Date: Jan. 13, 2010 Time: 11:50~12:10 Place: Lectrue Theater F

The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 1

Null-field boundary integral equation approach for

hydrodynamic scattering by multiple circular and elliptical cylinders

Date: Jan. 13, 2010Time: 11:50~12:10 Place: Lectrue Theater F

National Taiwan Ocean UniversityMSVLABDepartment of Harbor and River Engineering

Jai-Wei Lee and Jeng-Tzong Chen

Hong Kong 2010

8


Outline

• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions


Outline



The 8th ACFD Conference in HK, 2010.1. 10~14

Keelung

NTOU

HKUST


NTOU/MSV Group members (2010)

1983

1962 1955 19751962 19711959 19761972

1988198719861987198519851978


NTOU/MSV visitors

陳俊賢(J S Chen, UCLA)

程宏達(Alex H.-D. Cheng, USA)

Jeong-Guon Ih (KAIST, Korea)

姚振漢(Yao Z H, China)

( 黃晉 , China)

(M.Tanaka, Japan)

陳清祥(C. S. Chen, USA)

祝家麟(J. L. Zhu, China) 陳鞏

(USA, Texas A M)

吳鼎文(T. W. Wu, USA)

杜慶華(Q. H. Du,China)

吳漢津(H C Wu, Iowa, USA)

余德浩中國科學院

美國中國

日本南韓


Outline


2010/01/13 Page 8The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 2010

Introduction of water wave problem(single cylinder)

Analytical solution

circular elliptical MacCamy and Fuchs (1954) Goda and Yoshimura (1972)


Introduction of water wave problem(multiple cylinders)

Analytical solutions are not available

Linton and Evans (1990)Spring and Monkmeyer (1974)

Present methodBoundary type

Meshless method

(Null-field BIEM)

Chatjigeorgio and Mavrakos (2009)

Multipole expansion

74

2b

1

2

3

x

y

incAOR (2009)

Semi-analytical methods


Introduction of water wave problem(multiple cylinders)

To the authors’ best knowledge

OK

Multipole expansion

Multipole expansion ?(Null-field BIEM) OK


Problem statement (3D)

2 ( , ; ) 0, ( , ) ,T z t z D x x

( , ; ) ( ) ( ) ,i tT Tz t u f z e x x

cosh ( )( ) ,

cosh( )k z higAf z

kh

Governing equation

Linearized wave theory and method of separation variables

:h water depth constant

( , )x yx


Reduction to 2D Problem

2 2( ) ( ) 0, ,Tk u D x x

( ) ( ) 0, ,TT

ut B

n

x

x x x

cos( ) sin( )( ) ,ik x yIu e x

( ) ( ) ( ),T I Ru u u x x x

Governing equation

( ) :Iu x

( ) :Ru x

Incident wave field

Radiation field

Boundary condition

2 2( ) ( ) 0, ,Rk u D x x

( ) ( ) 0, ,R It t B x x x

Governing equation

Boundary condition


Outline



Interior case Exterior case

cDD

Dx

x

x

x

cD

x

Degenerate (separable) formDegenerate (separable) form

2 ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),B B

u T u dB U t dB D x s x s s s x s s x

( ) . . . ( , ) ( ) ( ) . . . ( , ) ( ) ( ),B B

u C PV T u dB R PV U t dB B x s x s s s x s s x

B0 ( , ) ( ) ( ) ( , ) ( ) ( ), c

B BT u dB U t dB D s x s s s x s s x

B

Boundary integral equation and null-field boundary integral equation

(1)0 ( )( , )2

( , )( , )

( )( )

i H krU

UTn

utn

s

s

s x

s xs x

ss

xB B


Degenerate (separable) form of fundamental solution (2D)

1

1

1ln c

1ln c

( , ) lnos

s ,

,

o

m

m

m

m

R mU

R mR

m

m

Rr

R

s x

, , , ,R r s x x s( , )R s

( , ) x

( , ) x

CircleExtension

Ellipse


Degenerate kernels

Modified Mathieu functions of the third kind

(1)0 ( )

