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

Adam Kos´ ık1 Miloslav Feistauer1 Jarom´ ır Hor´ aˇ cek2 Petr Sv´ aˇ cek3

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 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 ∂t + (v · ∇)v + ∇p − ν∆v = 0 in Ωf

t,

∇ · v = 0 in Ωf

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

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

t,

∇ · v = 0 in Ωf

t.

This system is equipped with the initial condition v(x, 0) = v0, x ∈ Ωf

0,

and boundary conditions.

Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments

The boundary conditions

v = vD

• n Γf

D,

v = w

• n ΓWt,

−(p − pref)n + ν ∂v ∂n = 0,

• n Γf

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

3

• j=1

∂τ b

ji

∂xj (x) + fi(x) = 0, i = 1, 2, 3, ∀x ∈ Ωb. Generalized Hooke’s law τ b

ij(x) = 3

• k,l=1

cijkl(x)ekl(x), i, j = 1, 2, 3, ∀x ∈ Ωb. Generalized Hooke’s law for isotropic material τ b

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

i, j = 1, 2, 3, ∀x ∈ Ωb.

Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Fluid flow Elasticity Numerical experiments

Dynamical problem of elasticity

̺b ∂2ui ∂t2 + C̺b ∂ui ∂t −

3

• j=1

∂τ b

ij

∂xj = fi,

• n Mb,

i = 1, 2, 3, u(0, ·) = u0, in Ωb, ∂u ∂t (0, ·) = z0, in Ωb, u = ud

• n (0, T) × Γb

D, 3

• j=1

τ b

ijnj = Tn i

• n (0, T) × ΓW,

i = 1, 2, 3.

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 Tn

i = − 3

• j=1

τ f

ijnj,

i = 1, 2, 3, τ f

ij = ̺f

• −pδij + ν

∂vi ∂xj + ∂vj ∂xi

• ,

i, j = 1, 2, 3.

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.

u0 ∈ H1(Ωb), z0 ∈ L2(Ωb), f ∈ L2 0, T; L2 Ωb . We want to find u ∈ L2 (0, T; V) weak solution of the dynamical problem of elasticity such that u satisfies u′ ∈ L2 0, T; L2(Ωb)

• , u′′ ∈ L2 (0, T; V∗),

d2 dt2 (̺bu(t), y)0,Ωb + d dt(C̺bu(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 form a(u, y; t) is defined a(u, y; t) =

• Ωb

2

• i,j=1

(λϑδij + 2µeij (u(t))) ∂yi ∂xj dx.

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) = y0, y′(0) = z0. The Newmark scheme yn+1 = yn + τnzn + τ 2

n

• βϕn+1 +

1 2 − β

• ϕn
• ,

zn+1 = zn + τn(γϕn+1 + (1 − γ)ϕn).

Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body

Time discretization

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

+

• C (γ − 1) ξnτn +

1 2 − β

• τ 2

n

M−1Gn − M−1Kdn − Czn

• .

Adam Kos´ ık Interaction of Fluid Flow and an Elastic Body