Transcript
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
2
Multiphase FSI Problems: Elastic Solids
Liquid – Rigid Solid
– Particulate Flow
– Robofish
Liquid – Elastic Solid
– Biomechanics
– Medical applications
– Aeroelasticity
3
Multiphase FSI Problems: Rigid Solids
Liquid – Rigid Solid
– Particulate Flow
– Robofish
Liquid – Elastic Solid
– Biomechanics
– Medical applications
– Aeroelasticity
4
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
5
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
66
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 ××∈=
1010
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 ××∈=
1111
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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3D Examples
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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
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