extending the scaled boundary finite-element method … · extending the scaled boundary...

EXTENDING THE SCALED BOUNDARY FINITE-ELEMENT METHOD TO WAVE

DIFFRACTION PROBLEMS

by

Boning Li

A thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

The University of Western Australia

March 2007

- I -

ABSTRACT

The study reported in this thesis extends the scaled boundary finite-element method to first-

order and second-order wave diffraction problems.

The scaled boundary finite-element method is a newly developed semi-analytical technique

to solve systems of partial differential equations. It works by employing a special local co-

ordinate system, called scaled boundary coordinate system, to define the computational

field, and then weakening the partial differential equation in the circumferential direction

with the standard finite elements whilst keeping the equation strong in the radial direction,

finally analytically solving the resulting system of equations, termed the scaled boundary

finite-element equation. This unique feature of the scaled boundary finite-element method

enables it to combine many of advantages of the finite-element method and the boundary-

element method with the features of its own. For instance, since only the boundaries of

computational fields are discretized, the spatial dimensions can be reduced by one.

Consequently the data preparation effort can be significantly decreased. Due to its

analytical nature in the radial direction, the singularity of field gradients near sharp re-

entrant corners can be modelled with ease and the radiation condition of wave diffraction

problems at infinity can be satisfied rigorously. The scaled boundary finite-element method

was originally developed for solving problems of elasto-statics and elasto-dynamics in

solid mechanics. It has been employed successfully for solving problems with singularities

and unbounded domains. However, to date there has been no report of its applications to

surface wave problems.

It is well known that there are many mathematical similarities between the solid mechanics

and fluid mechanics. However, due to differences in the physical characteristics of the solid

and fluid mediums, the existing computational procedure of the scaled boundary finite-

element method for solid mechanics can not be applied to ocean wave problems directly.

The present thesis aims to develop a suite of computational procedures of the scaled

boundary finite-element method suitable for calculating two-dimensional surface wave

diffraction problems, and to examine the accuracy and advantages of this method for

solving surface wave diffraction problems.

- II -

In this thesis, both first-order and second-order solutions of wave diffraction problems are

presented in the context of scaled boundary finite-element analysis. In the first-order wave

diffraction analysis, the boundary-value problems governed by the Laplace equation or by

the Helmholtz equation are considered. The solution methods for bounded domains and

unbounded domains are described in detail. The solution process is implemented and

validated by practical numerical examples. The numerical examples examined include well

benchmarked problems such as wave reflection and transmission by a single horizontal

structure and by two structures with a small gap, wave radiation induced by oscillating

bodies in heave, sway and roll motions, wave diffraction by vertical structures with

circular, elliptical, rectangular cross sections and harbour oscillation problems. The

numerical results are compared with the available analytical solutions, numerical solutions

with other conventional numerical methods and experimental results to demonstrate the

accuracy and efficiency of the scaled boundary finite-element method. The computed

results show that the scaled boundary finite-element method is able to accurately model the

singularity of velocity field near sharp corners and to satisfy the radiation condition with

ease. It is worth nothing that the scaled boundary finite-element method is completely free

of irregular frequency problem that the Green’s function methods often suffer from. For the

second-order wave diffraction problem, this thesis develops solution schemes for both

monochromatic wave and bichromatic wave cases, based on the analytical expression of

first-order solution in the radial direction. It is found that the scaled boundary finite-

element method can produce accurate results of second-order wave loads, due to its high

accuracy in calculating the first-order velocity field.

- III -

STATEMENT OF CANDIDATE CONTRIBUTION

I certify that except where references are made in the text to the work of others, the

contents of this thesis are original and have not been submitted to any other university.

This thesis is the result of my own work.

Boning Li

March, 2007.

V

TABLE OF CONTENTS

Chapter 1 Introduction

1.1 Background ................................................................................................. 1

1.2 Aim of research ........................................................................................... 5

1.3 Publications ................................................................................................. 5

1.4 Thesis structure ........................................................................................... 6

Chapter 2 Literature review

2.1 General ........................................................................................................ 8

2.2 Mathematical development of water wave diffraction theory ................... 9

2.2.1 Wave diffraction theory .............................................................................. 9 2.2.2 Analytical and semi-analytical solutions.................................................. 11

2.3 Numerical methods for wave diffraction problems ................................. 15

2.3.1 Finite-element method .............................................................................. 15 2.3.2 The Green’s function method ................................................................... 17

2.4 Development of the scaled boundary finite-element method .................. 19

2.5 Summary ................................................................................................... 21

Chapter 3 Linear SBFEM solution of Laplace equation

3.1 General ...................................................................................................... 23

3.2 Mathematical formulation......................................................................... 24

3.3 Local co-ordinate system .......................................................................... 25

3.3.1 Standard scaled boundary co-ordinate system for bounded domains...... 25 3.3.2 Modified scaled boundary co-ordinate system for unbounded domains. 26

3.4 Scaled boundary finite-element equations................................................ 29

3.5 Solution process ........................................................................................ 33

3.5.1 Bounded domain solution ......................................................................... 33

VI

3.5.2 Unbounded domain solution..................................................................... 37 3.5.3 Coupling solutions of bounded and unbounded domains ........................ 40

