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M M S S V V 0 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods 研 究 生 : 李應德 指導教授 : 陳正宗 教授 陳義麟 博士 時 間 : 2004 年 06.

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Presentation on theme: "M M S S V V 0 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods 研 究 生 : 李應德 指導教授 : 陳正宗 教授 陳義麟 博士 時 間 : 2004 年 06."— Presentation transcript:

1 M M S S V V 0 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods 研 究 生 : 李應德 指導教授 : 陳正宗 教授 陳義麟 博士 時 間 : 2004 年 06 月 16 日 Free vibration of membrane and plate problems by using meshless methods 國立臺灣海洋大學河海工程研究所結構組 碩士班畢業論文口試

2 M M S S V V 1 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statements 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

3 M M S S V V 2 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statements 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

4 M M S S V V 3 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Literature review (MFS) Kupradze (1964) --- MFS J. T. Chen et al. (2003) --- Annular membrane T. Tsangaris et al. (2004)--- Annular plate (statics) 1. 3. 5. 2.Der-Liang Young. (2003) --- Clamped plate (Real-part) 4.J. T. Chen et al. (2004) --- Plate vibration (Imaginary-part) (Circulant)

5 M M S S V V 4 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Literature review (Spurious eigenequation of BEM) 1.Tai and Shaw 1974 (complex-valued BEM) 2.De Mey 1976, Hutchinson and Wong 1979 (real-part kernel) 3.Wong and Hutchinson (real-part direct BEM program) 4.Shaw 1979, Hutchinson 1988, Niwa et al. 1982 (real- part kernel) 5.Tai and Shaw 1974, Chen et al. Proc. Roy. Soc. Lon. Ser. A, 2001, 2003 (multiply-connected problem) 6.Chen et al. (dual formulation, domain partition, SVD updating technique, CHEEF method)

6 M M S S V V 5 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Successful experiences of spurious eigenvalue (BEM) RealImaginaryComplex Saving CPU timeYes No Avoid singular integralNoYesNo Spurious eigenvaluesAppear No Simply-connected problem Multiply-connected problem Complex Spurious eigenvalues Appear (Membrane and plate)

7 M M S S V V 6 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Comparison M. H. Chang (2001) S. Y. Lin (2003) Present thesis (2004) Imaginary-part MFS 2-D and 3-D acoustics Imaginary-part, real-part and complex- valued BEM 2-D plate Imaginary-part, real-part and complex- valued MFS 2-D membrane and plate Simply and multiply-connected problem simply-connected problem Simply and multiply-connected problem

8 M M S S V V 7 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Comparison Real-partImaginary-partComplex-valued Simply-connectedMultiply-connected Plate Acoustics (membrane) MFSBEM M. H. Chang (2001) S. Y. Lin (2003) Present thesis (2004)

9 M M S S V V 8 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Method of fundamental solutions Field representation: xixi sjsj r ij s j+1 Interior problemExterior problem xixi sjsj r ij s j+1 Source point Observation point

10 M M S S V V 9 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statements 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

11 M M S S V V 10 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Topics in the thesis PlateMembrane Simply- connected Multiply- connected Problem Domain Chapter 2 Chapter 3 Chapter 4 Chapter 5

12 M M S S V V 11 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Governing equation Plate vibrationMembrane vibration

13 M M S S V V 12 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Fundamental solution Plate vibration Membrane vibration

14 M M S S V V 13 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Influence matrices L ’ Hôpital ’ s rule Invariant method Indeterminate forms ( ) The diagonal terms of imaginary-part MFS RBF for

15 M M S S V V 14 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected problem real boundary fictitious boundary

16 M M S S V V 15 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected problem real boundaryfictitious boundary

17 M M S S V V 16 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Topics in the thesis PlateMembrane Simply- connected Multiply- connected Chapter 2 Chapter 3 Chapter 4 Chapter 5

18 M M S S V V 17 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Methods of solution Single-layer potential approachDouble-layer potential approach

19 M M S S V V 18 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected membrane For the Dirichlet problem (u=0) By using single-layer potential approach By using double-layer potential approach det[U ij ]=0 det[T ij ]=0

20 M M S S V V 19 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected membrane For the Neumann problem (t=0) By using single-layer potential approach By using double-layer potential approach det[L ij ]=0 det[M ij ]=0

21 M M S S V V 20 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Topics in the thesis PlateMembrane Simply- connected Multiply- connected Chapter 2 Chapter 3 Chapter 4 Chapter 5

22 M M S S V V 21 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected membrane For the Dirichlet-Dirichlet problem (u=0) By using single-layer potential approach det[SM D1 ]=0 By using double-layer potential approach det[SM D2 ]=0

