A conservative FE-discretisation of the Navier-Stokes equation

A conservative FE-discretisation of the Navier-Stokes equation

JASS 2005, St. Petersburg

Thomas Satzger

2

Overview

• Navier-Stokes-Equation– Interpretation– Laws of conservation

• Basic Ideas of FD, FE, FV

• Conservative FE-discretisation of Navier-Stokes-Equation

3

Navier-Stokes-Equation

• The Navier-Stokes-Equation is mostly used for the numerical simulation of fluids.

• Some examples are- Flow in pipes- Flow in rivers- Aerodynamics- Hydrodynamics

4


The Navier-Stokes-Equation writes:

gupuuut

1

Equation of momentum

0 u

Continuity equation

with u

Velocity field

Pressure fieldp

Density

Dynamic viscosity

5


The interpretation of these terms are:

gupuuut

1

Derivative of velocity field

Convection

Pressure gradient

Diffusion

Outer Forces

6


The corresponding for the components is:

1122

2

121

2

11

221

111

1gu

xu

xp

xu

xuu

xuu

t

2222

2

221

2

22

222

112

1gu

xu

xp

xu

xuu

xuu

t

022

11

ux

ux

for the momentum equation, and

for the continuity equation.

7


With the Einstein summation jiji

m

jjiji xayxay

1

and the abbreviation we get:tt

i

i x

iijjiijjii gupuuu

1

)2,1( i

0 jju

for the momentum equation, and

for the continuity equation.

8


Now take a short look to the dimensions:

iijjiijjii gupuuu

1

2s

m

2

1

s

m

ss

m

2

1

s

m

ms

m

s

m

22

1

3 s

m

msm

kgm

m

kg

22

1

3 s

m

ms

m

m

kgsm

kg

9

Navier-Stokes-Equation - Interpretation

We see that the momentum equations handles with accelerations. If we rewrite the equation, we get:

iijjiijjii gupuuu

1

This means:

Total acceleration is the sum of the partial accelerations.

10


Interpretation of the Convection

1x

2x

fluid particle

iiu pi

1ijj u

igijj uu

Transport of kinetic energy by moving the fluid particle

11


Interpretation of the pressure Gradient

1x

2x

fluid particle

iiu ijj uu ijj u

igpi

1

Acceleration of the fluid particle by pressure forces

12


Interpretation of the Diffusion

iiu ijj uu pi

1ig

ijj u

1x

2x

fluid particle

Distributing of kinetic Energy by friction

13


Interpretation of the continuity equation 0 jju

• Conservation of mass in arbitrary domain

h

h

11u 12

u

21u

22u

this means:

influx = out fluxhuhu

11 21 huhu 22 21

01212 2211

h

uu

h

uu

for 0h we get

02211 uu

14

Navier-Stokes-Equation - Laws of conservation

Conservation of kinetic energy:

We must know that the kinetic energy doesn't increase, this means:

Proof:

duuduE iikin 212

21

duu

duuuuduuduudt

dE

dt

d

iti

iititiiitiikin 21

21

21

0kinEdt

d

15


With the momentum equation it holds

Using the relations (proof with the continuity equation)

and

0Re

1 puuuu iijjijjii

puuuuuu

puuuuduuEdt

d

iiijjiijji

iijjijjiitikin

Re

1

Re

1

ijjiji uuuu

ijjiijijijji

ijijijii

product

ruleijii

Gauß

ii

uuuuuuuuu

uuuuuuuuuuuu

2

0

16


Additionally it holds

Therefore we get

Due to Greens identity we have

00

Re

1

puuuuuuE

dt

diiijjiijjikin

jj

continuity

equationjjjj

product

rulejj

Gauß

upupuppuup0

0

0 ijij

Green

ijji uuuu

17


This means in total

We have also seen that the continuity equation is very important for energy conservation.

