1 m. darbani, a. ouahsine, p. villon university of technology of compiègne roberval laboratory...
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1
M. Darbani, A. Ouahsine, P. Villon
University of Technology of Compiègne Roberval laboratory UMR-CNRS 6253, France
MULTIPHYSICS-200909-11 December 2009, LILLE ,France
Natural Elements Method for ShallowWater Equations
2
Contents
1)1) Problem FormulationProblem Formulation
2)2) Shape FunctionsShape Functions
3)3) Shallow water Equations and Time discretizationShallow water Equations and Time discretization
4)4) Nodal Integration Nodal Integration
5)5) Numerical tests and results application to Dam BreakNumerical tests and results application to Dam Break
Problem Shape Function SWE & Time discretizatio Nodal Integration Numerical test Appllication
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Dam Break
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Breaking waves
Large deformation
Finite Elements method with meshing is not always convenient.
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Why meshless method ?
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test
Appllication
In NEM method the possibility of
treating the problems is easier for large
deformations than in the finite elements method
Ability to insert or remove the nodes easier
Example: Domain enrichment
Shape functions depend only on the positions
of the nodes.
5
Voronoi Diagram and Delaunay triangles
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Delaunay Triangulations
and Circumscribed circles
Voronoi diagram
Cloud of the nodes
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Mathematical formulation
ijkXXdXXdXXdRxT kjim
ij ),,(),(),(:
jixxdxxdRxT jim
i ),,(),(:
The Natural Neighbors of a node are the nodes associated with neighboring Voronoi cells. They are connected to the node by an edge of Delaunay triangle.
The first order and second order cells of the Voronoi diagram are defined mathematically by :
Cells 1st and 2d order
Natural Neighbor
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Example
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Sibsonian Shape Function
Example
nxi
i x xii 1x
K(x) ,K K
K
3
Area(cdef )(P)
Area(abcd)
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Non-Sibsonian Shape Function
1
( ) ( )( ) , ( )
( )( )
i i
i ini
jj
x l xx x
d xx
di(x): Euclidian distance between
points x and node ni
li(x): length of the Voronoi edge
associated to x and node ni
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Properties of NEM Shape Function
property of the Kronecker delta
Partition of unity
Local co-ordinate property
Linear consistency
i j ij
1 , i j(x )
0 , i j
n
ii 1
(x) 1
n
i ii 1
x (x)x
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Properties of NEM Shape Function
is C1 in every points of Circumscribed circles
is only continuous in every points of Circumscribed circles
sibi (x) :
non sibi (x):
Remark :
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shallow Water Equations
0(u, v) f 2 sin v
),0(),(),(),( TttVt DD xxxv
D Stand for the Dirichlet
portion of the boundary
Governing and Continuty Equations for Incompressible Fluid
0
d(hv)f (h ) g h .(h ) in *(0,T)
dt
.(h ) 0 in *(0,T)t
k v v
v
Boundary Conditions
h(x,y,t)=η(x,y,t)+H0(x,y)
H(x,y,t)=h(x,y,t)+hb(x,y)
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Time discretization
n n 1
d( ( . ) ) Material derivative
dt t
(x) (X)Lagrangian formulation
t
v vv v
v v
dt
Xd
t
xd
t
Xx nnnn*
1**
1 )()()()(v
vv
vv
vv
kk
kknn
kk
kknn
td
t
X
td
t
x
)()()(
)()()(
*1*
1
**
vvv
v
vvv
v
n 1
n
X Position of Point at t
x Position of Point at t
n 1
n
*
k
Gauss point at t
Gauss point at t
v Test function
Weight function
of Gauss
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Iterative process
n
n
tatPtGauss
tatPtGauss
.
. 1
Determine vertex of triangle at the time step t (n-1) :X(n-1)=x(t)-v(n-1)t
Finding velocity at the point old with interpolation :
V(old )=ΣNi(old )vi
Find New at t (n-1) step withNew=ξ-v (old )
If | old - New |/ | old |<10-5
New = old
New=ξ-v (old )
15
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
The weak form of the Diffusion Term leads to the following integral :
The above integral can be approximated by the following assumption with considering the method SCNI [Chen 2001] :
i
ij i jDif N N d
i
ij app i jΩi σΩ σΩ
1(Dif ) = nN dγ nN dγ
mes( ) j
With method of Gauss for numerical integration we have:
gi
Nedge Nedge
i i g g i gk 1 k 1 x edgeσΩ edge
nN d nN d n(x )ω N (x )
Diffusion Term
n i: External normal
Xg &ω i :Integration point and weight of Gauss
Mes(ω i): Area of ω i
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllicatio
i j i iΩj
N N dΩv =mes(Ω)v
The weak form of the Coriolis Term leads to the following integral:
i j i i j iΩ Ωj j
N N dΩv = N ( N )dΩv
iN d mes( )
Finally, the approximated Coriolis term will become :
By using we obtain :ii
N =1
and by using the second propriety we will obtain i j ijN (x )=δ
Coriolis Term
i j i i ij
N N d v N d v
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
The global matricial form is :
n 1 n n n 1
(n 1)
(n)
[M ] : Mass matrix, at t
[M] : Mass matrix, at t
[k] :Stiffness matrix
F : Residual v
[M] [M ]U U [K] (1 )U U
ector
Ft t
),,( hvuU
Algebraic form of Sallow Water Equations
α =0 : Euler Explicit
0< α <1 : Crank–Nicolson
α=1 : Euler Implicit
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Numerical tests application of Dam Break
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
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Numerical tests application for Dam break
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Magnitude of elevation for transect 1
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Numerical tests application for Dam break
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
magnitude of elevation for transect 2
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MULTIPHYSICS-200909-11 December 2009, LILLE ,France
Any questions?
