研究生：吳清森 指導教授：陳正宗 教授 陳義麟 博士

1. Degenerate scale analysis for membrane and plate problems using the meshless method and boundary element method 研究生：吳清森 指導教授：陳正宗　教授 　　　　　 陳義麟　博士 國立台灣海洋大學河海工程學系 結構組 碩士班論文口試 日期: 2004/06/16 09:00-10:20

2. Frame of the thesis Chapter 1 Introduction Free term and Jump term Degenerate kernel Chapter 5 Free terms for plate problem (Biharmonic problem) Chapter 6 Conclusions and further research Chapte3 BIEM and BEM for degenerate scale problem (Laplace and biharmonic problem) Chapter 2 Green’s function and Poisson integral formula (Laplace problem) Chapter 4 Meshless method for degenerate scale problem (Laplace and biharmonic problem)

3. Literature review (Engineering background)

4. Literature review (Mathematical background)

5. Motivation Methods Techniques (1) BIEM, BEM (2) MFS, Trefftz Method • Degenerate kernel • Circulants Statics 1-D case (Euler beam) Membrane (Laplace equation) Plate (biharmonic equation) Degenerate scale problem

6. Degenerate kernel x s x s O1 R R O1 x O2 O2 x (field point): variable s (source point): fixed r S

7. Alternative derivations for the Poisson integral formula

8. Derivation of the Poisson integral formula Methods G. E.: B. C. : Free of image concept Image concept Poisson integral formula Searching the image point Null-field integral equation method Reciprocal radii method Degenerate kernel a Image source Traditional method

9. Null-field integral equation in conjunction with degenerate kernels Green’s identity Fundamental solution B Boundary densities: specified unknown Degenerate kernel Unknown coefficients

10. Degenerate scale for plate analysis using the BIEM and BEM

11. Engineering problem governed by biharmonic equation 1. Plane elasticity: 2.Slow viscous flow (Stokes’ Flow): 3.Solid mechanics (Plate problem):

12. Problem statement w=constant uniform pressure a B Governing equation: Boundary condition: Splitting method Governing equation: Boundary condition: : deflection of the circular plate : flexure rigidity : uniform distributed load : domain of interest

13. Boundary integral equations for plate (1) Displacement (2) Slope (3) Normal moment (4) Effective shear force

14. Operators Slope Normal moment Effective shear force

15. Kernel functions Fundamental solution: Kernel functions:

16. Degenerate kernels for biharmonic operator

17. Mathematical analysis --- Discrete model formulation: For the clamped circular plate (u and  are specified):

18. Circulant 5 4 3 2 1 a 2N 2N-1 2N-2 2N-3

19. Eigenvalues of the four matrices kernel kernel kernel kernel

20. Determinant a Degenerate scale

21. Degenerate scales for the clamped case Degenerate scales for the simply-supported case 6 options

22. Degenerate scale formulation formulation a a

23. Degenerate scales for the free case

24. Relationship between the Laplace problem and biharmonic problem (a) translation: constant (b) rotation:

25. Nontrivial modes in FEM and BEM Q4 or Q8 Q4 or Q8

26. Number of degenerate scales (Laplace problem) Laplace problem: UT formulation: LM formulation: No degenerate scale

27. Number of degenerate scales (biharmonic problem) formulation formulation formulation

28. Number of degenerate scales (biharmonic problem) formulation formulation No degenerate scale occurs formulation

29. Illustrative example (JFM, Mill 1977) a We adopt the null-field integral equation in conjunction with degenerate kernel to derive the analytic solution. M=20 M=50 Exact solution :

30. On the equivalence of the Trefftz method and MFS for Laplace and biharmonic equations

31. r s u(x) u(x) D D Trefftz method and MFS is the number of complete functions is the number of source points in the MFS

32. Statement for Laplace problem D D B B Two-dimensional Laplace problem with a circular domain: B.C. : G.E. : Exterior : Interior : Analytical solution:

33. Derivation of unknown coefficients(Trefftz method) T-complete set functions : Interior: Exterior: Fieldsolution: Interior : Exterior : By matching the boundary condition at Interior problem: Exterior problem:

34. Derivation of unknown coefficients(MFS) Degenerate kernel : Fieldsolution: Interior : Exterior : Exterior problem: Interior problem:

35. Relationship between the two methods By setting = = Trefftz method MFS Trefftz MFS Interior: Exterior:

36. Matrix

37. Matrix ill-posed problem Degenerate scale problem ill-posed problem ? Degenerate scale problem

38. Circulants a Exterior: R=1 fail Nonunique problem(a=1) R fail a=1 where Interior: Degenerate scale problem(R=1)

39. Numerical Examples Y y a X x D B B a D

40. Numerical Example 1

41. Numerical Example 2

42. Numerical Example 3

43. Numerical Example 4

44. Trefftz method and MFS for biharmonic equation Trefftz method: Field solution : T-complete functions: MFS: Field solution : Analytical solution:

45. Relationship between the Trefftz method and MFS Mapping matrix [K] Coefficients of the MFS Coefficients of the Trefftz method

46. Decomposition of the K matrix

47. Diagonal matrix TR Existence of the degenerate scales O. K.! Nonuniqueness (in numerical aspect) Degenerate scale problem

48. The occurrence of the degenerate scales using the MFS Degenerate scale problem Mathematics: rank-deficiency problem (nonuniqueness problem) a Numerical failure Special size: : position of the source points

49. On the complete set of the Trefftz method and the MFS using the degenerate kernel m=1 m=1 m=0 m=0 m=2, 3….. m=2, 3….. T-complete functions of the Trefftz method: Degenerate kernel of the MFS:

50. Free terms for the biharmonic equation using the dual boundary integral equation