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National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering. Null-field boundary integral equation approach for hydrodynamic scattering by multiple circular and elliptical cylinders. Jai-Wei Lee and Jeng-Tzong Chen. Date: Jan. 13, 2010 Time: 11:50~12:10 - PowerPoint PPT Presentation
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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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 2
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 3
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 4
The 8th ACFD Conference in HK, 2010.1. 10~14
Keelung
NTOU
HKUST
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 5
NTOU/MSV Group members (2010)
1983
1962 1955 19751962 19711959 19761972
1988198719861987198519851978
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 6
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 7
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
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)
2010/01/13 Page 9The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 2010
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
2010/01/13 Page 10The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 2010
Introduction of water wave problem(multiple cylinders)
To the authors’ best knowledge
OK
Multipole expansion
Multipole expansion ?(Null-field BIEM) OK
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 11
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 12
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 13
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 14
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 15
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 16
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 17
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 18
Degenerate kernels
(elliptic coordinates)
(polar coordinates)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 19
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 20
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 21
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
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 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
Se q So qi Se q Je q He q So q Jo q Ho qM q M 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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 22
Adaptive observer systems and linear algebraic equations
{} { }[ ] [ ] ,U t T u=
Collocation point
Boundary contour integration
( , ) x xx
( , ) x xxdB
dB
x dB
dB
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 23
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 24
Illustrative examples
• Case 1: A single elliptical cylinder
• Case 2: Two parallel identical elliptical cylinders
• Case 3: One circular and one elliptical cylinders
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 25
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 26
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 27
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 28
Case 2: Two parallel identical elliptical cylinders
1 2
1 2
10.25
21.5
a ab bdh
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 29
Resultant forces of two parallel identical elliptical cylinders
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 30
Case 3: One circular and one elliptical cylinders
1
1
2
10.250.5
21.5
abadh
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 31
Resultant forces of two cylinders containing one circular and one elliptical cylinder
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 32
Outline
• Introduction of NTOU/MSV group• Motivation and problem statement• Method of solution• Illustrative examples• Conclusions
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 33
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 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 34
The endThanks for your kind attentions
http://msvlab.hre.ntou.edu.tw/Welcome to visit the web site of MSVLAB/NTOU
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 35
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 36
Degenerate kernels(polar coordinates)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 37
Four degenerate kernels(elliptic coordinates)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 38
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 39
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國立台灣海洋大學力學聲響振動實驗室 (NTOU/MSV Lab)負責老師:陳正宗 終身特聘教授 ( 海洋大學河海工程學系 ) 地點:河工二館 HR2306 室 陳義麟 副教授 ( 高雄海洋科技大學造船學系 ) 聯絡電話: 886-2-24622192 ext.6177 or 6140 李為民 副教授 ( 中華技術學院機械系 ) URL : http://ind.ntou.edu.tw/~msvlab 呂學育 助理教授 ( 中華技術學院航空機械系 ) E-mail: jtchen@mail.ntou.edu.tw 陳桂鴻 副教授 ( 國立宜蘭大學土木系 ) Fax: 886-2-24632375 徐文信 助理教授 ( 屏東科技大學教學資源中心 ) 范佳銘 助理教授 ( 海洋大學河海工程學系 )
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The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 40
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 41
Resultant forces of a circular cylinder
Number of degree of freedom
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 42
Degenerate (separable) form of fundamental solution (1D)
-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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 43
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 44
Difference between the 33rd CTAM and present work
33rd CTAM Present work
Interior problemEigenproblems
Interior problemWater wave problems
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 45
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.
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 46
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 47
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 48
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 49
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 50
Elliptic coordinates and Mathieu function2
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 51
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 S
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 52
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 53
Why spurious solution occurs
• FDM for ODE
• Real-part BEM & MRM (Simply-connected problem)
• Complex-valued BEM (Multiply-connected problem)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 54
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 55
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 56
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
0
0
( ) ( ), ,( , )
( ) ( ), ,
s
s
j jj
j jj
F GU
F G
x ss x
s x
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 57
Degenerate cases in mathematics and mechanics
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 58
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 59
Other applications
Electromagnetics
Acoustics1. Hermetic compressor
2. Small automotive muffler
1. Waveguides
TM mode (Dirichlet BC)
TE mode (Neumann BC)
Water wave1. Harbor resonance
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 60
Literature review
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 61
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 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 62
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 63
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 64
Adaptive observer system
collocation pointcollocation point
0 , 01 , 1k , k2 , 2
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 65
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=
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 66
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 67
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
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 68
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 69
Other degenerate kernels-2
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 S
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 70
Other degenerate kernels-3
( )
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)
The 8th Asian Computational Fluid Dynamics ConferenceHong Kong, 10-14 January, 20102010/01/13 Page 71
Orthogonal coordinate systems2D 3D
Cartesian Cartesian
Polar Sphere
EllipticOblate spheroidalProlate spheroidal
BipolarBispherical
Toroidal
Parabolic
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