Multigrid Semismooth Newton Methods for Elastic Contact Problems - PowerPoint PPT Presentation

Multigrid Semismooth Newton Methods for Elastic Contact Problems Stefan Ulbrich Department of Mathematics TU Darmstadt ICCP 2014, Berlin Joint work with Michael Ulbrich, TU München, and Daniela Bratzke, TU Darmstadt. SFB 666 ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 1

Outline ◮ Contact problem in 3D elasticity ◮ Regularized dual problem and error estimates ◮ Application of semismooth Newton methods ◮ Multigrid method for discrete semismooth Newton system ◮ Convergence result and condition number estimate ◮ Numerical results ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 2

Elastic 3D Contact Problem (Signorini Problem) Obstacle n Γ C Γ D Ω Γ N ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 3

Elastic 3D Contact Problem (Signorini Problem) Elastic 3D Contact Problem as Optimization Problem (P): � µǫ ( u ) : ǫ ( u ) + λ � 2 div( u ) 2 − f T � � f T dx − min J ( u ) := V u S u dS ( x ) u ∈ V Ω Γ N u T n ≤ g s. t. on Γ C Ω ⊂ R 3 reference domain of an elastic body, Γ D , Γ N ⊂ ∂ Ω Dirichlet boundary, Neumann boundary, Γ C ⊂ ∂ Ω possible contact boundary on Ω , u ∈ H 1 ( Ω ) 3 ; u | Γ D = 0 � � u ∈ V displacement, V = 2 ( ∇ u + ∇ u T ) ǫ ( u ) = 1 linearized strain, λ , µ Lamé material constants, u T n normal displacement on Γ C , g ∈ H 1 / 2 ( Γ C ) normal distance of the body to the obstacle, f V ∈ L 2 ( Ω ) 3 , f S ∈ L 2 ( Γ N ) 3 volume / surface forces. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 4

Related Work ◮ Semismooth Newton methods for contact problems: Christensen, Hoppe, Hüeber, Ito, Kunisch, Pang, Stadler, M. Ulbrich, S. U., Wohlmuth, . . . ◮ Multilevel methods for contact problems: Dostal, Hüeber, Kornhuber, Krause, Schöberl, Stadler, Wohlmuth, . . . ◮ Abstract multilevel theory (only the references we build on): Bornemann, Yserentant (. . . and many more) ◮ Multilevel trust region methods: Gratton, von Loesch, Toint, . . . ◮ Regularization of obstacle and state constrained problems: Hintermüller, Ito, Kunisch, Meyer, Prüfert, Rösch, Schiela, Tröltzsch, Weiser, . . . ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 5

A Class of Nonlinear Elastic 3D Contact Problems � Φ ( x , ǫ ( u ) : ǫ ( u )) + 1 � � 2 Ψ ( x , div( u ) 2 ) − f T � f T dx − min J ( u ) := V u S u dS ( x ) u ∈ V Ω Γ N u T n ≤ g on Γ C s. t. cf. Necas, Hlavacek 81; Axelsson, Padiy 00; Blaheta 97 Φ ( x , s ) = µ s , Ψ ( x , s ) = λ s recovers the linear case. Assumptions: ◮ 0 < µ 0 ≤ Φ ′ ( s ) ≤ µ 1 ◮ 0 < λ 0 ≤ Ψ ′ ( s ) ≤ λ 1 ◮ 0 < µ ′ ∂ s ( Φ ′ ( s 2 ) s ) ≤ µ ′ ∂ 0 ≤ 1 ∂ ◮ 0 < µ ′ ∂ s ( Ψ ′ ( s 2 ) s ) ≤ µ ′ 0 ≤ 1 Several results of the talk can be extended to this case (current work). ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 6

