Post on 07-Jul-2020
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A Monolithic ALE-FEM Technique for Numerical Benchmarking and
Optimization of
Fluid-Structure Interaction Problems
M. Razzaq, S. Turek,
Institute of Applied Mathematics, LS III, TU Dortmund
http://www.featflow.de
M. Razzaq | ALE-FEM for FSI
Recent Developments in Fluid Mechanics August 3-5, 2010
Isamabad, Pakistan
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Aim: Featflow with optimization
M. Razzaq | ALE-FEM for FSI
FEM-Multigrid for FEM-Multigrid for Multiphase (FSI) ProblemsMultiphase (FSI) ProblemsWith optimizationWith optimization
Aim:Aim:FSI-modelFSI-modelBenchmark Benchmark OptimizationOptimizationExtensions Extensions
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Fluid Structure Interaction (FSI)
M. Razzaq | ALE-FEM for FSI
large deformation of a structure in internal/external flow
aeorelasticity
bioengineering
polymer, food, paper processing
. . .
physical models
viscous fluid flow
elastic body under large
deformations
interaction between the two parts
numerical tasks involved space and time discretization nonlinear system large linear systems
testing and validation accuracy, efficiency, robustnes benchmarking
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Problem description
M. Razzaq | ALE-FEM for FSI
Structure part Fluid part
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Governing Equations
M. Razzaq | ALE-FEM for FSI
structure part fluid part
interface conditions
( ) sTss
fJt
v Ω+=∂∂ − in F div σ
( ) 1Fdet = sΩin
0 =su
0 =nsσ 3on Γ
( ) f
t
fff
f
fvvt
v Ω+=∇+∂
∂in div σ
0 div =fv
0 vv f =
0 or =nfσ
sf vv = 0on tΓ
0on tΓnn sf σσ =
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Arbitrary Lagrange-Euler Formulation
M. Razzaq | ALE-FEM for FSI
Lagrangian description: :
[ ] tT Ω×Ω 0, :χ , t
v∂∂= χ , F
X∂∂= χ
[ ] tR RTR 0, : ×ζ ttR Ω⊂ [ ], 0, Tt∈∀ tv R
R ∂∂= ζ
X
RR ∂
∂= ζ F
RRJ F det =
( ) 0 =∫ ⋅−+∫∂∂
∂ t
t
t RRR
R
danvvdvt
ρρ
( ) ( )( ) 0 F div =−+∂∂ −T
RRRR vvJJt
ρρ
Eulerian description
vvJJ RRRR , F, F ===⇒= χζ
( ) 0 =∂∂
Jt
ρ
0 ,1 I, F Id ===⇒= RRRR vJζ
( ) 0 div =+∂∂
vt
ρρ
Fdet=J
Mesh nodes are fixedFluid mechanicsLarge distortion handleNumerical instable for convective term
Mesh nodes are not fixedStructure mechanicsMesh tangleexpensive
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Coupling strategies
M. Razzaq | ALE-FEM for FSI
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Constitutive equations
M. Razzaq | ALE-FEM for FSI
incompressible Newtonian fluid
hyperelastic material, incompressible
where and
or St. Venant--Kirchhoff material, compressible
where
D2 I νσ +−= pf
,FF
2F I Tf p∂Ψ∂+−=σ 1 det =F
( ) ( ) Hook-Neo3 I F C −=Ψ α
( ) ( ) ( ) ( ) canisotropi Rivlin -Mooney 1 Fe 3 I 3 I F2
3C2C1 +−+−+−=Ψ αααTFF C = trC, IC = ( )( )22
C trC trC2
1 I −=
( )( ) Tsss
JFE ItrEF
1 µλσ +=
( )I FF2
1 E −= T
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Discretization in space and time
M. Razzaq | ALE-FEM for FSI
Discretization in space: FEM
Discretization in time: Crank-Nicholson, BE, FS, schemes
discPQQ 122 //
( )[ ] ( )[ ] ,on 0 ,Q | , 12
2
2 Γ=∈∀∈Ω∈= hhThhhh uTTuCuU τ
