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Universität DortmundFakultät für Mathematik
IAM
technische universität
dortmund
FluidFluid--Structure Interaction Problems:Structure Interaction Problems:
FEM FEM MultigridMultigrid Techniques and BenchmarkingTechniques and Benchmarking
S. Turek with support by the FEAST Group
Institut für Angewandte Mathematik, TU Dortmundhttp://www.mathematik.uni-dortmund.de/LS3
http://www.featflow.de
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Multiphase FSI Problems: Elastic Solids
Liquid – Rigid Solid
– Particulate Flow
– Robofish
Liquid – Elastic Solid
– Biomechanics
– Medical applications
– Aeroelasticity
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Multiphase FSI Problems: Rigid Solids
Liquid – Rigid Solid
– Particulate Flow
– Robofish
Liquid – Elastic Solid
– Biomechanics
– Medical applications
– Aeroelasticity
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Required: Special Numerics for FSI
Computational mesh (can be) independent of ‘internal objects’
Special FEM Techniques
Space-Time AdaptivityImplicit Approaches
Stabilization for high Re, Pe, We,… Numbers
Multigrid Solvers
GPU Computing
Grid Deformation Methods Fictitious Boundary Methods
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Challenges for Numerics
5
Special FEM discretization techniques to handle thefollowing challenging points Stable FE spaces for velocity and pressure fields, and velocity and
extra-stress fields
Q2/P1/Q2, Q1(nc)/P0/Q1(nc) (new: Q2(nc)/P1/Q2(nc))
Special treatment of the „convective“ terms
edge-oriented/interior penalty EO-FEM, TVD/FCT
Special treatment of the „reactive" terms in viscoelastic problems
LCR + EO-FEM
Special (nonlinear) solvers to deal with different sources
of nonlinearity nonlinear operators Newton method via divided differences
stiff coupling of equations monolithic/operator splitting multigrid
complex geometries and meshes
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Nonlinear Solvers
Solve for the residual of the nonlinear system algebraic equations
Use Newton method with damping results in iterations of the form
Continuous Newton: on variational level (before discretization)
The continuous Frechet operator can be analytically calculated
Inexact Newton: on matrix level (after discretization)
The Jacobian matrix is approximated using finite differences as
( ) ( )pR , , u, x ,0 x σΘ==
( ) ( )n
1n
nn1n xx
x x x R
R−
+
∂
∂+= ω
( ) ( ) ( )ε
εε
2
e x e x
xnnn
jiji
ij
RR
x
R −−+≈
∂
∂
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Multigrid Solvers
Standard geometric multigrid approach with full FEM grid transfer
Smoother: Local/Global MPSC
Local MPSC via Vanka-like smoother
Monolithic multigrid solver
Global MPSC
solve for an intermediate u (generalized momentum equation)
solve for p (pressure Poisson equation)
update of u and p
solve for (tracer equation)
solve for (constitutive equation)
Decoupled multigrid solver
[ ]
Tp
TT
l
l
l
l
l
l
l
l
l
JK
pp
h
|
u
1
|
1
1
1
1
Res
Res
Res
Res
u
u
+Σ+
Θ=
Θ Θ
−
∈
+
+
+
+
σ
τωσσ
Θ
σ
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1) Aspects of (Elastic) FSI Problems
incompressible Newtonian fluid (with nonlinear extensions)
hyperelastic material, incompressible
where and
or St. Venant-Kirchhoff material, compressible
where
D2 I νσ +−= pf
,FF
2F I Tfp
∂
Ψ∂+−=σ 1 det =F
( ) ( ) Hook-Neo3 I FC
−=Ψ α
( ) ( ) ( ) ( ) canisotropi Rivlin -Mooney 1 Fe 3 I 3 I F2
3C2C1+−+−+−=Ψ ααα
TFF C = trC, IC
= ( )( )22
CtrC trC
2
1 I −=
( )( ) Tsss
JFE ItrEF
1 µλσ +=
( )I FF2
1 E −= T
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Monolithic ALE-FEM Approach
( ) 0 x =R
( ) ( )n
h
n
hh
f
h
s
hvuuLvM
kMu ,rhs
2 =+−
( ) ( ) ( ) ( ) ( )( ) ( )n
h
n
h
n
hhh
f
h
s
hhhhh
sfpvukBpvSuS
kuvNuvN
kvMM , ,rhs
2,
2
1,
2
21=−+++++ β
( ) 1=+h
Tf
hvBuC
( ) ( ) ( )
∂
∂+
∂
+∂+
∂
∂++
∂
∂+
∂
++∂
−
=∂
∂
0
22
1
2
1
02
2
xx
2
121
Tf
h
h
fs
h
f
h
fs
h
hh
fs
sf
Bvu
BB
kBv
SNk
v
NMMp
u
Bk
u
SSN
Mk
Lk
M
R
T
T
β
⇓
( ) hhhhhh PVUpvu , , x ××∈=
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Typical discrete saddle-point problem
