An all-floating formulation of the BETI method G. Of O. Steinbach - PowerPoint PPT Presentation

Institut f¨ ur Numerische Mathematik An all-floating formulation of the BETI method G. Of O. Steinbach Institute of Computational Mathematics Graz University of Technology 17th International Conference on Domain Decomposition Methods July 4th 2006 G. Of An all-floating formulation of the BETI method DD17 1 / 17

Institut f¨ ur Numerische Mathematik Outline Linear elastostatics and boundary element method Dirichlet domain decomposition method Boundary Element Tearing and Interconnecting method Inversion of Steklov Poincar´ e operators Floating subdomains All–floating BETI formulation Numerical examples G. Of An all-floating formulation of the BETI method DD17 2 / 17

� Institut f¨ ur Numerische Mathematik Linear elastostatics Mixed boundary value problem: 3 , − div ( σ ( u )) = 0 for x ∈ Ω ⊂ u i ( x ) = g D , i ( x ) for x ∈ Γ D , i , i = 1 , . . . , 3 , t i ( x ) := ( T x u ) i ( x ) = ( σ ( u ) n ( x )) i = g N , i ( x ) for x ∈ Γ N , i , i = 1 , . . . , 3 . G. Of An all-floating formulation of the BETI method DD17 3 / 17

� Institut f¨ ur Numerische Mathematik Linear elastostatics Mixed boundary value problem: 3 , − div ( σ ( u )) = 0 for x ∈ Ω ⊂ u i ( x ) = g D , i ( x ) for x ∈ Γ D , i , i = 1 , . . . , 3 , t i ( x ) := ( T x u ) i ( x ) = ( σ ( u ) n ( x )) i = g N , i ( x ) for x ∈ Γ N , i , i = 1 , . . . , 3 . The stress tensor σ ( u ) is related to the strain tensor e ( u ) by Hooke’s law � � E ν E σ ( u ) = tr e ( u ) I + (1 + ν ) e ( u ) . (1 + ν )(1 − 2 ν ) Young modulus E > 0, Poisson ratio ν ∈ (0 , 1 2 ), strain tensor e ( u ) = 1 2( ∇ u ⊤ + ∇ u ) . G. Of An all-floating formulation of the BETI method DD17 3 / 17

� Institut f¨ ur Numerische Mathematik Linear elastostatics Mixed boundary value problem: 3 , − div ( σ ( u )) = 0 for x ∈ Ω ⊂ u i ( x ) = g D , i ( x ) for x ∈ Γ D , i , i = 1 , . . . , 3 , t i ( x ) := ( T x u ) i ( x ) = ( σ ( u ) n ( x )) i = g N , i ( x ) for x ∈ Γ N , i , i = 1 , . . . , 3 . The stress tensor σ ( u ) is related to the strain tensor e ( u ) by Hooke’s law � � E ν E σ ( u ) = tr e ( u ) I + (1 + ν ) e ( u ) . (1 + ν )(1 − 2 ν ) Young modulus E > 0, Poisson ratio ν ∈ (0 , 1 2 ), Γ 12 strain tensor Ω 1 Ω 2 e ( u ) = 1 2( ∇ u ⊤ + ∇ u ) . Γ 13 Γ 24 Bounded Lipschitz domain Ω given by a domain Γ 34 decomposition into p non–overlapping subdomains. Ω 3 Ω 4 Assumptions: piecewise constant: E i > 0 and ν i ∈ (0 , 1 / 2). G. Of An all-floating formulation of the BETI method DD17 3 / 17

Institut f¨ ur Numerische Mathematik Boundary integral formulation Representation formula: � � [ U ∗ ( x , y )] ⊤ t ( y ) ds y − [ T y U ∗ ( x , y )] ⊤ u ( y ) ds y u ( x ) = for x ∈ Ω . Γ Γ Fundamental solution of linear elastostatics: � � 1 + ν | x − y | + ( x k − y k )( x l − y l ) δ kl U ∗ kl ( x − y ) = (3 − 4 ν ) . | x − y | 3 8 π E (1 − ν ) G. Of An all-floating formulation of the BETI method DD17 4 / 17

Institut f¨ ur Numerische Mathematik Boundary integral formulation Representation formula: � � [ U ∗ ( x , y )] ⊤ t ( y ) ds y − [ T y U ∗ ( x , y )] ⊤ u ( y ) ds y u ( x ) = for x ∈ Ω . Γ Γ Fundamental solution of linear elastostatics: � � 1 + ν | x − y | + ( x k − y k )( x l − y l ) δ kl U ∗ kl ( x − y ) = (3 − 4 ν ) . | x − y | 3 8 π E (1 − ν ) Calderon projector for the Cauchy data u ( x ) and t ( x ) on the boundary Γ : � u � � 1 � � u � 2 I − K V = on Γ 1 2 I + K ′ t D t Boundary integral operators: � � [ U ∗ ( x , y )] ⊤ t ( y ) ds y , ( Du )( x ) = − T x [ T y U ∗ ( x , y )] ⊤ u ( y ) ds y , ( Vt )( x ) = Γ Γ � � [ T y U ∗ ( x , y )] ⊤ u ( y ) ds y , ( K ′ t )( x ) = ⊢ [ T x U ∗ ( x , y )] ⊤ t ( y ) ds y . ( Ku )( x ) = ⊢ ⊣ ⊣ Γ Γ G. Of An all-floating formulation of the BETI method DD17 4 / 17

Institut f¨ ur Numerische Mathematik Dirichlet domain decomposition method Solve the global system of linear equations iteratively: p p � � � i � A ⊤ A ⊤ S h u = S i , h A i u = i f i , i =1 i =1 R M i × M mapping from the global nodes with some connectivity matrices A i ∈ I to the local nodes such that v i = A i v . G. Of An all-floating formulation of the BETI method DD17 5 / 17

