Optimal Control of Perfect Plasticity Christian Meyer TU Dortmund, Faculty of Mathematics joint work with Stephan Walther (TU Dortmund) supported by DFG Priority Program SPP 1962 Special Semester on Optimization, RICAM, Linz, Oct., 14–18, 2019
Outline Introduction to Perfect Plasticity Stress Tracking Existence of Optimal Solutions and their Approximation Optimality System Displacement Tracking Existence of Optimal Solutions Reverse Approximation Conclusion and Outlook Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Outline Introduction to Perfect Plasticity Stress Tracking Existence of Optimal Solutions and their Approximation Optimality System Displacement Tracking Existence of Optimal Solutions Reverse Approximation Conclusion and Outlook Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Stress-Strain Relation Plasticity in a nutshell σ ε Linear elasticity Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Stress-Strain Relation Plasticity in a nutshell σ σ 0 ε Plasticity with hardening Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Stress-Strain Relation Plasticity in a nutshell σ σ 0 ε Perfect plasticity Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Formal Strong Formulation Linear elasticity − div σ = 0 in Ω × ( 0 , T ) , σ = C ∇ s u in Ω × ( 0 , T ) , u = u D on Γ D × ( 0 , T ) , on Γ N × ( 0 , T ) , σν = 0 u ( 0 ) = u 0 , σ ( 0 ) = σ 0 in Ω with � u : Ω → R d displacement, σ : Ω → R d × d sym stress � C linear and coercive elasticity tensor � Γ D ∪ Γ N = ∂ Ω , Γ D ∩ Γ N = ∅ , Γ D � = ∅ , ν outward normal � u D given Dirichlet boundary data, u 0 , σ 0 initial data Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Formal Strong Formulation Perfect plasticity − div σ = 0 in Ω × ( 0 , T ) , σ = C ( ∇ s u − z ) in Ω × ( 0 , T ) , ∂ t z ∈ ∂ I K ( σ ) in Ω × ( 0 , T ) , u = u D on Γ D × ( 0 , T ) , on Γ N × ( 0 , T ) , σν = 0 u ( 0 ) = u 0 , σ ( 0 ) = σ 0 in Ω with � u : Ω → R d displacement, σ : Ω → R d × d sym stress, z plastic strain � C linear and coercive elasticity tensor � Γ D ∪ Γ N = ∂ Ω , Γ D ∩ Γ N = ∅ , Γ D � = ∅ , ν outward normal � u D given Dirichlet boundary data, u 0 , σ 0 initial data � K set of admissible stresses, closed and convex Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Challenges in Perfect Plasticity � Displacement and plastic strain are in general not unique � Lack of regularity : • Time derivative of the displacement field only in L 2 w ( 0 , T ; BD (Ω)) • Space of bounded deformation, not Bochner measureable • Plastic strain is only a regular Borel measure � Existence only under a safe load condition : Applied loads must admit an elastic solution not obeying the Dirichlet boundary conditions such that the associated stress is in the interior of K Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Challenges in Perfect Plasticity � Displacement and plastic strain are in general not unique � Lack of regularity : • Time derivative of the displacement field only in L 2 w ( 0 , T ; BD (Ω)) • Space of bounded deformation, not Bochner measureable • Plastic strain is only a regular Borel measure � Existence only under a safe load condition : Applied loads must admit an elastic solution not obeying the Dirichlet boundary conditions such that the associated stress is in the interior of K BUT, if the safe load condition is fulfilled, then ... For every Dirichlet displacement u D there exists a unique stress field Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Outline Introduction to Perfect Plasticity Stress Tracking Existence of Optimal Solutions and their Approximation Optimality System Displacement Tracking Existence of Optimal Solutions Reverse Approximation Conclusion and Outlook Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Notation and Standing Assumptions Spaces � Stress space: H p := L p (Ω; R d × d sym ) , H := H 2 � Test space for displacements: V p := W 1 , p (Ω; R d ) , V := V 2 , W 1 , p (Ω; R n ) , V p V D := V 2 D := { ψ | Ω : ψ ∈ C ∞ 0 ( R n ) , supp( ψ ) ∩ Γ D = ∅} D Standing assumptions � K ⊂ R d × d sym nonempty, closed, and convex � C : R d × d sym → R