reporter: 陳柏源 authors : 陳柏源、陳正宗博士 ...

Nation Taiwan Ocean University

Department of Harbor and River

April 21, 2023 pp.1

A semi-analytical approach for solving surface motion of multiple alluvial valleys for incident plane SH-waves

半解析法求解多山谷沉積物入射 SH波之地表位移

Reporter: 陳柏源Authors : 陳柏源、陳正宗博士

國立台灣海洋大學河海工程學系

Mechanics Sound Vibration Laboratory HRE. NTOUhttp://ind.ntou.edu.tw/~msvlab/ April 21, 2023 pp.2

Outlines

MotivationPresent method

Expansions of fundamental solution and boundary density

Adaptive observer system

Linear algebraic equation

Image technique for solving scattering problems of half-plane

Numerical examplesConclusions


Outlines








Conventional BEM

Interior case Exterior case

cD

D D

x

xx

xcD

s

s

(s, x) ln x s ln

(s, x)(s, x)

n

(s)(s)

n

U r

UT

jy

= - =

¶=

¶

¶=

¶

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U dB Dj y= - Îò ò

(x) . . . (s, x) (s) (s) . . . (s, x) (s) (s), xB B

C PV T dB R PV U dB Bpj j y= - Îò ò

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB Dpj j y= - Îò ò

x x

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB D Bpj j y= - Î Èò ò

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U d D BBj y= - Î Èò ò


Motivation

BEM / BIEMBEM / BIEM

Improper integralImproper integral

Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity

Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary

Collocation Collocation pointpoint

Fictitious BEMFictitious BEM

Null-field approachNull-field approach

CPV and HPVCPV and HPVIll-posedIll-posed

Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)

Waterman (1965)Waterman (1965)

Achenbach Achenbach et al.et al. (1988) (1988)


Present method

(s, x)iK

(s, x)eK

(s, x(x) (s) (s))B

dBKj y=ò

Fundamental solutionFundamental solution

(s, x), s x

(s, x), x s

i

i

K

K

ìï ³ïíï >ïîln x s-

No principal valueNo principal value

Advantages of degenerate kernel1. No principal value2. Well-posed3. Exponential convergence4. Free of boundary-layer effect

Degenerate kernelDegenerate kernel

CPV and HPVCPV and HPV


Outlines








Separable form of fundamental solution (1D)

-10 10 20

2

4

6

8

10

Us,x

2

1

2

1

(x) (s), s x

(s,x)

(s) (x), x s

i ii

i ii

a b

U

a b

=

=

ìïï ³ïïïï=íïï >ïïïïî

å

å

1(s x), s x

1 2(s, x)12

(x s), x s2

U r

ìïï - ³ïïï= =íïï - >ïïïî

-10 10 20

-0.4

-0.2

0.2

0.4

Ts,x

s

Separable property

continuouscontinuous jumpjump

1, s x

2(s, x)1

, x s2

T

ìïï >ïïï=íï -ï >ïïïî


Degenerate (separate) form of Degenerate (separate) form of fundamental solution (2-D)fundamental solution (2-D)

s( , )R q

R

r

rx( , )r f

x( , )r f

o

iU

eU

s

x

2

s x

(s, x)(s, x)

n

(s, x)(s, x)

n

(s, x)(s, x)

n n

UT

UL

UM

¶º

¶

¶º

¶

¶º

¶ ¶

(1

(1)

0

(1)

0

)0

( , ; , ) ( ) ( ) cos( ( )), 2

(s,x)

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

( )

2

> 2

im m

m

em m

m

iU R J k H kR m R

Ui

U R H

i

k J kR m

H r

R

k


1

1

(s,x) (x) (s), s x

(s,x)

(s,x) (s) (x), x s

ii i

i

ei i

i

U a b

U

U a b

¥

=

¥

=

ìïï = ³ïïïï=íïï = >ïïïïî

å

å

1

1

(s,x) (x) , s x

(s,x)

(s,x) (x), x

(s)