( ) ( )n
h
n
hh
f
h
s
hvuuLvM
kMu ,rhs
2 =+−
( ) ( ) ( ) ( ) ( )( ) ( )n
h
n
h
n
hhh
f
h
s
hhhhh
sfpvukBpvSuS
kuvNuvN
kvMM , ,rhs
2,
2
1,
2
21=−+++++ β
( ) 1=+h
Tf
hvBuC
⇓
=
p
T
fv
T
su
vvvu
uvuu
fp
v
u
BcBc
kBSS
SS
u
u
f
f
0
0
Monolithic ALE-FEM Approach
( ) 0 x =R ( ) hhhhhh PVUpvu , , x ××∈=
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Multigrid Solver for Q2/Q2/P1
standard geometric multigrid approach
smoother by local MPSC-Ansatz (Vanka-like smoother
full inverse of the local problems by LAPACK (39 x39 systems)
alternatives: simplified local problems (3x3 systems) or ILU(k)
combination with GMRES/BiCGStab methods possible
full (canonical) FEM prolongation, restriction by
Very accurate, flexible and highly efficient FSI solver
( FSI Benchmarks)
TP R =
∑
−
=
Ω
−
ΩΩ
ΩΩΩ
ΩΩ
+
+
+
iPatch
1
||
|||
||
1
1
1
def
def
0
0
l
p
l
v
l
u
T
fv
T
su
vvvu
uvuu
l
l
l
l
l
l
defBcBc
kBSS
SS
p
v
u
p
v
u
ii
iii
ii
ω
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2) Aspects of Particulate Flow
Fluid flow is modelled by the Navier-Stokes equations:
,fuut
u=⋅∇−
∇⋅+
∂
∂σρ 0=⋅∇ u ( ) ( ) ][,
TuupItX ∇+∇+−= µσ
Motion of particles is described by the Newton-Euler equations, i.e., the
translational velocities and angular velocities of the p-th particle satisfy:
( ) ,'gMFF
dt
dUM ppp
p
p ∆++= ( ).pppp
p
p TIdt
dI =×+ ωω
ω
,
and are the hydrodynamical forces and the torque at mass center
acting on the p-th particle and are the collision forcespF pT
'
pF
∫Γ Γ⋅−=p
ppp dnF ,σ ( ) ( )∫Γ Γ⋅×−−=p
pppp dnXXT σ
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No slip boundary conditions at interface between particles and fluid
i.e., for any , the velocity u(X) is defined by:
The position of the p-th particle and its angle are obtained
by integration of the kinematic equations:
pΓ
pX Γ∈
( ) ( )ppp XXUXu −×+= ω
pX pθ
,p
pU
dt
dX= p
p
dt
dω
θ=
Particle-Fluid Interaction
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Idea: ‘Replace the surface integral by a volume integral’
and use indicator functions ( )
+Fictitious Boundary Method on Generalized
Tensorproduct Meshes
Hydrodynamic forces and torque acting on the i-th particle
∫∂ Γ⋅−=iP
iii dnF ,σ ( ) ( )∫∂ Γ⋅×−−=iP
iiii dnXXT σ
How to Calculate the Forces?
∫∫ ΩΓΩ∇⋅−=Γ⋅−=
TpTpppp ddnF ασσ
ppn α∇≈
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Idea : construct transformation with
local mesh area
1. Compute monitor function and
3. Solve the ODE system
new grid points:
Grid deformation preserves the (local) logical structure of the grid
( )tx ,, ξφφ = f=∇φdet
f≈
( ) 1,0, Cftxf ∈>
( ) ,,1 Ω=∫Ω− dxtxf
( )( )
,,
1,
∂
∂−=∆
tfttv
ξξ
]1,0[∈∀ t
2. Solve ])1,0[( ∈t
0=∂
∂
Ω∂n
v
( ) ( )( ) ( )( )ttvttftt
,,,,, ξφξφξφ ∇=∂
∂
( )1,iix ξφ=
Grid Deformation Methods
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(Semi-explicit) Operator-Splitting Approach
→ Required: efficient calculation of hydrodynamic forces
→ Required: efficient treatment of (many) particle interaction
→ Required: efficient (dynamic) grid alignment
→ Required: fast (nonstationary) Navier-Stokes solver FEASTFLOW
1.
2.
4.
3.
Fluid velocity and pressure :
Calculate hydrodynamic forces:
Calculate velocity of particles:
Update position of particles:
The algorithm for consists of the following 5 substeps
5. Align new mesh
1+→ nntt
( ) ( )n
p
n
p
nn
f uBCpuNSE ,, 11 Ω=++
1+n
pF
( )11 ++ = n
p
n
p Fgu
( )11 ++ =Ω n
p
n
p uf
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Dynamic Adaptation: 2D Sedimentation
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3) Benchmarking of Multiphase CFD
0 0.5 1 1.5 2 2.5 3
0.9
0.92
0.94
0.96
0.98
1
1.02
TP2D
FreeLIFE
MooNMD
Comsol
Fluent
• Initiative “Rising Bubble”
quantitative validation and comparison of
multiphase codes
1.75 1.8 1.85 1.9 1.95 2 2.05
0.896
0.898
0.9
0.902
0.904
0.906
TP2D
FreeLIFE
MooNMD
Comsol
Fluent• Initiative “Elastic FSI”
quantitative validation and
comparison of monolithic vs.
decoupled approaches
• Initiative “Particulate Flow”
quantitative validation and
comparison with experimental
configurations