3.6 Results and discussions............................................................................. 41

3.6.1 Wave reflection and transmission............................................................. 41 3.6.1.1 Wave scattering by a single surface obstacle (Example 1)...................... 42 3.6.1.2 Wave scattering by twin surface obstacles (Example 2).......................... 43

3.6.2 Wave radiation .......................................................................................... 45 3.6.2.1 Wave radiation by an oscillating rectangular structure (Example 3)....... 48 3.6.2.2 Wave radiation by an oscillating twin-hull structure (Example 4) .......... 49

3.7 Summary ................................................................................................... 50

Chapter 4 Linear SBFEM solution of Helmholtz equation

4.1 General ...................................................................................................... 75

4.2 Mathematical formulation......................................................................... 76

4.3 Scaled boundary finite-element equation ................................................. 77

4.4 Solution process ........................................................................................ 80

4.4.1 Bounded domain solution ......................................................................... 80 4.4.2 Unbounded domain solution..................................................................... 85

4.5 Assembly of subdomains .......................................................................... 90

4.6 Results and discussions............................................................................. 91

4.6.1 Wave diffraction by piercing-surface structures ...................................... 91 4.6.1.1 Wave diffraction by a circular cylinder (Example 1)............................... 91 4.6.1.2 Wave diffraction by an elliptical cylinder with an aspect ratio of 2:1

(Example 2) ............................................................................................... 94 4.6.1.3 Wave diffraction by an elliptical cylinder with an aspect ratio of 4:1

(Example 3) ............................................................................................... 94 4.6.1.4 Wave diffraction by a square cylinder (Example 4) ................................ 95 4.6.1.5 Wave diffraction by twin caissons with a small gap (Example 5)........... 96

4.6.2 Harbour oscillations .................................................................................. 98 4.6.2.1 Wave diffraction by a rectangular narrow bay (Example 6).................... 98

VII

4.6.2.2 Wave diffraction by a square harbour with two straight breakwaters

(Example 7) ............................................................................................... 98 4.7 Summary ................................................................................................... 99

Chapter 5 Second-order solution to monochromatic wave diffraction problems

5.1 General .................................................................................................... 126

5.2 Mathematical formulation....................................................................... 127

5.3 Scaled boundary finite-element equations.............................................. 128

5.3.1 Scaled boundary finite-element equations

for a bounded subdomain........................................................................ 129 5.3.2 Scaled boundary finite-element equations

for an unbounded subdomain.................................................................. 131 5.4 Solution process ...................................................................................... 133

5.4.1 General solution ...................................................................................... 134 5.4.2 Boundary condition at infinity................................................................ 137 5.4.3 Determination of wave forces................................................................. 138

5.5 Results and discussions........................................................................... 139

5.5.1 Wave diffraction by a rectangular obstacle (Example 1)....................... 139 5.5.2 Wave diffraction by a trapezoidal obstacle (Example 2) ....................... 143 5.5.3 Wave diffraction by twin rectangular obstacles (Example 3)................ 145

5.6 Summary ................................................................................................. 147

Chapter 6 Second-order solution to bichromatic wave diffraction problems

6.1 General .................................................................................................... 188

6.2 Mathematical formulation....................................................................... 189

6.2.1 Second-order incident potential of bichromatic wave ........................... 189 6.2.2 Second-order wave force ........................................................................ 195 6.2.3 Second-order wave surface elevation ..................................................... 197

6.3 Scaled boundary finite-element equation ............................................... 197

6.4 Solution process ...................................................................................... 200

VIII

6.4.1 General solution ...................................................................................... 200 6.4.2 Determination of integration constants................................................... 203

6.5 Results and discussions........................................................................... 204

6.5.1 Wave diffraction by a rectangular obstacle (Example 1)....................... 205 6.5.1.1 Second-order wave forces....................................................................... 205 6.5.1.2 Second-order wave reflection and transmission coefficients................. 206

6.5.2 Wave diffraction by a trapezoidal obstacle (Example 2) ....................... 208 6.5.3 Wave diffraction by twin rectangular obstacles (Example 3)................ 210

6.6 Summary ................................................................................................. 212

Chapter 7 Conclusions

7.1 Summary ................................................................................................. 256

7.2 Future work ............................................................................................. 260

7.3 Concluding remarks ................................................................................ 261

References ..................................................................................................................261

IX

TABLE OF FIGURES

Figure 3.1. Configuration of the mathematical problem................................................... 52

Figure 3.2(a). The original boundary co-ordinate system..................................................... 52

Figure 3.2(b). The original boundary co-ordinate system .................................................... 53

Figure 3.3. The translated boundary co-ordinate system................................................. 53

Figure 3.4. Substructured model and meshes for Example 1, consisting of two

unbounded domains and two bounded domains ........................................ 54

Figure 3.5. Horizontal wave force (amplitude) for Example 1........................................ 55

Figure 3.6. Vertical wave force (amplitude) for Example 1............................................ 55

Figure 3.7. Moment around y-axis (amplitude) for Example 1 ....................................... 56

Figure 3.8. Reflection coefficient for Example 1............................................................. 56

