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”.

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

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

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

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

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

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

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

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

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

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 Dec. 10, 2009

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 Dec. 10, 2009

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 Dec. 10, 2009

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 Dec. 10, 2009

Linear algebraic system x 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 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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 Dec. 10, 2009

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

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 Dec. 10, 2009

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 4.08482 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

Near-trapped mode for the four cylinders at ka=4.08482 (a/d=0.8) (a) Contour by the present method (M=20) 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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 Dec. 10, 2009

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

Four cylinders a/d=0.8, ka=5.797, present Evans and Porter, JEM ,1999. When we change the incident wave to 12.25 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

ka=4.08482 We can discover the trapped mode at 0 degree and 33 degree. 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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

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

Sketch of four cylinders 1 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 Dec. 10, 2009

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=4.08482 ka=4.08482 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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

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

You can get more information from our website. Thanks for your kind attentions You can get more information from our website. http://msvlab.hre.ntou.edu.tw/ 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 Dec. 10, 2009

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

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

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 Dec. 10, 2009

Adaptive Mesh FEM BEM DtN interface (natural BEM) 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

Nonuniform radiation：Dirichlet problem Numerical solution: BEM Numerical solution: FEM 64 ELEMENTS 2791 ELEMENTS 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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 Dec. 10, 2009

Effect of multiple scattering Contour by the present method 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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

Addition theorem = + Addition theorem Q r a b P Subtraction theorem O 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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

Degenerate cases in mathematics and mechanics 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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

Singular and hypersingular integrals Conventional approach (bump contour) x s x 3D s Present approach x s x s 3D 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

Singular and hypersingular integrals Conventional approach (bump contour) s x x s 2D Present approach s s 2D x 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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 Dec. 10, 2009

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

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

Trapped mode（nonuniqueness in physics ） 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

Two cylinders subject to an incident wave 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

Force ratio of two equal cylinders 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

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 Dec. 10, 2009

Trapped and near-trapped modes

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

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 11.0647 N=6 8.7715 12.3386 kNi i N 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009

Variation of incident angle k=4.08482 and a/d=0.8 Angle ka Angle ka=4.08482 2009 International Workshop on Students' Exchanges of Nano and Computational Mechanics Dec. 10, 2009