interaction of fluid flow and an elastic body€¦ · fluid ﬂow elasticity numerical experiments...

Fluid flowElasticity

Numerical experiments

Interaction of Fluid Flow and an Elastic Body

Adam Kosık1 Miloslav Feistauer1 Jaromır Horacek2

Petr Svacek3

1Faculty of Mathematics and Physics, Charles University in Prague

2Institute of Thermomechanics, Czech Academy of Sciences

3Faculty of Mechanical Engineering, Czech Technical University Prague

Workshop Numerical Analysis Dresden-Prague 2010

Adam Kosık Interaction of Fluid Flow and an Elastic Body



Outline

Formulation of a flow problem in a moving domain

Formulation of the problem of elasticity





IntroductionModel of the vocal folds




Navier-Stokes equations

Incompressible viscous flow is described by the system of theNavier-Stokes equations

∂v∂t

+ (v · ∇)v + ∇p − ν∆v = 0 in Ωft ,

∇ · v = 0 in Ωft ,

equipped with the initial and boundary conditions.




The Lagrangian and Arbitrary-Lagrangian-Eulerianmapping




The Navier-Stokes equations in the ALE form

DA

Dtv + ((v − w) · ∇) v + ∇p − ν∆v = 0 in Ωf

t ,

∇ · v = 0 in Ωft .

This system is equipped with the initial condition

v(x, 0) = v0, x ∈ Ωf0,

and boundary conditions.




The boundary conditions

v = vD onΓfD, v = w onΓWt ,

−(p − pref )n + ν∂v∂n

= 0, onΓf0.




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




Equations od equilibrium, Generalized Hooke’s lawEquations of equilibrium

3∑

j=1

∂τ bji

∂xj(x) + fi(x) = 0, i = 1, 2, 3, ∀x ∈ Ωb.

Generalized Hooke’s law

τ bij(x) =

3∑

k,l=1

cijkl(x)ekl(x), i, j = 1, 2, 3, ∀x ∈ Ωb.

Generalized Hooke’s law for isotropic material

τ bij (x) = λ(x)div u(x)δij + 2µ(x)eij(x),

i, j = 1, 2, 3, ∀x ∈ Ωb.




Dynamical problem of elasticity

b ∂2ui

∂t2+ Cb ∂ui

∂t−

3∑

j=1

∂τ bij

∂xj= fi, onMb, i = 1, 2, 3,

u(0, ·) = u0, in Ωb,

∂u∂t

(0, ·) = z0, in Ωb,

u = ud on (0, T) × ΓbD,

3∑

j=1

τ bij nj = Tn

i on (0, T) × ΓW , i = 1, 2, 3.




Discretization

time discretization: the Newmark scheme- suitable for the second order system of the ordinary differentialequations

space discretization: the finite element method




Coupled problem

Trasmission conditions, fluid stress tensor

Tni = −

3∑

j=1

τfijnj, i = 1, 2, 3,

τfij = f

(

−pδij + ν

(

∂vi

∂xj+

∂vj

∂xi

))

, i, j = 1, 2, 3.




Tension test

AVI 400x MPEG 400x vMPEG 400x AVI 10x MPEG 10x vMPEG 10x




Press test

AVI 100x MPEG 100x vMPEG 100x AVI RT MPEG RT vMPEG RT




Model of vocal folds

AVI 1000x MPEG 1000x vMPEG 1000x AVI 100x MPEG 100x vMPEG 100x




Interaction

AVI MPEG vMPEG




Conclusion

mathematical model of 2D viscous flow

non-stationary incompressible Navier-Stokes equations in theALE form

mathematical model of the elastic body movement

Generalized Hooke’s law

Coupled problem



Weak formulation of the dynamical problem of elasticity.u0 ∈ H1(Ωb), z0 ∈ L2(Ωb), f ∈ L2

(

0, T; L2(

Ωb))

. We want to findu ∈ L2 (0, T; V) weak solution of the dynamical problem of elasticitysuch thatu satisfiesu′ ∈ L2

(

0, T; L2(Ωb))

, u′′ ∈ L2 (0, T; V∗),

d2

dt2(bu(t), y)0,Ωb +

ddt

(Cbu(t), y)0,Ωb + a(u, y; t) =

(f(t), y)0,Ωb + (Tn(t), y)0,ΓW ,

∀y ∈ V, t ∈ [0, T]

V = V2, where

V =

ϕ ∈ H1(Ωb)∣

∣ ϕ|Γb

D= 0

.

The forma(u, y; t) is defined

a(u, y; t) =

∫

Ωb

2∑

i,j=1

(λϑδij + 2µeij (u(t)))∂yi

∂xjdx.


Newmark method

Second order initial problem

y′′(t) = ϕ(t, y(t), y′(t)),

y(0) = y0,

y′(0) = z0.

The Newmark scheme

yn+1 = yn + τnzn + τ2n

(

βϕn+1 +

(

12− β

)

ϕn

)

,

zn+1 = zn + τn(γϕn+1 + (1− γ)ϕn).


Time discretization

(

I + ξnM−1

K)

dn+1 = dn + (τn − Cξn) zn + ξnM−1Gn+1+

+

(

C (γ − 1) ξnτn +

(

12− β

)

τ2n

)

(

M−1Gn − M

−1Kdn − Czn

)

.


interaction of fluid flow and an elastic body€¦ · fluid ﬂow elasticity numerical experiments...

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