Figure 3.9. Transmission coefficient for Example 1........................................................ 57


unbounded domains and four bounded domains.......................................... 58

Figure 3.11(a). Horizontal wave forces (amplitude) on the block B1................................... 59

Figure 3.11(b). Horizontal wave forces (amplitude) on the block B1................................... 59

Figure 3.12(a). Horizontal wave forces (amplitude) on the block B2................................... 60

Figure 3.12(b). Horizontal wave forces (amplitude) on the block B2................................... 60

Figure 3.13(a). Vertical wave forces (amplitude) on the block B1 ....................................... 61

Figure 3.13(b). Vertical wave forces (amplitude) on the block B1....................................... 61

Figure 3.14(a). Vertical wave forces (amplitude) on the block B2 ....................................... 62

Figure 3.14(b). Vertical wave forces (amplitude) on the block B2....................................... 62

Figure 3.15(a). Squared reflection coefficient for Example 2 ............................................... 63

Figure 3.15(b). Squared reflection coefficient for Example 2 ............................................... 63

Figure 3.16(a). Squared transmission coefficient for Example 2 .......................................... 64

Figure 3.16(b). Squared transmission coefficient for Example 2.......................................... 64

Figure 3.17. Summation of squared reflection and transmission coefficients

for Example 2................................................................................................ 65

Figure 3.18. Cartesian co-ordinate system and substructured computational domain...... 65

Figure 3.19(a). Dimensionless added mass coefficient for a rectangular structure

heaving in calm water ................................................................................... 66

X

Figure 3.19(b). Dimensionless damping coefficient for a rectangular structure



swaying in calm water .................................................................................. 67




rolling in calm water ..................................................................................... 68



Figure 3.22(a). Dimensionless added mass coefficient for a twin-hull structure


Figure 3.22(b). Dimensionless damping coefficient for a twin-hull structure




















XI

Figure 3.27(b). Dimensionless damping coefficient for a twin-hull structure rolling in calm

water .............................................................................................................. 74

Figure 4.1. Definition sketch of wave diffraction around obstacles............................. 100

Figure 4.2. Substructuring configuration and scaled boundary

co-ordinate definition.................................................................................. 100

Figure 4.3. Circular cylinder (ka=2).............................................................................. 101

Figure 4.4. Scaled boundary finite-element meshes for a circular cylinder................. 101

Figure 4.5(a). Variation of scattered wave elevation (real part) around

circular cylinder for Example 1 .................................................................. 102

Figure 4.5(b). Variation of scattered wave elevation (imaginary part) around circular

cylinder for Eample 1.................................................................................. 102

Figure 4.6(a). Variation of total wave elevation (real part) around


Figure 4.6(b). Variation of total wave elevation (imaginary part) around


Figure 4.7(a). Variation of tangential velocity (real part) around


Figure 4.7(b). Variation of tangential velocity (imaginary part) around


Figure 4.8 (a). Contour plots of wave elevation (real part, ka=4) for Example 1 .............. 105

Figure 4.8 (b). Contour plots of wave elevation (imaginary part, ka=4) for Example 1 ... 105

Figure 4.9(a). Vector plots of velocity at the water surface (real part, ka=4)

for Example 1.............................................................................................. 106

Figure 4.9(b). Vector plots of velocity at the water surface (imaginary part, ka=4) for

Example 1.................................................................................................... 106

Figure 4.10(a). Horizontal wave force on circular cylinder for Example 1 ......................... 107

Figure 4.10(b). Wave moment on circular cylinder for Example 1 ..................................... 107

Figure 4.11. Elliptical cylinder of aspect ratio 2:1 (ka=4) for Example 2........................ 108

Figure 4.12. Scaled boundary finite-element meshes for an elliptical cylinder

for Example 2................................................................................................ 108


elliptical cylinder for Example 2 .................................................................. 109

XII


elliptical cylinder for Example 2 ................................................................ 109

Figure 4.14. Elliptical cylinder of aspect ratio 4:1 (ka=4) for Example 3...................... 110

Figure 4.15. Scaled boundary finite-element meshes for an elliptical cylinder for

Example 3. 110


elliptical cylinder for example 3 ................................................................. 111


elliptical cylinder for example 3 ................................................................. 111


bounded subdomains and one unbounded subdomain............................... 112

Figure 4.18(a). Computed variation of wave elevation (real part)

along the surface of the single square cylinder for Example 4 .................. 113

Figure 4.18(b). Computed variation of wave elevation (imaginary part)


Figure 4.19(a). Computed variation of tangential velocity (real part)


Figure 4.19(b). Computed variation of tangential velocity (imaginary part)


Figure 4.20. Horizontal wave force (amplitude) on the single

square cylinder for Example 4.................................................................... 115

Figure 4.21. Configuration of wave diffraction by twin rectangular

caissons for Example 5 ............................................................................... 115

Figure 4.22. Substructured model and mesh for Example 5, consisting of four bounded

subdomains and one unbounded subdomain.............................................. 116

Figure 4.23(a). Computed variation of wave elevation (real part) along the surface of the

twin caissons for Example 5 ....................................................................... 116

