National Taiwan Ocean University MSVLAB Department of Harbor

National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Trapping and near-trapping by arrays of porous cylinders in water waves using the addition theorem and superposition technique 台灣海洋大學陳正宗特聘教授 Dr. Jeng-Tzong Chen Date: July 12, 2009 Place: 南京河海大學 pp.

Outline Motivation and literature review Unified formulation of null-field approach Numerical examples Concluding remarks 2

Outline Motivation and literature review Engineering problems Motivation Present approach Unified formulation of null-field approach Numerical examples Concluding remarks 3

Engineering problems Platform (Offshore structure) 4

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

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

Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV 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 Kress, 1989 7

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 8

Boundary integral equation and nullfield integral equation Interior case Exterior case Degenerate (separate) form 9

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

Degenerate kernel Expand fundamental solution by using degenerate kernel Ｄegenerate kernel （Helmholtz） x s O U(s, x) T(s, x) L(s, x) M(s, x) x 11

Degenerate kernel and Fourier series Expand boundary densities by using Fourier series kth circular boundary cosnθ, sinnθ boundary distributions 12

Adaptive observer system Source point Collocation point 13

Linear algebraic system y x 14

Flowchart of present method Collocation on the real boundary Original problem Linear algebraic system Decompose two parts Free field Radiation field Calculation of the unknown Fourier Expansion BIE for the domain point Degenerate kernel for fundamental solution Fourier series of boundary densities 0 = òB T ( s, x)u ( s)d. B( s) - òB U (s, x)t (s)d. B( s), x Î D c Superposing the solution of two parts Total field 15

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 16

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

Problem statement Boundary condition: , Dispersion relationship: . Dynamic pressure: Force: Original problem 18

Numerical examples Case 1. Four-cylinders array for one sets (ka=4. 08482, a/b=0. 8) Case 2. Five sets (ka=4. 08482, a/b=0. 8) y 5 y 37 1 4 1 1 11 00 15 9 4 3 2 b 16 2 14 8 x 13 2 7 7 12 1 2 b 6 x 19

Effect of impermeable case for contour plots (a) BEM ( , (b) Null-field BIEM (M=20 Chen) Contour plots of free-surface elevation of the four impermeable cylinders (G=0. 0, ) ) 20

Effect of impermeable case for free-surface elevation (a) Williams and Li (c) Null-field BIEM (M=20) (b) BEM (Chen) Free-surface elevation of the arrays of four impermeable cylinders. (G=0. 0, ) 21

Effect of disorder case for contour plots (a) BEM ( , (b) Null-field BIEM (M=20 Chen) Contour plots of free-surface elevation of the four porous cylinders (G=1. 0, ) ) 22

Effect of disorder case for free-surface elevation (a) Williams and Li (c) Null-field BIEM (M=20) (b) BEM (Chen) Free-surface elevation of the arrays of four impermeable cylinders. (G=0. 0, ) 23

Near-trapped mode for the four cylinders at ka=4. 08482 (a/b=0. 8, G=0. 0, ) (no disorder and no porosity) (a) Contour by the present method (M=20) 24

Near-trapped mode for the four cylinders at ka=4. 08482 (a/b=0. 8, G=0. 0, ) 3 1 4 2 54 (b) Free-surface elevations by the present method (M=20) (c) Horizontal force on the four cylinders against wavenumber 25

Near-trapped modes versus incident angle 54 2 1 3 4 26

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

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

Outline Motivation and literature review Unified formulation of null-field approach Numerical examples Concluding remarks 29

Concluding remarks -1/2 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. 30

Concluding remarks -2/2 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. 31

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 32

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

Trapped and near-trapped modes Trapped modes Near-trapped modes Dirichlet or Neumann modes pp. Trapped and near-trapped modes. ppt

Irregular fictitious and spurious frequency and trap modes Exterior acoustics (fictitious) mathematics Irregular frequencies Interior acoustics (spurious) Floating body (water wave) physics trap modes Acoustics Physical Water wave resonance Mathematical Nonuniqueness problem pp. Trap 2009. ppt

Further studies -1/4 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. SH wave In the further research, the Helmholtz problems with circular boundaries may be extended to other shapes instead of incident plane wave, shorecrested incident wave can be also considered. 37

Further studies -2/4 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. Failure of Linton and Evans method 38

Further studies -3/4 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. Crack In the further research, we may extend to mixed-type BCs by using the null-field integral equation approach. Dirichlet Neumann 39

Further studies -4/4 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 F k versus incident angle θ 90° 40

Decompose two parts ＝ Original Problem ＋ Free field Radiation field (typical BVP) 41