1. Study of near-trapped modes and fictitious frequencies using null-field integral equation for water wave
Chine-Feng Wu(吳建鋒), Yi-Jhou Lin(林羿州) ,
I-Lin Chen(陳義麟), Jeng-Tzong Chen(陳正宗)
National Taiwan Ocean University
December 10, 2009
13:00~13:20
It is a pleasure to be in Lumg Hwa University.
I come from National Taiwan Ocean University in Keelung, Taiwan.
This is my first time to present in the international conference.
My name is Chine-Feng Wu; Prof. Jeng-Tzong Chen is my advisor.
Now let us begin this talk.
The title of the presentation is “Study of near-trapped modes and fictitious frequencies using null-field integral equation for water wave”.

2. Outline Motivation Problem statements and formulation
Numerical examples
Conclusions
I will introduce four parts.
First one is the motivation.
I will describe the Problem statements and formulation that we used.
Next I will take three numerical examples to demonstrate the present method.
Finally, I will make some conclusions.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

3. Engineering problems Platform (Offshore structure) Array of cylinders
Ringing, Rodney Eatock Taylor
In this approach, we focus on the null-field integral equations to solve the scattering of water waves.
This is BHP’s oilfield. Such as oil platforms which consist of four legs, it is important to understand the interaction between the vertical cylinders and plane water wave.
BHP Billiton's giant Atlantis oilfield in the deepwater Gulf of Mexico
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

4. Engineering problems Array of cylinders Keelung, Taiwan. Keelung
This phenomena is common at everywhere. If you come to Keelung, you will see the wharf.
This is the wharf building photo, we can see the wharf consist of a number of legs.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

5. Numerical and physical resonance
Fictitious frequency
(BEM/BIEM)
Present
t(a,0)
We can talk two parts in this approach.
Physical resonance
When damping is small, the frequency of free vibrations is approximately equal to the natural frequency of the system, which is the resonant frequency.
Fictitious frequency
The fictitious frequency (mathematics) is due to the integral formulation for exterior Helmholtz problems.
So, it’s interesting issue on two part would be happened in the same time.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

6. Water wave interaction with surface-piercing cylinders
Governing equation:
Separation variable :
Seabed boundary conditions :
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

7. Problem statement 2 1 3 j Governing equation: Dispersion relationship:,
Dynamic pressure: j .
Force:
Original problem
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

8. Present approach Advantages of present approach Mesh-free generation
Degenerate kernel
Fundamental solution
No principal value
CPV and HPV
Advantages of present approach
Mesh-free generation
Well-posed model
Principal value free
Elimination of boundary-layer effect
Exponential convergence
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

9. Outline Introduction Problem statements and formulation
Numerical examples
Conclusions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

10. Convergence rate between present method and conventional BEM
Degenerate kernel
Two-point function
Fundamental solution
Constant, linear, quadratic elements
Fourier series expansion
Boundary density
In the conventional BEM is using the constant element, linear element and quadratic element.
However, we want to solve a circular boundary, so we are expanding the boundary densities by using Fourier series to calculate.
So, we get the exponential convergence, it’s better than conventional BEM.
Convergence rate
Exponential convergence
Linear convergence
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

11. BIE and null-field integral equation
Interior case
Exterior case
Conventional BEM
Domain point integral equation.
Boundary integral equation.
Null-field integral equation.
When we want to solve the Boundary integral equation, we mast have to solve the Principal value.
However, We don’t want to touch this problem. So, what can I do?
In fact, we can used the Degenerate form to solve the problem.
No matter inside the domain or outside the domain, even on the boundary, we can explain very well.
Degenerate (separable) form
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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12. Degenerate kernel L(s,x) M(s,x) U(s,x) T(s,x)
Expand fundamental solution by using degenerate kernel
L(s,x)
M(s,x)
U(s,x)
T(s,x) s x
Source point
A source point is here, and draw a circle.
When the collocation point is inside the circle, we used the first formulation.
When the collocation point is outside the circle, we used the second formulation.
Collocation point x
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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13. Degenerate kernel and Fourier series
Ｄegenerate kernel （Helmholtz）
Expanding boundary densities by using Fourier series
ith circular
boundary
The unknown boundary density is expanded in the Fourier series.
So, we can use the orthogonal property in the BIE.
cosnθ, sinnθ
boundary distributions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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14. Adaptive observer system
Source point
Collocation point
Adaptive observer system
A Collocation point is here, and we want to integral first circle.
The coordinate set on the center of the first circle.
And then, we want to integral second circle, the coordinate set on the center of the second circle.
And so on.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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15. Linear algebraic systemx
To fully use the orthogonal property, the BIE as well as a Linear algebraic equation.
It will be easily to deal with multiple circular holes.
Index of collocation circle
Index of routing circle
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16. Flowchart of present method
Collocation on the real boundary
Original problem
Decompose two parts
Free field
Radiation field
Linear algebraic system
Calculation of the unknown Fourier
Expansion
Degenerate kernel
for fundamental solution
Fourier series of boundary densities
BIE for the domain point
Next I introduce the flowchart of present.
This is called the null-field BIE.
The kernel functions are expanded to the degenerate kernel.
The unknown boundary conditions are expanded to the Fourier series.
Then we can obtain the linear algebraic system.
So, the Fourier coefficients can be obtained easily.
Finally, the potential function can be solved.
Superposing the solution of two parts
Total field c B D x s dB t U u T Î - = ò ), ( ) ,
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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17. Outline Introduction Problem statements and formulation
Numerical examples
Case 1: Near-trapped mode and fictitious frequency
Case 2: Effect of incident angle on the near-trapped mode
Case 3: Disorder of the periodical pattern for four cylinders
Conclusions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