Institut f¨ ur Numerische Mathematik Dirichlet domain decomposition method Solve the global system of linear equations iteratively: p p � � � i � A ⊤ A ⊤ S h u = S i , h A i u = i f i , i =1 i =1 R M i × M mapping from the global nodes with some connectivity matrices A i ∈ I to the local nodes such that v i = A i v . Local realization of the matrices of the Steklov Poincar´ e operators: S i , h = D i , h + (1 i , h (1 � 2 M ⊤ i , h + K ⊤ i , h ) V − 1 2 M i , h + K i , h ) , with the boundary element matrices realized by the Fast Multipole Method (integration by parts ⇒ single and double layer potentials of the Laplacian) � V i ϕ i k , ϕ i � K i ψ i n , ϕ i V i , h [ ℓ, k ] = ℓ � L 2 ( Γ i ) , K i , h [ ℓ, n ] = ℓ � L 2 ( Γ i ) , � D i ψ i n , ψ i � ψ i n , ϕ i D i , h [ m , n ] = m � L 2 ( Γ i ) , M i , h [ ℓ, n ] = ℓ � L 2 ( Γ i ) . G. Of An all-floating formulation of the BETI method DD17 5 / 17

Institut f¨ ur Numerische Mathematik Dirichlet domain decomposition method Solve the global system of linear equations iteratively: p p � � � i � A ⊤ A ⊤ S h u = S i , h A i u = i f i , i =1 i =1 R M i × M mapping from the global nodes with some connectivity matrices A i ∈ I to the local nodes such that v i = A i v . Local realization of the matrices of the Steklov Poincar´ e operators: S i , h = D i , h + (1 i , h (1 � 2 M ⊤ i , h + K ⊤ i , h ) V − 1 2 M i , h + K i , h ) , with the boundary element matrices realized by the Fast Multipole Method (integration by parts ⇒ single and double layer potentials of the Laplacian) � V i ϕ i k , ϕ i � K i ψ i n , ϕ i V i , h [ ℓ, k ] = ℓ � L 2 ( Γ i ) , K i , h [ ℓ, n ] = ℓ � L 2 ( Γ i ) , � D i ψ i n , ψ i � ψ i n , ϕ i D i , h [ m , n ] = m � L 2 ( Γ i ) , M i , h [ ℓ, n ] = ℓ � L 2 ( Γ i ) . Preconditioning: p � C − 1 A ⊤ = i V i , lin , h A i e S i =1 G. Of An all-floating formulation of the BETI method DD17 5 / 17

Institut f¨ ur Numerische Mathematik BETI method Starting from the equivalent minimization problem � 1 � p � 2( � F ( u ) = min S i , h A i v , A i v ) − ( f i , A i v ) v ∈ I R M i =1 one can derive the Boundary Element Tearing and Interconnecting method [Langer, Steinbach 2003] (FETI [Farhat, Roux 1991; Klawonn,Widlund 2001; . . . ]) : G. Of An all-floating formulation of the BETI method DD17 6 / 17

Institut f¨ ur Numerische Mathematik BETI method Starting from the equivalent minimization problem � 1 � p � 2( � F ( u ) = min S i , h A i v , A i v ) − ( f i , A i v ) v ∈ I R M i =1 one can derive the Boundary Element Tearing and Interconnecting method [Langer, Steinbach 2003] (FETI [Farhat, Roux 1991; Klawonn,Widlund 2001; . . . ]) : ◮ introducing local vectors u i = A i u . G. Of An all-floating formulation of the BETI method DD17 6 / 17

Institut f¨ ur Numerische Mathematik BETI method Starting from the equivalent minimization problem � 1 � p � 2( � F ( u ) = min S i , h A i v , A i v ) − ( f i , A i v ) v ∈ I R M i =1 one can derive the Boundary Element Tearing and Interconnecting method [Langer, Steinbach 2003] (FETI [Farhat, Roux 1991; Klawonn,Widlund 2001; . . . ]) : ◮ introducing local vectors u i = A i u . ◮ describing the connections across the interfaces by introducing the constraints p � e M × M i . B i u i = 0 where B i ∈ I R i =1 Ω 1 Ω 2 Ω 1 Ω 2 Ω 1 Ω 2 Ω 3 Ω 4 Ω 3 Ω 4 Ω 3 Ω 4 G. Of An all-floating formulation of the BETI method DD17 6 / 17

Institut f¨ ur Numerische Mathematik BETI method R e M , one gets After introducing Lagrange multipliers λ ∈ I       � − B ⊤ S 1 , h u 1 f 1 1  .   .   .  ... . . .       . . .  =  .           � − B ⊤ u p f p S p , h p λ 0 B 1 . . . B p 0 G. Of An all-floating formulation of the BETI method DD17 7 / 17

Institut f¨ ur Numerische Mathematik BETI method R e M , one gets After introducing Lagrange multipliers λ ∈ I       � − B ⊤ S 1 , h u 1 f 1 1  .   .   .  ... . . .       . . .  =  .           � − B ⊤ u p f p S p , h p λ 0 B 1 . . . B p 0 Dirichlet b.c., � u i = � S − 1 i , h ( f i + B ⊤ S i , h are invertible: i λ ) . Then the bottom line of the linear system leads to the Schur complement system: p p � � B i � S − 1 i , h B ⊤ B i � S − 1 i λ = − i , h f i . i =1 i =1 G. Of An all-floating formulation of the BETI method DD17 7 / 17