Download A finite element method for incompressible Navier

A finite element method for
incompressible Navier-Stokes equations
in a time-dependent domain
Yuri Vassilevski1,2
Maxim Olshanskii3
Alexander Lozovskiy1
Alexander Danilov1,2
1
2
Institute of Numerical Mathematics RAS
Moscow Institute of Physics and Technology
3
University of Houston
Ginzburg Centennial Conference on Physics
May 30, 2017, Moscow
The work was supported by the Russian Science Foundation
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
I
b := Ωf ∪ Ωs
v and u denote velocities and displacements in Ω
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
I
b := Ωf ∪ Ωs
v and u denote velocities and displacements in Ω
I
ξ(x) := x + u(x), F := ∇ξ = I + ∇u, J := det(F)
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
I
b := Ωf ∪ Ωs
v and u denote velocities and displacements in Ω
I
ξ(x) := x + u(x), F := ∇ξ = I + ∇u, J := det(F)
I
Cauchy stress tensors σ f , σ s
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
I
b := Ωf ∪ Ωs
v and u denote velocities and displacements in Ω
I
ξ(x) := x + u(x), F := ∇ξ = I + ∇u, J := det(F)
I
Cauchy stress tensors σ f , σ s
I
pressures pf ,ps
Fluid-Structure Interaction problem
Prerequisites for FSI
I
reference subdomains Ωf , Ωs
I
transformation ξ maps Ωf , Ωs to Ωf (t), Ωs (t)
I
b := Ωf ∪ Ωs
v and u denote velocities and displacements in Ω
I
ξ(x) := x + u(x), F := ∇ξ = I + ∇u, J := det(F)
I
Cauchy stress tensors σ f , σ s
I
pressures pf ,ps
I
densities ρs , ρf are constant
Fluid-Structure Interaction problem
Universal equations in reference subdomains
Dynamic equations

−T

ρ−1
) in Ωs ,
s div (Jσ s F
∂v 
=
∂u
−1
−T
−1

∂t
v−
in Ωf
 (Jρf ) div (Jσ f F ) − ∇v F
∂t
Fluid-Structure Interaction problem
Universal equations in reference subdomains
Dynamic equations

−T

ρ−1
) in Ωs ,
s div (Jσ s F
∂v 
=
∂u
−1
−T
−1

∂t
v−
in Ωf
 (Jρf ) div (Jσ f F ) − ∇v F
∂t
Kinematic equation
∂u
=v
∂t
in Ωs
Fluid-Structure Interaction problem
Universal equations in reference subdomains
Dynamic equations

−T

ρ−1
) in Ωs ,
s div (Jσ s F
∂v 
=
∂u
−1
−T
−1

∂t
v−
in Ωf
 (Jρf ) div (Jσ f F ) − ∇v F
∂t
Kinematic equation
∂u
=v
∂t
in Ωs
Fluid incompressibility
div (JF−1 v) = 0
in Ωf
or
J∇v : F−T = 0
in Ωf
Fluid-Structure Interaction problem
Universal equations in reference subdomains
Dynamic equations

