Page 1 Page 1 Page 1 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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Page 1Page 1Page 1
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
Page 2Page 2
Aim: Featflow with optimization
M. Razzaq | ALE-FEM for FSI
FEM-Multigrid for FEM-Multigrid for Multiphase (FSI) ProblemsMultiphase (FSI) ProblemsWith optimizationWith optimization
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
Page 4Page 4
Problem description
M. Razzaq | ALE-FEM for FSI
Structure part Fluid part
Page 5Page 5
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 =
Page 6Page 6
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
Page 7Page 7
Coupling strategies
M. Razzaq | ALE-FEM for FSI
Page 8Page 8
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
Page 9Page 9
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∈=
Page 10Page 10
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 , , ××∈=
Page 11Page 11
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 , , ××∈=
Page 12Page 12
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 −−+≈
∂∂
Page 13Page 13
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
Page 14Page 14
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
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).
Page 22Page 22
Nelder Mead algorithm
M. Razzaq | ALE-FEM for FSI
B= Best
G=Good
W=Worst
Mid, reflect point Reflect, extend point
Contraction pointshrink
Page 23Page 23
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+
Page 24Page 24
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
Page 25Page 25
Summary-Outlook
M. Razzaq | ALE-FEM for FSI
Numerical benchmarks (tests, comparisons) Optimization test case definition and results