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

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

MSVLABHRE, NTOU

Outlines


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

MSVLABHRE, NTOU

Introduction

xxuxu ),()( 44

D

h2

4 )1(12 2

3

Eh

D

circular frequency

G.E.

u

Poisson ratio

DE

h

domain

frequency parametersurface density

flexural rigidity plate thickness

Young's modulus

lateral displacement


MSVLABHRE, NTOU

Free vibration of plate

-0.003

-0.002

-0.001

0

0.001

0.003


MSVLABHRE, NTOU

Literature review1. 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)


MSVLABHRE, NTOU

Motivation

Real Imaginary ComplexSaving CPU time Yes Yes NoAvoid singular integral No Yes NoSpurious eigenvalues Appear Appear No

Simply-connected problem

Multiply-connected problem

ComplexSpurious eigenvalues Appear


MSVLABHRE, 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.


Real-part BEM (N.G.)Imaginary-part BEM (N.G.)Complex-valued (OK)

Complex-valued (N.G.)

MSVLABHRE, NTOU

1. Introduction







MSVLABHRE, NTOU

Boundary integral equations for plate eigenproblems

),()}(),()(),()(),()(),({)( sdBsuxsVsxsMsmxssvxsUxuB

),()}(),()(),()(),()(),({)( sdBsuxsVsxsMsmxssvxsUxB

),()}(),()(),()(),()(),({)( sdBsuxsVsxsMsmxssvxsUxm mmB mm

),()}(),()(),()(),()(),({)( sdBsuxsVsxsMsmxssvxsUxv vvB vv

)(K

)(mK

)(vK

(2) Slope

(3) Normal moment

(4) Effective shear force

(1) Displacement


x

MSVLABHRE, NTOU

Operators

nK

)(

)(

2

2

2 )()1()()(

nK

m

))(

()1()(

)(22

tntnK

v


Slope

Normal moment

Effective shear force

t

n

MSVLABHRE, NTOU

Kernel functions

)(),(),( 44 sxxsUxsU cc

)),((),( xsUKxs

)),((),( xsUKxsM m

)),((),( xsUKxsV v

]Re[),(c

UxsU

)))()((2

)()((81

),(00002

riIrKriJrYxsUc

Kernel functions

Fundamental solution


MSVLABHRE, NTOU

Mathematical analysis (Discrete system)

}]{[}]{[0 mvU

}]{[}]{[0 mvU

0][

m

vSM

NNU

USM

44

][

For clamped circular plate (u=0 and =0)

)(sv

)(sma


MSVLABHRE, NTOU

Expansion formulae

mmm

mmm

mJY

mJYrY

)),(cos()()(

)),(cos()()()(

0

mmm

mmm

mIK

mIKrK

)),(cos()()(

)),(cos()()()(

0

Degenerate kernels (separable kernels)


MSVLABHRE, NTOU

Circulant

NNN

NNN

NN

N

zzzzz

zzzz

zzzz

zzzz

U

22012321

3201222

221012

12210

][

)2

1(

)2

1(

m

m


12,...,2,1,0

,),,,()],,,([

Nm

aaaUdaaaU m mz

MSVLABHRE, NTOU

The properties of the circulant

NN

ezzN

m

Nm

i

m

N

m

m

m

U

),1(,...,2,1,0

,12

0

2212

0

][

12,..,2,1,0

,1,..,2,1,0,22

Nor

NNe Ni

daaaU

aaaUm m

N

mN

U

2

0

12

0

][

)]0,,,()[cos(

)]0,,,([)cos(lim


MSVLABHRE, NTOU

Eigenvalues of the four matrices

)]()()()([4

][ aIaKaJaYa

)]()()()([4

][ aIaKaJaYaU

)]()()()([4

][ aIaKaJaYa

)]()()()([4 2

][ aIaKaJaYaU

][

][ U

][

][U

NN ),1(,...,2,1,0


MSVLABHRE, NTOU

Eigenvalue decomposition (similar matrices)

