Coupling of BEM and FEM in the Time Domain for Fluid-Structure - PowerPoint PPT Presentation

Coupling of BEM and FEM in the Time Domain for Fluid-Structure Interaction . Stephan 1 Ernst P (joint with Heiko Gimperlein 2 ) 1: Leibniz Universität Hannover, Germany 2: Heriot–Watt University and Maxwell Institute, Edinburgh, UK RICAM Workshop Linz, 11. Nov. 2016 Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 1 / 23

Outline Fluid-structure interaction: Set-up & well-posedness FEM-BEM in time domain: Reduction to boundary, variational formulation, discretization A priori and a posteriori error analysis Some numerical experiments Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 2 / 23

Fluid-structure interaction d = 2 , 3 submersed in a fluid: u : R + × Ω → R d Elastic body Ω ⊂ R d , t u − div σ ( u ) = f in R + × Ω , ̺ 1 ∂ 2 u = ∂ t u = 0 for t = 0 Wave equation in Ω c = R d \ Ω : p : R + × Ω → R t p − ∆ p = 0 in R + × Ω c , ∂ 2 p = ∂ t p = 0 for t = 0 Transmission conditions on Γ = ∂ Ω : n exterior unit normal, data: ̺ 1 , ̺ 2 densities, f load, p I incident wave ̺ 2 ∂ t u · n + ∂ n p + ∂ n p I = 0 σ ( u ) n + ( ∂ t p + ∂ t p I ) n = 0 and Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 3 / 23

Space–time anisotropic Sobolev spaces σ ( R + , H s ( R d )) defined using Fourier–Laplace transform F for σ > 0 : H r � � R d d ξ | ω | 2 r ( | ω | 2 + | ξ | 2 ) s |F ψ ( ω, ξ ) | 2 < ∞} { ψ : supp ψ ⊂ R + × R d , R + i σ d ω H r σ ( R + , H s (Ω)) , H r σ ( R + , H s (Γ)) with norms � · � r , s , Ω , � · � r , s , Γ . � + × R d ) t ) r / 2 ( σ − ∂ 2 t − ∆) s / 2 ψ ( t , x ) ∈ L 2 ( R ( σ − ∂ 2 Fourier–Laplace transform: t �→ ω = η + i σ , x �→ ξ Ha-Duong, Bamberger 1986, . . . Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 4 / 23

Well-posedness of fluid-structure interaction Theorem (Felipe 1994 + ε , cf. also Hsiao-Sayas-Weinacht 2015) Assume s ≥ 0 f ∈ H 1 + s ( R + , H − 1 (Ω)) d σ p I ∈ H 3 + s 1 2 (Γ)) , ∂ n p I ∈ H 3 + s ( R + , H − 1 ( R + , H 2 (Γ)) σ σ Then there exists a unique solution ( R + , H 1 (Ω)) d × H s ( u , p ) ∈ H 1 + s σ ( R + , H 1 (Ω c )) , σ which depends continuously on the data. Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 5 / 23

Green’s functions G / Huygens potential H ( t − s − | x − y | ) G ( t − s , x , y ) = � (2d) ( t − s ) 2 + | x − y | 2 2 π G ( t − s , x , y ) = δ ( t − s − | x − y | ) (3d) 4 π | x − y | satisfy wave equation for point source in R t × R d x : t G ( t , x , x ′ ) − ∆ x G ( t , x , x ′ ) = δ ( t , x − x ′ ) ∂ 2 Single layer ansatz for wave equation / Huygens potential � x ∈ R d \ Γ p ( t , x ) = S q ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , R + × Γ is continuous and solves wave equation on R d \ Γ : ∂ 2 t p − ∆ p = 0 . Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 6 / 23

Huygens potential / integral operators Single layer ansatz for wave equation / Huygens potential � x ∈ R d \ Γ , p ( t , x ) = S q ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , R + × Γ is continuous and solves wave equation on R d \ Γ : ∂ 2 t p − ∆ p = 0 . Single layer operator � Vq ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , x ∈ Γ . R + × Γ S and V : there are no differences – it’s just where x is. Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 7 / 23

Huygens potential / integral operators Single layer ansatz for wave equation / Huygens potential � x ∈ R d \ Γ , p ( t , x ) = S q ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , R + × Γ is continuous and solves wave equation on R d \ Γ : ∂ 2 t p − ∆ p = 0 . Single layer operator � Vq ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , x ∈ Γ . R + × Γ Adjoint of double layer operator � ∂ G K ′ ϕ ( t , x ) = ( t − τ, x , y ) ϕ ( τ, y ) d τ ds y , x ∈ Γ . ∂ n x R + × Γ Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 7 / 23

Fluid-structure interaction: reduce wave equation to Γ Elastic body Ω ⊂ R d submersed in a fluid: t u − div σ ( u ) = f in R + × Ω , ̺ 1 ∂ 2 u = ∂ t u = 0 for t = 0 Wave equation in Ω c = R d \ Ω : � x ∈ R d \ Γ p ( t , x ) = S q ( t , x ) = G ( t − τ, x , y ) q ( τ, y ) d τ ds y , R + × Γ Transmission conditions on Γ = ∂ Ω : ̺ 2 ∂ t u · n + ∂ n p + ∂ n p I = 0 σ ( u ) n + ( ∂ t p + ∂ t p I ) n = 0 , 2 + K ′ ) q + ∂ n p I = 0 � σ ( u ) n + ( ∂ t Vq + ∂ t p I ) n = 0 , ̺ 2 ∂ t u · n + ( − 1 Solve the coupled red system with FEM / BEM in the time domain. Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 8 / 23

