A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches

1. A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches M.I.G. BLOOR, M.J.WILSON Department of Applied Mathematics Leeds University

2. The PDE Method Parameter space Physical space is a partial differential operator (usually elliptic) of order m in the independent variables u and v.

3. Usual partial differential equation: (1) a = ‘smoothing parameter’ Solve over finite region subject to boundary conditions on function and parametric derivatives:

4. Basic PDE Surface

5. The PDE Method Often periodic boundary conditions are used: physical space

6. Basic PDE Surface

7. PDE Yacht Hull

8. Marine Propeller (each blade a single patch)

9. Wine Glass (three patches)

11. Lipid Membranes in Two Component Systems (Doubly Periodic Lawson Surface)

12. When the solution is periodic, we can, in principle, express it in the form: where

13. IN THIS SITUATION WE CAN FIND AN ANALYTIC APPROXIMATION TO THE SURFACE, AND THIS RESULTS IN A VERY FAST METHOD FOR GENERATING AND REGENERATING A PDE. HOWEVER, WHEN WE NEED A FOUR-SIDED PATCH, WE ARE BASICALLY FACED WITH SOLVING THE BIHARMONIC EQUATION IN A RECTANGULAR DOMAIN

14. THE SOLUTION OF TWO-DIMENSIONAL BIHARMONIC • EQUATION IS A CLASSICAL PROBLEM, WITH MANY • APPLICATIONS IN MECHANICS, • E.G. • CREEPING, VISCOUS FLOW IN A RECTANGULAR CAVITY • EQUILIBRIUM OF ELASTIC MEMBRANE • BENDING OF CLAMPED THIN ELASTIC PLATE SUBJECT • TO A NORMAL LOAD. ACCORDING TO MELESHKO (1998) ‘IT REPRESENTS A BENCHMARK PROBLEM FOR VARIOUS ANALYTICAL AND NUMERICAL METHODS’.

15. WE COULD USE A STANDARD NUMERICAL METHOD SUCH A FINITE-DIFFENCE OR FINITE-ELEMENT. BUT THE SURFACE PATCH WOULD BE REPRESENTED DISCRETELY. WE ARE SEEKING A FAST METHOD OF SOLUTION THAT PRODUCES A CONTINUOUS, ANALYTICAL APPROXIMATION TO THE SURFACE.

16. Approximating 4-sided PDE Surface Patches

17. BEFORE DEALING WITH 3 & 5 SIDED PATCHES, LET US CONSIDER A 4 SIDED PATCH: LETBE A REGULAR 4-SIDED PATCH BOUNDED BY 4 REGULAR SPACE CURVES SUCH THAT:

18. APPROACH:WE SEEK AN ANALYTIC APPROXIMATION OF THE FORM: WHERE REPRESENTS THE SUM OF SEPARABLE EIGENSOLUTIONS OF THE 4-ORDER OPERATOROF EQ (1) REPRESENTS A POLYNOMIAL SOLUTION OF EQ (1) THAT TO ENSURE THAT CORNER CONDITIONS ARE SATISFIED (SMALL) REMAINDER TERM TO ENSURE CONTINUITY AT PATCH BOUNDARIES

19. BOUNDARY CONDITIONS: Positional continuity at the corners implies: Boundary conditions on normal derivatives: Note: functions on RHS may be chosen to ensure tangent-plane continuity with adjacent patches.

20. Regularity at corners implies :

21. Continuity of twist vectors at corners implies :

22. WE NOW SEEK A POLYNOMIAL ‘CORNER’ SOLUTION OF THE FORM: (Note 28 vector constants to be determined) WHICH SATISFIES THE 12 CORNER CONDITIONS:

23. AND WHICH MATCHES THE 4 TWIST VECTORS: AND WHICH IS A SOLUTION OF EQ (1):

24. THIS GIVES 22 CONDITIONS WITH WHICH TO FIND THE 28 THE REMAINING 6 ARE OBTAINED FROM THE CONDITIONS:

25. FINDING THE EIGENSOLUTION The eigensolution is defined by and satisfies Eq (1) and also the modified (homogenous) boundary conditions:

26. Important to note that are all zero at the 4 corners of the patch LOOK FOR A SEPARABLE SOLUTION OF THE FORM THAT SATISFIES THE ABOVE HOMOGENEOUS BOUNDARY CONDITIONS

27. IT TURNS OUT THAT IS OF THE FORM A SO-CALLED PAPKOVICH-FADLE FUNCTION WHERE SATIFIES THE EIGENVALUE EQUATION (WITH COMPLEX ROOTS)