1

][

][

)1(

][

1

][

1

][

1

][

0

1

00000

00000

00000

00000

00000

00000

][

U

N

U

N

U

N

U

U

U

UU

1

][

][

)1(

][

1

][

1

][

1

][

0

1

00000

00000

00000

00000

00000

00000

][

N

N

N

1

][

][

)1(

][

1

][

1

][

1

][

0

1

00000

00000

00000

00000

00000

00000

][

U

N

U

N

U

N

U

U

U

UU

1

][

][

)1(

][

1

][

1

][

1

][

0

1

00000

00000

00000

00000

00000

00000

][

N

N

N


MSVLABHRE, NTOU

Determinant

1

11

11

0

0

0

0

][

U

U

U

USM

N

N

UU

U

USM

)1(

][][][][ )(det]det[


MSVLABHRE, NTOU

Eigenequations (real-part BEM for clamped plate)

)}()()()({

)]()()()([16

)]}()()()([)]()()()([

)]()()()([)]()()()({[16

]det[

11

)1(112

2

)1(2

2

aJaIaJaI

aYaKaYaKa

aIaKaJaYaIaKaJaY

aIaKaJaYaIaKaJaYa

SM

N

N

N

N

)}()()()({ 11 aJaIaJaI

Spurious eigenequation True eigenequation


0)]()()()([ 11 aYaKaYaK

MSVLABHRE, NTOU

Comparisons of Leissa and present method

Leissa (Kitahara) Present method

Clamped

Simply-supported

Free

011 JIJI 011 JIJI

)1(

211

I

I

J

J02))(1( 11 JIJIJI

][)1(

][)1(

])[1(

])[1(

23

23

22

22

III

JJI

III

JJJ

0)]()1()1([

)()1)(1(2

]2)1(4)[1(

114222

112

1122

JIJI

JIJI

JIJI


J

,......3,2,1,0 where

MSVLABHRE, NTOU

Spurious eigenequations using the real-part BEM

011 YKYK

02))(1( 11 YKYKYK

0)(2))(1( 1122

112 YKYKYKYKYK

0))](1(2)3(2[

)(2)(2

112

1122

11222

YKYK

YKYKYKYK

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)

,......3,2,1,0

0)(2))(1( 1122

112 YKYKYKYKYK

where


0)]()1()1([)(

)1)(1(2]2)1(4)[1(

1144222

11

2211

222

YKYKaYKYK

aYKaYKa

MSVLABHRE, 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)

,......3,2,1,0 where

011 JIJI

02))(1( 11 JIJIJI

0)(2))(1( 1122

112 JIJIJIJIJI

0)(2))(1( 1122

112 JIJIJIJIJI

0))](1(

)(2)(2

112

1122

11222

JIJI

JIJIJIJI

0)]()1()1([)(

)1)(1(2]2)1(4)[1(

1144222

11

2211

222

JIJIJIJI

JIJI


MSVLABHRE, NTOU

True and spurious eigenequations

True Spurious

Real (u, )

Imaginary (u, )

Complex (u, ) -

011 JIJI

011 JIJI 011 JIJI

True SpuriousReal (UT)Imaginary (UT)Complex (UT) -

011 JIJI

011 YKYK

0J

0J 0J

0J

0Y


Membrane (Dirichlet)

Plate (clamped)

MSVLABHRE, NTOU

Determinant v.s frequency parameter (real-part BEM)

0)]()()()([11

aYaKaYaK

Spurious eigenequation

True eigenequation

0)}()()()({11

aJaIaJaI


ma 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8fre q u e n cy p a ra m e te r