Thank youfor your attention
22
M. Darbani, A. Ouahsine, P. Villon
Université de Technologie de Compiègne, laboratoire Roberval UMR-CNRS 5263, France
MULTIPHYSICS-200909-11 December 2009, LILLE ,France
Natural Elements Method for ShallowWater Equations
UTCUTC
23
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication 5/5
The global matricial form reads :
)(
)1(
11
,:][
,:][
)1(][][][
n
n
nnnn
tatmatrixmassM
tatmatrixmassM
FUUKUt
MU
t
M
=0 : Euler Explicite
=1 : Euler Implicite
Numerical resolution :
0( **20
dp.dfp)).(t
ρ vvvvkvvv
),,( hvuU ),,( wvuv
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication 1/3
Shallow water equations Integration over the total water depth
0
1
1
0
0
[vH]y
[uH]xt
h
vFy
P
ρy
hgfu
y
vv
x
uu
t
v
uFx
P
ρx
hgfv
y
uv
x
uu
t
u
yatm
xatm
Numerical resolution
25
Why meshless method ?
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test
Appllication
Shape functions depends only on the position of nodes
In Meshless Methods the possibility of treating the problems is easier in large deformation than in the finite element method
Ability to insert or remove the nodes easily Example: Domain enrechissement
26
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shape Function of NEM (Sibsonian)
n
ixix
x
xii KK
K
Kx
1
,)(
)(
)()(
bcefAir
abcdAirxi
Example
is at least C1 in every points but only
continuous at the nodal position
)(xi
27
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Mathematical formulation
Cells 1st and 2d order
ijkXXdXXdXXdRxT kjim
ij ),,(),(),(:
jixxdxxdRxT jim
i ),,(),(:
The Natural Neighbors of a node are the nodes associated with neighboring Voronoi cell, or which are connected to the node by a edge of Delaunay triangle.
The first order and second order cells of the Voronoï diagram are defined mathematically by :
Natural Neighbor
28
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Iterative process
ii
oldi
old vNv
3
1
)()(
t vxXDetermine vertex of triangle at t (n-1) step :
oldFinding velocity at the point with interpolation :
tv oldnew )(Find at t (n-1) step with:new
n
n
tatPtGauss
tatPtGauss
.
. 1
Adjusting the point position in the old triangle
29
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
d
x
NNAd
x
NAN
k
ji
jik
j k
jki vv i
May be approximatted by
vanishesofoncontributithejiwhenj ,
Thus, for any arbitrary vector b we can write
dn
NdN
xd
x
NN kk
iik
kj k
jik b
2)b
2
1(
22b
0
dx
NN
k
ji
onN i 0
dvx
NAN j
j k
jki
30
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
dx
N
x
NN
kkki
2i2
1i1 )v(v)v(vA
12
11 2
1,
2
ik2ikNN
jiwhen
ij k
iii
k
jijj
j k
ji d
x
NNd
x
NNd
x
NN vvAvA kk
13
11
dN i
By taking into account of
1 2: 1
j
jj k k k
N N NPartitionof unity N
x x x
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Nodal Integration
ii
dnNdN ii
i
dnNL
N ii
appi
)(
1)(
Divergence theorem :
Based on the substitution of the gradient term by an average gradient of each node in an area surrounding the representative node
Method of Stabilized Conforming Nodal Integration (SCNI)
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Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
Shape Function of NEM (Sibsonian)
Example
nxi
i x xii 1x
K(x) ,K K
K
3
Air(cdef )(x)
Air(abcd)
33
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
The pressure term leads to the following integral:
dvx
NAN j
j k
jki
For example: Consider the nodes 1,2,3,,..7 as natural neighbors of the node i
Pressure Term
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Solution Stability
Evolution of the particle nodes
Initial
Final
35
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
221
11 )2
()2
( eeie
keeie
eki
k k
jki n
vvAn
vvAdv
x
NAN
36
Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication
The weak form of pressure term leads to the following integral:
ji k jΩ
j k
NN A v dΩ
x
Pressure Term
221
11 )2
()2
( eeie
keeie
eki
k k
jki n
vvAn
vvAdv
x
NAN