Optimization of phase-field damage Robert Haller-Dintelmann 1 , Hannes Meinlschmidt 2 , Masoumeh Mohammadi 1 , Ira Neitzel 3 , Thomas Wick 4 , Winnifried Wollner 1 RICAM Special Semester on Optimization Workshop 1: New trends in PDE constrained optimization Linz – Oct. 14, 2019 1 TU-Darmstadt, 2 RICAM, 3 U-Bonn, 4 U-Hannover Oct. 14, 2019 | W. Wollner | 1
Outline of the Talk The Phase-Field Model The Crack-Propagation Model and its Regularizations Improved Differentiability An Optimization Problem for Regularized Crack Propagation Limit in γ → ∞ Oct. 14, 2019 | W. Wollner | 2
Content The Phase-Field Model The Crack-Propagation Model and its Regularizations Improved Differentiability An Optimization Problem for Regularized Crack Propagation Limit in γ → ∞ Oct. 14, 2019 | W. Wollner | 3
Content The Phase-Field Model The Crack-Propagation Model and its Regularizations Improved Differentiability An Optimization Problem for Regularized Crack Propagation Limit in γ → ∞ Oct. 14, 2019 | W. Wollner | 4
The Forward Griffith’s Model Minimize the total energy in each time-step E ( q ; u , C ) = 1 E ε ( q ; u , ϕ ) = 1 � � 2( C e ( u ), e ( u )) Ω \C g ( ϕ ) C e ( u ), e ( u ) 2 − ( q , u ) ∂ N Ω + H d − 1 ( C ), 2 ε � 1 − ϕ � 2 + ε − ( q , u ) ∂ N Ω + 1 2 �∇ ϕ � 2 , 0 ≤ ϕ ( t i ) ≤ ϕ ( t i − 1 ) ≤ 1 ∀ i = 1, ... , N subject to Oct. 14, 2019 | W. Wollner | 5
Simulations from ongoing SPP1748 project Basava, Mang, Walloth, Wick, W. Questions (ongoing work): ◮ Incompressible materials ◮ Convergence (rates?) ◮ Pressure robust discretization ◮ Optimization Oct. 14, 2019 | W. Wollner | 6
Troubleshooting ◮ Minimizers of E ε not unique. ◮ Necessary conditions for minimizing E ε give a sequence of obstacle problems. ◮ Trouble in optimization, e.g., optimality conditions, (non-adapted) algorithms may converge to non-stationary limits... ◮ − → More regularization! Given q i and ϕ i − 1 solve ε ( u i , ϕ i ) := E ε ( q i ; u i , ϕ i ) + γ R ( ϕ i − 1 ; ϕ i ) + η � ϕ i − ϕ i − 1 � 2 E γ min (C γ , η ) u with 0 ≤ γ → ∞ and R ( ϕ i − 1 ; ϕ i ) = 1 4 � ( ϕ i − ϕ i − 1 ) + � 4 L 4 . Oct. 14, 2019 | W. Wollner | 7
The Problem (C γ , η ) Formally, any minimizer of (C γ , η ) satisfies for any ( v , ψ ) ∈ V = H 1 D ( Ω ; R 2 ) × H 1 ( Ω ). � � g ( ϕ i ) C e ( u i ), e ( v ) − ( q i , v ) ∂ N Ω = 0 ε ( ∇ ϕ i , ∇ ψ ) − 1 (EL γ , η ) ε (1 − ϕ i , ψ ) + (1 − κ )( ϕ i C e ( u i ) : e ( u i ), ψ ) + γ ([( ϕ i − ϕ i − 1 ) + ] 3 , ψ ) + η ( ϕ i − ϕ i − 1 , ψ ) = 0. for any ( v , ψ ) ∈ V = H 1 D ( Ω ; R 2 ) × H 1 ( Ω ). But: Not immediately clear, if well-defined! Theorem (Neitzel, Wick, W. 2017) Given some assumptions on the data, there are minimizers, solving (EL γ , η ) . Any solution ( u , ϕ ) to (EL γ , η ) satisfies (for some p > 2 ): ϕ ∈ H 1 ( Ω ) 0 ≤ ϕ ≤ 1 u ∈ W 1, p ( Ω ) ∩ H 1 D ( Ω ) � u � 1, p ≤ c � q � Oct. 14, 2019 | W. Wollner | 8