( , )2

i H krU

s x r s x

Addition theorem (Morse and Feshbach’s book)

0 1

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) , ,( ) ( )

( , )( , ) ( , )2 ( , ) ( , ) ( , )

( ) (

m mm m m m m me o

m mm m

m mm m me o

m m

Se q So qi Se q Je q He q So q Jo q Ho qM q M q

USe q So qi Se q Je q He q

M q M q

s sx s x x s x x s

s sx x s

s x

0 1

( , ) ( , ) ( , ) , ,) m m m

m m

So q Jo q Ho q

x x s x s

Normalized constants

Methods of Theoretical Physics, 1953, p.1421

Analytical study (norm)


Contour plots of the closed-form fundamental solution and the degenerate kernel

0

0

( ) ( )(

, ,

( ) (,

, ,)

)

j j s

j j sj

j

F G

F GU

x

x

sx

xs

s

x

(1)0 ( )

( , )2

i H krU

r

s x

s x

Closed-form fundamental

solution

Degenerate kernel

Abs Re Im

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

0 1

( , ) ( , )2 ( , ) (

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) , ,( ) ( )

, ) ( ,)

( , )

)( (

m mm m m m m

m mm m me o

m m

me om mm m

Se q So qi Se q Je q He q So q Jo q Ho

Se q So qi Se q Je q He qM q M q

M qU

qM q

s s

x s x x s x x s

s sx x s

s x

0 1

( , ) ( , ) ( , ) , ,) m m m

m m

So q Jo q Ho q

x x s x s


Degenerate kernels

(elliptic coordinates)

(polar coordinates)


Expansions of boundary densities and incident plane wave for circular boundaries

0 1

( ) cos( ) sin( ), ,n nn n

u g n h n B

s ss s

0 1

( ) cos( ) sin( ), ,n nn n

t p n q n B

s ss s

Boundary densities

0

( ) ( ) ( )cos( ( )),nI n n

n

u i J k n

x xxIncident plane wave

Fourier series

Polar coordinates


Expansions of boundary densities and incident plane wave for elliptical boundaries

0 1

( ) ( , ) ( , ), ,n n n nn n

u g Se q h So q B

s ss s

0 1

1( ) ( , ) ( , ) , ,n n n nn n

t p Se q q So q BJ

s ss

s s 2cEigenfunction expansion (Mathieu functions)

0 1

( , ) ( , )( ) 8 ( ) ( , ) ( , ) ( ) ( , ) ( , ) ,

( ) ( )n nn n

I n n n ne on nn n

Se q So qu i Se q Je q i So q Jo q

M q M q

x x x xx

Boundary densities

Incident plane wave

Elliptic coordinates


Keypoint for solving the problem with elliptical boundaries

0 ( , ) ( ) ( ) ( , ) ( ) ( )B BT u dB U t dB s x s s s x s s

( ) s sd s JB d

0 1

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,( ) ( )

( , )( , ) ( , )2 ( , ) ( , ) ( , )

( ) ( )

m mm m m m m me o

m mm m

m mm m me o

m m


USe q So qi Se q Je q He q S

M q M q

s ss s s

s ss

s x

0 1

0 1

( , ) ( , ) ( , ) ,

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,( ) ( )

( , )(

1

12

m m mm m

m mm m m m m me o

m mm m

m

o q Jo q Ho q


TS qi

J

Je

s

s

s s

s ss s s

s x

0 1

, ) ( , )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,( ) ( )

mm m m m m me o

m mm m

So qSe q Je q He q So q Jo q Ho qM q M q

s s

s s s

0 1

( ) ( , ) ( , ), ,n n n nn n

u g Se q h So q B

s ss s

0 1

( ) ( , ) ( , ) , ,1n n n n

n n

t p Se q q So qJ

B

s

s ss s

Orthogonal relations are reserved


Adaptive observer systems and linear algebraic equations

{} { }[ ] [ ] ,U t T u=

Collocation point

Boundary contour integration

( , ) x xx

( , ) x xxdB

dB

x dB

dB


Outline



Illustrative examples

• Case 1: A single elliptical cylinder

• Case 2: Two parallel identical elliptical cylinders

• Case 3: One circular and one elliptical cylinders


Case 1: A single elliptical cylinder

1 1

1

1

11

101.5

tanh

10

ba

b

h

a

1a1b

1

1

0 1

1

1 1

( , ) ( , )2( ) 4 ( ) ( , ) ( , )( ) ( , )