−T

ρ−1
) in Ωs ,
s div (Jσ s F
∂v 
=
∂u
−1
−T
−1

∂t
v−
in Ωf
 (Jρf ) div (Jσ f F ) − ∇v F
∂t
Kinematic equation
∂u
=v
∂t
in Ωs
Fluid incompressibility
div (JF−1 v) = 0
in Ωf
or
J∇v : F−T = 0
in Ωf
Constitutive relation for the fluid stress tensor
σ f = −pf I + µf ((∇v)F−1 + F−T (∇v)T )
in Ωf
FSI problem
User-dependent equations in reference subdomains
Constitutive relation for the solid stress tensor
σ s = σ s (J, F, ps , λs , µs , . . . )
in Ωs
FSI problem
User-dependent equations in reference subdomains
Constitutive relation for the solid stress tensor
σ s = σ s (J, F, ps , λs , µs , . . . )
in Ωs
Monolithic approach: Extension of the displacement field to the fluid
domain
G (u) = 0
in Ωf ,
u = u∗ on ∂Ωf
FSI problem
User-dependent equations in reference subdomains
Constitutive relation for the solid stress tensor
σ s = σ s (J, F, ps , λs , µs , . . . )
in Ωs
Monolithic approach: Extension of the displacement field to the fluid
domain
G (u) = 0
in Ωf ,
u = u∗ on ∂Ωf
for example, vector Laplace equation or elasticity equation
FSI problem
User-dependent equations in reference subdomains
Constitutive relation for the solid stress tensor
σ s = σ s (J, F, ps , λs , µs , . . . )
in Ωs
Monolithic approach: Extension of the displacement field to the fluid
domain
G (u) = 0
in Ωf ,
u = u∗ on ∂Ωf
for example, vector Laplace equation or elasticity equation
+ Initial, boundary, interface conditions (σ f F−T n = σ s F−T n)
Numerical scheme
I
b
Conformal triangular or tetrahedral mesh Ωh in Ω
I
LBB-stable pairs for velocity and pressure P2 /P1 or P2 /P0
I
Fortran open source software Ani2D, Ani3D
(Advanced numerical instruments
2D/3D)
http://sf.net/p/ani2d/
I
I
I
mesh generation
FEM systems
algebraic solvers
http://sf.net/p/ani3d/:
Numerical scheme
Find {uk+1 , vk+1 , p k+1 } ∈ V0h × Vh × Qh s.t.
∂u
k+1
= vk+1 on Γfs
v
= gh (·, (k + 1)∆t) on Γf 0 ,
∂t k+1
Numerical scheme
Find {uk+1 , vk+1 , p k+1 } ∈ V0h × Vh × Qh s.t.
∂u
k+1
= vk+1 on Γfs
v
= gh (·, (k + 1)∆t) on Γf 0 ,
∂t k+1
where
b 3 , Qh ⊂ L2 (Ω),
b V0 = {v ∈ Vh : v|Γ ∪Γ = 0}, V00 = {v ∈ V0 : v|Γ = 0}
Vh ⊂ H 1 (Ω)
h
h
h
s0
f0
fs
∂f
∂t
:=
k+1
3f k+1 − 4f k + f k−1
2∆t
Numerical scheme
Z
Z
∂v
ψ dΩ +
Jk F(e
uk )S(uk+1 , e
uk ) : ∇ψ dΩ +
∂t k+1
Ωs
g !
Z
∂v
∂u
k+1 −1
k
k
ψ dΩ +
ρf J k
ψ dΩ +
ρf Jk ∇v F (e
u ) e
v −
∂t k+1
∂t k
Ωf
Z
p k+1 Jk F−T (e
uk ) : ∇ψ dΩ = 0 ∀ψ ∈ V0h
2µf Jk Deuk vk+1 : Deuk ψ dΩ −
ρs
Ωs
Z
Ωf
Z
Ωf
Jk := J(e
uk ),
Ω
e
f k := 2f k − f k−1 ,
Du v := {∇vF−1 (u)}s ,
{A}s :=
1
(A + AT )
2
Numerical scheme
Z
∂v
ψ dΩ +
Jk F(e
uk )S(uk+1 , e
uk ) : ∇ψ dΩ +
∂t k+1
Ωs
g !
Z
∂v
∂u
k+1 −1
k
k
ψ dΩ +
ρf J k
ψ dΩ +
ρf Jk ∇v F (e
u ) e
v −
∂t k+1
∂t k
Ωf
Z
p k+1 Jk F−T (e
uk ) : ∇ψ dΩ = 0 ∀ψ ∈ V0h
2µf Jk Deuk vk+1 : Deuk ψ dΩ −
Z
ρs
Ωs
Z
Ωf
Z
Ω
Ωf
Z
Ωs
∂u
∂t
Jk := J(e
uk ),
Z
φ dΩ −
k+1
vk+1 φ dΩ +
Ωs
e
f k := 2f k − f k−1 ,
Z
G (uk+1 )φ dΩ = 0
∀φ ∈ V00
h
Ωf
Du v := {∇vF−1 (u)}s ,
{A}s :=
1
(A + AT )
2
Numerical scheme
Z
∂v
ψ dΩ +
Jk F(e
uk )S(uk+1 , e
uk ) : ∇ψ dΩ +
∂t k+1
Ωs
g !
Z
∂v
∂u
k+1 −1
k
k
ψ dΩ +
ρf J k
ψ dΩ +
ρf Jk ∇v F (e
u ) e
v −
∂t k+1
∂t k
Ωf
Z