( ) ss

ii

i

ei i

i

iT a

T

T b

b

a

¥

=

¥

=

ìïï = ³ïïïï¢

¢=íïï = >ïïïïî

å

å

( )1

x s

(s(s,x) (s,x) (x) (x)

(s), ( )

) (s)

s

i ei i

i

i i

i iT T a b

W

b

b

a

a

¥

=

Þ

¢ ¢-

¾ ¾®

-=

¾

å

sn

U(s,x)U(s,x) T(s,x)T(s,x)

0

0

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

(

> 2

( , )

( , ) ( ) ( ) cos( ( )), ) > 2

im m

m

em m

m

m

m

mm

kiT s x J k m RY kR iJ kR

J kR

T s xki

T s x Y k iJ k m R


Jump behavior across the boundary

2

0

2 2

(s,x) (s,x) cos

cos cos

cos

2 cos , x

i e

n n n n n n

m m m m

T T n Rd

kR J kR Y kR iJ kR n kR J kR Y kR iJ kR n

kR Y kR J kR Y kR J kR n

n B

2,m m m m m mW J kR Y kR Y kR J kR Y kR J kR

kR

x BÎ WÈ

x c BÎ W È

cW

W

2 (x) (s,x) (s) (s) (s,x) (s) (s)i i

B Bu T u dB U t dBp = -ò ò

0 (s,x) (s) (s) (s,x) (s) (s)B

e

B

eT u dB U t dB= -ò ò

2

0

2 2

(s,x) (s,x) cos

cos cos

0, x

i e

n n n n n n

U U n Rd

R J kR Y kR iJ kR n R J kR Y kR iJ kR n

B


Outlines









collocation pointcollocation point

0 , 01 , 1k , k2 , 2


Outlines









{ }

0

1

2

N

ì üï ïï ïï ïï ïï ïï ïï ïï ï=í ýï ïï ïï ïï ïï ïï ïï ïï ïî þ

M

y

y

y y

y

[ ]

00 01 0

10 11 1

0 1

N

N

N N NN

é ùê úê úê ú= ê úê úê úê úë û

U U U

U U UU

U U U

L

L

M M O M

L

Column vector of Fourier coefficientsColumn vector of Fourier coefficients((NthNth routing circle) routing circle)

0B

1B

Index of collocation circleIndex of collocation circle

Index of routing circle Index of routing circle

2B

NB

[ ]{ } { }[ ]=U Ty j


Explicit form of each submatrix and Explicit form of each submatrix and vectorvector

0 1 11 1 1 1 1

0 1 12 2 2 2 2

0 1 13 3 3 3 3

0 1 12 2 2 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

c c s Lc Lsjk jk jk jk jkc c s Lc Ls

jk jk jk jk jkc c s Lc Ls

jk jk jk jk jkjk

c c s Lc Lsjk L jk L jk L jk L jk

U U U U U

U U U U U

U U U U U

U U U U U

ff ff f

ff ff f

ff ff f

ff ff

é ù=ê úë ûU

L

L

L

M M M O M M

L 20 1 1

2 1 2 1 2 1 2 1 2 1

( )

( ) ( ) ( ) ( ) ( )L

c c s Lc Lsjk L jk L jk L jk L jk LU U U U U

f

ff ff f+ + + + +

é ùê úê úê úê úê úê úê úê úê úê úê úê úë ûL

{ } { }0 1 1

Tk k k k kk L Lp p q p q= Ly

1f

2f

3f

2Lf

2 1Lf +

Fourier coefficientsFourier coefficients

Truncated terms of Truncated terms of Fourier seriesFourier series

Number of collocation pointsNumber of collocation points


Outlines








Image technique for solving half-plane problem

Alluvial

h a

SH-Wave

Matrix

-4 -2 2 4

-1

-0.5

0.5

1

-4 -2 2 4

-1

-0.5

0.5

1

Matrix

SH-Wave

SH-Wave

Free surface


Take free body

,M I M M I It tw w t t

I

I

w

t

Inclusion

Matrix

SH-Wave

SH-Wave

i r

i r

w w

t t

Matrix

SH-Wave

SH-Wave

Matrix

M M i rt

M M i rt

w w w w

t t t t

M

M

w

t


Linear algebraic system of the inclusion problem

Matrix field

Inclusion field

Two constrains

{ } { }M M i r M M i rt tU t t = T u u+ +é ù é ù- -ê ú ê úë û ë û

{ } { }I I I IU t = T ué ù é ùê ú ê úë û ë û

{ } { }M Itu = u

{ } { }M M I Itμ t = - μ té ù é ùê ú ê úë û ë û

M M M i rt

I I MtI

M I I

T -U 0 0 u u(x)

