MSVLAB HRE, NTOU Mathematical analysis and numerical study for free vibration of plate using BEM 研究生：林盛益 指導教授：陳正宗 教授 陳義麟 博士 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 日期： 2003/6/6 Mathematical analysis and numerical study for free vibration of plate using BEM- 1

MSVLAB HRE, NTOU Outlines Mathematical analysis and numerical study for free vibration of plate using BEM- 2 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research

MSVLAB HRE, NTOU Introduction circular frequency G.E. Poisson ratio domain frequency parameter surface density flexural rigidityplate thickness Young's modulus lateral displacement Mathematical analysis and numerical study for free vibration of plate using BEM- 3

MSVLAB HRE, NTOU Free vibration of plate Mathematical analysis and numerical study for free vibration of plate using BEM- 4

MSVLAB HRE, NTOU Literature review 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 (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) Mathematical analysis and numerical study for free vibration of plate using BEM- 5

MSVLAB HRE, NTOU Motivation RealImaginaryComplex Saving CPU time Yes No Avoid singular integral NoYesNo Spurious eigenvalues Appear No Simply-connected problem Multiply-connected problem Complex Spurious eigenvalues Appear Mathematical analysis and numerical study for free vibration of plate using BEM- 6

MSVLAB HRE, NTOU Spurious eigenvalues Spurious eigenvalues occur in two aspects: 1.Simply-connected eigenproblem by using the real-part or imaginary- part BEM. 2.Multiply-connected eigenproblem by using the complex-valued BEM. Mathematical analysis and numerical study for free vibration of plate using BEM- 7 Real-part BEM (N.G.) Imaginary-part BEM (N.G.) Complex-valued (OK) Complex-valued (N.G.)

MSVLAB HRE, NTOU 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research Mathematical analysis and numerical study for free vibration of plate using BEM- 8

MSVLAB HRE, NTOU Boundary integral equations for plate eigenproblems (2) Slope (3) Normal moment (4) Effective shear force (1) Displacement Mathematical analysis and numerical study for free vibration of plate using BEM- 9

MSVLAB HRE, NTOU Operators Mathematical analysis and numerical study for free vibration of plate using BEM- 10 Slope Normal moment Effective shear force

MSVLAB HRE, NTOU Kernel functions Fundamental solution Mathematical analysis and numerical study for free vibration of plate using BEM- 11

MSVLAB HRE, NTOU Mathematical analysis (Discrete system) For clamped circular plate (u=0 and  =0) Mathematical analysis and numerical study for free vibration of plate using BEM- 12

MSVLAB HRE, NTOU Expansion formulae Degenerate kernels (separable kernels) Mathematical analysis and numerical study for free vibration of plate using BEM- 13

MSVLAB HRE, NTOU Circulant Mathematical analysis and numerical study for free vibration of plate using BEM- 14

MSVLAB HRE, NTOU The properties of the circulant Mathematical analysis and numerical study for free vibration of plate using BEM- 15

MSVLAB HRE, NTOU Eigenvalues of the four matrices Mathematical analysis and numerical study for free vibration of plate using BEM- 16

MSVLAB HRE, NTOU Eigenvalue decomposition (similar matrices) Mathematical analysis and numerical study for free vibration of plate using BEM- 17

MSVLAB HRE, NTOU Determinant Mathematical analysis and numerical study for free vibration of plate using BEM- 18

MSVLAB HRE, NTOU Eigenequations (real-part BEM for clamped plate) Spurious eigenequationTrue eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 19

MSVLAB HRE, NTOU Comparisons of Leissa and present method Leissa (Kitahara)Present method Clamped Simply- supported Free Mathematical analysis and numerical study for free vibration of plate using BEM- 20 where

MSVLAB HRE, NTOU Spurious eigenequations using the real-part BEM Eqs. number Spurious eigenequation using the real-part BEM u, θ (1) and (2) u,m (1) and (3) u,v (1) and (4) θ,m (2) and (3) θ,v (2) and (4) m,v (3) and (4) where Mathematical analysis and numerical study for free vibration of plate using BEM- 21