1E-080

1E-075

1E-070

1E-065

1E-060

1E-055

1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

de

t|SM

|

C lam ped circu lar p lateusing u, rea l-part form ulation

S: Spurious e igenvalues

S <1.42>

S <2.63>

S <3.17>

S <3.78>

S <4.64>

S <4.90>

S <6.01>

S <6.30>

S <7.32>

S <7.81>

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>

MSVLABHRE, NTOU

1. Introduction







MSVLABHRE, NTOU

Treatments

1. SVD updating term2. The Burton & Miller concept3. The complex-valued BEM (conventional BEM)4. The CHEEF concept


MSVLABHRE, NTOU

SVD updating term (clamped plate)


NN

c

U

USM

44

1

NNvv

mmc

U

USM

44

2

0][

m

vC

02

m

vSM c

01

m

vSM c

NN

c

c

SM

SMC

482

1

u, formulation

m,v formulation

SVD technique of updating term

MSVLABHRE, NTOU

Determinant

1

1

2

1

0

0

000

000

000

000

VV

MM

UU

c

c

SM

SMC

])()()(

)()()[(det

2][][][][2][][][][2][][][][

2][][][][2][][][][2][][][][

)1(

UUUUUU

UUUUUUN

N

T CC

The only possibility for zero determinant

,0)( 2][][][][ UU,0)( 2][][][][

UU

,0)( 2][][][][ UU

,0)( 2][][][][ UU .0)( 2][][][][

UU

,0)( 2][][][][ UU

at the same time for the same .


MSVLABHRE, NTOU

The Burton & Miller concept (clamped plate)

0)(21

m

vSMiSM cc

0)}()()()()]{()([

)det(

11

21

aJaIaJaIiBA

SMiSM cc

0)( B

42

1

2222224422242

42

1

33222222442224

1

32

})]}1(221[)1()1(2)1({)1()1(4{

32

})1(2)]}1(221[)1()1(2)1({{)(

nnn

nnn

YnnnYnnK

YYnnnKA

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.


MSVLABHRE, NTOU

The complex-valued BEM (clamped plate)

NNU

USM

44

][

0)}()()()()]{()([)det(11

aJaIaJaIiBASM

),(),( xsUxsU cNotes:

Since the term [A()+i B()] is never zero for any , we can obtain the true eigenvalues

2

1111

32)1()1(

)(

nnnnnn

n

nn

n YKYKIJIJA

2

1111

32)()1(

)(

nnnn

n

nnnnYIYIKJKJ

B

0)}()()()({11

aJaIaJaI True eigenequation


MSVLABHRE, NTOU

The CHEEF concept (clamped plate)

01444

NNNm

v

U

U

01442

NNN

CC

CC

m

v

U

U

c

0][ *

m

vC

u, formulation

CHEEF points

NNN

CC

CC

cU

U

U

U

C

4)2(2

* ][


MSVLABHRE, NTOU

Comparison of the matrix dimension

Dimension Matrix type

SVD updating term 8N4N Real (Imag.)

Burton & Miller method 4N4N Complex

Complex-valued BEM 4N4N Complex

CHEEF method 2(2N+Nc)4N Real (Imag.)


MSVLABHRE, NTOU

SVD updating term (real-part BEM)


1E-025

1E-024

1E-023

1E-022

1E-021

1E-020

1E-019

1E-018

1E-017

1E-016

1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

de

t|CT C

C lam ped circu lar p la teSVD technique of updating term

using u, + m , v rea l-part form ulations

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>

T: T rue e igenvalues


ma 1

MSVLABHRE, NTOU

The Burton & Miller concept (real-part BEM)


1E-030

1E-029

1E-028

1E-027

1E-026

1E-025

1E-024

1E-023

1E-022

1E-021

1E-020

1E-019

1E-018

1E-017

1E-016

1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

de

t|SM

|

C lam ped circu lar p la teBurton & M iller m ethod

using (u , + i (m , v) real-part form ulations

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


S

S S

S

S

S


ma 1

Fail(u, ) + i (m, v)

MSVLABHRE, NTOU

The Burton & Miller method (real-part BEM)