0 0 T -U t 0=

I 0 -I 0 u 0

0 μ 0 μ t 0

+é ùì ü ì üï ï ï ïï ï ï ïê úï ï ï ïê úï ï ï ïï ï ï ïï ï ï ïê úí ý í ýê úï ï ï ïï ï ï ïê úï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïë ûî þ î þ

{ }i r

i r M M

i r

uu(x) T -U

t

++

+

ì üï ïï ï= í ýï ïï ïî þ


Flowchart of present method

0 [ (s, x) (s) (s, x) (s)] (s)B

T u U t dB= -òDegenerate kernel Fourier series

Collocation point and matching B.C.



Fourier coefficientsPotential of

domain pointSurface

amplitude

Stress field

Vector decomposition

Numerical

Analytical


Outlines








A half-plane problem with a semi-circular alluvial valley subject to the SH-wave

AlluvialMatrix

h a

SH-Wave

x

y

kac

a

2 2( ) ( ) 0,k w x x

Wave number

Dimensionless frequency

Governing equation

Velocity of shear wave

c


Surface amplitudes of the alluvial valley problem

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

5

10

15

20

Am

plit

ud

e

0

5

10

15

20-4 -3 -2 -1 0 1 2 3 4

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

5

10

15

20

Am

plit

ud

e

0

5

10

15

20-4 -3 -2 -1 0 1 2 3 4

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

5

10

15

20

Am

plit

ud

e

0

5

10

15

20-4 -3 -2 -1 0 1 2 3 4

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

5

10

15

20

Am

plit

ud

e

0

5

10

15

20-4 -3 -2 -1 0 1 2 3 4

Present methodPresent method

Manoogian’s results [60]Manoogian’s results [60]

2, / 1/ 6, / 2 /3I M I M

0 30 60 90

2 2Re ImAmplitude w w

vertical horizontal


Limiting case of a canyon


Manoogian’s results [60]Manoogian’s results [60]

8/ 102, , / 2 /3IM MI

0 30 60 90

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

1

2

3

4

5

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4

0

1

2

3

4

5

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

1

2

3

4

5A

mp

litu

de

- 4 - 3 - 2 - 1 0 1 2 3 4

0

1

2

3

4

5

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

1

2

3

4

5

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4

0

1

2

3

4

5

- 4 - 3 - 2 - 1 0 1 2 3 4

x/a

0

1

2

3

4

5

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4

0

1

2

3

4

5

vertical horizontal


Limiting case of a rigid alluvial valley

- 4 - 3 - 2 - 1 0 1 2 3 4x / a

0

2

4

6

8

1 0

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4x / a

0

2

4

6

8

1 0

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4x / a

0

2

4

6

8

1 0

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4x / a

0

2

4

6

8

1 0

Am

plit

ud

e

Present methodPresent method4/ 102, , / 2 /3I MI M

0 30

60 90



Soft-basin effect

/ 2 /3, 2I M

/ 3/ 6I Mc c / 2 / 6I Mc c / 12 / 6I Mc c

/x a /x a /x a

14 18

3


No boundary-layer effect


A half-plane problem with two alluvial valleys subject to the incident SH-wave

Canyon

Matrix

3aSH-Wave

房 [93] 將正弦和餘弦函數的正交特性使用錯誤，以至於推導出錯誤的聯立方程，求得錯誤的結果。

-- 亞太學報曹 2004


Limiting case of two canyons


Tsaur et al.’s results [103]Tsaur et al.’s results [103]