23 M M S S V V 22 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected membrane For the Neumann-Neumann problem (u=0) By using single-layer potential approach det[SM N1 ]=0 By using double-layer potential approach det[SM N2 ]=0

24 M M S S V V 23 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Topics in the thesis PlateMembrane Simply- connected Multiply- connected Chapter 2 Chapter 3 Chapter 4 Chapter 5

25 M M S S V V 24 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Kernel functions of plate vibration L L L Fundamental solution

26 M M S S V V 25 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Operators L L L

27 M M S S V V 26 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Displacement, Slope, Moment and Shear Displacement Slope Moment Shear L L L

28 M M S S V V 27 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Matrix form Clamped Free Simply-supported

29 M M S S V V 28 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected plate For the clamped boundary condition ( u=0, ) By using formulation

30 M M S S V V 29 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Topics in the thesis PlateMembrane Simply- connected Multiply- connected Chapter 2 Chapter 3 Chapter 4 Chapter 5

31 M M S S V V 30 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected plate For the clamped-clamped boundary condition ( u=0, ) By using formulation B1B1 B2B2 B1B1 B2B2

32 M M S S V V 31 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statement 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

33 M M S S V V 32 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Degenerate kernels for circular case (0,0) X1X1 Y1Y1 R1R1  1 Y2Y2 (0,0) X2X2 R2R2 22 I U E U

34 M M S S V V 33 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Degenerate kernels Membrane Plate

35 M M S S V V 34 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Circulant Discritization into 2N nodes on the circular boundary

36 M M S S V V 35 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Circulant : eigenvalue of C 2N

37 M M S S V V 36 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Eigenvalues Riemann sum reduces to integral: Membrane: Plate:

38 M M S S V V 37 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Eigenvalues of matrices is an unitary matrix

39 M M S S V V 38 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected membrane For Dirichlet problem (u=0) By using single-layer potential approach det[U]=0 Never zero True eigenequation:

40 M M S S V V 39 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Results of analytical derivation Boundary condition Dirichlet problem Neumann problem Eigenequation True eigenequation Spurious eigenequation True eigenequation Spurious eigenequation Single-layer potential approach Double-layer potential approach No Complex-valued MFS

41 M M S S V V 40 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Results of analytical derivation Boundary condition Dirichlet problem Neumann problem Eigenequation True eigenequation Spurious eigenequation True eigenequation Spurious eigenequation Single-layer potential approach Double-layer potential approach Imaginary-part MFS

42 M M S S V V 41 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Results of analytical derivation Boundary condition Dirichlet problem Neumann problem Eigenequation True eigenequation Spurious eigenequation True eigenequation Spurious eigenequation Single-layer potential approach Double-layer potential approach Real-part MFS

43 M M S S V V 42 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected membrane For Dirichlet-Dirichlet problem (u=0) By using single-layer potential approach det[SM D1 ]=0 True eigenequation: Spurious eigenequation: Never zero

44 M M S S V V 43 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Results of analytical derivation Boundary condition Dirichlet- Dirichlet Dirichlet- Neumann Eigenequation True eigenequation Spurious eigenequation True eigenequation Spurious eigenequation Single-layer potential approach Double-layer potential approach Multiply-connected membrane

45 M M S S V V 44 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Results of analytical derivation Neumann- Dirichlet Multiply-connected membrane Boundary condition Neumann- Neumann Eigenequation True eigenequation Spurious eigenequation True eigenequation Spurious eigenequation Single-layer potential approach Double-layer potential approach

46 M M S S V V 45 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Simply-connected plate True eigenequation: For clamped boundary condition ( u=0, ) By using formulation Never zero

47 M M S S V V 46 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True and spurious eigenequations Boundary condition Clamped Simply- supported Free True Eigenequation Spurious No True Spurious True Spurious Complex-valued MFS

48 M M S S V V 47 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Comparisons of Leissa and present method Leissa (Kitahara)Present method Clamped Simply- supported Free

49 M M S S V V 48 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True and spurious eigenequations Boundary condition Clamped Simply- supported Free True Eigenequation Spurious True Spurious True Spurious Imaginary-part MFS

50 M M S S V V 49 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiplicity Boundary condition Clamped Simply- supported Free Eigenequation Imaginary-part MFS Formulation

51 M M S S V V 50 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True and spurious eigenequations Boundary condition Clamped Simply- supported Free True Eigenequation Spurious True Spurious True Spurious Real-part MFS

52 M M S S V V 51 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True and spurious eigenequations TrueSpurious Real Imaginary ComplexNo TrueSpurious Real Imaginary ComplexNo Membrane (Dirichlet) --- Single-layer potential approach Plate (Clamped) --- formulation

53 M M S S V V 52 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected plate For clamped-clamped boundary condition ( u=0, ) By using formulation True eigenequation Spurious eigenequation

