numerical simulation of the incompressible navier-stokes ...acannas/talentos/... · numerical...
TRANSCRIPT
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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
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Outline
1 Motivation and examples
2 Differential model
3 Numerical method
4 Application to hemodynamics
Gonalo Pena (CMUC) 16 de Julho de 2010 2 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 3 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 3 / 17
-
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
Gonalo Pena (CMUC) 16 de Julho de 2010 3 / 17
-
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
Gonalo Pena (CMUC) 16 de Julho de 2010 3 / 17
-
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
Gonalo Pena (CMUC) 16 de Julho de 2010 3 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 4 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 5 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 5 / 17
-
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
Gonalo Pena (CMUC) 16 de Julho de 2010 5 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 6 / 17
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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
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 8 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 8 / 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
Gonalo Pena (CMUC) 16 de Julho de 2010 8 / 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
Gonalo Pena (CMUC) 16 de Julho de 2010 8 / 17
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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)
Gonalo Pena (CMUC) 16 de Julho de 2010 9 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 10 / 17
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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,
Gonalo Pena (CMUC) 16 de Julho de 2010 11 / 17
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Numerical method
Steps in the construction of A,
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,
Gonalo Pena (CMUC) 16 de Julho de 2010 11 / 17
-
Numerical method
Steps in the construction of A,
1 Generate a straight edge mesh2 Given the description of the boundary t, calculate the discrete
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,
Gonalo Pena (CMUC) 16 de Julho de 2010 11 / 17
-
Numerical method
Steps in the construction of A,
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 , ANh
4 Change the values of the degrees of freedom of ANh to fit the boundary,A,
Gonalo Pena (CMUC) 16 de Julho de 2010 11 / 17
-
Numerical method
Steps in the construction of A,
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,
Gonalo Pena (CMUC) 16 de Julho de 2010 11 / 17
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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
].
Gonalo Pena (CMUC) 16 de Julho de 2010 12 / 17
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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
Gonalo Pena (CMUC) 16 de Julho de 2010 13 / 17
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Application to hemodynamics
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
Gonalo Pena (CMUC) 16 de Julho de 2010 14 / 17
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Application to hemodynamics
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
Gonalo Pena (CMUC) 16 de Julho de 2010 14 / 17
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Application to hemodynamics
A carregar...
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Go
Gonalo Pena (CMUC) 16 de Julho de 2010 15 / 17
FSI_compliant_tube_magnified.aviMedia File (video/avi)
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Application to hemodynamics
Thank you
Gonalo Pena (CMUC) 16 de Julho de 2010 16 / 17
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Application to hemodynamics
t = 0ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
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Application to hemodynamics
t = 2ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
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Application to hemodynamics
t = 4ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
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Application to hemodynamics
t = 6ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
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Application to hemodynamics
t = 8ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
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Application to hemodynamics
t = 10ms
Figura: Pressure wave propagating through the blood vessel. The displacement ismagnified five times.
Back
Gonalo Pena (CMUC) 16 de Julho de 2010 17 / 17
Motivation and examplesDifferential modelNumerical methodApplication to hemodynamics