MSVLAB HRE, NTOU Spurious eigenequations using the imaginary-part BEM Eqs. number Spurious eigenequation using the imaginary-part BEM u, θ (1) and (2) u,m (1) and (3) u,v (1) and (4) θ,m (2) and (3) θ,v (2) and (4) m,v (3) and (4) where Mathematical analysis and numerical study for free vibration of plate using BEM- 22

MSVLAB HRE, NTOU True and spurious eigenequations TrueSpurious Real (u,  ) Imaginary (u,  ) Complex (u,  )- TrueSpurious Real (UT) Imaginary (UT) Complex (UT)- Mathematical analysis and numerical study for free vibration of plate using BEM- 23 Membrane (Dirichlet) Plate (clamped)

MSVLAB HRE, NTOU Determinant v.s frequency parameter (real-part BEM) Spurious eigenequation True eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 24

MSVLAB HRE, NTOU 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research Mathematical analysis and numerical study for free vibration of plate using BEM- 25

MSVLAB HRE, NTOU Treatments 1.SVD updating term 2.The Burton & Miller concept 3.The complex-valued BEM (conventional BEM) 4.The CHEEF concept Mathematical analysis and numerical study for free vibration of plate using BEM- 26

MSVLAB HRE, NTOU SVD updating term (clamped plate) Mathematical analysis and numerical study for free vibration of plate using BEM- 27 u,  formulation m,v formulation SVD technique of updating term

MSVLAB HRE, NTOU Determinant The only possibility for zero determinant at the same time for the same. Mathematical analysis and numerical study for free vibration of plate using BEM- 28

MSVLAB HRE, NTOU The Burton & Miller concept (clamped plate) Since the term [A( )+i B( )] is never zero for any, we can obtain the true eigenvalues The combination of (u,  ) and (m, v) real-part BEM in conjunction with the Burton & Miller concept fails. Mathematical analysis and numerical study for free vibration of plate using BEM- 29

MSVLAB HRE, NTOU The complex-valued BEM (clamped plate) Notes: Since the term [A( )+i B( )] is never zero for any, we can obtain the true eigenvalues True eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 30

MSVLAB HRE, NTOU The CHEEF concept (clamped plate) u,  formulation CHEEF points Mathematical analysis and numerical study for free vibration of plate using BEM- 31

MSVLAB HRE, NTOU Comparison of the matrix dimension DimensionMatrix type SVD updating term 8N  4NReal (Imag.) Burton & Miller method 4N  4NComplex Complex-valued BEM 4N  4NComplex CHEEF method 2(2N+Nc)  4NReal (Imag.) Mathematical analysis and numerical study for free vibration of plate using BEM- 32

MSVLAB HRE, NTOU SVD updating term (real-part BEM) Mathematical analysis and numerical study for free vibration of plate using BEM- 33

MSVLAB HRE, NTOU The Burton & Miller concept (real-part BEM) Mathematical analysis and numerical study for free vibration of plate using BEM- 34 Fail (u,  ) + i (m, v)

MSVLAB HRE, NTOU The Burton & Miller method (real-part BEM) Mathematical analysis and numerical study for free vibration of plate using BEM- 35 Success (u, m) + i ( , v)

MSVLAB HRE, NTOU The complex-valued BEM Mathematical analysis and numerical study for free vibration of plate using BEM- 36

MSVLAB HRE, NTOU The CHEEF concept (real-part BEM) Mathematical analysis and numerical study for free vibration of plate using BEM- 37

MSVLAB HRE, NTOU 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research Mathematical analysis and numerical study for free vibration of plate using BEM- 38

MSVLAB HRE, NTOU Mathematical analysis (Discrete system) Mathematical analysis and numerical study for free vibration of plate using BEM- 39 For clamped-clamped annular plate Notes:

MSVLAB HRE, NTOU Eigenequations (C-C annular plate) Spurious eigenequation True eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 40

MSVLAB HRE, NTOU Physical meaning of the spurious eigenequation Spurious eigenequation of the u,  formulation (multiply-connected: The radii of the outer and inner circles are a and b) True eigenequation of the clamped case (simply-connected plate: The radius is b) Mathematical analysis and numerical study for free vibration of plate using BEM- 41

MSVLAB HRE, NTOU True eigenequations for the annular plate B.C. C-C S-S F-F Mathematical analysis and numerical study for free vibration of plate using BEM- 42

MSVLAB HRE, NTOU Spurious eigenequations for the annular plate Eqs. number Spurious eigenequations for the annular plate u, θ (1) and (2) u,m (1) and (3) u,v (1) and (4) θ,m (2) and (3) θ,v (2) and (4) m,v (3) and (4) Mathematical analysis and numerical study for free vibration of plate using BEM- 43

MSVLAB HRE, NTOU Determinant v.s frequency parameter (C-C) Spurious eigenequation True eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 44 imply

MSVLAB HRE, NTOU Physical meaning of the spurious eigenequation Spurious eigenequation (u,  ) (multiply-connected) True eigenequation (clamped) (simply-connected plate) Mathematical analysis and numerical study for free vibration of plate using BEM- 45

MSVLAB HRE, NTOU 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research Mathematical analysis and numerical study for free vibration of plate using BEM- 46

MSVLAB HRE, NTOU Treatments SVD updating term Burton & Miller method CHIEF method Mathematical analysis and numerical study for free vibration of plate using BEM- 47

MSVLAB HRE, NTOU SVD updating term Mathematical analysis and numerical study for free vibration of plate using BEM- 48

MSVLAB HRE, NTOU The Burton & Miller concept Mathematical analysis and numerical study for free vibration of plate using BEM- 49 Success (u, m) + i ( , v)

MSVLAB HRE, NTOU The CHIEF concept Mathematical analysis and numerical study for free vibration of plate using BEM- 50

MSVLAB HRE, NTOU 1.Introduction 2.BEM for the free vibration of simply-connected plate 3.Treatments of the spurious eigenvalues for simply- connected plate 4.BEM for the free vibration of multiply-connected plate 5.Treatments of the spurious eigenvalues for multiply- connected plate 6.Conclusions and further research Mathematical analysis and numerical study for free vibration of plate using BEM- 51

MSVLAB HRE, NTOU Conclusions 1.The true and spurious eigenequations depend on the B. C. and formulation, respectively. 2.The spurious eigenvalue in multiply-connected plate is the true eigenvalue of the associated simply- connected problem. 3.We provide the general form of the true eigenequation for the annular plates instead of the separate form. 4.The SVD updating term, Burton & Miller method and CHEEF (CHIEF) method, were successfully applied to suppress the spurious eigenvalues. Mathematical analysis and numerical study for free vibration of plate using BEM- 52

MSVLAB HRE, NTOU Further research 1.A general-purpose program for arbitrarily-shaped plate. The principal value P.V. and the free term. 2.Extend to other biharmonic problems (Stokes' flow or bulcking of beam and plate). 3.The boundary element method for the static problem (  0 ). 4.A plate with multiple holes (a truly multiply-connected problems). 5.The extension of the BEM approach to meshless formulation. Mathematical analysis and numerical study for free vibration of plate using BEM- 53

MSVLAB HRE, NTOU Thanks for your kind attention The End Mathematical analysis and numerical study for free vibration of plate using BEM- 54

MSVLAB HRE, NTOU Real-part BEM for simply-connected plate Mathematical analysis and numerical study for free vibration of plate using BEM- 55

MSVLAB HRE, NTOU Imaginary-part BEM for simply-connected plate Mathematical analysis and numerical study for free vibration of plate using BEM- 56

