1. Numerical Conformal Mapping of an Irregular Area
BY TARUN GEHLOTS

2. 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

3. Mapping between two orthogonal coordinates
Forward mapping:
w f (z) ( x, y ) i ( x, y ) , z x iy
f (z) df ( z ) z
i
x dz x x x
f (z) df ( z ) z
i
y dz y y y
df ( z )
i
dz x x
df ( z )
i
dz y y
; Cauchy-Riemann Condition
x y y x
2 2 2 2
2 2
0 ; 2 2
0
x y x y

4. Mapping between two orthogonal coordinates
Backward mapping:
w i
z g (w) x( , ) iy ( , )
g dg w x y
i
dw
g dg w x y
i
dw
dg x y
i
dw
dg x y
i
dw
x y x y
; Cauchy-Riemann Condition
2 2 2 2
x x y y
2 2
0 ; 2 2
0

5. One-to-one Mapping from a hyper-rectangle onto a rectangle
• Hyper-rectangle : • Rectangle :
Four corner right angles Four corner right angles
Four smooth curvilinear lines Four smooth straight lines
C
z x iy w i
D
0 '
D
' C
f (z)
y
0
A
B g (w ) 0
x '
A 0 B
'
' '
AB A B : 0
' '
BC B C : 0
' '
CD C D : 0
' '
DA D A : 0

6. One-to-one Mapping from a hyper-rectangle onto a rectangle
Local orthogonal coordinates ( s , n )
Cauchy Riemann Condition : ;
s n n s
Along AB : 0
0
s n
Along BC : 0
0
s n
Along CD : 0
0
s n
Along DA : 0
0
s n

7. Numerical Formulations of Mapping (BEM)
n
0 0
C
D
W i
2
0 2 2 2
2
0 W i 0
0 0 0
n
y
n
0
B
A
n
0
0
x
• Boundary integral element method (Liggett ＆ Liu,
1983):
ln r W
CW ( p ) W ln r d , C: interior angle
n n

8. Mapping an irregular area onto a hyper-rectangle
e.g.
y 2
w z
B i 2
( re )
2 i2
r e
0 A x B
'
O
'
A
'
Property : conformal except the origin
n
w z , n
α: original corner angle
β =π/2 or π

9. 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

10. Grid generation Applications
(A) step1
step2:
mapping onto a rectangle
'
' D
A
r0
'
y ' C
ri B
A B x C D
step4 step3:
establish the grids
'
A D
'
' '
B C
ln r / ri
Ref : Analytical mapping : 0
ln r0 / ri
, 0
/

11. Grid generation Applications
(B) step1
F
C
A B D E
step3:∠B 90º step2:∠A 90º

12. Grid generation Applications
(B) step4:∠C 180º
step5:∠D 90º
step6:∠E 90º
step7:∠F 180º

13. Grid generation Applications
(B)
step8: step9:
mapping onto a rectangle construct the grids
step11:∠F 90º step10:
transform to original domain

14. Grid generation Applications
(B) step12:∠E 45º step13:∠D 135º
step15:∠B 135º step14:∠C 270º

15. Grid generation Applications
(B)
step16:∠A 45º

16. Grid generation Applications
(C) D step1
A C
B
step3:∠B 90º
step2:∠A 90º

17. Grid generation Applications
(C) step4:∠C 90º step5:∠D 90º
step7: construct the grids step6:
mapping onto a rectangle

18. Grid generation Applications
step8:
(C) transform to original domain step9:∠D 180º
step11:∠B 180º step10:∠C 180º

19. Grid generation Applications
(C) step12:∠A 180º

20. Grid generation Applications
(D)
r0
ri
A B
O D C
Ref : Analytical mapping :
ln r / ri
0
/( 2 ) , 0
ln r0 / ri

21. Grid generation Applications
(E) Parallel sin-wave
Symmetrical sin-wave

22. Grid generation Applications
(F)

23. A bite of a moon cake

24. Two bites of a moon cake

25. A present of four moon cakes

28. The End
Thank you very much !