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 −−+≈

∂

∂

77

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

88

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

99

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

12

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

13

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

14

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∇≈

15

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

16

(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

17

Dynamic Adaptation: 2D Sedimentation

18

3D Examples

19

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