Weak formulation in space � � � � ∂ 2 ̺ 1 ̺ 2 t u ˙ v dx + ̺ 2 σ ( u ) : ǫ (˙ v ) dx − ̺ 2 σ ( u ) n · ˙ v ds = ̺ 2 f ˙ v dx Ω Ω Γ Ω � � � ∂ t p I ˙ ̺ 2 σ ( u ) n · ˙ v ds + ̺ 2 V ∂ t q ˙ vn ds = − ̺ 2 vn ds Γ Γ Γ � � � � V ∂ t q ′ � � � ∂ n p I � V ∂ t q ′ � V ∂ t q ′ ds = ( − 1 / 2 + K ′ ) q ˙ − ̺ 2 un ds − ds Γ Γ Γ Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 9 / 23

Space-time variational formulation Bilinear form, with ε ( v ) = 1 2 ( ∇ v + ( ∇ v ) T ) and σ > 0 : � � � R + e − 2 σ t � B (( u , q ) , ( v , q ′ )) = � ( ∂ 2 ̺ 1 ̺ 2 t u ) · ( ∂ t v ) dx + ̺ 2 σ ( u ) : ∂ t ε ( v ) dx Ω Ω � � ( ∂ t u · n )( V ∂ t q ′ ) ds + ̺ 2 ( V ∂ t q )( ∂ t v · n ) ds − ̺ 2 Γ Γ � � (( − 1 2 + K ′ ) q )( V ∂ t q ′ ) ds − dt Γ σ ( R + , H 1 (Ω)) d × D ( V )) 2 , where defined on ( H 1 σ ( R + , H − 1 σ ( R + , H − 1 D ( V ) = { q ∈ H 1 2 (Γ)) : Vq ∈ H 1 2 (Γ)) } . σ ( R + , H − 1 1 σ ( R + , H V : H 1 2 (Γ)) → H 0 2 (Γ)) Note: Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 10 / 23

Space-time variational formulation Bilinear form, with ε ( v ) = 1 2 ( ∇ v + ( ∇ v ) T ) and σ > 0 : � � � R + e − 2 σ t � B (( u , q ) , ( v , q ′ )) = � ( ∂ 2 ̺ 1 ̺ 2 t u ) · ( ∂ t v ) dx + ̺ 2 σ ( u ) : ∂ t ε ( v ) dx Ω Ω � � ( ∂ t u · n )( V ∂ t q ′ ) ds + ̺ 2 ( V ∂ t q )( ∂ t v · n ) ds − ̺ 2 Γ Γ � � (( − 1 2 + K ′ ) q )( V ∂ t q ′ ) ds − dt Γ Linear functional: � � R + e − 2 σ t � � � ( ∂ t p I )( ∂ t v · n )+ ∂ p I F ( v , q ) = − ̺ 2 ∂ n ( V ∂ t q ) ds + ρ 2 Ω f ˙ dt . v dx Γ σ ( R + , H 1 (Ω)) d × D ( V ) s.t. Variational formulation: Find ( u , q ) ∈ H 1 � B (( u , q ) , ( v , q ′ )) = F ( v , q ′ ) σ ( R + , H 1 (Ω)) d × D ( V ) . for all ( v , q ′ ) ∈ H 1 Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 10 / 23

Coercivity / continuity of the bilinear form � � � R + e − 2 σ t � B (( u , q ) , ( v , q ′ )) = � ( ∂ 2 ̺ 1 ̺ 2 t u ) · ( ∂ t v ) dx + ̺ 2 σ ( u ) : ∂ t ε ( v ) dx Ω Ω � � ( ∂ t u · n )( V ∂ t q ′ ) ds + ̺ 2 ( V ∂ t q )( ∂ t v · n ) ds − ̺ 2 Γ Γ � � (( − 1 2 + K ′ ) q )( V ∂ t q ′ ) ds − dt Γ Theorem (Felipe 1994) 2 , Γ � � � u � 2 1 , 1 , Ω + � q � 2 2 , Γ + � Vq � 2 B (( u , q ) , ( u , q )) � σ � u � 2 0 , 1 , Ω + � q � 2 1 , − 1 1 , − 1 0 , − 1 2 , Γ Difference of time derivatives between upper and lower bounds: Wave equation is hyperbolic. V ∂ t coercive with loss Parseval: Im ω = σ > 0 � � 1 ˆ R + e − 2 σ t f ( t ) g ( t ) dt f ( ω )ˆ g ( ω ) d ω = 2 π R + i σ Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 11 / 23

Discretization Ω = ∪ N i = 1 T i , Γ = ∪ M i = 1 Γ i regular triangulations each Γ i is a face of one T j W p h piecewise polynomial functions of degree p on Ω = ∪ N i = 1 T i (continuous if p ≥ 1 ) V p h restrictions of functions in W p h to Γ [ 0 , T ) = ∪ L n = 1 [ t n − 1 , t n ) , t n = n (∆ t ) V q ∆ t piecewise polynomial functions of degree q in time (continuous and vanishing at t = 0 if q ≥ 1 ) tensor products in space-time: W p , q h , ∆ t = W p h ⊗ V q ∆ t , V p , q h , ∆ t = V p h ⊗ V q ∆ t Abboud, Joly, Rodriguez, Terrasse 2011: DGFEM Sayas, Hsiao et al. 2015–: convolution quadrature Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 12 / 23