0kinEdt

d

18

Basic Ideas of FD, FE, FV

We can solve the Navier-Stokes-Equations only numerically.Therefore we must discretise our domain. This means, we regard our Problem only at finite many points.There are several methods to do it:

•Finite Difference (FD)One replace the differential operator with the difference operator, this mean you approximate by

or an similar expression.

f

h

xfhxfxf

19


• Finite Volume (FV)- You divide the domain in disjoint subdomains- Rewrite the PDE by Gauß theorem- Couple the subdomains by the flux over the

boundary• Finite Elements (FE)

- You divide the domain in disjoint subdomains- Rewrite the PDE in an equivalent variational

problem- The solution of the PDE is the solution of the

variational problem

20


Comparison of FD, FE and FV

Finite Difference

Finite Element

Finite Volume

21


Advantages and DisadvantagesFinite Difference:

+ easy to programme- no local mesh refinement- only for simple geometries

Finite Volume:+ local mesh refinement+ also suitable for difficult geometries

Finite Element:+ local mesh refinement+ good for all geometries

BUT:Conservation laws aren't always complied by the discretisation. This can lead to problems in stability of the solution.

22

Conservative FE-Elements

We use a partially staggered grid for our discretisation.

h

h

u

v

u

v

u

v

u

v

p

We write: N for the number of grid pointsiu

ivfor the horizontal velocity in the i-th grid point

for the vertical velocity in the i-th grid point

23


The FE-approximation is an element of an finite-dimensional function space with the basis

The approximation has the representation

Nff 21,...,

2: IRfi whereby

N

iNiiiih fvfuu

1

24


If we use a Nodal basis, this means

we can rewrite the approximation

0

1point grid th-iif

0

0points gridother allNif

1

0point grid th-iNif

0

1points gridother allif

and

and

N

i vNi

uNii

vi

uii

h

h

f

fv

f

fu

v

u

1 ,

,

,

,

25


Every approximation should have the following properties:continuousconservative

In the continuous case the continuity equation was very important for the conservation of mass and energy.

If the approximation complies the continuity pointwise in the whole area, e.g. , then the approximation preserves energy.

hu

0 hu

26


Now we search for a conservative interpolation for the velocities in a box.

We also assume that the velocities complies the discrete continuity equation.

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

27




1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

hvv

huu

224331

28




1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

hvv

huu

224331 h

vvh

uu

22

2142

29



We also assume that the velocities complies the discrete continuity equation:

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

hvv

huu

224331 h

vvh

uu

22

2142

043314321 vvvvuuuu (1)

30


The bilinear interpolation isn't conservative

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

1 2 3 4

1 2 3 4

, 1 1 1 1

, 1 1 1 1

h

h

x y x y x y x yu x y u u u u

h h h h h h h h

x y x y x y x yv x y v v v v

h h h h h h h h

31



1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

1 2 3 4

1 2 3 4

, 1 1 1 1

, 1 1 1 1

h

h


h h h h h h h h

x y x y x y x yv x y v v v v

h h h h h h h h

It is easy to show that

0general in

hh vy

ux

32



1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

, , ,,1, 3, 4,2,

,2 4,,1 4,

2 3 4

2

3

1

1 3

, 1 1 1 1

, 1 1 1

f x y f x y f x yf x yu u uu

f x yvxv

h

y

h

f f


h h h h h h h h

x y x y x yv x y v v v

h h h h h h

4

, ,4, 4 4,

1

x y f x yv v

x yvh h

Basis on the box

33


These basis function for the bilinear interpolation are calledPagoden.

The picture shows the function on the whole support.

, i iu v

h

h

34


Now we are searching a interpolation of the velocities which complies the continuity equation on the box.

How can we construct such an interpolation?

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

35




1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

Divide the box in four triangles.

36




1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

Divide the box in four triangles.Make on every triangle an linear interpolation.

37


What's the right velocity in the middle?

