1. Dr. Jeng-Tzong Chen Date: September, 2009 Place: City London University Trapping and near-trapping by arrays of porous cylinders in water waves using the addition theorem and superposition technique National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering National Taiwan Ocean University

2. June 16, 2015 pp. 2 Outline Motivation and literature review Unified formulation of null-field approach Numerical examples Concluding remarks

3. June 16, 2015 pp. 3 Outline Motivation and literature review Engineering problems Motivation Present approach Unified formulation of null-field approach Numerical examples Concluding remarks

4. June 16, 2015 pp. 4 Engineering problems Platform (Offshore structure)

5. June 16, 2015 pp. 5 Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Mesh-less method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect effect Convergence rate rate Ill-posed model Motivation Mesh generation Constant, linear, quadratic elements Free C.P.V., H.P.V., M.P.V.

6. June 16, 2015 pp. 6 BEM / BIEM Improper integral Singularity & hypersingularity Regularity Bump contour Limit process Fictitious boundary Collocation point Fictitious BEM Null-field approach CPV and HPV Ill-posed Guiggiani (1995) Gray and Manne (1993) Waterman (1965) Achenbach et al. (1988) Motivation

7. June 16, 2015 pp. 7 Fundamental solution No principal value Advantages of present approach 1.mesh-free generation 2.well-posed model 3.principal value free 4.elimination of boundary-layer effect 5.exponential convergence Degenerate kernel CPV and HPV Present approach Kress, 1989

8. June 16, 2015 pp. 8 Outline Motivation and literature review Unified formulation of null-field approach Boundary integral equation and null-field integral equation Convergence rate between present method and conventional BEM Degenerate kernel and Fourier series Adaptive observer system Linear algebraic system Flowchart of present method Numerical examples Concluding remarks

9. June 16, 2015 pp. 9 Boundary integral equation and null- field integral equation Interior case Degenerate (separate) form Exterior case

10. June 16, 2015 pp. 10 Convergence rate between present method and conventional BEM Degenerate kernel Fourier series expansion Fundamental solution Boundary density Convergence rate Present method Conventional BEM Two-point function Constant, linear, quadratic elements Exponential convergence Linear convergence

11. June 16, 2015 pp. 11 Expand fundamental solution by using degenerate kernel Degenerate kernel L(s,x) M(s,x) U(s,x) T(s,x) s O x x Ｄ egenerate kernel （ Helmholtz ）

12. June 16, 2015 pp. 12 kth circular boundary cosnθ, sinnθ boundary distributions Expand boundary densities by using Fourier series Degenerate kernel and Fourier series

13. June 16, 2015 pp. 13 Adaptive observer system Source point Collocation point

14. June 16, 2015 pp. 14 Linear algebraic system x y

15. June 16, 2015 pp. 15 c BB DxsdBstxsUs suxsT   ),()(),()()(),(0 Flowchart of present method Original problem Decompose two parts Free fieldRadiation field Expansion Degenerate kernel for fundamental solution Fourier series of boundary densities Collocation on the real boundary Linear algebraic system Calculation of the unknown Fourier Superposing the solution of two parts BIE for the domain point Total field

16. June 16, 2015 pp. 16 Outline Motivation and literature review Unified formulation of null-field approach Numerical examples Water wave interaction with surface-piercing porous cylinders Concluding remarks Further studies

17. June 16, 2015 pp. 17 Water wave interaction with surface-piercing porous cylinders Governing equation: Separation variable : where Seabed boundary conditions : Free-surface conditions : kinematic boundary condition at free surface (KFSBC) dynamic boundary condition at free surface (DFSBC)

18. June 16, 2015 pp. 18 Problem statement Boundary condition:,. Dispersion relationship: Dynamic pressure: Force: Original problem

19. June 16, 2015 pp. 19 Case 1. Four-cylinders array for one sets (ka=4.08482, a/b=0.8) Numerical examples Case 2. Five sets (ka=4.08482, a/b=0.8) 7 4 2b2b 1 2 3 x y 7 12 2b2b 1010 1 2 3 4 5 13 14 15 16 6 7 8 9 1010 1 x y

20. June 16, 2015 pp. 20 Effect of impermeable case for contour plots (a) BEM (, Chen) (b) Null-field BIEM (M=20 ) Contour plots of free-surface elevation of the four impermeable cylinders (G=0.0, )

21. June 16, 2015 pp. 21 (a) Williams and Li (b) BEM (Chen) (c) Null-field BIEM (M=20) (G=0.0, ) Free-surface elevation of the arrays of four impermeable cylinders. Effect of impermeable case for free-surface elevation

22. June 16, 2015 pp. 22 Effect of disorder case for contour plots (a) BEM (, Chen) (b) Null-field BIEM (M=20 ) Contour plots of free-surface elevation of the four porous cylinders ( G=1.0, )