1E-030

1E-029

1E-028

1E-027

1E-026

1E-025

1E-024

1E-023

1E-022

1E-021

1E-020

1E-019

1E-018

1E-017

1E-016

1E-015

1E-014

1E-013

1E-012

de

t|SM

|

C lam ped circu lar p la teBurton & M iller m ethod

using (u , m + i ( , v) real-part form ulations

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>



ma 1

Success

(u, m) + i (, v)

MSVLABHRE, NTOU

The complex-valued BEM


1E-065

1E-045

1E-025

1E-005

1E+015

de

t|SM

|

C lam ped circu lar p la teusing u , com plex-vauled form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>



ma 1

MSVLABHRE, NTOU

The CHEEF concept (real-part BEM)


1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

C lam ped circu lar p latetw o C H EEF poin ts

using u, rea l-part form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>



ma 1

MSVLABHRE, NTOU

1. Introduction







MSVLABHRE, NTOU

Mathematical analysis (Discrete system)


For clamped-clamped annular plate

22212221

12111211

22212221

12111211

UU

UU

UU

UU

SM cc

}]{12[}]{11[}]{12[}]{11[0 2121 mmvUvU

}]{22[}]{21[}]{22[}]{21[0 2121 mmvUvU

}]{12[}]{11[}]{12[}]{11[0 2121 mmvUvU

}]{22[}]{21[}]{22[}]{21[0 2121 mmvUvU

2

1

2

1

0

0

0

0

m

m

v

v

SM cc

a

b

1B

2B


MSVLABHRE, NTOU

Eigenequations (C-C annular plate)


True eigenequation


0

)()()()(

)()()()(

)()()()(

)()()()(

det]det[

bKaKbKaK

bIaIbIaI

bYaYbYaY

bJaJbJaJ

T cc

0

)()()()(

0))()1()((0))()((

)()()1()()(

0)()1()(0)()(

det]det[

bIbIibJbJi

aIiaKaJiaY

bIbIibJbiJ

aIiaKaiJaY

Sn

n

u

N

N

ccSM)1(

(det

]det[ uS ]det[ ccT 0)

MSVLABHRE, NTOU

Physical meaning of the spurious eigenequation

)()(

)()(][

bIbJ

bIbJSb

nn

nnu

n

0)}()()()({ 11 bJbIbJbI nnnn

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)


]det[]det[]det[ un

un

un SbSaS

))()1()(())()((

)()1()()()(][

aIiaKaJiaY

aIiaKaiJaYSa

n

n

nnn

n

n

nnnu

n 0]det[ u

nSa

0]det[ unSb

0)}()()()({]det[ 11 bJbIbJbISb nnnn

u

n

MSVLABHRE, NTOU

True eigenequations for the annular plate

][n

T

)()()()(

)()()()(

)()()()(

)()()()(

bKaKbKaK

bIaIbIaI

bYaYbYaY

bJaJbJaJ

nnnn

nnnn

nnnn

nnnn

)()()()(

)()()()(

)()()()(

)()()()(

babKaK

babIaI

babYaY

babJaJ

Kn

Knnn

In

Innn

Yn

Ynnn

Jn

Jnnn

)(1

)()(1

)()()(

)(1

)()(1

)()()(

)(1

)()(1

)()()(

)(1

)()(1

)()()(

bb

bab

aba

bb

bab

aba

bb

bab

aba

bb

bab

aba

Kn

Kn

Kn

Kn

Kn

Kn

In

In

In

In

In

In

Yn

Yn

Yn

Yn

Yn

Yn

Jn

Jn

Jn

Jn

Jn

Jn

B.C.