Elastic 3D Contact Problem Elastic 3D Contact Problem as Optimization Problem (P): � µǫ ( u ) : ǫ ( u ) + λ � 2 div( u ) 2 − f T � � f T dx − min J ( u ) := V u S u dS ( x ) u ∈ V Ω Γ N u T n ≤ g s. t. on Γ C Ω ⊂ R 3 reference domain of an elastic body, Γ D , Γ N ⊂ ∂ Ω Dirichlet boundary, Neumann boundary, Γ C ⊂ ∂ Ω possible contact boundary on Ω , u ∈ H 1 ( Ω ) 3 ; u | Γ D = 0 � � u ∈ V displacement, V = 2 ( ∇ u + ∇ u T ) ǫ ( u ) = 1 linearized strain, λ , µ Lamé material constants, u T n normal displacement on Γ C , g ∈ H 1 / 2 ( Γ C ) normal distance of the body to the obstacle, f V ∈ L 2 ( Ω ) 3 , f S ∈ L 2 ( Γ N ) 3 volume / surface forces. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 7

KKT-System of the Elastic Contact Problem We define a : V × V → R , A ∈ L ( V , V ∗ ), N ∈ L ( V , H 1 / 2 ( Γ C )), f ∈ V ∗ by � � � a ( v , w ) = � v , Aw � V , V ∗ = 2 µǫ ( v ) : ǫ ( w ) + λ div( v )div( w ) dx , Ω � � Nu = u T n | Γ C , f T f T � f , u � V ∗ , V = V u dx + S u dS ( x ). Ω Γ N Contact problem (P) in abstract form: 1 min 2 a ( u , u ) − � f , u � V ∗ , V s. t. Nu ≤ g . u ∈ V The problem is uniformly convex and quadratic. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 8

KKT-System of the Elastic Contact Problem Contact problem (P) in abstract form: 1 min 2 a ( u , u ) − � f , u � V ∗ , V s. t. Nu ≤ g . u ∈ V The problem is uniformly convex and quadratic. Optimality conditions: u ∈ V solves (P) if and only if there exists z ∈ H 1 / 2 ( Γ C ) ∗ such that Au − f + N ∗ z = 0 z ≥ 0, Nu − g ≤ 0, � z , Nu − g � ( H 1 / 2 ) ∗ , H 1 / 2 = 0. ∀ v ∈ H 1 / 2 ( Γ C ), v ≥ 0. Here, z ≥ 0 means � z , v � ( H 1 / 2 ) ∗ , H 1 / 2 ≥ 0 ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 9

Dual Problem Applying Lagrange duality yields the Equivalent dual problem (D): − 1 2 � z , NA − 1 N ∗ z � ( H 1 / 2 ) ∗ , H 1 / 2 + � z , NA − 1 f − g � ( H 1 / 2 ) ∗ , H 1 / 2 max z ∈ H 1 / 2 ( Γ C ) ∗ s. t. z ≥ 0. In the following, we assume sufficient regularity of the problem data and the solution u of (P) to ensure the following: Assumption: The optimal solution of (D) satisfies z ∈ L 2 ( Γ C ) (Necas). Idea: Replace the numerically inconvenient space H 1 / 2 ( Γ C ) ∗ by L 2 ( Γ C ). But: Objective function of (D) is coercive in H 1 / 2 ( Γ C ) ∗ but not in L 2 ( Γ C ). Remedy: We introduce an L 2 -regularization. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 10

Regularization of the Dual Problem Dual problem (D): z ∈ H 1 / 2 ( Γ C ) ∗ − 1 2 � z , NA − 1 N ∗ z � ( H 1 / 2 ) ∗ , H 1 / 2 + � z , NA − 1 f − g � ( H 1 / 2 ) ∗ , H 1 / 2 max z ≥ 0. s. t. We add an L 2 -regularization and obtain the following Regularized dual problem (D γ ): 2 ( z , NA − 1 N ∗ z ) L 2 − γ 2 � z − z r � 2 z ∈ L 2 ( Γ C ) − 1 L 2 + ( z , NA − 1 f − g ) L 2 max z ≥ 0. s. t. Here, γ > 0 and z r ∈ L 2 ( Γ C ) are suitably chosen. Problem is uniformly concave and quadratic (variant of normal compliance reg.). ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 11