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

38



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

We must have at every point in the box the following relations:

, , 0 , ,

, ,

u x y v x y u x y v x yx y x y

v x y u x yy x

39




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

40




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

41




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

42




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

43




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

44




, , 0 , ,

, ,


v x y u x yy x

1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

45



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

5 33u u

46



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 35 33 2

u uhu u

h

47



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 33 2 2

u u v vh hu u

h h

48



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2

u u v vh hu u u u u v v u u v v

h h

49



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2


h h

5 42u u

50



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2


h h

2 45 42 2

h u uu u

h

51



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2


h h

2 4 2 45 42 2 2

h u u h v vu u

h h

52



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2


h h

2 4 2 45 4 4 2 4 2 4 2 4 2 42

1 1

2 2 2 2

h u u h v vu u u u u v v u u v v

h h

53



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

1 3 1 35 3 3 1 3 1 3 1 3 1 33

1 1

2 2 2 2


h h

2 4 2 45 4 4 2 4 2 4 2 4 2 42

1 1

2 2 2 2

h u u h v vu u u u u v v u u v v

h h

free are and

41 55 uu

54



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

4 3 4 35 3 3 4 3 4 3 3 4 3 44

1 1

2 2 2 2

v v u uh hv v v v v u u u u v v

h h

55



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

51u

52u53

u

54u

51v

52v53

v

54v

1

ux

1

ux

4

ux

4

ux

3

vy

3

vy

2

vy

2

vy

4 3 4 35 3 3 4 3 4 3 3 4 3 44

1 1

2 2 2 2

v v u uh hv v v v v u u u u v v

h h

2 1 2 15 1 1 2 1 2 1 1 2 1 21

1 1

2 2 2 2

h v v h u uv v v v v u u u u v v

h h

free are and

23 55 uv

56


Till now we have:

5 1 3 1 33

1

2u u u v v 5 2 4 2 42

1

2u u u v v

5 3 4 3 44

1

2v u u v v 5 1 2 1 21

1

2v u u v v free are and

23 55 uv

free are and 41 55 uu

57


Till now we have:

With the discrete continuity equation

we get

5 1 3 1 33

1

2u u u v v 5 2 4 2 42

1

2u u u v v

5 3 4 3 44

1

2v u u v v 5 1 2 1 21

1


23 55 uv


1 2 3 4 1 2 3 4 0 (1)u u u u v v v v

5 53 2u u 5 54 1

v v

58


Till now we have:

With the discrete continuity equation

we get

Therefore we choose

5 1 3 1 33

1

2u u u v v 5 2 4 2 42

1

2u u u v v

5 3 4 3 44

1

2v u u v v 5 1 2 1 21

1


23 55 uv


1 2 3 4 1 2 3 4 0 (1)u u u u v v v v

5 53 2u u 5 54 1

v v

5 5 5 5 5 1 2 3 4 1 2 3 41 2 3 4

5 5 5 5 5 1 2 3 4 1 2 3 41 2 3 4

1: : : : :

41

: : : : :4

u u u u u u u u u v v v v

v v v v v u u u u v v v v

59



1u 2u

3u 4u

h

h

1v 2v

3v 4v

h

h

12

34

12

34

5u 5v

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

60


Now we calculate the basis., ,

, ,1

Ni u i N uh

i ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

61


Now we calculate the basis.

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

, ,

, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

i i

62



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

h

h

h

h

,i uf ,i vf

0 0 0

0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

63


Now we calculate the basis.

