numerical simulation of complex and multiphase flows 18 th – 22 nd april 20005 porquerolles 1/24...
Post on 11-Dec-2015
215 Views
Preview:
TRANSCRIPT
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 1/24
Finite volumes and finite elements for the numerical simulation of wave breaking
F. Golay
University of Toulon, FranceANAM/MNC
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 2/24
• Numerical simulation of wave breaking
• Finite volume and finite element code
• Mesh refinement
Plan
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 3/24
• Mathematical model
• Numerical model
• Numerical results
Numerical simulation of wave breaking
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 4/24
Numerical simulation of wave breaking: Mathematical model
0ut
ugu)pE(divt
E
g)Ipuu(divt
u
0)u(divt
2
(x,y,t) is the density
u(x,y,t) is the velocity
g is the gravity
1E u is the energy
2is the internal energy
0 1 is the fraction of fluid
p( , , ) is the pressure
where
Equation Of State: stiffened gaz
(Abgrall-Saurel, 1996)
1)1(
11)(
)()(
1
1)1(
1
1
1)(
1
)()(1)(p
a
aa
w
ww
aw
)p(
c
Sound velocity
P. Helluy, F. Golay:”Mathematical and Numerical aspects of Low Mach Number Flows”, Porquerolles 2004
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 5/24
The system has the form of a system of conservation laws
11 2
1 2
( ).
( )
: approximation of in the volume at time ,
: numerical flux (exact Godunov),
( ), ( ) : length of , volume of .
i
n n ii i
i C
ni i n
i i i i
L Cw w t F G
V C
w w C t
F G
L C V C C C
1 2
21 1 1 2 1 1
22 1 2 2 2 2
2
( ) ( ) ( ),
( , , , , ),
( ) ( , , , , ( ) , ),
( ) ( , , , , ( ) , ),
( ) (0,0, , ,0).
t x yw F w G w H w
w u u E
F w u u p u u E p u u
G w u u u u p E p u u
H w g gu
We solve it by a standard finite volume scheme
Ci
Cj
•Second order extension:MUSCL•No pressure oscillation thanks to a special non-conservative discretisation of the fraction evolution.
Numerical simulation of wave breaking: Numerical model
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 6/24
Numerical simulation of wave breaking: Test case
In the air sound velocity c=20m/s, p=105 Paa=-99636 Pa, a=1.1
In the water sound velocity c=20m/s, p=105 Paw=263636 Pa, w=1.1
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 7/24
Mesh: 2000x150
Numerical simulation of wave breaking: Numerical results wave propagation
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 8/24
Numerical simulation of wave breaking: Numerical results wave breaking
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 9/24
• Simple and efficient method: no interface tracking• The same code can be used for compressible multifluid flows
Improvements:• Unstructured mesh, automatic mesh refinement• A posteriori error• Physical interaction• Mixed numerical method
Numerical simulation of wave breaking: Partial conclusion
Integration in a finite element code
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 10/24
• Finite element formulation
• Finite volume formulation
• Software architecture
• Validation
Finite volume in a finite element code
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 11/24
Finite volume/element formulation
e
e
e e
e
e
i i
e e
dUdivF S dans
dtdU
dx divFdx Sdx ...dt
dUdx Fdx div( F)dx Sdx
dt
dUdx
..dx ..dx U(x)=N
Fdx Fndx Sdxdt
Udx Fdx Sdx
..
(
.
)U
t
x
e e e
i ij j j
j
e e e
N UN dx N Fdx N Sdx ...
t
=N (x)
Finite element formulation
Finite volume formulation
e
e
ee
e
is an indicatrice function over each ele
f (U
men
,
t
dUdx Sdx U
t,n x
d)d
Discontinuous finite element formulation
e e e
e
e e e e
e
e
e e
e e e e
dUdx divFdx Sdx
dt
dUdx Fdx Fndx ... Sdx
d
..dx .
