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 The Elastoplastic Forward Problem 1 Introduction The Plastic Multiplier Comparison to Obstacle Problem An Elastoplastic Control Problem 2 MPCCs C-Stationarity B-Stationarity Roland Herzog Optimality Conditions in Plasticity Control

Outline The Elastoplastic Forward Problem 1 Introduction The Plastic Multiplier Comparison to Obstacle Problem An Elastoplastic Control Problem 2 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 g f Γ D Γ N Material laws and boundary conditions Variables C − 1 σ = ε ( u ) σ stress tensor in Ω Hooke’s Law u displacement vector ∇ · σ = − f in Ω equilibrium condition ε ( u ) lin. strain tensor u = 0 on Γ D displacement b/c σ · n = g on Γ N stress b/c ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) C ijkl = λ δ ij δ kl + µ ( δ ik δ jl + δ il δ jk ) Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization Linear elasticity 1 Minimize 2 a ( 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 1 Minimize 2 a ( σ , σ ) s.t. b ( σ , v ) = � ℓ, v � for all v ∈ V Bilinear and linear forms � σ : C − 1 τ d x a ( σ , τ ) = Ω � ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) b ( σ , v ) = − σ : ε ( v ) d x , Ω � � � ℓ, v � = − f · v d x − g · v d s Ω Γ N Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization Linear elasticity 1 Minimize 2 a ( σ , σ ) s.t. b ( σ , v ) = � ℓ, v � for all v ∈ V Bilinear and linear forms � σ : C − 1 τ d x a ( σ , τ ) = Ω � ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) b ( σ , v ) = − σ : ε ( v ) d x , Ω � � � ℓ, v � = − f · v d x − g · v d s Ω Γ N � � σ ∈ S = L 2 (Ω; R d × d u ∈ V = H 1 Γ D (Ω; R d ) , sym ) , u = 0 on Γ D Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization Linear elasticity 1 Minimize 2 a ( σ , σ ) s.t. b ( σ , v ) = � ℓ, v � for all v ∈ V Bilinear and linear forms � σ : C − 1 τ d x a ( σ , τ ) = σ Ω � ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) b ( σ , v ) = − σ : ε ( v ) d x , Ω � � � ℓ, v � = − f · v d x − g · v d s ε ( u ) Ω Γ N � � σ ∈ S = L 2 (Ω; R d × d u ∈ V = H 1 Γ D (Ω; R d ) , sym ) , u = 0 on Γ D Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization Linear elasticity 1 Minimize 2 a ( σ , σ ) s.t. b ( σ , v ) = � ℓ, v � for all v ∈ V Bilinear and linear forms � σ : C − 1 τ d x a ( σ , τ ) = σ Ω � ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) b ( σ , v ) = − σ : ε ( v ) d x , Ω � � � ℓ, v � = − f · v d x − g · v d s ε ( u ) Ω Γ N � � σ ∈ S = L 2 (Ω; R d × d u ∈ V = H 1 Γ D (Ω; R d ) , sym ) , u = 0 on Γ D Roland Herzog Optimality Conditions in Plasticity Control

Elasticity & Plasticity: Energy Minimization Static plasticity (linear kinematic hardening) 1 Minimize 2 a ( Σ , Σ ) , Σ = ( σ , χ ) s.t. b ( Σ , v ) = � ℓ, v � for all v ∈ V and Σ ∈ K (convex) Bilinear and linear forms � � σ : C − 1 τ d x + χ : H − 1 µ d x a ( Σ , T ) = σ Ω Ω � ε ( u ) = 1 2 ( ∇ u + ∇ u ⊤ ) b ( Σ , v ) = − σ : ε ( v ) d x , Ω � � � ℓ, v � = − f · v d x − g · v d s ε ( u ) Ω Γ N � � χ , σ ∈ S = L 2 (Ω; R d × d u ∈ V = H 1 Γ D (Ω; R d ) , sym ) , u = 0 on Γ D Roland Herzog Optimality Conditions in Plasticity Control

Kinematic Hardening Model: Yield Condition Von Mises yield condition (linear kinematic hardening) � � � sym : | σ D + χ D | Frob ≤ � ( σ , χ ) ∈ R d × d K = σ 0 := 2 / 3 σ 0 K = { Σ = ( σ , χ ) ∈ S × S : ( σ ( x ) , χ ( x )) ∈ K a.e. in Ω } A D = 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) � � � ( σ , χ ) ∈ R d × d K = sym : |D Σ | ≤ � σ := 2 / 3 σ 0 K = { Σ = ( σ , χ ) ∈ S × S : ( σ ( x ) , χ ( x )) ∈ K a.e. in Ω } A D = A − 1 D Σ = σ D + χ D d (trace A ) I 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 ∈ L 2 (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 � � |D Σ | 2 − � 1 σ 2 0 ≤ λ ⊥ ≤ 0 a.e. in Ω 2 0 � c ( λ, Σ , T ) = λ D Σ : D T d x Ω [Herzog, Meyer, Wachsmuth (GAMM, 2011)] 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 ∈ L 2 (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 � � |D Σ | 2 − � 1 σ 2 0 ≤ λ ⊥ ≤ 0 a.e. in Ω 2 0 � c ( λ, Σ , T ) = λ D Σ : D T d x Ω L 2 L ∞ L 2 [Herzog, Meyer, Wachsmuth (GAMM, 2011)] 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 Γ N ˘ ¯ K = Σ = ( σ , χ ) : |D Σ | ≤ e σ 0 Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem Static Plasticity Problem Obstacle Problem a ( y , z − y ) ≥ ( f , z − y ) Ω ∀ z ∈ K a ( Σ , T − Σ ) + b ( τ − σ , u ) ≥ 0 ∀ T ∈ K b ( σ , v ) = � ℓ, v � ∀ v ∈ V ` ´ ` ´ ` ´ with a ( y , z ) = ∇ y , ∇ z with � ℓ, v � = − f , v Ω − g , v Ω Γ N ˘ ¯ ˘ ¯ y ∈ H 1 K = Σ = ( σ , χ ) : |D Σ | ≤ e σ 0 K = 0 (Ω) : y ≥ 0 Roland Herzog Optimality Conditions in Plasticity Control

Comparison to the Obstacle Problem Static Plasticity Problem Obstacle Problem a ( y , z − y ) ≥ ( f , z − y ) Ω ∀ z ∈ K a ( Σ , T − Σ ) + b ( τ − σ , u ) ≥ 0 ∀ T ∈ K b ( σ , v ) = � ℓ, v � ∀ v ∈ V ` ´ ` ´ ` ´ with a ( y , z ) = ∇ y , ∇ z with � ℓ, v � = − f , v Ω − g , v Ω Γ N ˘ ¯ ˘ ¯ y ∈ H 1 K = Σ = ( σ , χ ) : |D Σ | ≤ e σ 0 K = 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 Obstacle Problem a ( y , z − y ) ≥ ( f , z − y ) Ω ∀ z ∈ K a ( Σ , T − Σ ) + b ( τ − σ , u ) ≥ 0 ∀ T ∈ K b ( σ , v ) = � ℓ, v � ∀ v ∈ V ` ´ ` ´ ` ´ with a ( y , z ) = ∇ y , ∇ z with � ℓ, v � = − f , v Ω − g , v Ω Γ N ˘ ¯ ˘ ¯ y ∈ H 1 K = Σ = ( σ , χ ) : |D Σ | ≤ e σ 0 K = 0 (Ω) : y ≥ 0 VI in mixed form elliptic VI a algebraic, b 1st order a 2nd order Roland Herzog Optimality Conditions in Plasticity Control