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

3. 3 力學聲響振動研究室 (MSVLAB) Literature review (Engineering background) EngineersYearProblemJournalTreatment He et al. (China) 1996 2-D potential and plane elasticity problem Computer & Structures CNME Necessary and sufficient boundary integral formulation (NSBIE) Zhou et al. (China) 19992-D elasticity problem CNME Boundary contour method (BCM) He, W. J. (China) 2000BIE---thin plate Computer & Structures Equivalent BIE Chen et al. (Taiwan) 2002Plane elasticity (Dual BIEM) IJNMEHypersingular formulation Chen et al. (Taiwan) 20022-D Laplace equation (Degenerate kernel) EABE Combined Helmholtz exterior integral equation formulation (CHEEF) Chen et al. (Taiwan) 20032-D Laplace and Navier problem IJNME Addition of rigid body term Hypersingular formulation CHEEF concept

4. 4 力學聲響振動研究室 (MSVLAB) Literature review (Mathematical background) MathematicianYearProblemJournal Treatments and logarithm capacity Soren Christiansen (Denmark) 1982 Detect non-unique solution Applicable Analysis Scaling method Restriction method Constanda (U. K.) 1995 Non-unique solution in plane elasticity problem Quart. Appl. Math. First kind integral equations Martin et al. (France) 1996 Invertibility of single layer potential operator Integr. Equat. Oper. Th. Logarithm capacity Soren Christiansen (Denmark) 1998 Investigation of direct BIE for biharmonic problem JCAMLogarithm capacity Soren Christiansen (Denmark) 2001 Detecting non- uniqueness of solution through SVD JCAMLogarithm capacity

5. 5 力學聲響振動研究室 (MSVLAB) Motivation (1) BIEM, BEM (2) MFS, Trefftz Method MethodsTechniques (1)Degenerate kernel (2)Circulants Membrane (Laplace equation) Plate (biharmonic equation) Statics Degenerate scale problem 1-D case (Euler beam)

6. 6 力學聲響振動研究室 (MSVLAB) S x r x (field point): variable s (source point): fixed Degenerate kernel x s O1O1 R O2O2 x s R O1O1 O2O2

7. 7 力學聲響振動研究室 (MSVLAB) Alternative derivations for the Poisson integral formula

8. 8 力學聲響振動研究室 (MSVLAB) G. E.: B. C. : a Derivation of the Poisson integral formula Traditional method Image source Null-field integral equation method Reciprocal radii method Poisson integral formula Image concept Methods Free of image concept Searching the image point Degenerate kernel

9. 9 力學聲響振動研究室 (MSVLAB) Null-field integral equation in conjunction with degenerate kernels B Boundary densities: Degenerate kernel Unknown coefficients unknown specified Fundamental solution Green’s identity

10. 10 力學聲響振動研究室 (MSVLAB) Degenerate scale for plate analysis using the BIEM and BEM

11. 11 力學聲響振動研究室 (MSVLAB) Engineering problem governed by biharmonic equation 1. Plane elasticity: 2. Slow viscous flow (Stokes’ Flow): 3. Solid mechanics (Plate problem):

12. 12 力學聲響振動研究室 (MSVLAB) Problem statement uniform pressure a B w=constant : flexure rigidity : uniform distributed load : domain of interest Governing equation: Boundary condition: Splitting method Governing equation: Boundary condition: : deflection of the circular plate

13. 13 力學聲響振動研究室 (MSVLAB) Boundary integral equations for plate (2) Slope (3) Normal moment (4) Effective shear force (1) Displacement

14. 14 力學聲響振動研究室 (MSVLAB) Operators Slope Normal moment Effective shear force

15. 15 力學聲響振動研究室 (MSVLAB) Kernel functions Fundamental solution: Kernel functions:

16. 16 力學聲響振動研究室 (MSVLAB) Degenerate kernels for biharmonic operator

17. 17 力學聲響振動研究室 (MSVLAB) Mathematical analysis --- Discrete model For the clamped circular plate (u and  are specified): formulation:

18. 18 力學聲響振動研究室 (MSVLAB) Circulant a 1 2 3 45 2N-1 2N-2 2N-3 2N2N

19. 19 力學聲響振動研究室 (MSVLAB) Eigenvalues of the four matrices kernel

20. 20 力學聲響振動研究室 (MSVLAB) Determinant Degenerate scale a

21. 21 力學聲響振動研究室 (MSVLAB) Degenerate scales for the clamped case Degenerate scales for the simply-supported case 6 options FormulationEquation of the degenerate scale in the BEM Never zero

22. 22 力學聲響振動研究室 (MSVLAB) Degenerate scale aa formulation

23. 23 力學聲響振動研究室 (MSVLAB) Degenerate scales for the free case FormulationEquation of the degenerate scale in the BEM Never zero except three rigid body modes