18. Sketch of four cylinders
We consider the water wave problem by an array of four vertical rigid circular cylinders.
Radius is equal a
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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19. Physical phenomenon and fictitious frequency
Near-trapped mode
Near-trapped mode
Fictitious frequency
Near-trapped mode
Near-trapped mode
Fictitious frequency
Near-trapped mode
Fictitious frequency
X axis means wavenumber ka.
Y axis means Xj over F
Xj means force on the jth cylinder are given by integrating the pressure over the circular boundary.
F is the analytic solution of single cylinder.
Two kinds of peaks for the resultant force on the cylinder versus the wavenumber are observed.
2.4
3.8
5.1
5.5
6.4 ka
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20. Near-trapped mode for the four cylinders at ka=4.08482 (a/d=0.8)
(a) Contour by the present method (M=20)
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21. Near-trapped mode for the four cylinders at ka=4.08482 (a/d=0.8)1 2 3 4 54
(b) Free-surface elevations by the present method (M=20)
(c) Horizontal force on the four cylinders against wavenumber
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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22. Outline Introduction Problem statements and formulation
Numerical examples
Case 1: Near-trapped mode and fictitious frequency
Case 2: Effect of incident angle on the near-trapped mode
Case 3: Disorder of the periodical pattern for four cylinders
Conclusions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

23. Four cylinders a/d=0.8, ka=5.797, present Evans and Porter, JEM ,1999.
When we change the incident wave to degree.
And then, we compare with Evans and Porter, publish on JEM in 1999.
We have obtained a good agreement.
present
Evans and Porter, JEM ,1999.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

24. ka=
We can discover the trapped mode at 0 degree and 33 degree.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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25. Outline Introduction Problem statements and formulation
Numerical examples
Case 1: Near-trapped mode and fictitious frequency
Case 2: Effect of incident angle on the near-trapped mode
Case 3: Disorder of the periodical pattern for four cylinders
Conclusions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

26. By perturbing the radius of one cylinder (a1/d≠0
By perturbing the radius of one cylinder (a1/d≠0.8) to destroy the periodical setup
a1/d=0.82
When we change the radius to disorder the periodical setup.
And then, we compare with Evans and Porter, publish on JEM in 1999.
We also have obtained a good agreement.
Evans and Porter, JEM ,1999.
Present method
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

27. Sketch of four cylinders1 1
So, we have two ideas to disorder the periodical array.
One is change one of radius, and the other is moving the center of one cylinder.
Changing radius
Moving the center of one cylinder
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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28. Disorder of the periodical pattern
Changing radius
Moving the center of one cylinder
i=2,3,4
Cylinder 1
Cylinder 3
Force
0.86
0.8
1.15
0.25
0.84
1.20
0.82
1.30
0.27
54.1
0.78
1.02
0.34
0.76
1.13
0.30
0.74
1.19
i=2,3,4
Cylinder 1
Cylinder 3
Force
0.86
0.8
1.15
0.29
0.84
1.20
0.28
0.82
1.27
0.27
54.1
0.78
1.12
0.76
1.17
0.26
0.74
1.16
When the periodical array, the cylinder of force is jump to 54.
And we can discover, no matter we change radius or moving the center of one cylinder.
The trapped mode are suppressed very quickly.
So, we can get some conclusions, the periodical array is an important condition to effect the trapped mode.
ka=
ka=
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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29. Outline Introduction Problem statements and formulation
Numerical examples
Conclusions
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

