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# Numerical conformal mapping of an irregular area

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### Numerical conformal mapping of an irregular area

1. 1. Numerical Conformal Mapping of an Irregular Area BY TARUN GEHLOTS
2. 2. Contents1. Mapping between two orthogonal coordinates2. One-to-one Mapping from a hyper-rectangle onto a rectangle3. Numerical Formulations of Mapping (BEM)4. Mapping an irregular area onto a hyper-rectangle5. Grid generation Applications
3. 3. Mapping between two orthogonal coordinatesForward 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. 4. Mapping between two orthogonal coordinatesBackward 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. 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 Cz 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. 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. 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. 8. Mapping an irregular area onto a hyper-rectanglee.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. 9. Mapping an Irregular area into a Rectangle1. Forward mapping the irregular area into a hyper-rectangle, w=zn2. Forward mapping the hyper-rectangle into a rectangle, ▽2ξ=0; ▽2η=03. Backward mapping the rectangle into the hyper-rectangle, ▽2x=0; ▽2y=04. Backward mapping the hyper-rectangle into the irregular area, z=w1/n
10. 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. 11. Grid generation Applications(B) step1 F C A B D E step3:∠B 90º step2:∠A 90º
12. 12. Grid generation Applications(B) step4:∠C 180º step5:∠D 90º step6:∠E 90º step7:∠F 180º
13. 13. Grid generation Applications(B) step8: step9: mapping onto a rectangle construct the grids step11:∠F 90º step10: transform to original domain
14. 14. Grid generation Applications(B) step12:∠E 45º step13:∠D 135º step15:∠B 135º step14:∠C 270º
15. 15. Grid generation Applications(B) step16:∠A 45º
16. 16. Grid generation Applications(C) D step1 A C B step3:∠B 90º step2:∠A 90º
17. 17. Grid generation Applications(C) step4:∠C 90º step5:∠D 90º step7: construct the grids step6: mapping onto a rectangle
18. 18. Grid generation Applications step8:(C) transform to original domain step9:∠D 180º step11:∠B 180º step10:∠C 180º
19. 19. Grid generation Applications(C) step12:∠A 180º
20. 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. 21. Grid generation Applications(E) Parallel sin-wave Symmetrical sin-wave
22. 22. Grid generation Applications(F)
23. 23. A bite of a moon cake
24. 24. Two bites of a moon cake
25. 25. A present of four moon cakes
26. 26. The EndThank you very much !