28. THUS IS OF THE FORM WHERE ARE CONSTANTS DETERMINED FROM THE BOUNDARY CONDITIONS BY A LEAST-SQUARES FIT Note that in practice we truncate the above series so that

29. NOW OUR APPROXIMATE SOLUTION IS GIVEN BY WHICH IS APPROXIMATE IN THE SENSE THAT BOUNDARY CONDITIONS ARE NOT EXACTLY SATISIFIED AT ALL POINTS ON BOUNDARIES. TO ENSURE GEOMETRIC CONTINUITY ADD IN A REMAINDER TERM THUS TO MAKE SURE THAT SATISFIES THE BOUNDARY CONDITIONS.

30. NOTE THAT AS THE NUMBER OF TERMS N INCLUDED IN THE SERIES FOR INCREASES, THEN GENERALLY DECREASES. NOTE THAT IN THIS WORK IT IS CONVENIENT TO CHOOSE TO BE A COON’S PATCH. NOTE THAT WE HAVE AN ANALYTIC EXPRESSION FOR EVERYWHERE.

31. EXAMPLE: Section of blend between circular cylinder and a flat plane at to the cylinder axis

32. Second example of approximation to 4-sided PDE surface patch Corresponding polynomial corner solution

33. Third example of approximation to 4-sided PDE surface patch

34. Approximating 5-sided PDE Surface Patches

35. PROCEED BY ASSUMING THAT PATCH IS PRODUCED BY MAPPING FROM RECTANGULAR REGION OF PARAMETER SPACE AS BEFORE. AND THAT 4 OF THE 5 VERTICES COINCIDE WITH THE CORNERS OF WITHOUT LOSS OF GENERALITY CHOOSE THE FIFTH VERTEX TO LIE ALONG U=1, THUS:

36. Positional boundary conditions along edges as before: where is continuous in v but may have a discontinuous derivative at singularity. Derivative conditions as before: Where could be discontinuous at singularity.

37. ASSUME THAT ALL CONDITIONS ON THE FUNCTION AND ITS DERIVATIVES AT CORNERS OF HOLD AS FOR THE 4-SIDED PATCH NOW LOOK FOR A SINGULARITY SOLUTION WHICH WILL GIVE THE FORM OF THE SOLUTION IN THE NEIGHBOURHOOD OF THE SINGULARITY

38. USING LOCAL POLAR COORDINATES EQUATION (1) SATISFIED BY BECOMES

39. EXPAND BOUNDARY CONDITIONS ABOUT (1,vs) FOR SMALL VALUES OF ALONG AND DENOTING COORDINATE(S) WITH SINGULARITY LOOK FOR SOLUTION OF THE FORM THE VALUE OF DETERMINED FROM DEPENDENCE OF BOUNDARY CONDITIONS, AND THE 4 ARBITRARY CONSTANTS IN CAN BE FIXED FROM THE 4 BOUNDARY CONDITIONS

40. REPEAT FOR ALL COORDINATES WITH A SINGULARITY TO FIND THE COMPLETE LOCAL SOLUTION NOW INTRODUCE A SOLUTION DEFINED BY WHICH SATISFIES MODIFIED BOUNDARY CONDITIONS

41. NOTE THAT SATISFIES EQ (1) AND IS REGULAR. THUS CAN BE FOUND BY WRITING AND USING THE METHOD FOR THE 4-SIDED PATCH, I.E.

42. EXAMPLE: are cubics chosen so that consistency conditions are satisfied at corners Singularity at (1,0.5)

43. FOLLOWING METHOD OUTLINED ABOVE, AND IDENTIFYING WITH Z COORDINATE, THE FOLLOWING BOUNDARY CONDITIONS ON APPLY: A solution for can be found Where Hence the solution can be found

44. EXAMPLE: 5 -sided patch with remainder term.

45. 5 -sided patch without remainder term.

46. Polynomial Corner Solution

47. Singularity Solution

48. Approximating 3-sided PDE Surface Patches

49. PROCEED AS BEFORE USING PARAMETRIC MAPPING FROM 4-SIDED DOMAIN CONSTANT

50. Boundary conditions on normal derivatives as before: But note not readily available from adjacent patches and so must be chosen with care to satisfy regularity conditions on parametric derivatives. OTHERWISE PROCEED AS BEFORE TO SEEK SOLUTION OF THE FORM