( , ) ( , )( ) ( , ) ( , ) ,

( ) ( , )

n n nR n ne

n n n

n n nn no

n n n

Se q Je qu i Se q He q

M q He q

So q Jo qi So q Ho q

M q Ho q

x x

x x

x


Number of degree of freedom

Resultant forces of an elliptical cylinder

[3] Au M. C. and Brebbia C. A., “Diffraction of water waves for vertical cylinders using boundary elements”, Applied Mathematical Modelling, Vol. 7, (1983), pp 106-114.

n u ll-f ie ld B IE MB E M





n u ll-f ie ld B IE MB E M


Case 2: Two parallel identical elliptical cylinders

1 2

1 2

10.25

21.5

a ab bdh


Resultant forces of two parallel identical elliptical cylinders


Case 3: One circular and one elliptical cylinders

1

1

2

10.250.5

21.5

abadh


Resultant forces of two cylinders containing one circular and one elliptical cylinder


Outline



Conclusions

1. The higher accurate and faster convergence rate of the present method over the EBM is observed

2.Null-field BIEM in conjunction with adaptive observer system and the degenerate kernel can solve water wave problems containing circular and elliptical cylinders in a semi-analytical way.

3. This method also belongs to a meshless method since collocation points on the boundaries are only required.


The endThanks for your kind attentions

http://msvlab.hre.ntou.edu.tw/Welcome to visit the web site of MSVLAB/NTOU


Extension (circle to ellipse)Expand fundamental solution by using the degenerate kernel

0

0

( ) ( ), ,( , )

( ) ( ), ,

s

s

j jj

j jj

F GU

F G

x

x

x ss x

s x

x( , ) x x

x( , ) x x

s ( , )s s

s ( , )s s


Degenerate kernels(polar coordinates)


Four degenerate kernels(elliptic coordinates)


Adaptive observer systems and linear algebraic equations

{} { }[ ] [ ] ,U t T u={} { }[ ] [ ] .L t M u=

Collocation point Collocation point

Boundary contour integration Boundary contour integration


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Elliptic coordinates and Mathieu function

?

2

2ckq

:

:

angular coordinateradial coordinate

2cMathieu function

Modified Mathieu function

( , ), 0,1 ,( )

( , ), 1, 2 ,m

m

Se q mA

So q m

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

( , ) ( , ), 1, 2 ,m m m m

m m m m

Je q Ye q mB

Jo q Yo q m

( ) )) (( BAu x


Resultant forces of a circular cylinder




-10 10 20

2

4

6

8

10

Us,x

1 ( s),1 2( , )

12 (s ),2

Ux x s

s x x sx s x

ìïï - ³ïïï= - =íïï - >ïïïî

-10 10 20

-0.4

-0.2

0.2

0.4

Ts,xs

continuouscontinuous jumpjump1 ,2( , )

1 ,2

Tx s

s xs x

ìïï - >ïïï=íïï >ïïïî


Elliptic coordinates and Mathieu function

?