t
.dx
Baumann, Oden (2000)
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 12/24
FV & FE: Finite Volume formulation
0ut
ugu)pE(divt
E
g)Ipuu(divt
u
0)u(divt
x
ye
u
uU
E
e
ee
1U(x) U U(x)dx
e
e e
l rlree e
ef (U ,U
tU (t t) ,n )dU (t) x x Sd
Geometrical node with no dof
Centroid node with 5 dof
1
2
4 3
Compute numerical flux exact Godunov schemeHelluy, Barberon, Rouy 2003
1
24
3
5N+1
Compute nodal load vector Estimation of U with slope limiter Display the result
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 13/24
FV & FE: Software architecture
Object oriented finite element code: SIC (Systeme Interactif de Conception)Touzot, Aunay, Breitkopf 1985
ObjectNameIdentifieur
Template:Character arrayReal arrayInteger array……
An object could be:- created- duplicated- listed- modified- …
Exemple of object :- a node- a element- a kinematic condition- a matrix- a vector- a command- a model- …
Object element
Identifieur
modelnumberzoneId material propertiesId geometric propertiesId element propertiesId interpolation functionId save vectorList of nodesList of load caseMesh refinement parameter edges numberList of neighbour elements
Object node
Identifieur
Id kinematic condition Id load casenumber X coordinateY coordinateZ coordinateDegree of freedomNodal propertiesEquation numbersList of elements
http://sic.univ-tln.fr
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 14/24
FV & FE: Validation
x y3 2 4
, p , u , u 04 3 3
x y1 , p 1 , u 1 , u 0
Stationnary choc
Test 1
x y2 , p 2 , u 0 , u 0 x y1 , p 1 , u 0 , u 0
Test 2
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 15/24
1,00
1,20
1,40
1,60
1,80
2,00
-0,50 -0,25 0,00 0,25 0,50
espace °1 time °1
espace °2 time °1
espace °2 time °2
exact
1,50
1,60
1,70
1,80
1,90
2,00
- 0,30 - 0,25 - 0,20 - 0,15 - 0,10
espace °1 time °1
espace °2 time °1
espace °2 time °2
exact
1,20
1,30
1,40
1,50
1,60
0,00 0,02 0,04 0,06 0,08 0,10
espace °1 time °1
espace °2 time °1
espace °2 time °2
exact
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 16/24
Mesh refinement / unrefinement / adaptation
• Finite element mesh refinement
• example: topologic optimization
• Quadtree mesh refinement
• Unrefinement
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 17/24
Mesh refinement: Finite element mesh refinement
e1e1 e2
e3e4e1
Refinement
e1e1
e2e3
e4
e1e1
e2
e1 e1e2
e3
e1 e1
e2
e3
e1
e2e3
e4
e1
conformity
e1 e1 e2
e1e1
e2
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 18/24
Mesh refinement: Mesh refinement Test
P=0,2,4 P=4,6,8 P=8,10,12 P=12,14,16
Criterion 1:
Criterion 2: Verfürth
Initial Mesh
Error
eface e
2
u dlD)u(h2
1e
2ece deR 22
e r
2
1
e
2eKerror
Criterion R. Verfürth (2000)
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 19/24
+1
+1
+1
?
ux=0uy=0 x
y
ux=0
+1/2
+1
P=0,2,4399 nodes130 elements
P=4,6,8708 nodes257 elements
P=8,10,121016 nodes389 elements
P=12,14,161472 nodes589 elements
Mesh refinement: Mesh refinement & topologic optimisation
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 20/24
• time cpu improved
• best precision
• « static » front captured
• but conformity!
• local unrefinement is difficult
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 21/24
Mesh refinement: Quadtree mesh refinement
Hierarchical approach on quadrilateral
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 22/24
1) Loop on volume to set a refinement criteria
2) Loop on nodes to find patch to unrefine- 4 volumes at same hierarchical
level- 4 edge at same hierarchical
level
Modification of the central nodeDestruction of the other central nodesDestruction of the central edge elementsModification of the peripheral edges
Loop on the nodes to merge edges if necessary
Mesh refinement: Quadtree mesh unrefinement
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 23/24
Mesh refinement: Wave breaking
To be continued …..New posteriori error criteria
Interface captured by the entropy jump
Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 24/24
Conclusion
• Compressible bi-fluid model• Finite volume formulation with exact Rieman solver (integration in FE code)• Validation: simulation of wave breaking (confrontation with others models)• Integration in a finite software architecture• Quadtree mesh (un)refinement•…•…• 3D• Parallel implementation• A posteriori error• Multiphysic simulation
top related