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

, ,

, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

3 3, u v 44, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

64



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

3 3, u v 44, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

45 1 2 3 1 2 3 4

5 1 2 3 4 1 2 3 4

:

1:

4

1

4u u u u v v v v

v

u

u u u u v v v v

65



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

3 3, u v 44, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

45 1 2 3 1 2 3 4

5 1 2 3 4 1 2 3 4

:

1:

4

1

4u u u u v v v v

v

u

u u u u v v v v

1

4

66



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

1 1, u v 2 2, u v

33, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

67



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

33, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

5 1 2 4 1 2 3 4

5 1 2 3 4 1 2

3

3 4

:

1

1

:

4

4

u u u u v v v v

v u u u u

u

v v v v

68



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

1 1, u v 2 2, u v

33, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

5 1 2 4 1 2 3 4

5 1 2 3 4 1 2

3

3 4

:

1

1

:

4

4

u u u u v v v v

v u u u u

u

v v v v

1

4

69



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

5 1 3 4 1 2 3 4

5 1 2 3 4 1 4

2

2 3

:

1

1

:

4

4

u u u u v v v v

v u u

u

u u v v v v

1 1, u v 22, u v

3 3, u v 4 4, u v

h

h

5 5, u v0

0

h

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

70



, ,1

hi

Ni u i N u

ii v i N vh i

f fuu v

f fv

5 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

1

1

:

4

4

u u u u v v v v

v u u u v

u

u v v v

11, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

0

h

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

71



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

72



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

5 1 2 3 4 1 2 3 4

5 1 2 3 1 2 3 44

1:

41

4:

u u u u u v v v v

v u u u v v v vu

1 1, u v 2 2, u v

3 3, u v 44, u v

h

h

5 5, u v0

0

h

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

1

4

73



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

5 1 2 3 4 1 2 3 4

5 1 2 4 13 2 3 4

1:

41

4:

u u u u u v v v v

v u u u v v v vu

1 1, u v 2 2, u v

33, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

1

4

1

4

74



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

1 1, u v 22, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

1

4

1

4

1

4

5 1 2 3 4 1 2 3 4

5 1 3 4 1 2 32 4

1:

41

4:

u u u u u v v v v

v u u u v v vu v

75



, ,1i

Ni u i N uh

ii i N vih v

f fu

u

vv

f f

5 1 2 3 4 1 2 3 4

5 2 3 4 1 31 2 4

1:

41

4:

u u u u u v v v v

v u u u v v vu v

11, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i uf ,i vf

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

1

4

1

4

1

4

1

4

76



, ,1

Ni u i N uh

ii v i N vh i

i

f fuu

f fvv

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i N uf ,i N vf

0 0 0

0 0

0 0 0

0

0 0 0

0 0

0 0 0

1

i i

77



, ,1

Ni u i N uh

ii v i N vh i

i

f fuu

f fvv

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

,i N uf ,i N vf

0 0 0

0 0

0 0 0

0

0 0 0

0 0

0 0 0

1

i i

1

4

1

4

1

4

1

4

1

4

1

4

1

4

1

4

78



, ,1

Ni u i N uh

ii v i N vh i

i

f fuu

f fvv

5 1 2 3 4 1 2 3 4

5 1 2 3 4 1 2 3 4

1:

41

:4

u u u u u v v v v

v u u u u v v v v

1 1, u v 2 2, u v

3 3, u v 4 4, u v

h

h

5 5, u v

h

h

h

h

, ,i N u i vf f , ,i N v i uf f

0 0 0

0 0

0 0 0

0

0 0 0

0 0

0 0 0

1

i i

1

4

1

4

1

4

1

4

1

4

1

4

1

4

1

4

79


h

h

h

h

, ,i u i N vf f , ,i v i N uf f

0 0 0

0 0

0 0 0

1

0 0 0

0 0

0 0 0

0

i i

1

4

1

4

1

4

1

4

1

4

1

4

1

4

1

4

Linear interpolation providesthe basis.

80


View on conservative elements in 3D

81


View on conservative elements in 3D

Partially staggered grid in 3D

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

p

hh

h

82


We also search for a conservative interpolation of the velocities.

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

p

hh

h

83


We also search for a conservative interpolation of the velocities.

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

, , u v w

hh

h

Divide every box into 24 tetrahedrons, on which you make a linear interpolation

Documents

A conservative FE-discretisation of the Navier-Stokes equation