Towards an Integrated NEM-Model Towards an Integrated NEM-Model Reduction Lagrangian Strategy in Reduction Lagrangian Strategy in

Fluid Structure Interaction Fluid Structure Interaction

F. Chinesta, D. Ryckelynck, F. Chinesta, D. Ryckelynck,

E. Cueto, D. GonzalezE. Cueto, D. Gonzalez

P. VillonP. Villon

LMSP UMR CNRS-ENSAM Paris (France)LMSP UMR CNRS-ENSAM Paris (France)I3A Universidad de Zaragoza (Spain)I3A Universidad de Zaragoza (Spain)

Laboratoire Roberval, UTC Compiegne (France)Laboratoire Roberval, UTC Compiegne (France)

hh

1

1

km Hm

k

H

h uh

Cuu

When the element is too distorted: When the element is too distorted:

0 , h mH

h uu

The Natural Element MethodThe Natural Element Method

Ix

Delaunay triangulation and Voronoï DiagramDelaunay triangulation and Voronoï Diagram

Empty circumcircle Empty circumcircle criterioncriterion

Properties:Properties: Linear interpolation on the boundary Linear interpolation on the boundary ((-NEM / C-NEM)-NEM / C-NEM)

Linear consistencyLinear consistency

Partition of UnityPartition of Unity

Better accuracy than linear FEMBetter accuracy than linear FEM Meshless character: no sensibility to the distorsion Meshless character: no sensibility to the distorsion

of the of the Delaunay Delaunay triangles triangles The shape functions are CThe shape functions are C1 1 everywhere except at everywhere except at

the nodes positions.the nodes positions. If one proceeds in an updated Lagrangian If one proceeds in an updated Lagrangian

framework and the internal variables are defined at framework and the internal variables are defined at the nodes, remeshing, stabilization and field the nodes, remeshing, stabilization and field projections will be no more required.projections will be no more required.

i

ii xxxN )(

iiiii bNaNu i

i xN 1)(

Generalized NewtonianGeneralized Newtonian FlowsFlows

EquationTranferHeat

dpvDiv

dDDTDIdp

t

t

f

f

0

0:),(2

)(

*

)(

*

)(

1

)(

1

)(

1)(

)()(

xNi

i

in

xNi

ii

Sibsoni

n

n

pxN

xp

vxNxv

LBB? No, but no LBB? No, but no locking !locking !The LBB problematic will be addressed later The LBB problematic will be addressed later

Power lawPower law

iiiii bNaNu

Vicoplasticity with Vicoplasticity with thermal couplingthermal coupling

Themo-Elasto-Plasticity in Large Transformations Themo-Elasto-Plasticity in Large Transformations involving localization and refinementsinvolving localization and refinements

Z-ZZ-Z

Interface TrackingInterface Tracking

1.0t 7.0t4.0t

1t

Vicsoelastic and two-phases flowsVicsoelastic and two-phases flows

DD

YY

NN

AA

MM

II

CC

SS

NEM-Hermite or NEM-Bubbles: NEM-Hermite or NEM-Bubbles:

A proper way to verify general LBB A proper way to verify general LBB conditions in mixed formulations.conditions in mixed formulations.

Hermite approximationHermite approximation

++

NEM-Hermite: Mixed formulationsNEM-Hermite: Mixed formulations

Diffuse -MLSDiffuse -MLS NEM shape function

22 ,,,,,1 yxyxyxp

NEM-Bubbles: Mixed formulationsNEM-Bubbles: Mixed formulations

b1 - NEM b2 - NEM

NEM or Bubble-NEM shape function

b1 – NEM+

xyyxp ,,,1

b2 – NEM+

22 ,,,,,1 yxyxyxp

About Model ReductionAbout Model Reduction

« To represent a linear function « To represent a linear function 1000000 degrees of freedom 1000000 degrees of freedom

seem to be too much !! »seem to be too much !! »

txu , ),( pi txu

ComputeCompute the best representation of the best representation of

x xu p

Ni

i

i

Pp

p

Ni

i

ip

i

x

xux

1

2

1

2

1

max

1u

2u

3u

4u

,0 ,0 ,, 11nn11 NNnn

ni

i

p

i

pi

p BU1

1 pp FUA

11 pppp FBAFUA 1 pTpT FBBAB

NxNNxN nxnnxnnx1nx1Nx1Nx1 nx1nx1

The Karhunen-Loève DecompositionThe Karhunen-Loève Decomposition

0

)(

1)0,(

10300

,1

,0

2

2

tx

tx

x

T

tqx

T

txT

, , x, tx

T

t

T

A Numerical ExampleA Numerical Example

)(tq

t

00 1010 3030

11

*

2

2**

T

dx

TTd

t

TT

01.0

0*1

0

*1

02

2*

dx

x

T

x

T

x

TTdx

x

TT

Taking into account the boundary condition at x=1Taking into account the boundary condition at x=1

0 **

0

**

0

*1

02

2* Tdx

x

T

x

TqTdx

x

T

x

T

x

TTdx

x

TT

xx

*

2

2**

T

dx

TTd

t

TT

*

0

**

* )(

T

tqTdx

T

x

Td

t

TT

x

iii-1i-1 i+1i+1 NN11

hh

)1(

101 ;1

1

ihx

NN

h

i

QTMTKM nnn 1

QTMTKM nnn 1

,, 2010 tt TT

st 1.0

tN

tN

tN

ttt

ttt

uuu

uuu

uuu

Q

3002010

3002

202

102

3001

201

101

TQQk

k

, ... , 11 NN

1

84

3

2

1

10

15.0

15.1

1790

)(

)(

)(

)(

03

2

1

xT

x

x

x

k

)(

)(

)(

)(

03

2

1

xT

x

x

x

NNNN xxxxT

xxxxT

xxxxT

B

3210

23222120

13121110

BT

Nx1Nx1

4x14x1

More than a significant More than a significant reduction !!reduction !!

QBBMBBKMB nTnTnT 1

44 !!fo d ...4

Solving « a similar » problemSolving « a similar » problem)(tq

t

00 1010 2020

11

3030

with the reduced with the reduced order approximation order approximation basis computed from basis computed from

the solution of the the solution of the previous problemprevious problem

)(tq

t

00 1010 3030

11

NNNN xxxxT

xxxxT

xxxxT

B

3210

23222120

13121110

QTMTKM nnn 1

QBBMBBKMB nTnTnT 1 dof 4

dof 101

Evaluating accuracy and Evaluating accuracy and enriching the approximation enriching the approximation

basisbasis)(tq

t

00 1010 2020

11

3030

-2-2

QTMTKM nnn 1

QTMTKM nnn 1

QBBMBBKMB nTnTnT 1 dof 4

dof 101

QBBMBBKMB nTnTnT 1

QBMBKMR nnn 1 5106.2 R

R

RR

nT 1nT

)(

)(

)(

3210

223222120

113121110

NNNNN xRxxxxT

xRxxxxT

xRxxxxT

B

QBBMBBKMB nTnTnT 1

QBMBKMR nnn 1* RR *

dof 5

nT 1nT

Enrichment Based on the use of the Enrichment Based on the use of the Krylov’s Subspaces: an “a priori” StrategyKrylov’s Subspaces: an “a priori” Strategy

controlt

IFIF R IFIF R continuecontinue

1)0()0(1 nn BBKM MR

1)0( )0()0(1 )0( nTnTBBBKM MB

RBB ,)0()1( 1.000.000 dof 1.000.000 dof ~10 rdof~10 rdof

PerpectivesPerpectives

Reduction and LBB ???Reduction and LBB ???