Post on 01-May-2018
Stabilized finite element method for heat transfer and fluid flow inside industrial furnaces
April 2009 elie.hachem@mines-paristech.fr
E. Hachem, H. Digonnet and T. Coupez*
Centre de Mise en Forme des Matériaux, Mines Paristech, France
GDR-FSI Fluid-Structure-InteractionApril 23-24, Nantes, France
elie.hachem@mines-paristech.fr
Outline
General context
Stabilized FEM (heat transfer - turbulent flow)
Immersed volume method (monolithic approach)
Numerical examples, results and discussion
elie.hachem@mines-paristech.fr
Bridging the gap: Academic research / industrial applications
Research team: Applied mathematics and Computational mechanics
elie.hachem@mines-paristech.fr
Bridging the gap: Academic research / industrial applications
Acknowledge:EDF - Arcelor Mittal - Snecma - Aubert & Duval - Terreal - Sfarsteel - Manoir Industries
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Where is a differential operator acting on weighting function
Stabilized Finite elementTransient Convection-diffusion-reaction equation
Variational formulation
Stabilized finite element method for convection dominated problem (high Peclet number)
0
:
(0, )
0 (0, )
( ,0)
t
Find u V such that
u u f in T
u on T
u u in
∈∂ + = Ω×
= ∂Ω×⋅ = Ω
L
: ( )u a u u uκ σ= ⋅∇ −∇ ⋅ ∇ +L
( , ) : ( , ) ( , ) ( , )
( ): ,
h h h h h h h h
h
b u v a u v u v u v
l v f v
κ σ= ⋅∇ + ∇ ∇ +=
1
0 /( ),h h h K his linear forV v H v K= ∈ Ω ∈T
( , ) ( , ) ( )t h h h h h h hu v b u v l v v V∂ + = ∀ ∈
( , ) ( , ) ( , ) ( )Kt h h h h h h K h h h
K
u v b u v u v l v v Vτ∂ + + = ∀ ∈∑ stabR L
stabL ( : )hSUPG a v= ⋅∇
stabL
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Stabilized Finite elementUsual stability coefficients
Where hK is an appropriate measure of the size of the mesh No time dependency
1. ’: Temporal discretization (e.g. Euler implicit)
2. hK : characteristic dimension of the triangle in the streamline direction
( )( )1 1 11/ , ( , ) ( , ) ( , ) ( , ) ( , )h
h h h
n
n n n
h h h h K h K h h
K
ut u v a u v k u v u a v v f v
tσ τ+ + ++ ∆ + ⋅∇ + ∇ ∇ + ⋅∇ = +
∆∑ R
1
2
24K
K K
ak
h hτ σ
−
′= + +
S. Micheletti, S. Perotto, M. Picasso, Some remarks on the stability coefficients and bubble stabilization of FEM on anisotropic grids, MOX Report 06, MOX--Modeliug and Scientific Computing, Department of Mathematics "F. Brioschi", Politecnico di Milano, 2002
Volker John, Ellen Schmeyer, Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusionComputer Methods in Applied Mechanics and Engineering, Volume 198, Issues 3-4, 15 December 2008, Pages 475-494
1
2en
iK
i
a Nh
a x
α
α
− ∂= ∂ ∑ ax
aay
h
The support length in the streamline direction
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Numerical example
An academic test: transient convection
Unstructured coarse gridSkew convection
0u =
0u
n
∂ =∂
0u =
0u
n
∂ =∂
(1,0.7)a =r
Case1. Profile of the solutions at t=0.5, ∆t=10-3: Galerkin (left), SUPG (centre) and SCPG (right)
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Stabilized Finite elementHeat transfer inside the solid
Thermal shock, instability when
Enriched finite element method + “Time interpolation”
Numerical test
( ) ] [. 0,p n
Tc k T f dans t
tρ ∂ − ∇ ∇ = Ω×
∂
1*
1*
(1 )
(1 )
1
p
h h K
n n
p
K
cw w
t
T T T
ctwhere
t t
ρτ
ξ ξ
ρξ τ
−
−
= −
∆ = + −
∆= = − ∆ ∆
%
.K
n
K h
K
k T w dξΩ
+ ∇ ∇ Ω∑∫
800 °°°°C
20 °°°°C
L.P. Franca and C. Farhat, On the limitations of bubble functions, Comput.Methods Appl. Mech. Engrg. 117 (1994) 225-230
Ilinca F. Hétu J-F, Galerkin gradient least-squares formulation for transientconduction heat transfer, Comput. Methods Appl. Mech. Engrg. 191 (2002) 3073-3097
Isaac Harari,Stability of semidiscrete formulations for parabolic problems at small timesteps, Computer Methods in Applied Mechanics and Engineering, Volume 193, Issues 15-16, 16 April 2004, Pages 1491-1516
E. Hachem, H. Digonnet, E. Massoni and T. Coupez, ”Enriched finite element spaces for transient conduction heat transfer” submitted in november 2008 to International Journal for Numerical Methods in Engineering