Figure 4.23(b). Computed variation of wave elevation (imaginary part)

along the surface of the twin caissons for Example 5................................ 117

Figure 4.24(a). Computed variation of tangential velocity (real part)


Figure 4.24(b). Computed variation of tangential velocity (imaginary part)

XIII


Figure 4.25. Horizontal wave force (amplitude) on the twin caissons for Example 5... 120

Figure 4.26. Configuration of the rectangular narrow bay for Example 6 ..................... 121

Figure 4.27. Mesh and substructure definition for Example 6 ....................................... 121

Figure 4.28. Variation of dimensionless wave elevation (amplitude)

at the point C with dimensionless wave number for Example 6 ............... 122

Figure 4.29. Configuration of the square harbor with straight breakwaters

for Example 7 ............................................................................................ 123

Figure 4.30. Mesh for Example 7, consisting of ten bounded domains and one

unbounded domain .................................................................................... 123

Figure 4.31(a). Variation of wave elevation (amplitude) at y/a=0 for Example 7.............. 124

Figure 4.31(b). Variation of wave elevation (amplitude) at y/a=0.5 for Example 7........... 124

Figure 4.31(c). Variation of wave elevation (amplitude) at x/a=1 for Example 7.............. 125

Figure 5.1. Definition of the boundary-value problem................................................. 151

Figure 5.2. Local co-ordinate systems in a bounded and an unbounded subdomain... 151

Figure 5.3. Substructure models and computational mesh ........................................... 152

Figure 5.4. Second-order components of wave loads on the horizontal rectangular

obstacle (B/H=1.0, D/H=0.4, L/B=1.0) ...................................................... 153


obstacle with different sizes of the bounded domain (B/H=1.0, D/H=0.4)153


obstacle (B/H=1.0, D/H=0.2, L/B=1.0) ...................................................... 154


obstacle (B/H=0.2, D/H=0.4, L/B=1.0) ...................................................... 154


obstacle (B/H=0.2, D/H=0.2, L/B=1.0) ...................................................... 155

Figure 5.9. Horizontal second-order wave loads, related to second-order velocity

potential, on a fixed rectangular cylinder (B/H=1, D/H=0.8).................... 155

Figure 5.10. Vertical second-order wave loads, related to second-order velocity potential,

on a fixed rectangular cylinder (B/H=1, D/H=0.8) .................................... 156

Figure 5.11. Horizontal second-order wave loads, related to second-order velocity


XIV

Figure 5.12. Vertical second-order wave loads, related to second-order velocity


Figure 5.13. Pressure on the bottom of a semi-submerged horizontal cylinder of

rectangular cross-section

(B=0.305m, D=0.3m, H=0.4m, A=0.014m, kH=3).................................... 158

Figure 5.14. Substructured model for problems of second-order wave diffraction by a

trapezoidal obstacle..................................................................................... 159

Figure 5.15. Second-order component of horizontal wave forces on the obstacle for

Example 2 (B/H=1.0, D/H=0.4, θ =30°) .................................................... 160

Figure 5.16. Second-order component of vertical wave forces on the obstacle for

Example 2 (B/H=1.0, D/H=0.4, θ =30°) .................................................... 160

Figure 5.17. Second-order component of moment about (B,-D)

for Example 2 (B/H=1.0, D/H=0.4, θ =30°)............................................... 161

Figure 5.18(a). Second-order component of horizontal wave force on obstacles with

various base angles (θ ≤ 90°) for Example 2 (B/H=1.0, D/H=0.4) ........... 161

Figure 5.18(b). Second-order component of horizontal wave force on obstacles with

various base angles (θ ≥90°) for Example 2 (B/H=1.0, D/H=0.4)........... 162

Figure 5.19(a). Second-order component of vertical wave force on obstacles with various

base angles (θ ≤90°) for Example 2 (B/H=1.0, D/H=0.4)........................ 162

Figure 5.19(b). Second-order component of vertical wave force on obstacles with various

base angles (θ ≥90°) for Example 2 (B/H=1.0, D/H=0.4) ......................... 163

Figure 5.20(a). Second-order component of moment about (B,-D) on obstacles with

various base angles (θ ≤90°) for Example 2 (B/H=1.0, D/H=0.4)........... 163

Figure 5.20(b). Second-order component of moment about (B,-D) on obstacles with various

base angles (θ ≥90°) for Example 2 (B/H=1.0, D/H=0.4) ......................... 164

Figure 5.21. Second-order component of horizontal wave forces on the obstacle

for Example 2 (B/H=1.0, D/H=0.2, θ =30°)............................................... 164

Figure 5.22. Second-order component of vertical wave forces on the obstacle for

Example 2 (B/H=1.0, D/H=0.2, θ =30°) .................................................... 165

Figure 5.23. Second-order component of moment about (B,-D) for Example 2

(B/H=1.0, D/H=0.2, θ =30°) ....................................................................... 165

XV

Figure 5.24(a). Second-order component of horizontal wave force on obstacles with various

base angles (θ ≤ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........................ 166

Figure 5.24(b). Second-order component of horizontal wave force on obstacles with

various base angles (θ ≥ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........... 166