Content The Phase-Field Model The Crack-Propagation Model and its Regularizations Improved Differentiability An Optimization Problem for Regularized Crack Propagation Limit in γ → ∞ Oct. 14, 2019 | W. Wollner | 9
The Problem ◮ Nice: u ∈ W 1, p is sufficient to have well-defined products (in 2d). ◮ Not so nice: In optimization - q varies (converges weakly!) then u does so in W 1, p , but products of weak-convergent sequences are not nice �→ trouble in the second equation! (Can be circumvented by compensated compactness) ◮ Not so nice: In numerics - approximation theory gives rates if a gap in differentiability is present (we only have integrability). �→ only qualitative convergence o (1) as h → 0 can be expected (not uniform in the data q , ϕ 0 ). ◮ If g ( ϕ ) ∈ L ∞ there is nothing we can do! Oct. 14, 2019 | W. Wollner | 10
The Problem ◮ Nice: u ∈ W 1, p is sufficient to have well-defined products (in 2d). ◮ Not so nice: In optimization - q varies (converges weakly!) then u does so in W 1, p , but products of weak-convergent sequences are not nice �→ trouble in the second equation! (Can be circumvented by compensated compactness) ◮ Not so nice: In numerics - approximation theory gives rates if a gap in differentiability is present (we only have integrability). �→ only qualitative convergence o (1) as h → 0 can be expected (not uniform in the data q , ϕ 0 ). ◮ If g ( ϕ ) ∈ L ∞ there is nothing we can do! ◮ For the ϕ -equation with ϕ ∈ L ∞ the right hand side − G c ε ∆ ϕ + G c ε ϕ = G c ε (1, · ) − (1 − κ )( ϕ C e ( u ), e ( u ) · ) − γ ([( ϕ − ϕ − ) + ] 3 , · ). is in L p / 2 , i.e., for a nice domain ϕ ∈ W 2, p / 2 . (better than L ∞ but not W 1, ∞ !) Oct. 14, 2019 | W. Wollner | 10
Improved Differentiability Theorem (Haller-Dintelmann, Meinlschmidt, W. 2019) Given some assumptions on the domain, assuming a Gårding inequality, and each coefficient (matrix) A i , j being a multiplier on H ε ( Ω ) d for some 0 ≤ ε < 1 2 . Then there exist γ ≥ 0 and 0 < δ ≤ ε such that for any | θ | < δ the elliptic system in H θ − 1 −∇ · A ∇ u + γ u = f ( Ω ) D has a unique solution u ∈ H θ +1 ( Ω ) satisfying D � u � H θ +1 ( Ω ) ≤ C � f � H θ − 1 ( Ω ) D D for some constant C ≥ 0 independent of f – C depends on multiplier norm of A but not A. – If coercive then γ = 0 can be chosen. Oct. 14, 2019 | W. Wollner | 11
Application to Phase-Fields Corollary Let q ∈ H θ 0 − 1 . Then there exists 0 < ¯ θ ≤ θ 0 such that the solution � W 1, p ( Ω ) ∩ H 1 � � H 1 ( Ω ) ∩ L ∞ ( Ω ) � of (EL γ , η ) admits the additional ( u , ϕ ) ∈ D ( Ω ) × regularity u ∈ H θ +1 ( Ω ) and ϕ ∈ H θ +1 ( Ω ) for any θ satisfying 0 < θ < ¯ θ . Moreover, we obtain the estimate � u � H 1+ θ ( Ω ) ≤ C � q � H θ 0 − 1 ( Ω ) D D with a constant C = C ( � q � 2 H − 1, p , γ , η , ε ) . Oct. 14, 2019 | W. Wollner | 12