2 /pt c h kρ∆ <
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Stabilized Finite element
static condensation
Incompressible Navier-Stokes problem
Variational formulation
Stable Mini-element (Arnold-Brezzi-Fortin) mixed formulation (Stokes problem)
:
( ) 2 ( ) in
0 in
0
t
Find u and p such that
u u u v p f
v
v on
ρ µ ε∂ + ⋅∇ − ∇ ⋅ + ∇ = Ω∇ ⋅ = Ω
= ∂Ω
( )( , ) ( , ) ( , ) ,
( , ) 0
Find v and p such that
a v w b w p c v w f w w V
b v q q Q
+ + = ∀ ∈
= ∀ ∈
( ) ( )( )( )
( , ) : 2 :
( , ) : ,
( , ) : ( , ) ( , )t
where
a v w v w
b v q v q
c v w v w v v w
ηε ε
ρ ρ
=
= − ∇ ⋅= ∂ + ⋅∇
( )( )
( , ) ( , ) ,
( , ) ( , ) ,
( , ) 0
Find v v v and p such that
a v w b w p f w w V
a v w b w p f w w V
b v q q Q
′= ++ = ∀ ∈
′ ′ ′ ′ ′+ = ∀ ∈
= ∀ ∈
%
%
%
%
( )( , ) ( , ) ,
( , ) ( , ) 0
Find v and p such that
a v w b w p f w w V
b v q v q q Q
+ = ∀ ∈
′− − ∇ = ∀ ∈
Build naturally by a matrix form
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Extension to high Reynolds flow
Two additional contributions:
1) Upwind bubble functions
2) Modelling the fine-scale pressure
( )( )
( , )
( , ); ( , ) ( , )
( , ); ( , ) ( , )
Find v v v and p p p such that w q V Q
A v v p p v q f w
A v v p p v q f w
′ ′= + = + ∀ ∈ ×′ ′+ + =′ ′ ′ ′ ′+ + =
% % % %
( ),h
C h h
K
grad div stabilization term
v wτ∈
−
+ ∇ ⋅ ∇ ⋅∑144424443T
( ),h
K
K
Upwind stabilization terms
v u wτ∈
+ ⋅∇∑14444244443T
R
A. Masud, R.A. Khurram, A multiscale finite element method for the incompressibleNavier–Stokes equations , Computer Methods in Applied Mechanics and Engineering,Volume 195, (2006)
T.J.R. Hugues et al., The variational multiscale method - a paradigm for computationalmechanics , Computer Methods in Applied Mechanics and Engineering, Nov (1998)
( ):
( , ) ( , ) ( , ) ,
convection term
fine scale equation
a v w b w p c v w v w w V
−′ ′ ′ ′ ′ ′+ + = ∀ ∈% %
14243R
L.P. Franca, A. Nesliturk, On a two-level finite element method for the incompressible Navier–Stokes equations, Int. J. Numer. Methods Engrg. (2001)
R. Codina, Stabilized finite element approximation of transient incompressible flows usingorthogonal subscales, Comput. Methods Appl. Mech. Engrg. (2002)
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boundary conditions (left) , coarse mesh 64x64(center) , 180x180 fine mesh (right)
Test case: driven cavity
Re=10000
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
0 0,5 1
y
u x
Re=20000
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
0 0,5 1
y
u x
Re=10000
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0 0,5 1
x
uy
Re=20000
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0 0,5 1
x
uy
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
0 0,5 1
y
u x
Re=33000
Re=50000
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0 0,5 1
x
u y
Re=33000
Re=50000
High Reynolds number
Comparisons with the reference
____ coarse mesh ___ fine mesh ◊ reference (601x601)
Velocity profile for
vx along x=0.5
Velocity profile for
vy along y=0.5
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Test case: driven cavity
High Reynolds number
- Increasing number of vortices
- Apparition of new resolved vortices
Streamline on colored velocity distribution from top-left to bottom-right: Re=1000, 5000, 10000, 20000, 33000 and 50000 with 180×180 mesh
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Test case: BFS (backward facing step)
Stabilized method using both
isotropic and anisotropic meshes
- High Reynolds number (44000)
- Heterogeneous anisotropic mesh (~19000 nodes)
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Application: hat-shaped disk
Problem settings: simple
Experimental dataLocation of the sensors
Temperature distribution
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Heat transfer coefficientsNeumann boundary conditions
4 4
( )
( )
c out
out
k T h T T on
k T T T onσε− ∇ ⋅ = − ∂Ω
− ∇ ⋅ = − ∂Ω
n
n
4 4( , ) ( , ) ( ) ( ) ( )t h h h h h c h out h out h hT v a T v l v h T T d T T d v Vσε∂Ω ∂Ω
∂ + = + − Γ + − Γ ∀ ∈∫ ∫
8 2 4
:
: 5.67 10 . .