C-C

S-S

F-F


,...2,1,0n

MSVLABHRE, NTOU

Spurious eigenequations for the annular plateEqs. 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)

))(())((

)()(

bIbJ

bIbJ

nn

nn

)()(

)()(

bb

bIbJIn

Jn

nn

)]()1(

)([)]()1(

)([

)()(

bb

bbb

b

bIbJIn

In

Jn

Jn

nn

)()(

)()(

bb

bIbJIn

Jn

nn

)]()1(

)([)]()1(

)([

)()(

bb

bbb

b

bIbJIn

In

Jn

Jn

nn

)()1(

)([)]()1(

)([

)()(

bb

bbb

b

bbIn

In

Jn

Jn

In

Jn

0,0 u

0,0 mu

0,0 vu

0,0 m

0,0 v

0,0 vm


b

b

b

b

b

b

MSVLABHRE, NTOU

Determinant v.s frequency parameter (C-C)


True eigenequation


0 1 2 3 4 5 6 7 8 9 10 11 12fre q u e n cy p a ra m e te r

1E-080

1E-060

1E-040

1E-020

de

t|SM

|

C -C annular p la teu, com plex-vauled form ulation

T<9.447>


T<10.370>

T<10.940>

T<9.499>

T<9.660>

T<9.945>

S<9.222>

S<6.392>

S<11.810>

S : Spurious e igenvalues

0

)()()()(

)()()()(

)()()()(

)()()()(

det

]det[

bKaKbKaK

bIaIbIaI

bYaYbYaY

bJaJbJaJ

T

nnnn

nnnn

nnnn

nnnn

ccn

0

)()()()(

0))()1()((0))()((

)()()1()()(

0)()1()(0)()(

det

]det[

bIbIibJbJi

aIiaKaJiaY

bIbIibJbiJ

aIiaKaiJaY

S

nnnn

nn

nnn

nnn

nn

nn

nnn

un

ma 1

mb 5.0

1B

2B

imply

0)()(

)()(det]det[

bIbJ

bIbJSb

nn

nnun

MSVLABHRE, NTOU

Physical meaning of the spurious eigenequation


1E-065

1E-045

1E-025

de

t|SM

|

u=0, =0 - s im ply-connected p la teu, com plex-vauled form ula tion

T<6.392>

T<9.222>

T<11.810>T: T rue e igenvalues

0)()(

)()(det

]det[

bIbJ

bIbJ

Sb

nn

nn

u

n

0)}()()()({ 11 bJbIbJbI nn

Spurious eigenequation (u, )(multiply-connected)

True eigenequation (clamped)(simply-connected plate)


a

b

1B

2B

b

mb 5.0

MSVLABHRE, NTOU

1. Introduction







MSVLABHRE, NTOU

Treatments

SVD updating term

Burton & Miller method

CHIEF method

NN

cc

cc

SM

SMC

8162

1

cccc SMiSM21

NNN cCCUCUC

CCUCUC

UU

UU

UU

UU

C

8)4(2

*

2121

2121

22212221

12111211

22212221

12111211

][


a

b

1B

2B

MSVLABHRE, NTOU

SVD updating term


1E-080

1E-060

1E-040

1E-020

1

1E+020

1E+040

1E+060

1E+080

1e+100

1e+120

1e+140

1e+160

1e+180

1e+200

1e+220

1e+240

1e+260

1e+280

1e+300

det|C

TC

|

F-F annular p la teSVD technique of updating term

u, + m , v com plex-va lued form ulations

T<8.139>

T: True e igenvalues

T<9.429>T

<9.800>

T<9.603>

T<3.037>

T<4.115>

T<2.050>

T<3.355>

T<4.557>

T<5.541>

T<5.704>

T<6.854>

T<10.342>

T<11.124>


ma 1

mb 5.0

1B

2B

MSVLABHRE, NTOU

The Burton & Miller concept


1e-100

1E-080

1E-060

1E-040

1E-020

1

1E+020

1E+040

1E+060

1E+080

1e+100

1e+120

1e+140

de

t|SM

|

F-F annular p la teBurton & M iller m ethod

(u, m + i ( , v) com plex-vau led form ulation

T<8.139>


T<9.429>T

<9.800>

T<9.603>

T<3.037>

T<4.115>

T<2.050>

T<3.355>

T<4.557>

T<5.541>

T<5.704>

T<6.854>

T<10.342>

T<11.124>


ma 1

mb 5.0

1B

2B

Success

(u, m) + i (, v)