( )[ ] ( )[ ] ,on 0 ,Q | , 12
2
2 Γ=∈∀∈Ω∈= hhThhhh vTTvCvV τ
( ) ( ) . | , 1
2
hThhhh TTPpLpP τ∈∀∈Ω∈=
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Discrete nonlinear system
M. Razzaq | ALE-FEM for FSI
( ) 0 x =R
( ) ( )n
h
n
hh
f
h
s
h vuuLvMk
Mu ,rhs 2
=+−
( ) ( ) ( ) ( ) ( )( ) ( )n
h
n
h
n
hhh
f
h
s
hhhhh
sf pvukBpvSuSk
uvNuvNk
vMM , ,rhs2
,2
1,
2 21 =−+++++ β
( ) 1=+ h
Tf
h vBuC
( ) ( ) ( )
∂∂+
∂+∂
+∂∂++
∂∂+
∂++∂
−
=∂∂
0
22
1
2
1
02
2
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
XX
R
T
T
β
⇓
( ) hhhhhh PVUpvux , , ××∈=
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Discrete nonlinear system
M. Razzaq | ALE-FEM for FSI
Typical discrete saddle-point problem
( ) ( )n
h
n
hh
f
h
s
h vuuLvMk
Mu ,rhs 2
=+−
( ) ( ) ( ) ( ) ( )( ) ( )n
h
n
h
n
hhh
f
h
s
hhhhh
sf pvukBpvSuSk
uvNuvNk
vMM , ,rhs2
,2
1,
2 21 =−+++++ β
( ) 1=+ h
Tf
h vBuC
⇓
=
pTfv
Tsu
vvvu
uvuu
fp
v
u
BcBc
kBSS
SS
u
u
f
f
0
0
( ) 0 x =R ( ) hhhhhh PVUpvux , , ××∈=
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Challenges Nonlinear Solvers
M. Razzaq | ALE-FEM for FSI
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 v,u, x ,0 x ==
( ) ( )n1n
nn1n xx
x x x R
R−
+
∂∂+= ω
( ) ( ) ( )ε
εε2
e x e x
xnnn
jiji
ij
RR
x
R −−+≈
∂∂
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Multigrid Solver
M. Razzaq | ALE-FEM for FSI
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
Very accurate, flexible and highly efficient FSI solver
( FSI Benchmarks)
∑
−
=
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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Applications
M. Razzaq | ALE-FEM for FSI
1st step: Identification of appropriate FSI settings for numerical benchmarking and calculate
If done 2nd step
2nd step: Extension to FSI-Optimisation benchmark settings
• Data files are available onhttp://featflow.de/beta/en/benchmarks.html
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Numerical FSI
M. Razzaq | ALE-FEM for FSI
Realistic materials
• incompressible Newtonian fluid, laminar flow regime• elastic solid, large deformations
Comparative evaluation• setup with periodical oscillations • non-graphically based quantities
Computable configurations• laminar flow• reasonable aspect ratios • simple geometry (2D)
Mainly based on validated CFD benchmarks, but also close to experimental set-up
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Material parameters
M. Razzaq | ALE-FEM for FSI
solid fluid
density density
Poisson ratio kinematic viscosity shear modulus
Parameter polybutadiene & glycerine polypropylene & glycerine
0.91 incompressible 0.50 0.53
1.1 compressible 0.42 317
1.26
1.13
1.26
1.13
Parameter FSI1 FSI2 FSI3
1 0.4 0.5
1 0.4 0.5
10.42.0
11
11
11
0.2 1 2
Parameter FSI1 FSI2 FSI3
10.4
10.4
10.4
20
0.2
100
1
200
2
sρ fρsν fνsµ
]kg/ms10[ 26sµsν
]kg/m10[ 33sρ
]kg/m10[ 33fρ]/m10[ 23 sf −ν
]kg/ms10[ 26sµ
sν]kg/m10[ 33sρ
]kg/m10[ 33fρ]/m10[ 23 sf −ν
]m/[ sU
f
s
ρρβ =
2
E Ae
Uf
s
ρ=
]m/s[U
f
dU
ν Re =
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Quantities of interest
M. Razzaq | ALE-FEM for FSI
The position of the end of the structure Pressure difference between the points A(t) and B
