numerical conformal mapping of an irregular area
TRANSCRIPT
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 !