MSVLABHRE, NTOU

The CHIEF concept


1E-010

de

t|SM

|

F-F annular p la tetw o C H IEF poin ts

u , com plex-vauled form ulation

T<8.139>


T<9.429>T

<9.800>

T<9.603>

T<3.037>

T<4.115>

T<2.050>

T<3.355>

T<4.557>

T<5.541>

T<5.704>

T<6.854>

T<10.342>

T<11.124>


ma 1

mb 5.0

1B

2B

MSVLABHRE, NTOU

1. Introduction







MSVLABHRE, 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.


MSVLABHRE, 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.


MSVLABHRE, NTOU

Thanks for your kind attention

The End


MSVLABHRE, NTOU

Real-part BEM for simply-connected plate

ma 1


1E-080

1E-075

1E-070

1E-065

1E-060

1E-055

1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

de

t|SM

|



S <1.42>

S <2.63>

S <3.17>

S <3.78>

S <4.64>

S <4.90>

S <6.01>

S <6.30>

S <7.32>

S <7.81>

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


MSVLABHRE, NTOU

Imaginary-part BEM for simply-connected plate


1E-080

1E-070

1E-060

1E-050

1E-040

1E-030

1E-020

1E-010

1

1E+010

1E+020

de

t|SM

|

C lam ped circu lar p la teusing u , m im aginary-part form ulation

S<2.23>

S<3.73>

S<5.06>

S<6.32>

S<6.96>

S<7.54>


S<5.45>T

<3.19>T

<4.61>T

<5.90>

T<6.30>

T<7.14>

T<7.80>

ma 1


MSVLABHRE, NTOU

Complex-valued BEM for simply-connected plate


1E-065

1E-045

1E-025

1E-005

1E+015

de

t|SM

|

C lam ped circu lar p la teusing u , com plex-vauled form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


ma 1


MSVLABHRE, NTOU

Complex-valued BEM for multiply-connected plate


1E-080

1E-060

1E-040

1E-020d

et|S

M|

C -C annular p la teu, com plex-vauled form ulation

T<9.447>


T<10.370>

T<10.940>

T<9.499>

T<9.660>

T<9.945>

S<9.222>

S<6.392>

S<11.810>


ma 1

mb 5.0

1B

2B


MSVLABHRE, NTOU

Real-part BEM (u, ) for simply-connected clamped plate


1E-080

1E-075

1E-070

1E-065

1E-060

1E-055

1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

de

t|SM

|


T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


0)( 2][][][][ UU


ma 1

MSVLABHRE, NTOU

Real-part BEM (u, m) for simply-connected clamped plate


1E-060

1E-055

1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

1E-020

de

t|SM

|

C lam ped circu lar p lateusing u, m rea l-part form ula tion

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


0)( 2][][][][ UU


ma 1

MSVLABHRE, NTOU

Real-part BEM (u,v) for simply-connected clamped plate


1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

1E-020

de

t|SM

|

C lam ped circu lar p la teusing u, v rea l-part form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


0)( 2][][][][ UU


ma 1

MSVLABHRE, NTOU

Real-part BEM (, m) for simply-connected clamped plate


1E-050

1E-045

1E-040

1E-035

1E-030

1E-025

1E-020

de

t|SM

|

C lam ped circu lar p lateusing , m rea l-part form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


0)( 2][][][][ UU


ma 1

MSVLABHRE, NTOU

Real-part BEM (, v) for simply-connected clamped plate


1E-035

1E-031

1E-027

1E-023

1E-019

1E-015

1E-011

de

t|SM

|

C lam ped circu lar p lateusing , v rea l-part form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


