Numerical Conformal Mapping of an Irregular AreaBY TARUN GEHLOTS

Contents

1. Mapping between two orthogonal coordinates

2. One-to-one Mapping from a hyper-rectangle onto a rectangle

3. Numerical Formulations of Mapping (BEM)

4. Mapping an irregular area onto a hyper-rectangle

5. Grid generation Applications

Mapping between two orthogonal coordinates

0;0

;

)(

)(

)()(

)()(

,),(),()(

2

2

2

2

2

2

2

2

yxyx

xyyx

yyi

dz

zdfx

ixdz

zdf

yi

yy

z

dz

zdf

y

zfx

ixx

z

dz

zdf

x

zf

iyxzyxiyxzfw

Forward mapping:

Cauchy-Riemann Condition

Backward mapping:

0;0

;

),(),()(

2

2

2

2

2

2

2

2

yyxx

yxyx

yxi

dw

dg

yi

x

dw

dg

yi

xw

dw

dgg

yi

xw

dw

dgg

iyxwgz

iw

Mapping between two orthogonal coordinates

Cauchy-Riemann Condition

One-to-one Mapping from a hyper-rectangle onto a rectangle

• Hyper-rectangle : Four corner right angles Four smooth curvilinear lines

iyxz

• Rectangle : Four corner right angles Four smooth straight lines

iw

0

0

0

0

'A 'B

'C'D

0:

:

:

0:

''

0''

0''

''

ADDA

DCCD

CBBC

BAAB

)(zf

)(wg

x

y

D

B

C

A

Local orthogonal coordinates ( s , n )

ns

DAAlongns

CDAlongns

BCAlongns

ABAlongsnns

ConditionRiemannCauchy

0

0:

0

:

0

:

0

0:

;:

0

0

One-to-one Mapping from a hyper-rectangle onto a rectangle

Numerical Formulations of Mapping (BEM)

0222

iW

iW

0 0

0n

0n

02

0n0n

0

0

02

• Boundary integral element method (Liggett ＆ Liu, 1983):

, C: interior angle

d

n

Wr

n

rWpCW ln

ln)(

x

y

D

B

C

A

Mapping an irregular area onto a hyper-rectangle

e.g.

A x0

B

y

'B 'A'O

22

2

2

)(i

i

er

re

zw

Property : conformal except the

origin

nzw n ,

α: original corner angleβ =π/2 or π

Mapping an Irregular area into a Rectangle

1. Forward mapping the irregular area into a hyper-rectangle, w=zn

2. Forward mapping the hyper-rectangle into a rectangle, ▽2ξ=0; ▽2η=0

3. Backward mapping the rectangle into the hyper-rectangle, ▽2x=0; ▽2y=0

4. Backward mapping the hyper-rectangle into the irregular area, z=w1/n

Grid generation Applications

/,/ln

/ln0

00

i

i

rr

rrRef : Analytical mapping :

(A) step1

'A

'B'C

'D

step2:mapping onto a rectangle

'A

'B 'C

'D

step3:establish the grids

step4

A B D C x

y ir

0r

(B)Grid generation Applications

A B D

C

E

F

step1

step2:∠A 90º step3: B 90º∠

(B)Grid generation Applications

step5: D 90º∠

step6: E 90º∠ step7: F 180º∠

step4: C 180º∠

step8:mapping onto a rectangle

step9:construct the grids

step10:transform to original domain

(B) Grid generation Applications

step11: F 90º∠

step12: E 45º∠(B) Grid generation Applications

step15: B 135º∠ step14: C 270º∠

step13: D 135º∠

step16: A 45º∠

(B)

Grid generation Applications

A

B

D

C

(C)

Grid generation Applications

step1

step2: A 90º∠step3: B 90º∠

step4: C 90º∠

step7: construct the grids step6: mapping onto a rectangle

(C)Grid generation Applications

step5: D 90º∠

step8: transform to original domain(C)

Grid generation Applications

step9: D 180º∠

step10: C 180º∠step11: B 180º∠

step12: A 180º∠(C)

Grid generation Applications

AO

0r

ir

CB

D

i

i

rr

rr

/ln

/ln,)2/(

000

Ref : Analytical mapping :

(D)

Grid generation Applications

Parallel sin-wave

(E)Grid generation Applications

Symmetrical sin-wave

Grid generation Applications(F)

A bite of a moon cake

Two bites of a moon cake

A present of four moon cakes

The End

Thank you very much !