24. 24 力學聲響振動研究室 (MSVLAB) Relationship between the Laplace problem and biharmonic problem (a) translation: (b) rotation: constant

25. 25 力學聲響振動研究室 (MSVLAB) Nontrivial modes in FEM and BEM FEMBEM Rigid body mode Spurious mode (Hour-glass mode) (zero-energy mode) Rigid body modes Spurious mode (Null-field) Physically realizable Mathematical realizable Physically realizableMathematical realizable Q4 or Q8

26. 26 力學聲響振動研究室 (MSVLAB) Number of degenerate scales (Laplace problem) Laplace problem: UT formulation: LM formulation: No degenerate scale

27. 27 力學聲響振動研究室 (MSVLAB) Number of degenerate scales (biharmonic problem) formulation

28. 28 力學聲響振動研究室 (MSVLAB) Number of degenerate scales (biharmonic problem) formulation No degenerate scale occurs

29. 29 力學聲響振動研究室 (MSVLAB) Illustrative example (JFM, Mill 1977) a We adopt the null-field integral equation in conjunction with degenerate kernel to derive the analytic solution. Exact solution : M=20M=50

30. 30 力學聲響振動研究室 (MSVLAB) On the equivalence of the Trefftz method and MFS for Laplace and biharmonic equations

31. 31 力學聲響振動研究室 (MSVLAB) Trefftz method and MFS Method Trefftz methodMFS Definition Base u j (x) (T-complete function), r=|x-s| G. E. L u(x)=0, L U(x,s)=0, (singularity at s) Match B. C.Determine c j Determine w j is the number of complete functions is the number of source points in the MFS s D u(x) r D

32. 32 力學聲響振動研究室 (MSVLAB) Statement for Laplace problem Two-dimensional Laplace problem with a circular domain: G.E. : B.C. : B D D B Interior : Exterior : Analytical solution:

33. 33 力學聲響振動研究室 (MSVLAB) By matching the boundary condition at Interior problem: Exterior problem: Derivation of unknown coefficients (Trefftz method) Field solution: Interior : Exterior : T-complete set functions : Interior: Exterior:

34. 34 力學聲響振動研究室 (MSVLAB) Degenerate kernel : Interior problem: Exterior problem: Field solution: Interior : Exterior : Derivation of unknown coefficients (MFS)

35. 35 力學聲響振動研究室 (MSVLAB) Trefftz MFS Relationship between the two methods Interior: Exterior: By setting Trefftz method MFS ==

36. 36 力學聲響振動研究室 (MSVLAB) Matrix

37. 37 力學聲響振動研究室 (MSVLAB) Matrix ill-posed problem Degenerate scale problem ill-posed problem Degenerate scale problem ?

38. 38 力學聲響振動研究室 (MSVLAB) Circulants where Interior: Degenerate scale problem (R=1) a R=1 fail Exterior: Nonunique problem (a=1) a=1 R fail

39. 39 力學聲響振動研究室 (MSVLAB) Numerical Examples D B a x y Simply-connected problemMultiply-connected problem X Y D B a

40. 40 力學聲響振動研究室 (MSVLAB) Numerical Example 1 Trefftz method for the simply-connected problem Interior problemExterior problem Exact solutionNumerical solutionExact solutionNumerical solution 5 Points: B.C. aliasing base deficiency 9 Points: a=15 Points: B.C. aliasing Failure ( ) 9 Points: Failure ( ) a=25 Points: B.C. aliasing 9 Points:

41. 41 力學聲響振動研究室 (MSVLAB) MFS for the simply-connected problem Interior problemExterior problem Exact solutionNumerical solutionExact solutionNumerical solution 5 Points: B.C. aliasing 9 Points: 20 Points: a=1: 5 Points: B.C. aliasing Failure ( ln a) 9 Points: Failure ( ln a) a=2:5 Points: B.C. Aliasing 9 Points: 55 Points: Numerical Example 2

42. 42 力學聲響振動研究室 (MSVLAB) Numerical Example 3 Trefftz method for the multiply-connected problem Concentric circleEccentric circle Exact solutionNumerical solutionExact solutionNumerical solution 26 Points 6 Points 14 Points 26 Points

43. 43 力學聲響振動研究室 (MSVLAB) Numerical Example 4 MFS for the multiply-connected problem Concentric circleEccentric circle Exact solutionNumerical solutionExact solutionNumerical solution Inner circle: 20 points outer circle: 60points Inner circle: a 1 =0.9 outer circle :a 2 =2.6 Inner circle: 20 points outer circle: 60points Inner circle: 20 points outer circle: 60points Inner: 20points; outer: 60points; inner a 1 =0.9 outer a 2 =2.6 outer a 2 =3.0 outer a 2 =4.0 outer a 2 =10.0 Inner: 20points; outer: 60points; outer a 2 2.6 inner a 1 =0.5 inner a 1 =0.3