:
ch convection heat transfer coefficient
Stephan Boltzmann constant W m K
emissivity
σε
− −−
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Immersed volume method
Immersed object inside an enclosure (10x10x15) Anisotropic mesh adaptation
air
Solid
Inconel 718
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Level-set function
Anisotropic mesh adaptation
Mixing materials properties
Immersed volume method
E. Hachem, T. Kloczko, E. Massoni, T. Coupez, ” Immersed volume technique for
solving natural convection, conduction and radiation of a hat-shaped disk inside
an enclosure” preprint for International Journal for Numerical Methods in
Engineering 2009
2
2 2
s / 2
( ) s / 2
I i e
M NB I i e
e
ε α
ε ε α
>=
− + <
2
T
where Bα α
α∇ ⊗ ∇=
∇
0
( ) 0
0
fluid
interface
solid
si x
x si x
si x
α> ∈ Ω= ∈ Γ> ∈ Ω
1
1 1( ) 1 sin
2
0
if e
H if ee e
if e
αα παα α
πα
> = + + ≤
< −
: ( ) (1 ( ))fluide solideexample H Hρ ρ α ρ α= + −
air
solid
air
solid
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Mixing thermo-physical propertiesZero isovalues
Results
Immersed volume methodMixing laws Anisotropic mesh adaptation
1
( ) (1 ( ))
air solide
H Hk
k k
α α−
−= +
1
/ (1 ) /air airair solide
tot tot
m mk k k
m m
−
= + −
( ) (1 ( ))air solidek H k H kα α= + −
.
( ) .
tot element
air element
m V
m H V
ρα ρ
== i.e. Density distribution along the interface
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Grid inside a furnace
Anisotropic mesh adaptation
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2D example
Thermal conductivity: air (0.02) , solid (175),
No interface Refined interfaceExact interface
air solidinterface
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Transient conduction :
Conjugate conduction-radiationConjugate convection-conduction
0.1%0.14%0.15%Appropriate law
1.73%4.1%6.4%Simple law
Adapted interfaceExact interfaceNo interface
t=0s t=1000s t=∞
Tair = 20°C Tsolid = 400°C
2D example
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Direct simulation air-solid
Coupled problem for air cooling inside an enclosure : (illustration of the method)
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- Velocity 2m/s
Outlet
Inlet
-Velocity vectors at different time step
2D heating ingot : (illustration of the method)
Direct simulation air-solid
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3D finite element solution:
Appropriate mixing law
Immersed volume methodVolumetric radiative source termStabilized finite element method
Magnified cutting plane:
Hat-shaped disk - Snecma
T: temperature G: radiation S: radiative source term!!!
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Hat-shaped disk - Snecma
Snapshot at different time stepStreamline and temperature distribution
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Comparisons with experimental data
Numerical resultssensors
Old results Recent results
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Industrial furnace: Aubert & Duval
Problem set up
Multidomain applicationAnisotropic adaptation
Coupled problemComplex geometryAdditional Turbulence models:
Standard & Low-Reynolds k-ε modelSmagorinsky LES model
Pro
blem
set
up
3 ingots10 burners1 outlet (in the top-center)2 doorsAveraged velocity (40m/s)Maximum temperature 800°C
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Industrial furnace: Aubert & DuvalT
emp
erat
ure
evo
luti
on
insi
de
the
furn
ace
Computation issuesHigh Reynolds numberParallel computation: 32 cores3 days of computation for 4 minutes of physical time~800000 elements
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Industrial furnace: Aubert & Duval
Computation issuesHigh Reynolds numberParallel computation: 32 cores3 days of computation for 4 minutes of physical time~800000 elements
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Thank you for your attention
ConclusionFully fluid-solid coupling simulationImmersed methodAnisotropic adaptation
Stabilized FEMTurbulent flows
Industrial applications
On going workTime reduction
A posteriori error driven anisotropic adaptation
Thermo-mechanical deformationMonolithic approach for FSI
Free surface turbulent flowG. François, E. Hachem, T. Coupez, ” Méthodes éléments finis pour le
remplissage à haut nombre de Reynolds” CFM, Marseille, 2009
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Coupled LES with Levelset
Experimental validation :
G. François, E. Hachem, T. Coupez, ” Méthodes éléments finis pour le
remplissage à haut nombre de Reynolds” CFM, Marseille, 2009