Forces exerted by the fluid on the whole body, i.e. lift and drag forces acting on the cylinder and the structure together
Frequency and maximum amplitude Compare results for one full period and 3 different levels of spatial
discretization h and 3 time step sizes
( ) ( ) ( )( )tytxtA , =
( ) ( ) ( ) ( )tptptp tABAB −=∆
( ) ∫ ∫=∫ +=∫=2 01
, |
S S
Sf
S
f
SLD dSnndSndSdSnFF σσσσ
t∆
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FSI1: steady, small deformations
M. Razzaq | ALE-FEM for FSI
Parameter FSI1 FSI2 FSI3
10.40.5
10.40.5
10.42.0
11
11
11
0.2 1 2
Parameter FSI1 FSI2 FSI3
10.4
10.4
10.4
200.2
1001
2002
FSI1 2.270493 8.208773 14.2942 0.76374
ux of A [ m] uy of A [ m] drag lift410 −×510 −×
]kg/m10[ 33sρsν
]kg/ms10[ 26sµ
]kg/m10[ 33sρ]s/m10[ 23−sν
]s/m[U
f
s
ρρβ =
sν2
E Ae
Uf
s
ρ=
f
dU
ν Re =
]s/m[U
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FSI2: large deformations
M. Razzaq | ALE-FEM for FSI
Test ux of A [ m] uy of A [ m] drag lift
FSI2
310 −× 310 −×]86.3[70.1285.14 ±− ]93.1[7.8130.1 ± ]86.3[65.7706.215 ± ]93.1[8.23761.0 ±
Mean ± amplitude[frequency]
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FSI3: large deformations
M. Razzaq | ALE-FEM for FSI
Test ux of A [ m] ux of A [ m] drag lift
FSI3
310 −×310 −×]93.10[72.288.2 ±− ]46.5[99.3447.1 ± ]93.10[74.275.460 ± ]46.5[91.15350.2 ±
Mean ± amplitude[frequency]
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FSI Optimization
M. Razzaq | ALE-FEM for FSI
The main design aims could be
I) Drag/Lift minimization
II) Minimal pressure loss
III) Minimal nonstationary oscillations To reach these aims, we might allow
1. Boundary control of inflow section
2. Change of geometry: elastic channel walls or length/thickness of elastic beam
3. Optimal control of volume forces Optimal control of nonstationary flow might be hard for the starting Results for the moment are combination of I)-III) with 1)-3).
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Nelder Mead algorithm
M. Razzaq | ALE-FEM for FSI
B= Best
G=Good
W=Worst
Mid, reflect point Reflect, extend point
Contraction pointshrink
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FSI Optimization
M. Razzaq | ALE-FEM for FSI
controlled flow
Aim:
w.r.t V1, V2.
V1 velocity from top
V2 velocity from below
ux of A [ m] uy of A [ m] drag lift310 −×310 −×
FSI1 0.0227 0.8209 14.295 0.7638
h=0.04
( )22 minimize Vlift α+
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FSI Opt1
M. Razzaq | ALE-FEM for FSI
TESTS for FSI 1 (Boundary control)
level 1 level 2
Iter steps
extreme point drag Lift
1e0 57 (3.74e-1,3.88e-1) 1.5471e+01 8.1904e-1
1e-2 60 (1.04e0,1.06e0) 1.5474e+01 2.2684e-2
1e-4 73 (1.06e0,1.08e0) 1.5474e+01 2.3092e-4
1e-6 81 (1.06e0,1.08e0) 1.5474e+01 2.3096e-6
Iter steps
extreme point drag Lift
59 (3.66e-1,3.79e-1) 1.5550e+01 7.8497e-1
59 (1.02e0,1.04e0) 1.5553e+01 2.1755e-2
71 (1.04e0,1.05e0) 1.5553e+01 2.2147e-4
86 (1.04e0,1.05e0) 1.5553e+01 2.2151e-6
α
Level 1 Level 2
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Summary-Outlook
M. Razzaq | ALE-FEM for FSI
Numerical benchmarks (tests, comparisons) Optimization test case definition and results
Biomedical application
like
Hemodynamics(aneurysm)