ∞ then every weak accumulation point g is a global minimizer of the unregularized problem (and in fact a strong accumulation point).

Strict Local Minimizers

If g is a strict local minimizer of the unregularized problem, then there exists a sequence {gk} of local optimal solutions to regularized problems which converges to g.

Roland Herzog Optimality Conditions in Plasticity Control

Approximability of Solutions

Global Minimizers

If {gk} are global solutions to regularized problems with γk → ∞ and εk → 0 as k → ∞ then every weak accumulation point g is a global minimizer of the unregularized problem (and in fact a strong accumulation point).

Strict Local Minimizers

If g is a strict local minimizer of the unregularized problem, then there exists a sequence {gk} of local optimal solutions to regularized problems which converges to g.

Local Minimizers

If g is a local minimizer of the unregularized problem then there exists a sequence {gk} of local optimal solutions to a perturbed and regularized problem which converges to g.

Roland Herzog Optimality Conditions in Plasticity Control

Passage to the Limit

AΣ + λ D⋆DΣ + B⋆u = 0 BΣ = R g 0 ≤ λ ⊥ φ(Σ) ≤ 0 DΣ := σD + χD, R g, v := −

• ΓN g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Passage to the Limit

AΥ + λ D⋆DΥ + θ D⋆DΣ + B⋆w = 0 LΣ = 0 BΥ = u − ud Lu = 0 DΥ : DΣ − µ = 0 Lλ = 0 AΣ + λ D⋆DΣ + B⋆u = 0 BΣ = R g 0 ≤ λ ⊥ φ(Σ) ≤ 0 DΣ := σD + χD, R g, v := −

• ΓN g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Passage to the Limit

AΥ + λ D⋆DΥ + θ D⋆DΣ + B⋆w = 0 LΣ = 0 BΥ = u − ud Lu = 0 DΥ : DΣ − µ = 0 Lλ = 0 AΣ + λ D⋆DΣ + B⋆u = 0 BΣ = R g 0 ≤ λ ⊥ φ(Σ) ≤ 0 ⊥ ⊥ µ θ DΣ := σD + χD, R g, v := −

• ΓN g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Passage to the Limit

AΥ + λ D⋆DΥ + θ D⋆DΣ + B⋆w = 0 LΣ = 0 BΥ = u − ud Lu = 0 DΥ : DΣ − µ = 0 Lλ = 0 AΣ + λ D⋆DΣ + B⋆u = 0 BΣ = R g 0 ≤ λ ⊥ φ(Σ) ≤ 0 ⊥ ⊥ µ θ R⋆w + ν g = 0 Lg = 0 DΣ := σD + χD, R g, v := −

• ΓN g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Passage to the Limit

AΥ + λ D⋆DΥ + θ D⋆DΣ + B⋆w = 0 LΣ = 0 BΥ = u − ud Lu = 0 DΥ : DΣ − µ = 0 Lλ = 0 AΣ + λ D⋆DΣ + B⋆u = 0 BΣ = R g 0 ≤ λ ⊥ φ(Σ) ≤ 0 ⊥ ⊥ µ · θ ≥ 0 C-stationarity R⋆w + ν g = 0 Lg = 0 DΣ := σD + χD, R g, v := −

• ΓN g · v ds

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Some Remarks on the Proof

derive bounds on adjoint states and ’regularized multipliers’: ΥγS2 + wγV ≤ C ((fγ, gγ)U + 1) θγL2(Ω) ≤

1 √ 2 γ+ε ˜ σ0 γ QγS2

where Qγ = −AΥγ − B⋆wγ θγ D⋆DΣγ2

S2 + λγ D⋆DΥγ2 S2 ≤ Qγ2 S2

λγ µγL1(Ω) ≤ C (ε + γ−1) (fγ, gγ)U

• DΥγS + QγS2
• θγ φ(Σγ)L1(Ω) ≤ C ε2

γ2 θγL2(A0

γ) + C γ−1(fγ, gγ)U QγS2

it is particularly hard to prove the C-stationarity relation µ θ ≥ 0 a.e. in Ω since only µk ⇀ µ and θk ⇀ θ in L2(Ω)

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (submitted)]

Outline

1

The Elastoplastic Forward Problem Introduction The Plastic Multiplier Comparison to Obstacle Problem

2

An Elastoplastic Control Problem MPCCs C-Stationarity B-Stationarity

Roland Herzog Optimality Conditions in Plasticity Control

Differentiability of the Control-to-State Map?