23. June 16, 2015 pp. 23 (a) Williams and Li (b) BEM (Chen) (c) Null-field BIEM (M=20) (G=0.0, ) Free-surface elevation of the arrays of four impermeable cylinders. Effect of disorder case for free-surface elevation

24. June 16, 2015 pp. 24 Near-trapped mode for the four cylinders at ka=4.08482 (a/b=0.8, G=0.0, ) (a) Contour by the present method (M=20) (no disorder and no porosity)

25. June 16, 2015 pp. 25 Near-trapped mode for the four cylinders at ka=4.08482 (a/b=0.8, G=0.0, ) (c) Horizontal force on the four cylinders against wavenumber (b) Free-surface elevations by the present method (M=20) 54 1 2 3 4

26. June 16, 2015 pp. 26 Near-trapped modes versus incident angle 1 4 2 3 54

27. June 16, 2015 pp. 27 Perturbation of ordered cylinder arrangements a random variable in the range [0,1]. maximum permissible displacement ( p=b-a ). global disorder parameter. 2b a

28. June 16, 2015 pp. 28 Effect of disorder and porosity (a) Contour by the present method (no disorder, impermeable) (b) Contour by the present method (disorder,, impermeable) (c) Contour by the present method (no disorder, porous, G =1) (d) Contour by the present method (disorder,, porous, G =1) Disorder cylinder Porous cylinderDisorder and porous cylinder

29. June 16, 2015 pp. 29 Outline Motivation and literature review Unified formulation of null-field approach Numerical examples Concluding remarks

30. June 16, 2015 pp. 30 A general-purpose program for solving the water wave problems with arbitrary number, different size and various locations of circular cylinders was developed. We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system. Concluding remarks -1/2

31. June 16, 2015 pp. 31 Near trapped mode is observed in this study. It is found that the disorder is more sensitive to suppress the occurrence of near-trapped modes than the porosity. Concluding remarks -2/2

32. June 16, 2015 pp. 32 The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab

33. June 16, 2015 pp. 33 Water waves containing circular and elliptical cylinders Analytical solutionSemi-analytical solutionNumerical solution Linton & Evan approach Bessel to Mathieu ? Null-field BIEM BEM ok ? MSVLab 陳正宗、李家瑋、李應德、 林羿州 岳景雲、陳一豪、賴瑋婷 Error

34. What I have done ( 林羿州 ) 佳聰 (2005) 羿州 (2009) 佳男 (2007) 多圓柱聲波散射 1. 入射平面波 2. 疊加法 3. 不含夾雜 多圓柱聲波散射 1. 點聲源 2. 疊加法 多圓柱水波分析 1. 入射平面波 2. 含透水圓 （等效夾雜） 3. 水波分析 （ trap modes ） 4. 錯位分析 多圓柱聲波散射 1. 點聲源 2. Green's third identity 3. 含夾雜 What I have done.doc 羿州 製

35. Trapped and near-trapped modes Trapped and near-trapped modes.ppt Trapped modesNear-trapped modes Dirichlet or Neumann modes

36. Irregular fictitious and spurious frequency and trap modes Irregular frequencies trap modes Exterior acoustics (fictitious) Interior acoustics (spurious) Floating body (water wave) Acoustics Water wave Trap2009.ppt physics mathematics Physical resonance Mathematical Nonuniqueness problem

37. June 16, 2015 pp. 37 The extension to Helmholtz problem with a hill can be studied by using the present approach in conjunction with the multi-domain technique by decomposing the original problem into one interior problem of circular domain and a half-plane problem with a semi-circular canyon. In the further research, the Helmholtz problems with circular boundaries may be extended to other shapes instead of incident plane wave, shore- crested incident wave can be also considered. Further studies -1/4 SH wave

38. June 16, 2015 pp. 38 For water-wave scattering with elliptical cylinders, it deserves further study by using our approach. We will deal with Laplace and Helmholtz problems containing circular and elliptical cylinders at the same time. Further studies -2/4 Failure of Linton and Evans method

39. June 16, 2015 pp. 39 The degenerate kernels are expanded in the polar coordinates and only problems with circular boundaries are solved. For boundary value problems with crack, further investigation should be considered. In the further research, we may extend to mixed-type BCs by using the null-field integral equation approach. Further studies -3/4 Crack Dirichlet Neumann

40. June 16, 2015 pp. 40 Following the success of applications in two-dimensional problems, it is straightforward to extend this formulation to 3-D problems with spherical boundaries by using the corresponding 3-D degenerate kernel functions for fundamental solutions and spherical harmonic expansions for boundary densities. Trapped modes versus incident angle Further studies -4/4 versus incident angle F k θ 90°

41. June 16, 2015 pp. 41 Decompose two parts ＝ Radiation field (typical BVP) Free field Original Problem ＋