p k+1 Jk F−T (e
uk ) : ∇ψ dΩ = 0 ∀ψ ∈ V0h
2µf Jk Deuk vk+1 : Deuk ψ dΩ −
Z
ρs
Ωs
Z
Ωf
Z
Ω
Ωf
Z
Ωs
∂u
∂t
Z
φ dΩ −
k+1
Z
vk+1 φ dΩ +
Ωs
Z
G (uk+1 )φ dΩ = 0
∀φ ∈ V00
h
Ωf
Jk ∇vk+1 : F−T (e
uk )q dΩ = 0
∀ q ∈ Qh
Ωf
Jk := J(e
uk ),
e
f k := 2f k − f k−1 ,
Du v := {∇vF−1 (u)}s ,
{A}s :=
1
(A + AT )
2
Numerical scheme
Z
... +
Ωs
I
ek ) : ∇ψ dΩ + . . .
Jk F(e
uk )S(uk+1 , u
St. Venant–Kirchhoff model
(geometrically nonlinear)
:
S(u1 , u2 ) = λs tr(E(u1 , u2 ))I + 2µs E(u1 , u2 );
E(u1 , u2 ) = {F(u1 )T F(u2 ) − I}s
I
inc. Blatz–Ko model:
S(u1 , u2 ) = µs (tr({F(u1 )T F(u2 )}s )I − {F(u1 )T F(u2 )}s )
I
inc. Neo-Hookean model:
S(u1 , u2 ) = µs I; F(e
uk ) → F(uk+1 )
{A}s := 12 (A + AT )
Numerical scheme
The scheme
I
provides strong coupling on interface
I
semi-implicit
I
produces one linear system per time step
I
second order in time
Numerical scheme
The scheme
I
provides strong coupling on interface
I
semi-implicit
I
produces one linear system per time step
I
second order in time
unconditionally stable (no CFL restriction), proved with
assumptions:
I
I
I
I
I
1st order in time
St. Venant–Kirchhoff inc./comp. (experiment: Neo-Hookean inc./comp.)
extension of u to Ωf guarantees Jk > 0
∆t is not large
A.Lozovskiy, M.Olshanskii, V.Salamatova, Yu.Vassilevski. An unconditionally stable
semi-implicit FSI finite element method. Comput.Methods Appl.Mech.Engrg., 297,
2015
2D validation: FSI3 2D benchmark problem
S. Turek and J. Hron. Proposal for numerical benchmarking of fluid-structure
interaction between an elastic object and laminar incompressible flow. In:
Fluid-structure interaction, Springer Berlin Heidelberg, 371–385, 2006.
I
fluid: 2D Navier-Stokes
I
stick: Saint Venant-Kirchoff
constitutive relation
I
inflow: parabolic profile
I
outflow: “do-nothing”
I
rigid walls: Dirichlet
2D validation: FSI3 2D benchmark problem
S. Turek and J. Hron. Proposal for numerical benchmarking of fluid-structure
interaction between an elastic object and laminar incompressible flow. In:
Fluid-structure interaction, Springer Berlin Heidelberg, 371–385, 2006.
0
×10-3
0.04
Y-displacement
X-displacement
-1
-2
-3
-4
-5
0.02
0
-0.02
-6
-0.04
7
I
7.2
7.4
7.6
Time
7.8
8
7
7.2
7.4
7.6
Time
7.8
8
Fortran open source software Ani2D
Advanced numerical instruments 2D, http://sf.net/p/ani2d/
2D validation: FSI3 2D benchmark problem
S. Turek and J. Hron. Proposal for numerical benchmarking of fluid-structure
interaction between an elastic object and laminar incompressible flow. In:
Fluid-structure interaction, Springer Berlin Heidelberg, 371–385, 2006.
0
×10-3
0.04
Y-displacement
X-displacement
-1
-2
-3
-4
-5
0.02
0
-0.02
-6
-0.04
7
7.2
7.4
7.6
Time
7.8
8
7
7.2
7.4
7.6
Time
7.8
8
I
Fortran open source software Ani2D
Advanced numerical instruments 2D, http://sf.net/p/ani2d/
I
Displacement in fluid equation: Laplace → mesh tangling,
heterogeneous elastisity (more stiff close-to-stick) → OK
2D validation: FSI3 2D benchmark problem
S. Turek and J. Hron. Proposal for numerical benchmarking of fluid-structure
interaction between an elastic object and laminar incompressible flow. In:
