Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Numerical conformal mapping of an irregular area

977 views

Published on

Published in: Education
  • Be the first to comment

  • Be the first to like this

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 !

×