Frontières implicites et adaptation anisotrope de maillage

Thierry Coupez ICI

Ecole Centrale de Nantes

Luisa Silva, Hugues Digonnet, ICI - Ecole Centrale de Nantes (ECN)

PLAN

• Multiphase modelling: • Anisotropic adaptive meshing: edge error analysis

and metric by length distribution tensor • Implicit Boundaryand multiphase meshing • Anisotropic Finite Element Solver - stabilization • Application Examples

Multiphase flow

Incompressible Navier-Stokes:

¯®­

� !

0001

)(DD

Dsisi

H ))(1()())(())(1()())((

21

21

DKDKDKDUDUDU

HHHHHH

�� ��

+ mixture law :

� �� �

� �

( ( )) 2 ( ( )) ( ( ))

0

. ( )

d H v H v p H gdt

v

v U s gt

U D K D H W U D

D O D O D

­��� � �� °

°°� � ®°w° � � � ° w¯

( ) ( ) tanh( )H HHD D D H|

Incompressible Navier Stokes and Multiphase flow

Smoothing :

*L. Ville and T. Coupez, Convective re-initialization of local Level Set function for moving surfaces and interfaces of incompressible flow, online in IJNMF

Convected Level Set method

� �� � � �0

. ( )

0,

( )

u v U u s g ut

u t x u x

u u

O O

D

­w� � � ° w®

° ¯

^ `( ) ( , ) ,

, ( ) 0x d x xx x

DD

r * �:­°®* �: °̄

The phase function as a signed distance function

Convected level set* : no need of redistancing stage

Ville L., Silva L., Coupez T. Convected level set method for the numerical simulation of fluid buckling International Journal for Numerical Methods in Fluids 66, 3 (2011) Pages 324-344

Falling droplet (milk/air) benchmark:

Viscosity ratio : 10-6/10-3

Density ratio : 1/103 Surface tension: 0.05

Computational Multiphase Material Dynamics

L= 0.1 m V ~ 2.0 m/s Re ~ 20000

7

. (2 ( ))

0

p Ft

U U KH U G*

w­ � � ��� �� �°w®

°�� ¯

v v v v g

v

Navier-Stokes incompressible with surface tension: add a surface dirac function

Surface tension

. . ( )n G uN � �

¸¹·

¨©§ ExExu )(tanh)( D

)( n uGi &

� �� �

¸̧¹

·¨̈©

§�� ¦

*� ij

ijh

i XuGuG2

.minarg)(

Gradient recovery from the error estimate Hyperbolic tangent

, . . ΓF w F w kn wdG G V* *: *

� ! ³ ³surface force to volume by a dirac function

• Dirac = derivative of heavyside • Heavyside as the hyperobolic tangent of the level set • The normal as the recovered gradient of the level set • Curvature as the divergence of the recoverd gradient

Droplet splashing with surface tension and crown phenomenon

Meshing and Anisotropic adaptive

Meshing by local mesh modification • Cut and paste :

Mi+1 = Mi – E + E’ • E = node or edge neighbouring, E’ Star operator • General 2D,3D,…:

o Conform meh: Minimal volume theorem

9 9

Mesh derivation process Mesh generation by local modification and volume minimization

^ `

2 2

1 1

122

( ) ( )

(2)

( ), ( )

' ( )( )( )

: Lagrange or Clement interpolation operator 1H(u) Hessian:the second derivative2

h h h K

h h K KL LK

h

V v H v P K

u u C u u C H u x x x x

u

: :

� : �

§ ·� d �3 d � �¨ ¸

© ¹3

� {

¦³

Approximation error and Interpolation error relationship

1 1

1 1

12

12

ˆ ˆ( ( ), ( )) ( , )ˆ ˆ( , )

1

K K K K K Kt

K K K

K K K

H x x x x H A x A xA H A x x

A H M HKK

� �

� �

� �

�

Metric field based on the unit mesh theory and linked to Hessian:

• Céa lemma for elliptic problem, remains an open question for Navier Stokes • Conjecture: Interpolation error is dominant for diffusion dominant problem • viscous boundary layers are diffusion dominant flow region

^ `

2 2

0,

1 1

12

2

( ) ( )

(2)

( ), ( )

' ( )( )( )

: Lagrange or Clement interpolation operator 1H(u) Hessian:the second derivative2

K

h h h K

h h K KL LK

h

V v H v P K

u u C u u C H u x x x x

u

: :

� : �

§ ·� d �3 d � �¨ ¸¨ ¸

© ¹3

� {

¦

Approximation error Interpolation error and Metric field

iMiX

jX

Node based metric

1 1

1 1

112

12

( ( ), ( ))ˆ ˆ( , )

ˆ ˆ( , )

1

K K K

K K Kt

K K K

K K K

H x x x xH A x A xA H A x x

A H M HKK

� �

� �

�

� �

�

K

anisotropic iX

jX

ijX

12

1/2

,

( ) ( )det( ) /

( ) ( , )

d

ij ij

i j K

c K Vol K h

h h K X X

MM

�

¦

1ij ijA X

� �, 1ij t ij ij i ij ijM A A M X X �

,iX i�Nij j iX X X �

� �( ) ( )