30. Conclusions
Physical phenomena of near-trapped mode as well as the numerical instability due to fictitious frequency in BIEM were both observed.
Fictitious frequency appears and is suppressed in sacrifice of higher number of Fourier terms.
The effect of incident angle and disorder on the near-trapped mode was examined.
Physical phenomena of near-trapped mode as well as the numerical instability due to fictitious frequency in BIEM were both observed.
Fictitious frequency appears and is suppressed in sacrifice of higher number of Fourier terms.
The effect of incident angle and disorder on the near-trapped mode was examined.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

31. You can get more information from our website.
Thanks for your kind attentions
You can get more information from our website.
I sincerely hope that you could give me some useful advice and valuable experiences on this important conference.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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32. 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

33. 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

34. Convergence rate between present method and conventional BEM
（OK）
（OK）
C.A. Brebbia
Song-Ping Zhu
Present
（OK）
（NO）
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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35. Adaptive Mesh FEM BEM DtN interface (natural BEM)
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36. Nonuniform radiation：Dirichlet problem
Numerical solution: BEM
Numerical solution: FEM
64 ELEMENTS
2791 ELEMENTS
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37. Water waves containing circular and elliptical cylinders
Analytical solution
Semi-analytical solution
Numerical solution
Linton & Evan approach
Null-field BIEM
BEM
MSVLab
陳正宗、李家瑋、李應德、林羿州
岳景雲、陳一豪、賴瑋婷 ok ok
Error
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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38. Effect of multiple scattering
Contour by the present method
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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39. Non-unique solution Near-trapped mode (physical) Non-unique solution:
t(a,0)
Fictitious frequency (Numerical)
(1) CHIEF method (Schenck, JASA , 1968)
Additional constraint (CHIEF point)
(2) Burton and Miller method (Burton and Miller, PRS , 1971)
(3) SVD updating term technique (Chen et al., JSV, 2002)
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40. Addition theorem = + Addition theorem Q r a b P Subtraction theorem O
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41. Literature review (Degenerate kernel )
Degenerate kernel approximation
Author
Applications
Sloan 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
(2007)
Link Trefftz method and method of fundamental solutions
(Schaback)
However, its applications in practical problems seem to have taken a back seat to other methods. ~ M. A. Golberg 1979
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42. Degenerate cases in mathematics and mechanics
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43. Singularity & hypersingularity
Motivation
Main idea
BEM / BIEM
路不轉 分內外核函數
Improper integral
Singularity & hypersingularity
Regularity
Fictitious BEM
Bump contour
exterior
Fictitious boundary
interior
Achenbach et al. (1988)
Null-field approach
Guiggiani (1995)
Collocation point
CPV and HPV
Ill-posed
山不轉 路轉
Waterman (1965)
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44. Singular and hypersingular integrals
Conventional approach (bump contour) x s x 3D s
Present approach x s x s 3D
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45. Singular and hypersingular integrals
Conventional approach (bump contour) s x x s 2D
Present approach s s 2D x
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46. Degenerate kernel L(s,x) M(s,x) U(s,x) T(s,x) x s O x
Expand fundamental solution by using degenerate kernel
Ｄegenerate kernel （Helmholtz） x
L(s,x)
M(s,x)
U(s,x)
T(s,x) s O x
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
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47. Degenerate kernel
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48. Eigenproblem Resonance
Incident wave
Transmitted wave
Reflected wave
Viscous effects on Bragg scattering of water wave by an array of piles.
A Tabaei, C C Mei
Resonance
Eigenproblem
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49. Trapped mode（nonuniqueness in physics ）
M.S. Longuet-higgins
JFM, 1967.
A.E.H. Love
1966.
Williams & Li
OE, 2000.
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50. Two cylinders subject to an incident wave
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51. Force ratio of two equal cylinders
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52. Moving the center of one cylinder
Changing radius
Moving the center of one cylinder
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53. Trapped and near-trapped modes

54. Trapped mode（nonuniqueness in physics ）
Hydraulic engineering, Earthquake engineering and Ocean engineering.
M.S. Longuet-higgins
JFM, 1967.
A.E.H. Love
1966.
Williams & Li
OE, 2000.
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009

55. Mechanism of fictitious frequency
2.4042
5.5201
N=2
3.8317
7.0156
N=3
5.1356
8.4172
N=4
6.3802
9.7610
N=5
7.5883
N=6
8.7715
kNi i N
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56. Variation of incident angle
k= and a/d=0.8
Angle ka
Angle
ka=
2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics
Dec. 10, 2009