2

2ckq

:

:


2cMathieu function


( , ), 0,1 ,( )

( , ), 1, 2 ,m

m

Se q mA

So q m

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

( , ) ( , ), 1, 2 ,m m m m

m m m m

Je q Ye q mB

Jo q Yo q m

( ) )) (( BAu x


Difference between the 33rd CTAM and present work

33rd CTAM Present work

Interior problemEigenproblems

Interior problemWater wave problems





Boundary densities

0 1

( ) cos( ) sin( ), ,n nn n

u g n h n B

s ss s

0 1

( ) cos( ) sin( ), ,n nn n

t p n q n B

s ss s

0 1

( ) ( , ) ( , ), ,n n n nn n

u g Se q h So q B

s ss s

0 1

1( ) ( , ) ( , ) , ,n n n nn n

t p Se q q So q BJ

s ss

s s

Expand boundary densities by using the Fourier series and eigenfunction expansion

2c

Circular boundaries

Elliptical boundaries


Expansions of incident plane wave using the polar and the elliptic coordinates

0

( ) ( ) ( )cos( ( )),nI n n

n

u i J k n

x xx

0 1

( , ) ( , )( ) 8 ( ) ( , ) ( , ) ( ) ( , ) ( , ) ,

( ) ( )n nn n

I n n n ne on nn n

Se q So qu i Se q Je q i So q Jo q

M q M q

x x x xx

Circular boundaries

Elliptical boundaries


Boundary densitiesExpand boundary densities by using the eigenfunction expansion

0 1

1( ) ( , ) ( , ) , ,n n n nn n

t p Se q q So q BJ

s ss

s s

0 1

( ) ( , ) ( , ), ,n n n nn n

u g Se q h So q B

s ss s

2 2

1.1

1.2

1.3

1.4

1.5

1.6Js

is a constants along the elliptical boundarys

11

c

s

2c


Successful experiences in 2-D eigenproblems with circular boundaries

0 ( ) ( )( , ) ( , ( ) ( ))B BT d tUu B dB s sxsx ss s

Kuo et al. Int. J. Numer. Meth. Engng. 2000

Key point Degenerate kernel(Polar coordinates)

0 ( ) ( )( , ) ( , ( ) ( ))B B

M d tLu B dB s sxsx ss s

UT equation

LM equation

Spurious eigenvalues Spurious eigenvalues

Chen et al. Proc. R. Soc. Lond., Ser. A, 2002 & 2003

KernelReal-part

Imaginary-part

Complex-valued kernelUT or LM

Inner boundary

(Found and treated)

(Singular)

(Hypersingular)


Elliptic coordinates and Mathieu function2

2ckq

:

:


2cMathieu function


( , ), 0,1 ,( )

( , ), 1, 2 ,m

m

Se q mA

So q m

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

( , ) ( , ), 1, 2 ,m m m m

m m m m

Je q Ye q mB

Jo q Yo q m

( ) )) (( BAu x


Degenerate kernels

Modified Mathieu functions of the third kind

(1)0 ( )

( , )2

i H krU

s x r s x

Addition theorem (Morse and Feshbach’s book)

0 1

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,( ) ( )

( , )( , ) ( , )2 ( , ) ( , ) ( , )

( ) ( )

m mm m m m m me o

m mm m

m mm m me o

m m



M q M q

s ss s s

s ss

s x

0 1

( , ) ( , ) ( , ) ,m m mm m

o q Jo q Ho q

s s

Orthogonal relations

Methods of Theoretical Physics, 1953, p.1421

Analytical study

(norm)


Successful experiences in 2-D problems with circular boundaries

using the present approach)()()( sdBsx

B ),( xsK

),( xsK e

Fundamental solutionFundamental solution

Advantages of present approach:1. No principal value2. Well-posed model3. Exponential convergence4. Free of mesh generation

Degenerate kernelDegenerate kernel

xsxsKxsxsK

e

i

),,(),,(

),( xsK i

sx ln

The proposed approach will be extended to

deal with 2-D problem with elliptic boundaries

)()1(0 sxkH

(Laplace)(Helmholtz)


Why spurious solution occurs

• FDM for ODE

• Real-part BEM & MRM (Simply-connected problem)

• Complex-valued BEM (Multiply-connected problem)


