Interaction of Fluid Flow and an Elastic Body k 1 Miloslav Feistauer - PowerPoint PPT Presentation

Fluid flow Elasticity Numerical experiments Interaction of Fluid Flow and an Elastic Body ık 1 Miloslav Feistauer 1 cek 2 Adam Kos´ Jarom´ ır Hor´ aˇ cek 3 Petr Sv´ aˇ 1 Faculty of Mathematics and Physics, Charles University in Prague 2 Institute of Thermomechanics, Czech Academy of Sciences 3 Faculty of Mechanical Engineering, Czech Technical University Prague Workshop Numerical Analysis Dresden-Prague 2010 Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Outline Formulation of a flow problem in a moving domain Formulation of the problem of elasticity Numerical experiments Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Introduction Model of the vocal folds Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Navier-Stokes equations Incompressible viscous flow is described by the system of the Navier-Stokes equations ∂ v in Ω f ∂ t + ( v · ∇ ) v + ∇ p − ν ∆ v = 0 t , in Ω f ∇ · v = 0 t , equipped with the initial and boundary conditions. Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments The Lagrangian and Arbitrary-Lagrangian-Eulerian mapping Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments The Navier-Stokes equations in the ALE form D A in Ω f Dt v + (( v − w ) · ∇ ) v + ∇ p − ν ∆ v = 0 t , in Ω f ∇ · v = 0 t . This system is equipped with the initial condition x ∈ Ω f v ( x , 0 ) = v 0 , 0 , and boundary conditions. Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments The boundary conditions on Γ f v = v D v = w on Γ W t , D , − ( p − p ref ) n + ν ∂ v on Γ f ∂ n = 0 , 0 . Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Discretization ◮ time discretization: the second-order two-step scheme ◮ space discretization: the finite element method ◮ stabilization of the FEM: the streamline-diffusion/Petrov-Galerkin technique Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Equations od equilibrium, Generalized Hooke’s law Equations of equilibrium ∂τ b 3 ji ∀ x ∈ Ω b . � ( x ) + f i ( x ) = 0 , i = 1 , 2 , 3 , ∂ x j j = 1 Generalized Hooke’s law 3 τ b ∀ x ∈ Ω b . ij ( x ) = � c ijkl ( x ) e kl ( x ) , i , j = 1 , 2 , 3 , k , l = 1 Generalized Hooke’s law for isotropic material τ b ij ( x ) = λ ( x ) div u ( x ) δ ij + 2 µ ( x ) e ij ( x ) , ∀ x ∈ Ω b . i , j = 1 , 2 , 3 , Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Dynamical problem of elasticity ∂τ b 3 ̺ b ∂ 2 u i ∂ t 2 + C ̺ b ∂ u i ij on M b , � = f i , i = 1 , 2 , 3 , ∂ t − ∂ x j j = 1 in Ω b , u ( 0 , · ) = u 0 , ∂ u in Ω b , ∂ t ( 0 , · ) = z 0 , u = u d on ( 0 , T ) × Γ b D , 3 τ b ij n j = T n � on ( 0 , T ) × Γ W , i = 1 , 2 , 3 . i j = 1 Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Discretization ◮ time discretization: the Newmark scheme - suitable for the second order system of the ordinary differential equations ◮ space discretization: the finite element method Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Coupled problem Trasmission conditions, fluid stress tensor 3 τ f T n � ij n j , i = 1 , 2 , 3 , i = − j = 1 � ∂ v i + ∂ v j � �� τ f ij = ̺ f − p δ ij + ν i , j = 1 , 2 , 3 . , ∂ x j ∂ x i Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Tension test AVI 400x MPEG 400x vMPEG 400x AVI 10x MPEG 10x vMPEG 10x Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Press test AVI 100x MPEG 100x vMPEG 100x AVI RT MPEG RT vMPEG RT Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Model of vocal folds AVI 1000x MPEG 1000x vMPEG 1000x AVI 100x MPEG 100x vMPEG 100x Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Interaction AVI MPEG vMPEG Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments Conclusion ◮ mathematical model of 2D viscous flow ◮ non-stationary incompressible Navier-Stokes equations in the ALE form ◮ mathematical model of the elastic body movement ◮ Generalized Hooke’s law ◮ Coupled problem ◮ Numerical experiments Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Weak formulation of the dynamical problem of elasticity. u 0 ∈ H 1 (Ω b ) , z 0 ∈ L 2 (Ω b ) , f ∈ L 2 � Ω b �� 0 , T ; L 2 � . We want to find u ∈ L 2 ( 0 , T ; V ) weak solution of the dynamical problem of elasticity such that u satisfies u ′ ∈ L 2 � , u ′′ ∈ L 2 ( 0 , T ; V ∗ ) , 0 , T ; L 2 (Ω b ) � d 2 d t 2 ( ̺ b u ( t ) , y ) 0 , Ω b + d d t ( C ̺ b u ( t ) , y ) 0 , Ω b + a ( u , y ; t ) = ( f ( t ) , y ) 0 , Ω b + ( T n ( t ) , y ) 0 , Γ W , ∀ y ∈ V , t ∈ [ 0 , T ] V = V 2 , where � � ϕ ∈ H 1 (Ω b ) V = � ϕ | Γ b � D = 0 . The form a ( u , y ; t ) is defined 2 ( λϑδ ij + 2 µ e ij ( u ( t ))) ∂ y i � a ( u , y ; t ) = � d x . ∂ x j Ω b i , j = 1 Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Newmark method Second order initial problem y ′′ ( t ) = ϕ ( t , y ( t ) , y ′ ( t )) , y ( 0 ) = y 0 , y ′ ( 0 ) = z 0 . The Newmark scheme � � 1 � � y n + 1 = y n + τ n z n + τ 2 βϕ n + 1 + 2 − β ϕ n , n z n + 1 = z n + τ n ( γϕ n + 1 + ( 1 − γ ) ϕ n ) . Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Time discretization d n + 1 = d n + ( τ n − C ξ n ) z n + ξ n M − 1 G n + 1 + I + ξ n M − 1 K � � � � 1 � � � C ( γ − 1 ) ξ n τ n + M − 1 G n − M − 1 K d n − C z n τ 2 � + 2 − β . n Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body