Fluid-structure interaction, Springer Berlin Heidelberg, 371–385, 2006.
0
×10-3
0.04
Y-displacement
X-displacement
-1
-2
-3
-4
-5
0.02
0
-0.02
-6
-0.04
7
7.2
7.4
7.6
Time
7.8
8
7
7.2
7.4
7.6
Time
7.8
8
I
Fortran open source software Ani2D
Advanced numerical instruments 2D, http://sf.net/p/ani2d/
I
Displacement in fluid equation: Laplace → mesh tangling,
heterogeneous elastisity (more stiff close-to-stick) → OK
I
Simulations are fast, stable, reproduce
I displacements
I drag, lift
I Strouhal numbers
3D benchmark: unsteady flow around silicon filament
A. Hessenthaler et al. Experiment for validation of fluid-structure interaction models and
algorithms. Int.J.Numer.Meth.Biomed.Engng.,2016.
Incompressible fluid flow in a moving domain
Navier-Stokes equations in reference domain Ωf
Let ξ mapping Ωf to Ωf (t), F = ∇ξ = I + ∇u, J = det(F) be given
Incompressible fluid flow in a moving domain
Navier-Stokes equations in reference domain Ωf
Let ξ mapping Ωf to Ωf (t), F = ∇ξ = I + ∇u, J = det(F) be given
Dynamic equations
∂v
∂u
−1
−T
−1
= (Jρf ) div (Jσ f F ) − ∇v F
v−
∂t
∂t
in Ωf
Incompressible fluid flow in a moving domain
Navier-Stokes equations in reference domain Ωf
Let ξ mapping Ωf to Ωf (t), F = ∇ξ = I + ∇u, J = det(F) be given
Dynamic equations
∂v
∂u
−1
−T
−1
= (Jρf ) div (Jσ f F ) − ∇v F
v−
∂t
∂t
in Ωf
Fluid incompressibility
div (JF−1 v) = 0
in Ωf
or
J∇v : F−T = 0
in Ωf
Incompressible fluid flow in a moving domain
Navier-Stokes equations in reference domain Ωf
Let ξ mapping Ωf to Ωf (t), F = ∇ξ = I + ∇u, J = det(F) be given
Dynamic equations
∂v
∂u
−1
−T
−1
= (Jρf ) div (Jσ f F ) − ∇v F
v−
∂t
∂t
in Ωf
Fluid incompressibility
div (JF−1 v) = 0
in Ωf
or
J∇v : F−T = 0
in Ωf
Constitutive relation for the fluid stress tensor
σ f = −pf I + µf ((∇v)F−1 + F−T (∇v)T )
in Ωf
Incompressible fluid flow in a moving domain
Navier-Stokes equations in reference domain Ωf
Let ξ mapping Ωf to Ωf (t), F = ∇ξ = I + ∇u, J = det(F) be given
Dynamic equations
∂v
∂u
−1
−T
−1
= (Jρf ) div (Jσ f F ) − ∇v F
v−
∂t
∂t
in Ωf
Fluid incompressibility
div (JF−1 v) = 0
in Ωf
or
J∇v : F−T = 0
in Ωf
Constitutive relation for the fluid stress tensor
σ f = −pf I + µf ((∇v)F−1 + F−T (∇v)T )
in Ωf
Mapping ξ does not define material trajectories → quasi-Lagrangian
formulation
Finite element scheme
Find {vhk , phk } ∈ Vh × Qh satisfying b.c.
(”do nothing” σF−T n = 0 or no-penetration no-slip vk = (ξk − ξk−1 )/∆t)
Finite element scheme
Find {vhk , phk } ∈ Vh × Qh satisfying b.c.
(”do nothing” σF−T n = 0 or no-penetration no-slip vk = (ξk − ξk−1 )/∆t)
Z
Ωf
Z
Ωf
Z
Ωf
vhk − vhk−1
· ψ dx +
Jk
∆t
Z
k −T
Jk ph Fk : ∇ψ dx +
Z
Ωf
Ωf
Jk ∇vhk F−1
k
vhk−1
ξ k − ξ k−1
−
∆t
Jk qF−T
: ∇vhk dx+
k
−T
k T −T
νJk (∇vhk F−1
+ F−T
k Fk
k (∇vh ) Fk ) : ∇ψ dx = 0
for all ψ and q from the appropriate FE spaces
!
· ψ dx−
Finite element scheme
The scheme
I
semi-implicit
I
produces one linear system per time step
I
first order in time (may be generalized to the second order)
Finite element scheme
The scheme
I
semi-implicit
I
produces one linear system per time step