Separation of variables in the elliptic coordinates

2 2 2 2( ) ( )cosh(2 ) cos(2 )( ) 2 ( ) 2

B k c A k c pB A

2 22 cos(2 ) 0,

42 cosh(2 ) 0

A q G c kqB q F

p

p

2 2 2 2

2 2 cosh(2 ) cos(2 02

u u k c u

2 2

22 22 2 2

1 0sinh ( ) sin ( )

u u k uc

2 2( , ) ( , ) 0u k u Elliptic coordinates 2 2( , ) ( , ) 0u x y k u x y Cartesian coordinates

( , ) ( ) ( )u A B separation of variables


Addition theorem

n

innn

n erkJbkJ

rbkJakJ

)(

00

1|)|(|)|()1(

|)|(|)|(

sincoscossin)sin(

ra

b

1

2

Q

O

P

a

sxsx eee

= + sinsincoscos)cos(

Addition theorem

s

xsxsxsx

eeeeee )(

n

innn ebkJakJ

bakJbakJrkJ

)(

000

2|)|(|)|(

|))(|(|)|(|)|(

Subtraction theorem

rb



1

1

1ln c

1ln c

( , ) lnos

s ,

,

o

m

m

m

m

R mU

R mR

m

m

Rr

R

s x

, , , ,R r s x x s

( , )R s

( , ) x

( , ) x

CircleExtension

Ellipse

0

0

( ) ( ), ,( , )

( ) ( ), ,

s

s

j jj

j jj

F GU

F G

x ss x

s x


Degenerate cases in mathematics and mechanics


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 kRkR

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


Other applications

Electromagnetics

Acoustics1. Hermetic compressor

2. Small automotive muffler

1. Waveguides

TM mode (Dirichlet BC)

TE mode (Neumann BC)

Water wave1. Harbor resonance


Literature review


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. 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)

Mathematical analysis and numerical study for free vibration of plate using BEM-61


The orthogonality of vector and function

1e

2e

3e

1,0,i j

if i je e

if i j

vectors

functions ( ) ( ) ( ) 0b

aw x A x B x dx ( )A x ( )B xand are orthogonal

1 2

2 3

3 1

e ee ee e

Mathieu function

Orthogonal relations (norm)


Jacobian

LJ arc length

AJ (Area)

2 2sinh( ) cos( ) cosh( )sin( )

ds Jd

J c

2 22 2sinh( )cos( ) cosh( )sin( )

dA Jd d

J c c

Elliptic coordinatesPolar coordinates

2 2ds dx dy

dA dxdy

ds JdJ

dA Jd d

J

, ,x y , ,x y

cosh cos

sinh sin

x c

y c

cossin

xy

,x y ,x y


Adaptive observer system

collocation pointcollocation point

0 , 01 , 1k , k2 , 2


Linear algebraic equation

{}

0

1

2

N

tt

t t

t

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

[ ]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 UU U U

U

U U U

LL

M M O ML

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

[ ]{} { }[ ]t uU T=


Literature review (Degenerate kernel )

Author ApplicationsSloan et al.(1975)

Prove that it is equivalent to iterated Petrov-Galerkin approximation

Kress(1989)

Prove that the integral equation combined with degenerate kernel has convergence of exponential order

Chen et al.(2005)

Applied it to solve engineering problems with circular boundaries

Chen et al.(2007)

Link Trefftz method and method of fundamental solutions

However, its applications in practical problems seem to have taken a back seat to other methods. ~ M. A. Golberg 1979

Degenerate kernel approximation

(Schaback)


Hypersingular integralH.P.V.(Hadamard principal value)

1

433

2

002

00

0sin14

)()(

2-sin4

1-

，Nn

nN

ddLLq

Principle value version Series summability version

x

1 1

2 21 10

1 1 2. . . lim = 2 H PV dx dxx x

1D2D

NTOU/MSV D. H. Yu


Other degenerate kernels-1

.,)(cos1ln21),;,(

,,)(cos1ln21),;,(

),(

1

1

RmRm

RU

RmRm

RRU

xsU

m

me

m

mi

0

)1(

0

)1(

,)),(cos()()(4

),;,(

,)),(cos()()(4

),;,(),(

mmmm

e

mmmm

i

RmkRJkHiRU

RmkRHkJiRUxsU

(2-D circular Laplace problem)