Optimal control problem

Minimize

1 2 u − ud2 L2(Ω;Rd) + ν 2g2 L2(ΓN;Rd)

s.t. the static plasticity problem with ℓ, v = −

• ΓN

g · v ds (P)

Unregularized forward problem

AΣ, T − Σ + B⋆u, T − Σ ≥ 0 for all T = (τ, µ) ∈ K BΣ, v = ℓ, v for all v ∈ V MPEC point of view (implicit approach): exploit properties of the control-to-state map ℓ → (Σ, u) to show that the reduced objective j is directionally differentiable then B-stationarity holds: δj(g; g − g) ≥ 0 for all g admissible

Roland Herzog Optimality Conditions in Plasticity Control

Weak Directional Differentiability

Theorem

For some p > 2, the map W −1,p

ΓD

(Ω; R3) ∋ ℓ → (Σ, u) ∈ S2 × V is weakly directionally differentiable (even for all directions δℓ ∈ V ′).

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (in revision)]

Weak Directional Differentiability

Theorem

For some p > 2, the map W −1,p

ΓD

(Ω; R3) ∋ ℓ → (Σ, u) ∈ S2 × V is weakly directionally differentiable (even for all directions δℓ ∈ V ′). This derivative is the unique solution (Σ′, u′) ∈ Sℓ × V of AΣ′, T − Σ′ + B⋆u′, T − Σ′ +

• λ, DΣ′ : D(T − Σ′)
• Ω ≥ 0

BΣ′ = δℓ for all T in the convex cone Sℓ := {T ∈ S2 : √ λ DT ∈ S, DΣ : DT ≤ 0 where φ(Σ) = λ = 0, DΣ : DT = 0 where λ > 0}.

Roland Herzog Optimality Conditions in Plasticity Control

[Herzog, Meyer, Wachsmuth (in revision)]

B-Stationarity

Theorem (B-stationariy)

Let ¯ g be a local optimal solution of (P). Then

• Ω
• u − ud
• · u′ dx + ν
• ΓN

g · (g − g) ds ≥ 0 for all g admissible, where (Σ′, u′) solves the derivative problem with δℓ generated by g − g.

Remark

purely primal concept equivalent to notion of B-stationarity, e.g., in [Scheel, Scholtes] weak directional derivative of ℓ → λ exists as well algorithmic exploitation unknown

Roland Herzog Optimality Conditions in Plasticity Control

Concluding Remarks

Conclusions

• ptimal control problem for a static plasticity problem

regularization (γ) and smoothing (ε) of the lower-level problem passage to the limit optimality conditions of C-stationary type analysis more involved than for obstacle control problems B-stationarity based on weak directional differentiability of the control-to-state map Both results required extra regularity for nonlinear elasticity systems shown in [Herzog, Meyer, Wachsmuth (JMAA, 2011)].

Roland Herzog Optimality Conditions in Plasticity Control

Concluding Remarks

Conclusions

• ptimal control problem for a static plasticity problem

regularization (γ) and smoothing (ε) of the lower-level problem passage to the limit optimality conditions of C-stationary type analysis more involved than for obstacle control problems B-stationarity based on weak directional differentiability of the control-to-state map Both results required extra regularity for nonlinear elasticity systems shown in [Herzog, Meyer, Wachsmuth (JMAA, 2011)]. This talk: static (incremental) setting See talk by Gerd Wachsmuth (Tue, 9:30) for quasi-static setting and numerics

Roland Herzog Optimality Conditions in Plasticity Control

References I

• R. Fletcher and S. Leyffer.

Solving mathematical programs with complementarity constraints as nonlinear programs. Optimization Methods and Software, 19(1):15–40, 2004.

• R. Herzog, C. Meyer, and G. Wachsmuth.

C-stationarity for optimal control of static plasticity with linear kinematic hardening. submitted, 2010.

• R. Herzog, C. Meyer, and G. Wachsmuth.

Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. Journal of Mathematical Analysis and Applications, 382(2):802–813, 2011a. doi: 10.1016/j.jmaa.2011.04.074.

• R. Herzog, C. Meyer, and G. Wachsmuth.

Existence and regularity of the plastic multiplier in static and quasistatic plasticity. GAMM Reports, 34(1):39–44, 2011b. doi: http://dx.doi.org/10.1002/gamm.201110006.

• R. Herzog, C. Meyer, and G. Wachsmuth.

B- and strong stationarity for optimal control of static plasticity with hardening. in revision, 2011c.

Roland Herzog Optimality Conditions in Plasticity Control

References II

• C. Kanzow and A. Schwartz.

A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. Technical Report Preprint 296, University of W¨ urzburg, Institute of Mathematics, 2010. Holger Scheel and Stefan Scholtes. Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity. Mathematics of Operations Research, 25(1):1–22, 2000.

Roland Herzog Optimality Conditions in Plasticity Control