8/ 102, , / 2 /3IM MI

0 30 60 90

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

2

4

6

8

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

2

4

6

8

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

2

4

6

8

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

2

4

6

8

Am

plit

ud

e

vertical horizontal


Surface displacements of two alluvial valleys

Present methodPresent method 2, / 1/ 6, / 2 /3I M I M

0 30

60 90

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

4

8

1 2

1 6

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

4

8

1 2

1 6

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

4

8

1 2

1 6

Am

plit

ud

e

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a

0

4

8

1 2

1 6

Am

plit

ud

e


Inclusion

Matrixh

SH-Wave

a

x

y

A half-plane problem with a circular inclusion subject to the incident SH-wave


Surface displacements of a inclusion problem under the ground surface

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

Present methodPresent method 2, / 1/ 6, / 2 /3I M I M

Tsaur et al.’s results [102]Tsaur et al.’s results [102]

Manoogian and Lee’s results [62]Manoogian and Lee’s results [62]

0 30 60 90

When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time.

--Private communication


Limiting case of a cavity problem


Lee and Manoogian’s [53] for the cavity case.Lee and Manoogian’s [53] for the cavity case.

8/ 102, , / 2 /3IM MI

0 30 60 90

- 3 - 2 - 1 0 1 2 3x/a

0

1

2

3

4

5

6

Am

plit

ud

e

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

- 3 - 2 - 1 0 1 2 3x/a

0

1

2

3

4

5

6A

mp

litu

de

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

- 3 - 2 - 1 0 1 2 3x/a

0

1

2

3

4

5

6

Am

plit

ud

e

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

- 3 - 2 - 1 0 1 2 3x/a

0

1

2

3

4

5

6

Am

plit

ud

e

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

vertical horizontal


A half-plane problem with two circular inclusions subject to the SH-wave

Matrix

Inclusionh

SH-Wave

a a

D

y

x


Limiting case of two-cavities problem


Jiang et al. result [95]Jiang et al. result [95]

8 ,/ / 2 /3,10 / 2.5IM MI D a

0.1 0.25 0.75 1.25

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

Am

plit

ud

e

h /a =1 .5 , D /a =2 .5

h 0 o

h 30 o

h 60 o

h 90 o

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

Am

plit

ud

e

h /a =1 .5 , D /a =2 .5

h 0 o

h 30 o

h 60 o

h 90 o

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

h /a =1 .5 , D /a =2 .5

h 0 o

h 30 o

h 60 o

h 90 o

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

Am

plit

ud

e

h/a=1.5, D /a=2.5

h 0 o

h 30 o

h 60 o

h 90 o


Surface amplitudes of two-inclusions problem

- 6 - 4 - 2 0 2 4 6

x/a

1

2

3

4

5

Am

plit

ud

e

h /a =1 .5 , D /a =2 .5

h

h

h

h

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

Am

plit

ud

e

h /a =1 .5 , D /a =2 .5

h

h

h

h

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

Am

plit

ud

e

h /a=1 .5 , D /a =2 .5

h

h

h

h

- 6 - 4 - 2 0 2 4 6

x/a

0

1

2

3

4

5

Am

plit

ud

e

h /a=1 .5 , D /a =2 .5

h

h

h

h

Present methodPresent method / 1/ 6, / 2 /3, / 2.5I M I M D a

0.1 0.25

0.75 1.25


Conclusions

A systematic way to solve the Helmholtz problems with circular boundaries was proposed successfully in this paper by using the null-field integral equation in conjunction with degenerate kernels and Fourier series.

The present method is more general for calculating the torsion and bending problems with arbitrary number of holes and various radii and positions than other approach.

Our approach can deal with the cavity problem as a limiting of inclusion problem with zero shear modulus.


Some findings

?房營光 1995

Analytical solution

Lee &

Manoogian1992

Tsaur et al.2004

Analytical solutionPresent method

Present method Tsaur et al.

Nation Taiwan Ocean University

Department of Harbor and River

April 21, 2023 pp.40

Thanks for your kind attentions.

You can get more information from our website.

http://msvlab.hre.ntou.edu.tw/

reporter: 陳柏源 authors : 陳柏源、陳正宗博士 ...

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