I
first order in time (may be generalized to the second order)
I
unconditionally stable (no CFL restriction), proved with
assumptions:
supQ (kFkF + kF−1 kF ) ≤ CF
I
inf Q J ≥ cJ > 0,
I
LBB-stable pairs (e.g. P2 /P1 or P2 /P0 )
I
∆t is not large
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
Energy equality for the weak solution
Let ∂Ω(t) = ∂Ωns (t) and ξ t be given on ∂Ωns (t). Then there exists
v1 ∈ C 1 (Q)d , v1 = ξ t , div (JF−1 v1 ) = 0 [Miyakawa1982]
and we can decompose the solution v = w + v1 , w = 0 on ∂Ωns
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
Energy equality for the weak solution
Let ∂Ω(t) = ∂Ωns (t) and ξ t be given on ∂Ωns (t). Then there exists
v1 ∈ C 1 (Q)d , v1 = ξ t , div (JF−1 v1 ) = 0 [Miyakawa1982]
and we can decompose the solution v = w + v1 , w = 0 on ∂Ωns
Energy balance for w:
1
1 d
kJ 2 wk2
|2 dt {z
}
variation of
kinetic energy
1
+2νkJ 2 Dξ (w)k2
|
{z
}
+(J(∇v1 F−1 w), w)
|
{z
}
= (ef, w)
| {z }
energy of
viscous dissipation
intensification
due to b.c.
work of
ext. forces
Dξ (v) = 12 (∇vF−1 + F−T (∇v)T )
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
Stability estimate for the FE solution
Energy equality for wh = vh − v1,h :
1
1 12 k 2
2
kJk wh k − kJk−1
whk−1 k2
|2∆t
{z
}
2
1
2
(∆t) 21
k
+2ν Jk2 Dk (whk ) +
Jk−1 [wh ]t {z
} | 2
|
{z
}
variation of
kinetic energy
energy of
viscous dissipation
O(∆t) dissipative
term
+(Jk (∇v1k F−1
)whk , whk )
{zk
}
|
intensification
due to b.c.
=
(ef k , wk )
| {z h }
work of
ext. forces
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
Stability estimate for the FE solution
Stability estimate:
C1 k∇v1k k ≤ ν/2:
n
n
k=1
k=1
X
X
1 n 2
1
kwh kn + ν
∆tkDk (whk )k2k ≤ kw0 k20 + C
∆tkef k k2
2
2
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
Stability estimate for the FE solution
Stability estimate:
C1 k∇v1k k ≤ ν/2:
n
n
k=1
k=1
X
X
1 n 2
1
kwh kn + ν
∆tkDk (whk )k2k ≤ kw0 k20 + C
∆tkef k k2
2
2
C1 k∇v1k k > ν/2:
n
X
2C2
1 n 2
∆tkDk (whk )k2k ≤ e α T
kwh kn +ν
2
k=1
n
X
1
∆tkef k k2
kw0 k20 + C
2
!
k=1
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
,
Stability estimate for the FE solution
Stability estimate:
C1 k∇v1k k ≤ ν/2:
n
n
k=1
k=1
X
X
1 n 2
1
kwh kn + ν
∆tkDk (whk )k2k ≤ kw0 k20 + C
∆tkef k k2
2
2
C1 k∇v1k k > ν/2:
n
X
2C2
1 n 2
∆tkDk (whk )k2k ≤ e α T
kwh kn +ν
2
k=1
n
X
1
∆tkef k k2
kw0 k20 + C
2
!
k=1
if (1 − 2C2 ∆t) = α > 0
A.Danilov, A.Lozovskiy, M.Olshanskii, Yu.Vassilevski. A finite element method for the
Navier-Stokes equations in moving domain with application to hemodynamics of the
left ventricle. Russian J. Numer. Anal. Math. Modelling, 32, 2017
,
Application to the left ventricle hemodynamics
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
∂u
∂t
and
Application to the left ventricle hemodynamics
∂u
∂t
and
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
I
Computational meshes contain 14033 nodes, 69257 tetrahedra,
88150 edges (320k unknowns)
100 meshes with varying node positions
Application to the left ventricle hemodynamics
∂u
∂t
and
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