Error Estimates Dual problem (D): z ∈ L 2 ( Γ C ) ∗ − 1 z , NA − 1 N ∗ z z , NA − 1 f − g � � � � max L 2 + s. t. z ≥ 0. 2 L 2 Regularized dual problem (D γ ): 2 � z − z r � 2 2 ( z , ( NA − 1 N ∗ z ) L 2 − γ L 2 + ( z , NA − 1 f − g ) L 2 s. t. z ≥ 0. z ∈ L 2 ( Γ C ) − 1 max Let z ∗ and z γ be solutions of (D) and (D γ ), with displacements u ∗ , u γ ∈ V , i.e., Au ∗ − f + N ∗ z ∗ = 0, Au γ − f + N ∗ z γ = 0. Then: � z γ − z ∗ � ( H 1 / 2 ) ∗ = o ( γ 1 / 2 ), � u γ − u ∗ � H 1 = o ( γ 1 / 2 ). (M. Ulbrich, S.U., Bratzke 13; see also Chouly, Hild 12) ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 12

Nonsmooth Reformulation Optimality conditions of the regularized dual problem (D γ ): u γ ∈ V and z γ ∈ L 2 ( Γ C ) satisfy Au γ − f + N ∗ z γ = 0 Nu γ − γ ( z γ − z r ) − g ≤ 0, z γ ( Nu γ − γ ( z γ − z r ) − g ) = 0. z γ ≥ 0, Using the NCP-Function min( a , γ − 1 b ) = a − max(0, a − γ − 1 b ) this can be rewritten as follows: Nonsmooth reformulation (R γ ): Au γ − f + N ∗ z γ = 0 z γ − max(0, γ − 1 ( Nu γ − g ) + z r ) = 0. This system is a semismooth equation. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 13

Semismooth Operators Let be given a continuous operator H : X → Y between Banach spaces and a setvalued generalized differential ∂ H : X ⇒ L ( X , Y ). The operator H is called ∂ H -semismooth at x ∈ X if sup � H ( x + s ) − H ( x ) − Ms � Y = o ( � s � X ) ( � s � X → 0). M ∈ ∂ H ( x + s ) (Kummer; Hintermüller, Ito Kunisch; M. Ulbrich) ◮ If H is semismooth and all M ∈ ∂ H ( x ) are uniformly bounded invertible near the solution then Newton’s method converges locally q-superlinearly. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 14

Semismoothness of the Nonsmooth Reformulation We use the following fact (Hintermüller, Ito, Kunisch; M. Ulbrich): For all p ∈ (2, ∞ ] and all b ∈ L 2 ( Γ C ), the operator S : L p ( Γ C ) → L 2 ( Γ C ), S ( w ) = max(0, w + b ) is ∂ S -semismooth with ∂ S ( w ) consisting of all operators  = 1 on { w + b > 0 } ,   D ∈ L ( L p ( Γ C ), L 2 ( Γ C )), Dv = d · v , d on { w + b < 0 } , = 0  ∈ [0, 1] on { w + b = 0 } .  Let p > 2 be such that the embedding H 1 / 2 ( Γ C ) ⊂ L p ( Γ C ) is continuous. Then u ∈ V �→ γ − 1 Nu ∈ L p ( Γ C ) is linear and continuous. ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 15

Semismoothness of the Nonsmooth Reformulation From the above considerations, we conclude: The operator � Au − f + N ∗ z � H ( u , z ) = z − max(0, γ − 1 ( Nu − g ) + z r ) is ∂ H -semismooth and ∂ H contains the operator � N ∗ � A M ∈ L ( V × L 2 ( Γ C ), V ∗ × L 2 ( Γ C )), M = − γ − 1 DN I with Dv = 1 { γ − 1 ( Nu − g )+ z r ≥ 0 } v . ICCP 2014, Berlin, August 7, 2014 | TU Darmstadt | S. Ulbrich | 16