Content The Phase-Field Model The Crack-Propagation Model and its Regularizations Improved Differentiability An Optimization Problem for Regularized Crack Propagation Limit in γ → ∞ Oct. 14, 2019 | W. Wollner | 13
Model Problem Given ( u 0 , ϕ 0 ) ∈ V with 0 ≤ ϕ 0 ≤ 1. Find ( q , u ) = ( q , ( u , ϕ )) ∈ ( Q × V ) M solving M M d � 2 + α q , u J ( q , u ) := 1 � u i − u i � � � q i � 2 min ∂ N Ω (NLP γ ) 2 2 i =1 i =1 s.t. ( q , u ) satisfy (EL γ , η ). where u d ∈ ( L 2 ( Ω )) M is a given desired displacement, α > 0. Theorem (Neitzel, Wick, W. (2017)) There exists at least one global minimizer ( q , u ) ∈ ( Q × V ) M to (NLP γ ) . Under the assumptions of the previous slides, any such minimizer satisfies the additional regularity u ∈ H 1+ s , ϕ ∈ W 2, p / 2 . Oct. 14, 2019 | W. Wollner | 14
Fredholm Property of Linearized Equation Lemma (Neitzel, Wick, W. (2017)) For any given ( u k , ϕ k ) ∈ ( V ∩ ( W 1, p ( Ω ; R 2 ) × L ∞ ( Ω )) M the linear operators A i : V → V ∗ corresponding to the the linearized fracture equation, i.e., � A i ( u , ϕ ), ( v , ψ ) � V ∗ , V = a i ( u , ϕ ; v , ψ ) � � g ( ϕ i + 2(1 − κ )( ϕ i k C e ( u i = k ) C e ( u ), e ( v ) k ) ϕ , e ( v )) + ε ( ∇ ϕ , ∇ ψ ) + (1 ε + η )( ϕ , ψ ) + (1 − κ )( ϕ C e ( u i k ) : e ( u i k ), ψ ) k − ϕ i − 1 + 3 γ ([( ϕ i ) + ] 2 ϕ , ψ ) + 2(1 − κ )( ϕ i k C e ( u i k ) : e ( u ), ψ ) k are Fredholm of index zero. The same is true for the A : V M → ( V ∗ ) M assembling all time-steps, thus injectivity is a constraint qualification. Oct. 14, 2019 | W. Wollner | 15
A QP-Approximation Let ( q k , u k ) = ( q k , u k , ϕ k ) ∈ Q M × ( V ∩ W 1, p ( Ω ; R 2 ) × L ∞ ( Ω ))) M . Find ( q , u ) ∈ Q M × V M solving M M ( q , u ) J lin ( q , u ) := 1 � u − ( u d − u k ) � 2 + α � � � q + q k � 2 min ∂ N Ω (+ ...) (QP γ ) 2 2 i =1 i =1 s.t. A u = B q . Where B : Q M → ( V ∗ ) M is M � ( q i , v i ) ∂ N Ω �B q , v � ( V ∗ ) M , V M := i =1 For suitably chosen p > 2 any solution u of the linear equation satisfies the desired regularity, e.g., the regularity does not degenerate. Oct. 14, 2019 | W. Wollner | 16
Existence of Solutions to (QP γ ) and optimality conditions Theorem (Neitzel, Wick, W. (2017)) Given ( u k , ϕ k ) ∈ ( V ∩ W 1, p ( Ω ; R 2 ) × L ∞ ( Ω )) M and q k ∈ Q M , the problem (QP γ ) has a unique solution ( q , u ) ∈ Q M × V M . Further, regardless of the invertibility of A , there exists a Lagrange multiplier, z ∈ V M satisfying in ( V ∗ ) M , A u = B q A ∗ z = u − ( u d − u k ) in ( V ∗ ) M , (KKT γ ) α ( q + q k ) + z = 0 on ∂ N Ω . Due to convexity, any such triplet gives rise to a solution of (QP γ ) . Oct. 14, 2019 | W. Wollner | 17
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