(2-D circular Helmholtz problem)

(2-D circular biHelmholtz problem)

32 2

2

222

32 2

2

22

1( , ) (1 ln ) ln [ (1 2ln ) ]cos( )2

1 1[ ]cos[ ( )],( 1) ( 1)

( , ) ln1( , ) (1 ln ) ln [ (1 2ln ) ]cos( )2

1 1[ ]co( 1) ( 1)

I

m m

m mm

E

m m

m mm

U s x R R R R RR

m Rm m R m m R

U s x r rRU s x R R

R Rm m m m

s[ ( )],m R

(2-D circular biharmonic problem)



0 1

( , ) ( , )2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ,( ) ( )

( , )( , ) ( , )2 ( , ) ( , ) ( , )

( ) ( )

m mm m m m m me o

m mm m

m mm m me o

m m



M q M q

s ss s s

s ss

s x

0 1

( , ) ( , ) ( , ) ,m m mm m

o q Jo q Ho q

s s

11 0

11 0

1 ( )!cos ( ) (cos ) (cos ) ,( )!1( , )

1 ( )!cos ( ) (cos ) (cos ) ,( )!

nni m m

m n n nn m

nne m m

m n n nn m

n mU m P P RR n m R

U s xr n m RU m P P R

n m

,)],(cos[)()()(cos)(cos)!()!()12(

4),,;,,(

,)],(cos[)()()(cos)(cos)!()!()12(

4),,;,,(

),()2(

0 0

)2(

0 0

mkhkjPPmnmnnikU

mkhkjPPmnmnnikU

xsUnn

mn

mn

n

n

mm

e

nnm

nm

nn

n

mm

i

(3-D spherical Lalace problem)

(2-D elliptical Laplace problem)

,,sinsinsinh2coscoscosh22

ln21),;,(

,,sinsinsinh2coscoscosh22

ln21),;,(

),(

11

11

m

m

m

me

m

m

m

mi

mmmem

mmmem

aU

mmmem

mmmem

cUxsU

(3-D spherical Helmholtz problem)

(2-D elliptical Helmholtz problem)



( )

1

( )

1

( )

1ln(2 ) cos ( ) cos( ) cos( ) , 0

1( , ) ln(2 ) cos ( ) cos( ) cos( ) , 0

1ln(2 ) cos ( ) cos( ) cos

s s

s s

s s

n nns s s s

n

n nns s s

n

n nns

c e n e n e nn

U s x c e n e n e nn

c e n e n en

1

( ) , 0s sn

n

1( )( )2

0 0

1 ( )!cos ( ) (cos( )) (cos( )) ,( )!1( , )

1 ( )!cos ( ) (co

cosh( ) cos( ) cosh( ) cos( )

cosh( ) s( )) (coscos( ) cosh( ) ( )cos( ) )( )!

sn

nm mm s n s n s

n m

m mm s n s n

s s

s s

n m m P P ec n m

U x sr n m m P P

c n m

1( )( )2

0 0

,sn

ns

n m

e

,,))1()1cos(()())2(cos()(121))(cos()(1ln)43(

)1(81),;,(

,,))1()1cos(()())2cos(()(121))(cos()(1ln)43(

)1(81),;,(

),(

0

1

0111

0

1

0111

11

RmmRmmRmRmG

RU

RmmR

mmR

mRm

RG

RUxsU

m

m

m

m

m

me

m

m

m

m

m

mi

(Circular Navier problem)

(Two circles(bipolar) Laplace problem)

(Two spheres(bispherical) Laplace problem)


Orthogonal coordinate systems2D 3D

Cartesian Cartesian

Polar Sphere

EllipticOblate spheroidalProlate spheroidal

BipolarBispherical

Toroidal

Parabolic

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Date: Jan. 13, 2010 Time: 11:50~12:10 Place: Lectrue Theater F

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