Figure 5.25(a). Second-order component of vertical wave force on obstacles with various


Figure 5.25(b). Second-order component of vertical wave force on obstacles with various

base angles (θ ≥ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........................ 167

Figure 5.26(a). Second-order component of moment about (B,-D) on obstacles with various


Figure 5.26(b). Second-order component of moment about (B,-D) on obstacles with various

base angles (θ ≥ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........................ 168


unbounded domains and four bounded domains........................................ 169

Figure 5.28(a). Second-order component of horizontal wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 170

Figure 5.28(b). Second-order component of horizontal wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 170

Figure 5.28(c). Second-order component of horizontal wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 171

Figure 5.29(a). Second-order component of vertical wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 171

Figure 5.29(b). Second-order component of vertical wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 172

Figure 5.29(c). Second-order component of vertical wave force on the obstacle B1 for

Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 172


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 173


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 173


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 174

XVI


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 174


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 175


Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 175


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 176


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 176


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 177


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 177


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 178


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 178


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 179


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 179


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 180


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 180


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 181


Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 181


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 182

XVII


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 182


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 183


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 183


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 184


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 184


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 185


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 185


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 186


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 186


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 187


Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 187

Figure 6.1 Variation of the dimensionless double-frequency horizontal wave force with

the dimensionless wave number ................................................................. 217

Figure 6.2 Variation of the dimensionless double-frequency vertical wave force with


Figure 6.3 Variation of the dimensionless double-frequency moment about (B, -D) with


Figure 6.4 Computed variation of the dimensionless sum-frequency wave loads with


Figure 6.5 Computed variation of the dimensionless difference-frequency wave loads

with the dimensionless wave number......................................................... 219

XVIII

Figure 6.6 Sum-frequency second-order reflection coefficients (kjH=0.01)............... 220

Figure 6.7 Sum-frequency second-order transmission coefficients (kjH=0.01)........... 220

Figure 6.8 Difference-frequency second-order reflection coefficients (kjH=0.01)...... 221

Figure 6.9 Difference-frequency second-order transmission coefficients (kjH=0.01) 221

Figure 6.10 Sum-frequency second-order reflection coefficients (kjH=1.0).................. 222

Figure 6.11 Sum-frequency second-order transmission coefficients (kjH=1.0)............. 222

Figure 6.12 Difference-frequency second-order reflection coefficients (kjH=1.0)........ 223

Figure 6.13 Difference-frequency second-order transmission coefficients (kjH=1.0)... 223

Figure 6.14 Sum-frequency second-order reflection coefficients (kjH=2.0).................. 224

Figure 6.15 Sum-frequency second-order transmission coefficients (kjH=2.0)............. 224

Figure 6.16 Difference-frequency second-order reflection coefficients (kjH=2.0)........ 225

Figure 6.17 Difference-frequency second-order transmission coefficients (kjH=2.0)... 225

Figure 6.18 Computed variation of the dimensionless sum-frequency horizontal wave

loads with the dimensionless wave number

for Example 2 (B/H=1.0, D/H=0.4, θ=30°)................................................ 226

Figure 6.19 Computed variation of the dimensionless sum-frequency vertical wave loads

with the dimensionless wave number

for Example 2 (B/H=1.0, D/H=0.4, θ=30°) 226

Figure 6.20 Computed variation of the dimensionless sum-frequency moment about (B,-

D) with the dimensionless wave number


Figure 6.21 Computed variation of the dimensionless difference-frequency horizontal

wave force with the dimensionless wave number


Figure 6.22 Computed variation of the dimensionless difference-frequency vertical wave

force with the dimensionless wave number for Example 2 (B/H=1.0,

D/H=0.4, θ=30°) ......................................................................................... 228

Figure 6.23 Computed variation of the dimensionless difference-frequency moment

about (B,-D) with the dimensionless wave number


XIX

Figure 6.24(a) Sum-frequency component of second-order horizontal wave force on

obstacles with various base angles for Example 2 (B/H=1.0, D/H=0.4, θ ≤

90°) .............................................................................................................. 229

Figure 6.24(b) Sum-frequency component of second-order horizontal wave force on

obstacles with various base angles for Example 2 (B/H=1.0, D/H=0.4, θ ≥

90°) .............................................................................................................. 229

Figure 6.25(a) Sum-frequency component of second-order vertical wave force on obstacles

with various base angles for Example 2 (B/H=1.0, D/H=0.4, θ ≤ 90°)..... 230

Figure 6.25(b) Sum-frequency component of second-order vertical wave force on obstacles

with various base angles

for Example 2 (B/H=1.0, D/H=0.4, θ ≥ 90°).............................................. 230

Figure 6.26(a) Sum-frequency component of second-order moment about (B,-D) on

obstacles with various base angles

for Example 2 (B/H=1.0, D/H=0.4, θ ≤ 90°).............................................. 231

Figure 6.26(b) Sum-frequency component of second-order moment about (B,-D) on



Figure 6.27(a) Difference-frequency component of second-order horizontal wave force on



Figure 6.27(b) Difference-frequency component of second-order horizontal wave force on



Figure 6.28(a) Difference-frequency component of second-order vertical wave force on