0)( 2][][][][ UU


ma 1

MSVLABHRE, NTOU

Real-part BEM (m,v) for simply-connected clamped plate


1E-025

1E-022

1E-019

1E-016

1E-013

1E-010

1E-007

de

t|SM

|

C lam ped circu lar p lateusing m , v rea l-part form ulation

T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


ma 1

0)( 2][][][][ UU


MSVLABHRE, NTOU

Membrane or acoustic problem

)()(00kriJkrY

)(0krY

)(),(),( 22 sxxsUkxsU

True and spurious eigenvalues

True eigenvalues

)(),(),( 44 sxxsUxsU

)(2

)(00

rKrY

))((2

)()(( 000 rKriJrY

True and spurious eigenvalues

De Mey

Tai and Shaw

Hutchinson

Kitahara


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Hilbert-transform The Hilbert transform is the constraint in the frequency

domain corresponding to the casual effect in the time-domain fundamental solution.


)(0krY )(

0krJ

Hilbert transform

du

utux

txΗ)(

)()]([

Definition as convolution integral

MSVLABHRE, NTOU

Fundamental solution


)(),(),( 44 sxxsUxsU cc

)(),(00

rKrY

)(),(00

rIrJ

)))()((2

)()((81

),(00002

riIrKriJrYxsUc

Singular solution

Regular solutionHilbert transform

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Burton & Miller method


0)(2))(1( 1122

112 YKYKYKYKYK

0)(2))(1( 1122

112 YKYKYKYKYK

Spurious eigenequation using the real-part BEM

u,v (1) and (4)

θ,m(2) and (3)

)()())((122121212211babaibbaaibaiba

MSVLABHRE, NTOU

Calderon projector

0]][[][][41 2 UMLI

]][[]][[ MLTM

]][[]][[ UTLU

0]][[][][41 2 MUTI

For Laplace and Helmholtz equation


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Without CHEEF point


1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

C lam ped circu lar p latew ithout C H EEF poin t


T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>T: T rue e igenvalues

S <3.78>

S <2.63>

S <3.17>

S <4.64>

S <4.90>

S <6.01>

S <6.30>

S <7.32>

S <7.81>

ma 1


MSVLABHRE, NTOU

One CHEEF point


1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

C lam ped circu lar p lateone C H EEF poin t


T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


S <3.78>

S <4.90>

S <6.01>


ma 1

MSVLABHRE, NTOU

Two CHEEF points


ma 1


1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

C lam ped circu lar p latetw o C H EEF poin ts


T<3.19>

T<4.61>

T<5.90>

T<6.30>

T<7.14>

T<7.80>


MSVLABHRE, NTOU

Flowchart for clamped plate using the real-part BEM

xxuxu ),()( 44 G.E.

BIE

BEM

True and spuriouseigenvalues

Construction of [SM]

B.C.

}]{[}]{[0 mvU

}]{[}]{[0 mvU

0u 0

NNU

USM

44][

0

)(]det[)1(

][][][][

N

N

UUSM


MSVLABHRE, NTOU

Flowchart for C-C plate using the complex-valued BEM

xxuxu ),()( 44 G.E.

BIE

BEM

True and spuriouseigenvalues

Construction of [SM]

B.C.

0

])det[](det[]det[)1(

N

N

u TSSM


}]{12[}]{11[}]{12[}]{11[0 2121 mmvUvU

}]{22[}]{21[}]{22[}]{21[0 2121 mmvUvU

}]{12[}]{11[}]{12[}]{11[0 2121 mmvUvU

}]{22[}]{21[}]{22[}]{21[0 2121 mmvUvU

01 u 02 u01 02

22212221

12111211

22212221

12111211

][

UU

UU

UU

UU

SM

MSVLABHRE, NTOU

Mathematical analysis (Continuous system)