44. 44 力學聲響振動研究室 (MSVLAB) Trefftz method and MFS for biharmonic equation Analytical solution: Field solution : T-complete functions: Trefftz method: MFS: Field solution :

45. 45 力學聲響振動研究室 (MSVLAB) Relationship between the Trefftz method and MFS Coefficients of the Trefftz method Coefficients of the MFS Mapping matrix [K]

46. 46 力學聲響振動研究室 (MSVLAB) Decomposition of the K matrix

47. 47 力學聲響振動研究室 (MSVLAB) Diagonal matrix T R Existence of the degenerate scales Nonuniqueness (in numerical aspect) Degenerate scale problem O. K.!

48. 48 力學聲響振動研究室 (MSVLAB) Special size: : position of the source points The occurrence of the degenerate scales using the MFS a Mathematics: rank-deficiency problem (nonuniqueness problem) Numerical failure Degenerate scale problem

49. 49 力學聲響振動研究室 (MSVLAB) On the complete set of the Trefftz method and the MFS using the degenerate kernel T-complete functions of the Trefftz method: Degenerate kernel of the MFS: m=0 m=1 m=2, 3…..

50. 50 力學聲響振動研究室 (MSVLAB) Free terms for the biharmonic equation using the dual boundary integral equation

51. 51 力學聲響振動研究室 (MSVLAB) History of free terms in the dual BEM 2-D and 3-D Laplace problem 2-D and 3-D elasticity problem W. C. Chen thesis 2-D biharmonic problem (1)Bump-contour method (2)Taylor series expansion Free terms Dual boundary integral equation Improper integrals

52. 52 力學聲響振動研究室 (MSVLAB) Bump-contour method For a smooth boundary: x y B+B+ B-B- B’B’ B’B’ B’B’ D Singular point Explicit forms for the sixteen kernel functions

53. 53 力學聲響振動研究室 (MSVLAB) Taylor expansion for boundary density functions Boundary Domain vector component:

54. 54 力學聲響振動研究室 (MSVLAB) Free terms of dual BIE for Laplace problem 2-D problem: 3-D problem: Half Singular point

55. 55 力學聲響振動研究室 (MSVLAB) Dual boundary integral equations F.P. denotes the finite part for a smooth boundary Sixteen improper integrals Density functions are expanded by the Taylor series

56. 56 力學聲響振動研究室 (MSVLAB) Singular behavior of the sixteen kernels

57. 57 力學聲響振動研究室 (MSVLAB) Free terms due to the bump integral for the biharmonic equation

58. 58 力學聲響振動研究室 (MSVLAB) Dual boundary integral equations for the biharmonic problem After deriving the sixteen improper integrals, we have for a smooth boundary

59. 59 力學聲響振動研究室 (MSVLAB) Conclusions

60. 60 力學聲響振動研究室 (MSVLAB) 1.New methods to derive the Poisson integral formula by using the degenerate kernels and the null-field integral equations. 2.The occurring mechanism of degenerate scales depends on the formulation instead of the boundary conditions. 3.It is interesting to find that the T-complete set in the Trefftz method is imbedded in the degenerate kernels of MFS. 4.We adopt the bump-contour method to derive the free terms surrounding the singularity. For a smooth boundary, the sum of free terms are half. Conclusions

61. 61 力學聲響振動研究室 (MSVLAB) Thanks for your kind attention

62. 62 力學聲響振動研究室 (MSVLAB) Image method knownunknown Image method B Closed-form Green’s function

63. 63 力學聲響振動研究室 (MSVLAB) Closed-form Green’s function (Interior problem) a Image point

64. 64 力學聲響振動研究室 (MSVLAB) Series-form Green’s function (degenerate kernels)

65. 65 力學聲響振動研究室 (MSVLAB) Closed-form and series-form Green’s functions for interior and exterior problems Closed-form Series-form Interior problem: Exterior problem:

66. 66 力學聲響振動研究室 (MSVLAB) Poisson integral formula Cosine theorem Poisson integral formula Series-form:

67. 67 力學聲響振動研究室 (MSVLAB) Degenerate scale Rigid body mode Solve u u is a null field u is solved to be the rigid body solution Discriminant Laplace problem s Biharmonic problems Dirichlet Neumann Free Clamped Simply-supported Mathematically realizable Physically realizable Flowchart of the nontrivial modes

68. 68 力學聲響振動研究室 (MSVLAB) By setting Interior problem: Exterior problem: Trefftz method MFS == Connection between the Trefftz method and MFS