Figure 6.28(b) Difference-frequency component of second-order vertical wave force on



Figure 6.29(a) Difference-frequency component of second-order moment about (B,-D) on



XX

Figure 6.29(b) Difference-frequency component of second-order moment about (B,-D) on


for Example 2 (B/H=1.0, D/H=0.4, θ ≥90°)............................................... 234

Figure 6.30(a) Sum-frequency component of second-order horizontal wave force on the

obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) ........................ 235

Figure 6.30(b) Sum-frequency component of second-order horizontal wave force on the


Figure 6.30(c) Sum-frequency component of second-order horizontal wave force on the


Figure 6.31(a) Sum-frequency component of second-order vertical wave force on the


Figure 6.31(b) Sum-frequency component of second-order vertical wave force on the


Figure 6.31(c) Sum-frequency component of second-order vertical wave force on the


Figure 6.32(a) Sum-frequency component of second-order horizontal wave force on the


Figure 6.32(b) Sum-frequency component of second-order horizontal wave force on the


Figure 6.32(c) Sum-frequency component of second-order horizontal wave force on the


Figure 6.33(a) Sum-frequency component of second-order vertical wave force on the


Figure 6.33(b) Sum-frequency component of second-order vertical wave force on the


Figure 6.33(c) Sum-frequency component of second-order vertical wave force on the



the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................. 241



XXI

Figure 6.34(c) Difference-frequency component of second-order horizontal wave force on


Figure 6.35(a) Difference-frequency component of second-order vertical wave force on the


Figure 6.35(b) Difference-frequency component of second-order vertical wave force on the


Figure 6.35(c) Difference-frequency component of second-order vertical wave force on the


Figure 6.35(d) Difference-frequency component of second-order vertical wave force on the






Figure 6.36(c) Difference-frequency component of second-order horizontal wave force on


Figure 6.37(a) Difference-frequency component of second-order vertical wave force on the


Figure 6.37(b) Difference-frequency component of second-order vertical wave force on the


Figure 6.37(c) Difference-frequency component of second-order vertical wave force on the


Figure 6.37(d) Difference-frequency component of second-order vertical wave force on the


Figure 6.38 Sum-frequency component of second-order horizontal wave force on the


Figure 6.39 Sum-frequency component of second-order vertical wave force on the






XXII

Figure 6.42 Difference-frequency component of second-order horizontal wave force on


Figure 6.43 Difference-frequency component of second-order vertical wave force on the







obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .......................... 252








the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................... 254

Figure 6.51 Difference-frequency component of second-order vertical wave force on






XXIII

ACKNOWLEDGMENTS

The research for this thesis was carried out in the School of Civil and Resource

Engineering at the University of Western Australia. It was generously supported by the

International Postgraduate Research Scholarship and the University Postgraduate Rewards

from the University of Western Australia, which are gratefully acknowledged.

My deepest heartfelt thanks and respects belong to my supervisors, Professor Liang Cheng

and Professor Andrew J. Deeks. Without their creativeness and patient way of sharing

knowledge, this thesis would not have been possible.

Professor Liang Cheng leads me to the research area of computational fluid dynamics.

Thanks to his great wisdom and vast knowledge, my research work could move forward

continuously. I have learned not only many academic skills but also an attitude of enjoying

life from Liang, during these years along with him. I feel that I have been very fortunate to

have a mentor like Liang.

I warmly thank Professor Andrew J. Deeks for his continuous support and encouragement.

I have been very honoured to share his long experience in the field of computational

mechanics. Many stimulating and instructive discussions with Andrew make me find a way

for tackling problems in my research.

I owe particular thanks to Professor Bin Teng, Dr. Chen Xiao-bo who have helped me

understand much of the concepts of wave mechanics, the boundary-element method and

boundary integration equation method.

I am deeply indebted to Dr. Chongmin Song who has given me many suggestions in the

application of the scaled boundary finite-element method.

Many thanks go to my fellows, Dr. Ming Zhao, Dr. Dongfang Liang, Dr. James Doherty,

Mr. Kervin Yeow, Mr. Steve Chidgezy, Dr. Hang Thu Vu, Mr. Hongwei An, Mr.

Muhammed Alam and my officemates Dr. Einav Itai, Mr. Chin Chai Ong, Mr. Matt

Helinski, Mr. Mark Richardson, Mr. Hongjie Zhou, Ms. Nina Levy, for a long friendship

and enjoyable refreshing moments during these years.

XXIV

I would also like to thank all the rest of the academic and support staff of School of Civil

and Resource Engineering and Centre for Offshore Foundation Systems (COFS) at the

University of Western Australia.

I feel incredibly grateful for my parents, whose understanding, support, strength and

generosity are the most valuable for ever in my life.

I would like to thank my wife, Lina Ding, for her love and patience.

Boning Li

Crawley, W.A.

March, 2007.

XXV

NOTATION

a : Characteristic body dimension.

A : Wave amplitude.

B : Width of a structure.

D : Draft.

H : Wave height or water depth.

fH : Horizontal wave force.

fV : Vertical wave force.

m : Moment.

g : Acceleration due to gravity.

h : Water depth.

i: Unit of imaginary number.

k : Wave number.

n : Normal to the boundary.