I
Computational meshes contain 14033 nodes, 69257 tetrahedra,
88150 edges (320k unknowns)
100 meshes with varying node positions
I
Time step between given frames 0.0127 s =⇒ interpolate
intermediate meshes with ∆t = 0.0127/20
Application to the left ventricle hemodynamics
∂u
∂t
and
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
I
Computational meshes contain 14033 nodes, 69257 tetrahedra,
88150 edges (320k unknowns)
100 meshes with varying node positions
I
Time step between given frames 0.0127 s =⇒ interpolate
intermediate meshes with ∆t = 0.0127/20
I
ρf = 103 kg/m , µf = 4 · 10−2 Pa ·s
3
(= 10µblood )
Application to the left ventricle hemodynamics
∂u
∂t
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
I
Computational meshes contain 14033 nodes, 69257 tetrahedra,
88150 edges (320k unknowns)
100 meshes with varying node positions
I
Time step between given frames 0.0127 s =⇒ interpolate
intermediate meshes with ∆t = 0.0127/20
I
ρf = 103 kg/m , µf = 4 · 10−2 Pa ·s
I
Large volume rate =⇒ extremely high influx velocities at the
diastole phase
3
(= 10µblood )
110
100
Volume (ml)
90
80
70
60
50
40
30
20
0
200
400
600
800
Time (ms)
and
1000
1200
1400
Application to the left ventricle hemodynamics
∂u
∂t
I
Boundary conditions: no-penetration no-slip v =
”do-nothing”
I
Computational meshes contain 14033 nodes, 69257 tetrahedra,
88150 edges (320k unknowns)
100 meshes with varying node positions
I
Time step between given frames 0.0127 s =⇒ interpolate
intermediate meshes with ∆t = 0.0127/20
I
ρf = 103 kg/m , µf = 4 · 10−2 Pa ·s
I
Large volume rate =⇒ extremely high influx velocities at the
diastole phase
(= 10µblood )
=⇒ need time smoothing for the wall displacements
110
100
90
Volume (ml)
I
3
and
80
70
60
A
B
C
D
E
original
50
40
30
20
0
200
400
600
800
Time (ms)
1000
1200
1400
Application to the left ventricle hemodynamics
Conclusions
I
We proposed unconditionally stable semi-implicit FE schemes for
FSI problem and NS eqs in moving domain
Conclusions
I
We proposed unconditionally stable semi-implicit FE schemes for
FSI problem and NS eqs in moving domain
I
The FSI scheme can incorporate diverse elasticity models
Conclusions
I
We proposed unconditionally stable semi-implicit FE schemes for
FSI problem and NS eqs in moving domain
I
The FSI scheme can incorporate diverse elasticity models
I
Only one linear system is solved per time step
Conclusions
I
We proposed unconditionally stable semi-implicit FE schemes for
FSI problem and NS eqs in moving domain
I
The FSI scheme can incorporate diverse elasticity models
I
Only one linear system is solved per time step
I
Ventricle hemodynamics: the NS scheme suffers from convection
instability due to large inflows
Conclusions
I
We proposed unconditionally stable semi-implicit FE schemes for
FSI problem and NS eqs in moving domain
I
The FSI scheme can incorporate diverse elasticity models
I
Only one linear system is solved per time step
I
Ventricle hemodynamics: the NS scheme suffers from convection
instability due to large inflows
I
We work on its stabilization