10 ))cos()cos(()(

nnn nqnppsm

10 ))cos()cos(()(

nnn nbnaasv

For clamped circular plate ( and )

e

B

xsdBsuxsVsxsM

smxssvxsU

),()}(),()(),(

)(),()(),({0

e

B

xsdBsuxsVsxsM

smxssvxsU

),()}(),()(),(

)(),()(),({0

0u 0

Null-field integral equations

)(sv

)(sma


MSVLABHRE, NTOU

Expansion formulae

)(]))cos()cos(()[,(]))cos()cos(()[,({01

01

0 sdBnqnppxsnbnaaxsUn

nnB

nnn

)(]))cos()cos(()[,(]))cos()cos(()[,({01

01

0 sdBnqnppxsnbnaaxsUn

nnB

nnn

mmm

mmm

mJY

mJYrY

)),(cos()()(

)),(cos()()()(

0

mmm

mmm

mIK

mIKrK

)),(cos()()(

)),(cos()()()(

0

Degenerate kernel


MSVLABHRE, NTOU

Relationship

,...,2,1,)()()()(

)()()()(1

Na

aIaKaJaY

aIaKaJaYp n

nnnn

nnnnn

,...,2,1,)()()()(

)()()()(1

NbaIaKaJaY

aIaKaJaYq n

nnnn

nnnnn

,...,2,1,)()()()(

)()()()(1

Na

aIaKaJaY

aIaKaJaYp n

nnnn

nnnnn

,...,2,1,)()()()(

)()()()(1

Nb

aIaKaJaY

aIaKaJaYq n

nnnn

nnnnn


MSVLABHRE, NTOU

Eigenequation

0)}()()()()]{()()()([ 1111 aJaIaJaIaYaKaYaK nnnnnnnn

)()()()(

)()()()(1

)()()()(

)()()()(1

aIaKaJaY

aIaKaJaY

aIaKaJaY

aIaKaJaY

nnnn

nnnn

nnnn

nnnn

Spurious eigenequation True eigenequation


MSVLABHRE, NTOU

Mathematical analysis (Continuous system)

For clamped-clamped annular plate


0,1,11 ))cos()cos(()(

n

ccn

ccn nqnpsm

0,2,22 ))cos()cos(()(

n

ccn

ccn nqnpsm

0,1,11 ))cos()cos(()(

n

ccn

ccn nbnasv

0,2,22 ))cos()cos(()(

n

ccn

ccn nbnasv

,01 u ,02 u,01 02 a

b

1B

2B


MSVLABHRE, NTOU

Relationship


ccni

ccni

ccni

ccni

p

p

a

a

TM

,

,

,

,

0

0

0

0

2121

2121

2121

2121

)()cos(),()()cos(),()()cos(),()()cos(),(

)()cos(),()()cos(),()()cos(),()()cos(),(

)()cos(),()()cos(),()()cos(),()()cos(),(

)()cos(),()()cos(),()()cos(),()()cos(),(

][

22212221

12111211

22212221

12111211

B BBB BBB BBB BB

B BBB BBB BBB BB

B BBB BBB BBB BB

B BBB BBB BBB BB

sdBnxssdBnxssdBnxsUsdBnxsU




TM

MSVLABHRE, NTOU

Eigenequations (C-C annular plate)


True eigenequation


0

)()()()(

)()()()(

)()()()(

)()()()(

det]det[

bKaKbKaK

bIaIbIaI

bYaYbYaY

bJaJbJaJ

T

nnnn

nnnn

nnnn

nnnn

cc

n

0

)()()()(

0))()1()((0))()((

)()()1()()(

0)()1()(0)()(

det]det[

bIbIibJbJi

aIiaKaJiaY

bIbIibJbiJ

aIiaKaiJaY

S

nnnn

n

n

nnn

nn

n

nn

n

n

nnn

u

n

0]det[]det[]det[ nun TSTM

Documents

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

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