P : Wave pressure.

t : Time.

w : Weighting function.

x,y,z : Cartesian co-ordinates.

η : Free surface elevation.

ε : Wave slope.

ρ : Density of water.

λ : Eigenvalue.

ξ : Radial co-ordinate.

s : Circumferential co-ordinate.

XXVI

Ω : Computational fluid domain.

⊕ : Direct sum.

∇ : Hamilton operator.

φ : Velocity potential only with spatial qualities.

Iφ : Incident velocity potential only with spatial qualities.

Sφ : Scattered velocity potential only with spatial qualities.

Ф: Velocity potential with respect to both time and spatial qualities.

ФI : Incident velocity potential with respect to both time and spatial qualities.

ФS : Scattered velocity potential with respect to both time and spatial qualities.

a(ξ): Nodal potential vector.

q(ξ): Nodal flow vector.

[Ф] : Eigenvector matrix.

[Λ] : Eigenvalue matrix.

[N(s)]: Shape function.

[E0], [E1], [E2]: Coefficient matrices.

Chapter One

- 1 -

CHAPTER 1

INTRODUCTION

1.1 Background

Nowadays, most engineering problems are solved numerically, since analytical

solutions are limited to relatively simple cases. The most widely used computational

procedures in continuous mechanics are the Finite-Element Method (FEM), the

Boundary-Element Method (BEM) and the Finite-Difference Method (FDM).

Generally speaking, which of the three methods is better depends on the specific

problem involved. This is because the three methods have their own specific features,

with both advantages and disadvantages.

The FEM works by using various geometrical elements to discretize the whole

computational domain, then transforming the weakened governing equations into a

set of algebraic equations and enforcing boundary conditions, and finally solving the

resulting algebraic system of equations. The attractive feature of the FEM is its ability

to handle complex geometries with relative ease. Furthermore, the coefficient matrix

of the global algebraic equation is usually sparse, banded, symmetric and positive

definite, which is of great benefit in improving the computational efficiency and

reducing memory requirements. Compared with the BEM, no fundamental solution is

required in the FEM. As a result, the FEM seems to have a larger scope of

applications. However, the FEM does not work as well as one would expect in certain

special problems. For example, the FEM has to truncate the computational domain

when dealing with unbounded domain problems, leading to a reduction in accuracy.

For modelling the stress singularity in the vicinity of sharp re-entrant corners, the

FEM solution converges rather slowly. As a consequence, some special methods have

Chapter One

- 2 -

been developed to address these special problems. For instance, the Infinite-Element

Method (IEM) was presented to handle unbounded domain problems and the Trefftz-

type Finite-Element Methods (TFEM) were found to be able to deal with singularities

well. Details of the development these methods will be presented in Chapter 2.

However, these methods need to be coupled with other approaches when solving

problems involving both unbounded domains and singularities.

The BEM is closely related to the Boundary Integral Equations Method (BIEM) in

the context of mathematical physics. For a boundary-value problem, the BIEM seeks

a solution as a linear combination of Green’s functions. The entire solution, termed

the region-dependent Green’s function, consists of a homogeneous solution and a

particular solution. The general homogeneous solution contains integration constants

which can be evaluated to satisfy the boundary conditions of the boundary-value

problem. The particular solution is termed the free-space Green’s function, and is also

referred to as the fundamental solution for the differential operator in the particular

problem area. The BEM can be viewed as a systematic way of constructing numerical

approximations to a region-dependent or exact Green’s function (Martin & Rizzo

2005). This method first reduces the partial differential equation for a boundary-value

problem within a domain to an integral equation on the boundary of the domain, then

implements an approximate discretization procedure to obtain a system of linear

algebraic equations and finally solves this system of equations. This computational

procedure means that spatial discretization only takes place on the boundary of the

domain and the spatial dimension of the problem is diminished by one, reducing the

computational and data preparation efforts. For unbounded domain problems, the

BEM can rigorously satisfy the radiation condition at infinity if the proper Green’s

function is applied. In spite of these advantages, the BEM can not completely replace

the FEM because it suffers from many numerical problems (Mikhailov 2005). Firstly,

the BEM needs to evaluate singular integrals, which can be very complicated at

times. Secondly, the matrix of the linear algebraic equation system is dense, resulting

in an increase in computation cost. Thirdly, the generation of the discrete matrix is

rather expensive computationally, compared with the FEM, unless the fundamental

solution is very simple. Fourthly, the BEM employing some certain Green’s functions

Chapter One

- 3 -

suffer from irregular frequency problems. Moreover, the BEM suffers numerical

difficulties near sharp re-entrant corners.

The FDM is the oldest technique and is much easier to implement than the FEM and

the BEM. The FDM approximates the derivatives of the solution at a set of mesh

points within the computational domain using the finite difference quotients to

transform the boundary-value problem to a system of algebraic equations. Although

this method is very simple, it usually requires, mainly for convenience, that the grid is

structured. Consequently, coordinate-mapping techniques or adaptive meshing

algorithms are needed to solve problems with complicated geometries. In addition,

there is no straight-forward way to test the accuracy of a solution, and the scheme is

prone to certain types of numerical instability, which require artificial correction.