d × d sym linear, symmetric, and coercive, A := C − 1 � u D ∈ H 1 ( 0 , T ; V ) , σ 0 ∈ H with − div σ 0 = 0, σ 0 ∈ K a.e. in Ω � Γ D relatively closed subset of ∂ Ω with positive measure, Ω ∪ Γ N regular in the sense of Gröger Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Reduction of the system Definition (Reduction to the stress only, Johnson’76) A function σ ∈ H 1 ( 0 , T ; H ) is called reduced solution (with respect to u D ), if fa.a. t ∈ ( 0 , T ) , it holds σ ( t ) ∈ E := { τ ∈ H : � τ, ∇ s ϕ � H = 0 ∀ ϕ ∈ V D } Equilibrium condition: (E) Yield condition: σ ( t ) ∈ K := { τ ∈ H : τ ( x ) ∈ K f.a.a. x ∈ Ω } (Y) � A ∂ t σ ( t ) − ∇ s ∂ t u D ( t ) , τ − σ ( t ) � H ≥ 0 Flow rule: ∀ τ ∈ E ∩ K (F) Initial condition: σ ( 0 ) = σ 0 (0) Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Yosida Regularization Yosida Regularization − div σ = 0 in Ω × ( 0 , T ) , σ = C ( ∇ s u − z ) in Ω × ( 0 , T ) , ∂ t z ∈ ∂ I K ( σ ) in Ω × ( 0 , T ) , u = u D on Γ D × ( 0 , T ) , σν = 0 on Γ N × ( 0 , T ) , u ( 0 ) = u 0 , σ ( 0 ) = σ 0 in Ω Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Yosida Regularization Yosida Regularization − div σ = 0 in Ω × ( 0 , T ) , σ = C ( ∇ s u − z ) in Ω × ( 0 , T ) , ∂ t z = ∂ I λ ( σ ) in Ω × ( 0 , T ) , u = u D on Γ D × ( 0 , T ) , σν = 0 on Γ N × ( 0 , T ) , u ( 0 ) = u 0 , σ ( 0 ) = σ 0 in Ω with ∂ I λ ( τ ) = 1 | ς − τ | 2 λ ( τ − π K ( τ )) and π K ( τ ) = arg min F ς ∈ K Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Existence and Uniqueness Proposition (Existence of a reduced solution) There exists a unique reduced solution σ ∈ H 1 ( 0 , T ; H ) . Proof: � Existence for the Yosida regularization by standard contraction arguments � A priori bounds for σ λ in H 1 ( 0 , T ; H ) ⇒ existence of a weak limit σ for λ ց 0 � Passage to the limit in (E) & (F), feasibility σ ( t ) ∈ K by Yosida regularization � Uniqueness of σ by coercivity of A � Theorem (Continuity properties of reduced solutions) Assume that u n D ⇀ u D in H 1 ( 0 , T ; V ) , u n D → u D in L 2 ( V ) , u D , n ( T ) → u D ( T ) in V . Then σ n ⇀ σ in H 1 ( 0 , T ; H ) and, if λ n ց 0 , then σ n λ ⇀ σ in H 1 ( 0 , T ; H ) . Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Regularization – Extension and Remarks � Instead of Yosida regularization, one could also use hardening to prove existence: ∂ t z ∈ ∂ I K ( σ − ε z ) with ε > 0 (and, of course, both, Yosida and hardening, together) � If u n D → u D in H 1 ( 0 , T ; V ) , then the convergence is strong, i.e., σ n → σ and σ n λ → σ in H 1 ( 0 , T ; H ) C. Johnson, Existence theorems for plasticity problems , Journal de Matématiques Pures et Appliquées, 55 (1976), pp. 431–444. P .-M. Suquet, Sur les équations de la plasticité: existence et régularité des solutions , J. Mécanique, 20 (1981), pp. 3–39. S. Bartels, A. Mielke, and T. Roubˇ cek, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation , SIAM Journal on Numerical Analysis, 50 (2012), pp. 951–976 Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Stress Tracking via Dirichlet Controls Optimal control of the stress 1 2 � σ ( T ) − σ d � 2 H + α 2 � ∂ t ℓ � L 2 ( 0 , T ; X c ) min (P σ ) s.t. σ is a reduced solution associated with u D = G ( ℓ ) + a and ℓ ( 0 ) = ℓ ( T ) = 0 with � α > 0 � Control space : X c ֒ − → X , X c Hilbert space, X Banach space ֒ � G : X → V linear and continuous, a ∈ V given offset, Example: • Λ ⊂ ∂ Ω , relatively closed, dist(Λ , Γ D ) > 0 • X := H − 1 Λ (Ω; R d ) , X c := L 2 (Ω; R d ) • G solution operator of linear elasticity with zero Dirichlet boundary conditions on Λ Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
Existence of Optimal Controls Theorem There exists at least one globally optimal control of (P σ ) . Proof: based on the continuity results by standard direct method � Possible extensions: � More general objectives (weakly lower semicontinuous functionals) � Directly use u D as control (boundary controls in H 1 / 2 ) Christian Meyer (TU Dortmund) · Optimal Control of Perfect Plasticity · RICAM Special Semester 2019
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