( )

, 1

: ( )

i ij ij

j i j i

i ij ij

j i

M X X

M X X i�* �*

�*

� � *

¦ ¦

¦

ijA maps to a unit vector ijX

Averaging process: find M such as

ijA

Unit Mesh node based Metric

1

Length distribution tensor from length of edges

• Length distribution tensor

• Length distribution tensor and new mesh Metric

( ): ( )i ij ij

j iM X X i

�*

� � *¦

jXiX

ijX

T. Coupez, Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing, Journal of Computational Physics 230 2391-2405, 2010

New metric by rescaling the edges : • Edge vector transformation:

• New length distribution tensor an new

Metric

ijX

T. Coupez, Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing, Journal of Computational Physics 230 2391-2405, 2010

Edge based error estimation Error analysis along the edge (1d analysis)

15

� � 1 ii iG�

X U

.ij ij ij j iije G X G G G �

Gradient operator:

Edge error estimate ( )

i ij ij

j iU X

�*

¦U

Stretching coefficient

Multiphase Meshing • Multiphase: interface • Explicit boundary:

o Lagrangian and ALE framework o Surface tracking o Coupling: surface and interface meshing constraint o Adaptive meshing nightmare o Parallel meshing ?

• Implicit Boundary

o Eulerian framework and Level Set representation o Moving or fixed geometry o Automatic and implicit coupling o Coupled adaptivity, parallel meshing o Volume meshing only

o Advanced Solver

• Narrow level set

^ `

^ `

( ) ( , ) ,, ( ) 0

smooth truncated level set:

( , ) tanh( )

, ( , ) 0 ,

1( , ) ( , )

x d x xx x

u

x uHeavyside

H u

DD

DH D HH

H D H

H D H DH

r * �:

* �:

� * �: �

Immersed volume by anisotropic mesh adaptation • flow past a cylinder

))(1()())(())(1()())((

21

21

DKDKDKDUDUDU

HHHHHH

�� �� � �� �( ( )) 2 ( ( )) ( ( ))

0

dvH H v p H gdt

v

U D K D H W U D­ ��� � �� °®°�� ¯

( ) ( ) ( ) ( , )H Tanh x d xDD H DH

| *

Aerodynamic at low Mach Stabilised Finite element: VMS = implicit LES - Immersed volume

- Multiphase to Fluid Structure - Anisotropic adaptive meshing:

- Geometry recovery - Boundary layer - Monolithic FSI

Fluid Structure Interaction

• Immersed Volume method: Direct multiphase modeling

• Monolithic approach • VMS FE Solver : immersed stress method

T. Coupez, L. Silva, E. Hachem, Implicit Boundary and Adaptive Anisotropic Meshing, to appear in SEMA SIMAI Springer Series, 2014 E. Hachem, S. Feghali, R. Codina and T. Coupez, Anisotropic Adaptive Meshing and Monolithic Variational Multiscale Method for Fluid-Structure Interaction, Computer and Structures, http://dx.doi.org/10.1016/j.compstruc.2012.12.004 , 2013 (available online)

The weak formulation reads the following:

The strong formulation of the fluid solid system:

:won

)()()( uHus HDH

� � � �� �� �� � f

fs

H

HH

KDK

UDUDU

�

��

1

1

� � � �� � fs HH KDKDK �� 1

FSI : fluid-structure interaction

� � ( )sH vW W D K Hm �Augmented Lagrangian +Uzawa iterations:

Stabilised Finite Elements VMS framework

Incompressible Navier-Stokes problem

( ) 2 ( ) in0 in0

tu u u u p fuu on

U P Hw � �� � �� �� :�� :

w:

The instabilities in convection-dominated regime (?)

The velocity-pressure compatibility condition (?)

� �� �

( , )

( , );( , ) ( , )

( , );( , ) ( , )

h h h hFind and Q Q such that Q

A u u p p v q f v

A u u p p v q f v

� �� � � � u

� �

� �

u p v qhV V V

Variational MultiScale Approach:

Small scale

Large scale

1- Large scale equations

2- Small scale equations

� � � � � � � �� � � �

� �

� � � � � � � �� � � �

� �

( ) , ( ( ), , 2 : ,

( ), 0

( ) , ( ) ( ), , 2 : ,

( ), 0

h hh h h h h h h h h h h h h h

h h h h h

h hh h h h h h h h h h h h h

h h h

u u v u u u u v p p v u v f v vt

u u q q Q

u u v u u u u v p p v u v f v vt

u u q q

U U KH H

U U KH H

w �§ · � � �� � � � �� � � �¨ ¸w© ¹� � � � �

w �§ · � � �� � � � �� � � �¨ ¸w© ¹� � � �

V

V

h Q�

• Stabilized Finite element

� �,h

C h hK

u vW�

� �� ��¦T

30

� �,h

K h c hK

u u vW�

� ��¦T

R

Matrix equivalent form: Three identified additional contributions

For convection dominated problems

For pressure instabilities

Incompressible flows at high Reynolds number

Stabilized finite element method for NS equations

0

T Tvv vb vp h v

Tbv bb bp h b

h pvp bp

A A A u B

A A A u B

p BA A

§ · § ·§ ·¨ ¸ ¨ ¸¨ ¸¨ ¸ ¨ ¸¨ ¸ ¨ ¸ ¨ ¸¨ ¸¨ ¸¨ ¸ ¨ ¸¨ ¸© ¹ © ¹© ¹

Tvv vp h v

h pvp pp

A A u B

p BA A

§ · § ·§ ·¨ ¸ ¨ ¸¨ ¸

¨ ¸¨ ¸ ¨ ¸© ¹ © ¹© ¹A. Masud, R.A. Khurram, A multiscale finite element method for the incompressible Navier–Stokes equations , Computer Methods in Applied Mechanics and Engineering, Volume 195, (2006) 1750-1777 T.J.R. Hugues et al., The variational multiscale method - a paradigm for computational mechanics , Computer Methods in Applied Mechanics and Engineering, Nov (1998)

V. Gravemeier, W.A. Wall, E. Ramm, A three-level finite element method for the instationary incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 193 (2004) 1323-1366. L.P. Franca, A. Nesliturk, On a two-level finite element method for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Engineering 52 (2001) 433-453 R. Codina, Stabilized finite element approximation of transient incompressible flows using orthogonal subscales, Comput. Methods Appl. Mech. Engrg. (2002)

� �,h

K h hK

u qW�

� �¦T

R

E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, T. Coupez, Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, Volume 229, Issue 23, 2010, 8643-8665

1

2en

iK

i

V NhV x

D

D

�§ ·w

¨ ¸¨ ¸w© ¹¦

V V

h

Anisotropic Finite Elements

Adaptive anisotropic mesh at high Re

Anisotropic vs Uniform isotropic Ref 1: J. L. Guermond and P. D. Minev, Start-up flow in a three-

dimensional lid-driven cavity by means of a massively parallel direction splitting algorithmInt. J. Numer. Meth. Fluids (2011)

Ref 2: Bruneau C-H, Saad M. The 2D lid-driven cavity problem

revisited. Computers & Fluids 2006; 35:326–348. Ref 3: Albensoeder S, Kuhlmann HC. Accurate three-dimensional lid-

driven cavity flow. Journal of Computational Physics 2005; 206:536–558.

Ref 4: E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, T. Coupez,

Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, 2010, 229, 8643-8665

2D Lid driven cavity: Re: 1000, 5000, 10000, 20000, 33000, 50000

Reynolds 20000:

Reynolds 100000 (~20000 elements)

(~50000 elements)

2D Lid Driven cavity with mesh adaptation Zoom x 100 RE = 5000

Industrial applications

Level-Set Function

Anisotropic mesh adaptation Mixing the material properties

0

( ) 0

0

fluid

interface

solid

si xx si x

si xD

! �:­°

� *®°! �:¯1

1 1( ) 1 sin20

if e

H if ee e

if e

D

D SDD DS

D

!­°

§ ·° § · � � d® ¨ ¸¨ ¸© ¹© ¹°° � �¯

( ) (1 ( ))air structureH HU U D U D � �

Implicit boundary for moving domain

E T. Coupez, L. Silva, E. Hachem, Implicit Boundary and Adaptive Anisotropic Meshing, to appear in SEMA SIMAI Springer Series, 2014

Naca0012 with Re ~500000 unique multiphase mesh fluid solid flow

Classical benchmark

Treated as a multiphase fluid

Wiping process

Drag out benchmark: an extrem simulation • Heavy Liquid (molten Zinc) moving up along a

vertical surface at 2 m/s • High surface tension coefficient (10 times the water

value) • geometry: 20 cm large, 60 cm high • Air jet impacting the liquid surface: 200 m/s • Liquid thickness above the impacting jet: less that

10 micron meter = 10-5 m) • Smallest edge of the mesh = 1 micron (10-6)

o Isotropic mesh would required : 120 billion of nodes o Present anisotropic mesh adaptation calculation: 50 000 mesh nodes

Droplet formation

Film of Liquid thinning less than 10 micron

Air jet speed: 200 m/s Re= 10^6

Space-time metric construction (Ghina Jannoun PhD)

New Space-time length distribution tensor:

Stretching the edges in their own direction

temporal edge

Importance of multi-criteria adaptation Forced and natural convection

In the vicinity of the object Temperature evolution over time

At the interface solid/air

Mesh adapted on : the levelset function, the temperature and the velocity fields Frequency of adaptation = 10 Nb nodes= 150,000 CPU time: 37.319 mins VS 5 days with a constant time-step size 0.005. On 64 cores

Temperature evolution over time

Example of the flow computation (to determine the permeability) across a microstructure done over 96 cores using a 10-million nodes adapted mesh on a 900x900x220-voxel image (tomography) of a real microstructure of a composite.

To conclude : numerical simulation from 3D images