M. Darbani, A. Ouahsine, P. Villon University of Technology of Compiègne

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

3

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.

4

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

6

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

7

Example

8

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

Sibsonian Shape Function

Example

nxi

i x xii 1x

K(x) ,K KK

3Area(cdef )(P)Area(abcd)

9

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

Non-Sibsonian Shape Function

1

( ) ( )( ) , ( )( )( )

i i

i ini

jj

x l xx xd xx

di(x): Euclidian distance between points x and node ni

li(x): length of the Voronoi edge associated to x and node ni

10

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 propertyLinear consistency

i j ij

1 , i j(x )

0 , i j

n

ii 1

(x) 1

n

i ii 1

x (x)x

11

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 :

12

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

0d(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)

13

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

Time discretization

n n 1

d ( ( . ) ) Material derivativedt t

(x) (X) Lagrangian formulationt

v v v v

v v

dtXd

txd

tXx nnnn

*1

**1 )()()()( vvvvvvv

kk

kknn

kk

kknn

td

tX

td

tx

)()()(

)()()(

*1*

1

**

vvvv

vvvv

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 functionWeight functionof Gauss

14

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 dγmes( )

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

16

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

17

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 matrixF : 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

18

Numerical tests application of Dam Break

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

19

Numerical tests application for Dam break

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

Magnitude of elevation for transect 1

20

Numerical tests application for Dam break

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

magnitude of elevation for transect 2

21

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

MUt

M

=0 : Euler Explicite

=1 : Euler Implicite

Numerical resolution :

0( **20

dp.dfp)).(t

ρ vvvvkvvv

),,( hvuU ),,( wvuv

24

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

Pρy

hgfuyvv

xuu

tv

uFx

Pρx

hgfvyuv

xuu

tu

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

KK

x1

,)(

)()()(

bcefAirabcdAirxi

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

xN

NAdxN

ANk

ji

jik

j k

jki vv i

May be approximatted by

vanishesofoncontributithejiwhenj ,

Thus, for any arbitrary vector b we can write

dnNdN

xd

xN

N kki

ikkj k

jik b

2)b

21(

22b

0

dxN

Nk

ji onN i 0

dvxN

AN jj k

jki

30

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

dxN

xNN

kkki

2i2

1i1 )v(v)v(vA

12

11 2

1,2

ik2ik NN

jiwhen

ij k

iii

k

jijj

j k

ji d

xNNd

xN

NdxN

N vvAvA kk

131

1

dNi

By taking into account of

1 2: 1

j

jj k k k

N N NPartition of unity Nx x x

31

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)

32

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 KK

3Air(cdef )(x)Air(abcd)

33

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

The pressure term leads to the following integral:

dvxN

AN jj k

jki

For example: Consider the nodes 1,2,3,,..7 as natural neighbors of the node i

Pressure Term

34

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

xN

AN

36

Problem Shape Function SWE&Time discretizatio Nodal Integration Numerical test Appllication

The weak form of pressure term leads to the following integral:j

i k jΩj k

NN A v dΩ

x

Pressure Term

221

11 )2

()2

( eeie

keeie

eki

k k

jki n

vvAn

vvAdv

xN

AN