54 M M S S V V 53 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Discussion of spurious eigenequation Never zero for anyλ =0

55 M M S S V V 54 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True eigenequation of clamped plate a Spurious eigenequation of clamped-clamped case:

56 M M S S V V 55 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods True eigenequations for the annular plate B. C. C-C S-S F-F

57 M M S S V V 56 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Spurious eigenequations for the annular plate Formulation Spurious eigenequations for the annular plate Formulation

58 M M S S V V 57 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statement 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

59 M M S S V V 58 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Treatment methods SVD updating document SVD updating term Burton & Miller method Extraction true Extraction spurious

60 M M S S V V 59 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statement 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

61 M M S S V V 60 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Complex-valued MFS) Single-layer potential approachDouble-layer potential approach

62 M M S S V V 61 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Imaginary-part MFS) Single-layer potential approachDouble-layer potential approach

63 M M S S V V 62 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Real-part MFS) Single-layer potential approachDouble-layer potential approach

64 M M S S V V 63 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Real-part MFS) Single-layer potential approach +SVD updating document Single-layer potential approach +Burton & Miller method

65 M M S S V V 64 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Real-part MFS) Single-layer potential approach +SVD updating term

66 M M S S V V 65 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 1 (k 1 =2.045) Present methodAnalytical solution

67 M M S S V V 66 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 2 (k 2 =3.083) Present methodAnalytical solution

68 M M S S V V 67 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 3 (k 3 =5.135) Present methodAnalytical solution

69 M M S S V V 68 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet-Dirichlet problem (Complex-valued MFS) Single-layer potential approachDouble-layer potential approach

70 M M S S V V 69 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet-Dirichlet problem (Complex-valued MFS) Single-layer potential approach + SVD updating document Single-layer potential approach + Burton & Miller method

71 M M S S V V 70 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Dirichlet problem (Real-part MFS) Single-layer potential approach + SVD updating term

72 M M S S V V 71 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 1 (k 1 =2.05) Present methodFEMBEM

73 M M S S V V 72 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 2 (k 2 =2.23) Present methodFEMBEM

74 M M S S V V 73 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Mode 3 (k 3 =2.66) Present methodFEMBEM

75 M M S S V V 74 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Clamped plate Complex-valued MFS

76 M M S S V V 75 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Clamped plate Real-part MFS Imaginary-part MFS

77 M M S S V V 76 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Clamped plate Real-part MFS +SVD updating document Real-part MFS +SVD updating term

78 M M S S V V 77 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Multiply-connected plate Clamped-clamped boundary Simply-supported-simply-supported boundary

79 M M S S V V 78 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Outlines 3. Mathematical analysis 2. Problem statement 1. Introduction 5. Numerical examples 6. Conclusions 4. Treatment methods

80 M M S S V V 79 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Conclusions We have verified that the true eigenequation depends on the boundary condition while spurious eigenequation is embedded in each formulation. For the imaginary-part MFS, the sources can be distributed on the real boundary without any difficulty. However, the sources must be distributed outside the domain to avoid the singularity when the real-part and complex- valued MFS are utilized. Spurious and true eigenequations for the simply-connected eigenproblems were analytically derived by using the degenerate kernels and circulants. 1. 2. 3.

81 M M S S V V 80 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Conclusions The spurious eigenvalues occurring in the multiply-connected eigenproblem for membrane and plate are the true eigenvalues of the associated problem bounded by the inner fictitious boundary where the sources are distributed. We provide the general form of the true eigenequation for the three cases of the eigenequations of annular plates instead of the separate form in the Leissa's book. Three remedies, the SVD updating document, the SVD updating term and the Burton & Miller method,were successfully employed to suppress the appearance of the spurious eigenvalues for simply-connected and multiply- connected eigenproblems not only in membrane vibration but also in the plate vibration. 4. 5. 6.

82 M M S S V V 81 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Thanks for your kind attention The End

83 M M S S V V 82 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Further research A general program for solving the arbitrarily-shaped plate eigenproblem needs to be developed If the wave number k in the membrane or the frequency parameter λ in the plate approach to zero, it may work for the determination of degenerate scale for static problem. Optimal location should be studied for general applications. Base on the present method, the possible extension to problem, e.g., corner, degenerate boundary, Stokes' flow, exterior acoustic problem and 3-D biharmonic problem may be studied in the further work. The property of Calderon projector for the sixteen kernel functions of biharmonic operator needs further investigation. 1. 2. 3. 4. 5.

84 M M S S V V 83 MSVLAB HRE, NTOU Free vibration of membrane and plate problems by using meshless methods Comparisons of the NDIF and present method KangPresent method RBF Clamped plate No Simply- supported plate Treatment Net approach CHEEF method Dual formulation with SVD updating

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