MSVLAB HRE, NTOU Complex-valued BEM for simply-connected plate Mathematical analysis and numerical study for free vibration of plate using BEM- 57

MSVLAB HRE, NTOU Complex-valued BEM for multiply-connected plate Mathematical analysis and numerical study for free vibration of plate using BEM- 58

MSVLAB HRE, NTOU Real-part BEM (u,  ) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 59

MSVLAB HRE, NTOU Real-part BEM (u, m) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 60

MSVLAB HRE, NTOU Real-part BEM (u,v) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 61

MSVLAB HRE, NTOU Real-part BEM ( , m) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 62

MSVLAB HRE, NTOU Real-part BEM ( , v) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 63

MSVLAB HRE, NTOU Real-part BEM (m,v) for simply-connected clamped plate Mathematical analysis and numerical study for free vibration of plate using BEM- 64

MSVLAB HRE, NTOU Membrane or acoustic problem True and spurious eigenvalues True eigenvalues True and spurious eigenvalues De Mey Tai and Shaw Hutchinson Kitahara Mathematical analysis and numerical study for free vibration of plate using BEM- 65

MSVLAB HRE, NTOU Hilbert-transform The Hilbert transform is the constraint in the frequency domain corresponding to the casual effect in the time- domain fundamental solution. Mathematical analysis and numerical study for free vibration of plate using BEM- 66 Hilbert transform Definition as convolution integral

MSVLAB HRE, NTOU Fundamental solution Mathematical analysis and numerical study for free vibration of plate using BEM- 67 Singular solution Regular solution Hilbert transform

MSVLAB HRE, NTOU Burton & Miller method Mathematical analysis and numerical study for free vibration of plate using BEM- 68 Spurious eigenequation using the real-part BEM u,v (1) and (4) θ,m (2) and (3)

MSVLAB HRE, NTOU Calderon projector For Laplace and Helmholtz equation Mathematical analysis and numerical study for free vibration of plate using BEM- 69

MSVLAB HRE, NTOU Without CHEEF point Mathematical analysis and numerical study for free vibration of plate using BEM- 70

MSVLAB HRE, NTOU One CHEEF point Mathematical analysis and numerical study for free vibration of plate using BEM- 71

MSVLAB HRE, NTOU Two CHEEF points Mathematical analysis and numerical study for free vibration of plate using BEM- 72

MSVLAB HRE, NTOU Flowchart for clamped plate using the real-part BEM G.E. BIE BEM True and spurious eigenvalues Construction of [SM] B.C. Mathematical analysis and numerical study for free vibration of plate using BEM- 73

MSVLAB HRE, NTOU Flowchart for C-C plate using the complex-valued BEM G.E. BIE BEM True and spurious eigenvalues Construction of [SM] B.C. Mathematical analysis and numerical study for free vibration of plate using BEM- 74

MSVLAB HRE, NTOU Mathematical analysis (Continuous system) For clamped circular plate ( and ) Null-field integral equations Mathematical analysis and numerical study for free vibration of plate using BEM- 75

MSVLAB HRE, NTOU Expansion formulae Degenerate kernel Mathematical analysis and numerical study for free vibration of plate using BEM- 76

MSVLAB HRE, NTOU Relationship Mathematical analysis and numerical study for free vibration of plate using BEM- 77

MSVLAB HRE, NTOU Eigenequation Spurious eigenequationTrue eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 78

MSVLAB HRE, NTOU Mathematical analysis (Continuous system) For clamped-clamped annular plate Mathematical analysis and numerical study for free vibration of plate using BEM- 79 Notes:

MSVLAB HRE, NTOU Relationship Mathematical analysis and numerical study for free vibration of plate using BEM- 80

MSVLAB HRE, NTOU Eigenequations (C-C annular plate) Spurious eigenequation True eigenequation Mathematical analysis and numerical study for free vibration of plate using BEM- 81