1 TECPLOT

September 2012

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Numerical Strategy

A Domain (Geometry)

B Governing equations (O.D.E., P.D.E.) and B.C.s

C Difference equations (Discretization) and B.C.s

D Finite difference / Finite element or other methods * Importance of boundary conditions and initial conditions

1If not exist, only general solutions can be obtained.

2For practical applications, they are necessary.

* Types of problems:

1Elliptice.g.

0yx 22

2

2

(Laplace equation)

2Parabolicboundary-layer flows (upstream-dominant)

3Hyperbolic

2

2

2

2

yC

xCk

tC

* Uniqueness and existence of particular solution.

* Types of boundary conditions

IDirichlet Problemsgiven values at boundaries IINeumann Problemsgiven normal derivatives at boundaries

19Finite Difference Method in Solving Differential Equations

To represent the quantity of xF

x xx

AForward (downwind) difference

xFF

xF i1i

i

* By Taylors expansion

xxFF 1j

......!3

xxF

!2)x(

xFx

xFxF

3

x3

32

x2

2

x

1

xOx

xFxxFxF

]x[OxFxxFxxF

x

2

x

BBackward (upwind) difference

xFF

xF 1ii

i

* By Taylors expansion

20

)xx(FF 1i

......!3x

xF

!2x

xFx

xFxF

3

3

32

x2

2

x

2

xOx

xxFxFxF

]x[OxxFxFxxF

x

2

x

CCentral Difference

x2FF

xF 1i1i

i

* 1 2

]x[Ox2

xxFxxFxF

]x[Ox2xFxxFxxF

2

x

3

x

Comments

(i) Better accuracy

(ii) Worse space resolution

(iii) Gradient independency of local values

21* Example 1 Dirichlet Type

0yx 22

2

2

0

1

1x

50

40

0

100y

B F

j+1

i+1ii-1

j

j-1

xxx

x

BF

j,i2

2

=> xx

j,ij,1iF

j,i

xxj,1ij,i

B

j,i

2

j,1ij,ij,1i

x2

Similarly, 21j,ij,i1j,i

j,i2

2

y2

y

= 0 40

= 40 80

= 0 80

= 0

22Substitution into the ordinary differential equation leads to

equation) e(differenc0y

2x

22

j,1ij,ij,1i2

j,1ij,ij,1i

If one takes ,yx then

)(41 n

1j,in

1j,in

j1in

j,1i1n

j,i

j+1

i+1ii-1

j

j-1

Procedure

(1) Make initial guesses at all interior points.

(2) Iterate until convergence is reached.

* Example 2Dirichlet + Neumann 0yx 22

2

2

0

1

1x

0

x

0

x

0

100y

= 0

= 40 100

= 0 100 100 0

23Image conditions at phantom points

0

x

Noted that at i = 0 , are unknowns.

n 1j,0n 1j,0n j,1n j,11n j,0 41

nj,1

nj1,- take, 0iat 0x

Finally, n 1j,0n 1j,0n j,11n j,0 241

0

1

11000

8020

0x

200

y

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25

( 22 1010 FORTRAN )

1X, Y, , .

dimension x(0:10,0:10), y(0:10,0:10), phi(0:10,0:10), oldphi(0:10,0:10)

2

do 100 i=0,10 do 100 j=0,10

100 x(i,j)=0.1*float(i)

do 200 i=0,10 do 200 j=0,10

200 y(i,j)=0.1*float(j)

3

do 300 i=0,10 phi(i,0)=0. () 300 phi(i,10)=40.+0.4*float(i) ()

do 400 j=0,10 phi(0,j)= 0.4*float(i) () 400 phi(10,j)= 0.8*float(i) ()

4

do 500 i=1,9 do 500 j=1,9

500 phi(i,j)= 0.

5 ( DO loop)

do 500 j=1,9 (Gauss-Seidel method) do 500 i=1,9

500 phi(i,j)=(phi(i-1,j)+ phi(i+1,j)+phi(i,j-1)+ phi(i,j+1))/4.

26do 500 j=1,9 do 500 i=1,9 (Jacobi method)

500 oldphi(i.j)=phi(i,j) do 600 j=1,9 do 600 i=1,9

600 phi(i,j)=(oldphi(i-1,j)+ phi(i+1,j)+oldphi(i,j-1)+ phi(i,j+1))/4.

6( tecplot )

5nj,i

1nj,i 101max

10,0j10,0i

32nj,i

1nj,i 10

27TECPLOT 9.0

1. Laplace equation R001R002

2. Tecplot macro

3. Tecplot play macro

Macro

1. Layout R001

2.

3. macro File -> macro -> record

macro recorder macro

File -> load datafile -> replace dataset and retain frame style R001

R002 R003 stop recording

28 .mcr macro

#!MC 900 $!VarSet |MFBD| = 'C:\tecplot avi'

C

$!READDATASET '"|MFBD|\R001" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"'

C

$!READDATASET '"|MFBD|\R002" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"'

C

$!READDATASET '"|MFBD|\R003" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"' $!RemoveVar |MFBD|

294. macro ()

macro output macro

macro

#!MC 900 $!VarSet |MFBD| = ' C:\tecplot avi '

$!EXPORTSETUP EXPORTFORMAT = AVI AVI

$!EXPORTSETUP ANIMATIONSPEED = 10

$!EXPORTSETUP IMAGEWIDTH = 600

$!EXPORTSETUP EXPORTFNAME = 'C:\tecplot avi\export.avi'

$!EXPORTSTART

$!READDATASET '"|MFBD|\R001" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"'

$!EXPORTNEXTFRAME

$!READDATASET '"|MFBD|\R002" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"'

$!EXPORTNEXTFRAME

30 $!READDATASET '"|MFBD|\R100" ' READDATAOPTION = NEW RESETSTYLE = NO INCLUDETEXT = NO INCLUDEGEOM = NO INCLUDECUSTOMLABELS = NO VARLOADMODE = BYNAME VARNAMELIST = '"X" "Y" "H"'

$!EXPORTFINISH

$!RemoveVar |MFBD|

5. macro 100

export.avi 10 600 File -> macro -> play

macro TEST1.mcr