A Monolithic ALE-FEM Technique for Numerical Benchmarking ...web.lums.edu.pk/~mudassar.razzaq/presentations/... · Page 17 Quantities of interest M. Razzaq | ALE-FEM for FSI The position

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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


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)


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


Structure part Fluid part

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Governing Equations


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


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


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Constitutive equations


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


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


( ) 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


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


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


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


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

http://featflow.de/beta/en/benchmarks.html

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Numerical 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


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


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


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



10.40.5

10.40.5

10.42.0

11

11

11

0.2 1 2


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


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


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


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


B= Best

G=Good

W=Worst

Mid, reflect point Reflect, extend point

Contraction pointshrink

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FSI Optimization


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


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


Numerical benchmarks (tests, comparisons) Optimization test case definition and results

Biomedical application

like

Hemodynamics(aneurysm)

Documents

A Monolithic ALE-FEM Technique for Numerical Benchmarking ...web.lums.edu.pk/~mudassar.razzaq/presentations/... · Page 17 Quantities of interest M. Razzaq | ALE-FEM for FSI The position