Numerical simulation of the incompressible Navier-Stokes ...acannas/Talentos/... · Numerical simulation of the incompressible Navier-Stokes equations in a moving domain: application

Numerical simulation of the incompressibleNavier-Stokes equations in a moving domain:

application to hemodynamics

Gonalo Pena

Center of Mathematics of the University of Coimbra

16 de Julho de 2010

Gonalo Pena (CMUC) 16 de Julho de 2010 1 / 17

Outline

1 Motivation and examples

2 Differential model

3 Numerical method

4 Application to hemodynamics


Motivation and examples

Incompressible Navier-Stokes equations in a moving domainModel for a viscous fluid with constant density in a domain that changesshape in time

Examples:Static domain: flow in a channelFluid-fluid: bubbles of air in waterFluid-structure: hemodynamics, bridges, sail design


Differential model

Navier-Stokes equations (Eulerian coordinate system)

ut u + (u )u +p = f, in t (0, T ) (1)

u = 0, in t (0, T ) (2)B(u, p) = g, on t [0, T ] (3)

Model for viscous fluidu is the velocity field, p is the pressure field is the viscosity constant, is the densityMomentum conservation: (1)Mass conservation: (2)B(u, p) is an operator with boundary conditions


Differential model

Coordinate systems:Eulerian coordinate system

Eulerian coordinate systemLagrangian coordinate systemArbitrary lagrangian-eulerian (ALE) coordinate system

tDt

Dt

t

t

tDt

Dt

t

t

t0 Dt0

Dt0

t0

t0

Lt At

u


Differential model

Coordinate systems:Eulerian coordinate systemLagrangian coordinate system

Lagrangian coordinate systemArbitrary lagrangian-eulerian (ALE) coordinate system

tDt

Dt

t

t

tDt

Dt

t

t

t0 Dt0

Dt0

t0

t0

Lt At

u


Differential model

Coordinate systems:Eulerian coordinate systemLagrangian coordinate systemArbitrary lagrangian-eulerian (ALE) coordinate system

tDt

Dt

t

t

tDt

Dt

t

t

t0 Dt0

Dt0

t0

t0

Lt At

u


Differential model

ALE approach

Navier-Stokes equations in ALE coordinates

ut

Y

+ [(uw) ]u +p u = f, in t u = 0, in t

0 D0

D0

0

0

tDt

Dt

t

t

ALE map: homeomorfism At : t0 tDomains deformation velocity w = Att A1tALE time derivative: ut

Y


Differential model

Weak formulationFor u,v, V(t) and p, q Q(t), let

(u,v)t=

t

u v dx a (u,v)t =

t

xu : xv dx

b (v, p)t=

t

divx(u) p dx c (u,v;)t =

t

[ x] u v dx.

ProblemFor t I, find u(t) V(t), with u(t0) = u0 in t0 and p(t) Q(t), such that

(u

t

Y

,v

)t

+ c (u,v; uw)t +

a (u,v)t+ b (v, p)t

= (f ,v)t, v V(t)

b (u, q)t= 0, q Q(t)

(4)

V(t) ={v : t I Rd, v = v A1t , v H1D (t0)

}Gonalo Pena (CMUC) 16 de Julho de 2010 7 / 17

Numerical method

Family of numerical methodsGet a weak formulation of the problemDiscretization in space

I Finite/spectral element methodI Construction of the ALE map

Discretization in timeI Finite differences/Runge-KuttaI Linearization of the convective term/NewtonI Discretization of the domains deformation velocity

Strategy to solve algebraic system of equationsI Direct methodI GMRES with preconditionerI Algebraic factorization methods


Numerical method

Build basis for the space

FN (Th) ={v C0() : v|e PN (e), e Th

}Plan: reference element approach

1 Construct a basis in PN ()I Lagrange polynomialsI Fekete/Gauss-Lobatto-Legendre points

Elemen

t 1

Elemen

t 2

2 Use geometrical transformation + glue similar functions (continuity)


Numerical method

How to build the discrete spaces?Define spaces in the reference configuration and use the ALE map:

V(t,) ={

v : t, I Rd, v = v A1t, , v H1D (t0) (FN (Tt0,))d}

Q(t,) ={q : t, I R, q = q A1t, , q FM (Tt0,)

}How to build the ALE map?

Usual approach: use finite elements and harmonic extensionLess usual approach: use spectral elements (Stokes or Laplace)

I Representation of the boundary with curved elementsI Keep inner elements with straight edges


Numerical method

Steps in the construction of A,

0 D0

D0

0

0

tDt

Dt

t

t

1 Generate a straight edge mesh2 Given the description of the boundary t, calculate the discrete

harmonic extension, A1h3 Project A1h in the space PN , ANh4 Change the values of the degrees of freedom of ANh to fit the boundary,A,


Numerical method


1 Generate a straight edge mesh

2 Given the description of the boundary t, calculate the discreteharmonic extension, A1h

3 Project A1h in the space PN , ANh4 Change the values of the degrees of freedom of ANh to fit the boundary,A,


Numerical method



harmonic extension, A1h

3 Project A1h in the space PN , ANh4 Change the values of the degrees of freedom of ANh to fit the boundary,A,


Numerical method



harmonic extension, A1h3 Project A1h in the space PN , ANh

4 Change the values of the degrees of freedom of ANh to fit the boundary,A,


Numerical method



harmonic extension, A1h3 Project A1h in the space PN , ANh4 Change the values of the degrees of freedom of ANh to fit the boundary,A,


Numerical method

Weak formulation (discrete problem)For each n > q 1, find (un+1 , p

n+1 ) V(tn+1,)Q(tn+1,), with u

0 = u0, in

t0,, such that

1t

(un+1 ,v

)tn+1,

+

a(un+1 ,v

)tn+1,

+ b(v, pn+1

)tn+1,

+

c(un+1 ,v;u

wn+1

)tn+1,

=(fn+1 ,v

)tn+1,

, v V(tn+1,)

b(un+1 , q

)tn+1,

= 0, q Q(tn+1,)

onde

fn+1 = fn+1 +

q1j=0

jt

unj

Assembling the matrices, we obtain[FN GNDN 0

] [Un+1NPn+1N

]=

[Fn+1N

0

].


Application to hemodynamics

Fluid-structure interaction (FSI) problem in hemodynamicsAn interaction exists between the blood flow and the arterial wall

tu

t wt

S2

S1

t represents the domain occupied by the bloodt represents the arterial wallwt is the interface between t and tS1 are S2 ficticious boundaries of the blood vessel



FSI problem

u

t

Y

+ ((uw) x)u 2D(u) +p = f , in t, t I

divx(u) = 0, in t, t I

wh2

t2 kGh

2

x2+

Eh

1 2

R20 v

3

x2t= r in (0, L), t I

u =( 1

)e2, in wt

r = (Tn e2)

Numerical method for the FSI problemModular approach: structure algorithm + fluid algorithm + interfaceoperatorsNon-modular approach



A carregar...

Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.

Go


FSI_compliant_tube_magnified.aviMedia File (video/avi)


Thank you



t = 0ms


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t = 2ms


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t = 4ms


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t = 6ms


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t = 8ms


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t = 10ms


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Motivation and examplesDifferential modelNumerical methodApplication to hemodynamics

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Numerical simulation of the incompressible Navier-Stokes ...acannas/Talentos/... · Numerical simulation of the incompressible Navier-Stokes equations in a moving domain: application