Also, the FDM is unable to handle sharp re-entrant corners very well.

In terms of the advantages and disadvantages of the numerical methods mentioned

above, engineers usually choose an appropriate numerical method as a solution tool

for a specific problem. However, it is not possible to solve problems involving both

unbounded domain and singularity very well using any single one of these existing

methods.

The Scaled Boundary Finite-Element Method (SBFEM) is a new numerical approach

and particularly suitable for problems with unbounded domains and certain types of

singularities. This method has been found to have some of the advantages of the FEM

and BEM and is able to avoid the corresponding disadvantages (Wolf 2003).

The SBFEM was originally established to model dynamic soil-structure interaction

problems in an unbounded medium by Wolf & Song (1994). Since then, the scaled

boundary finite-element method has been dramatically developed for analysis of

various solid mechanics problems, including elasto-statics, elasto-dynamics and

fracture mechanics. Furthermore, numerous examples have demonstrated that this

approach offers many numerical benefits, particularly for modelling an unbounded

computational domain and dealing with the stress singularity in the vicinity of sharp

re-entrant corners.

Chapter One

- 4 -

The key advantages of the scaled boundary finite-element method are summarized

by Song & Wolf (1997) as follows,

• Reduction of spatial dimension by one.

• No fundamental solution required.

• No singular integrals to be evaluated.

• Suitable for anisotropic materials.

• Radiation condition satisfied.

• Boundary conditions at free and fixed surfaces and interfaces between different

materials satisfied without discretization.

• Exact in radial direction permitting semi-analytical representation of stress

singularities for statics.

• Symmetric mass matrix obtained without assumptions.

Recently, Deeks & Cheng (2003) applied the SBFEM to the field of fluid-structure

interaction, solving problems of potential flow around two-dimensional obstacles.

Compared with the FDM, it was found that the SBFEM can calculate the velocity

singularity near the field of sharp re-entrant corners very effectively and accurately.

So far, there has been no attempt to apply the SBFEM to wave-structure interaction

problems. Generally speaking, wave problems become more challenging due to the

existence of free water surface and radiation conditions. The conventional numerical

methods suffer from some numerical problems (See Chapter 2 for details). The

objective of this study is to explore the suitability of using the SBFEM to overcome

the numerical difficulties frequently met in wave diffraction problems.

This thesis aims to improve understanding of the SBFEM, by investigating the

advantages and disadvantages of this method for solving wave diffraction problems.

Chapter One

- 5 -

1.2 Aim of research

The major purpose of this thesis is to develop the SBFEM solutions to wave

diffraction problems and to evaluate its advantages and disadvantages over

conventional numerical methods. Specifically the SBFEM solutions for the following

problems will be developed:

• First-order wave diffraction by horizontal cylinders

• First-order wave diffraction by vertical cylinders

• Second-order monochromatic wave diffraction

• Second-order bichromatic wave diffraction

1.3 Publications

Publications based on this thesis are as follows:

• B.Li, L.Cheng, A.J.Deeks. (2006) Scaled boundary finite-element solution to

second-order wave diffraction problems. Proceedings of The Tenth East Asia-

Pacific Conference on Structural Engineering and Construction, Emerging

Trends: Keynote Lectures and Symposia, Bangkok, Thailand. 437-442.

• B.Li, L.Cheng, A.J.Deeks and M.Zhao. (2006) A semi-analytical solution

method for two-dimensional Helmholtz Equation. J.Applied Ocean Research.

28, 193-207.

• B.Li, L.Cheng, A.J.Deeks and B.Teng. (2005) A modified scaled boundary

finite-element method for problems with parallel side-faces: Part I. Theoretical

developments. J. Applied Ocean Research. 27, 216-223.

• B.Li, L.Cheng, A.J.Deeks and B.Teng. (2005) A modified scaled boundary

finite-element method for problems with parallel side-faces: Part II. Application

and Evaluation. J. Applied Ocean Research. 27, 224-234.

Chapter One

- 6 -

• B.Li, L.Cheng and A.J.Deeks. (2004) Wave diffraction by vertical cylinder

using the scaled boundary finite-element method. The Sixth World Congress on

Computational Mechanics in conjunction with the Second Asian-Pacific

Congress on Computational Mechanics. Beijing, China, September.

1.4 Thesis structure

This thesis is organized as follows:

Chapter 1 first introduces the background of this thesis, indicating the motivation of

conducting this study. The aim of this thesis is introduced. Publications resulting

from the study described in this thesis are listed.

Chapter 2 reviews the development of mathematical theory of water wave theory and

widely used numerical methods. The development of the SBFEM also is reviewed.

Chapter 3 develops a modified scaled-boundary finite-element method to extend the

original SBFEM to solving problems with parallel side-faces. A new local co-

ordinate system is proposed for the unbounded domain solution, and a modified

bounded domain solution of Laplace equation with linear free water surface boundary

conditions. This method is then applied to calculate wave diffraction by horizontal

fixed and free-floating structures. Comparisons are made with the analytical solution

and the BEM (Green’s Function Method (GFM)).

Chapter 4 presents a semi-analytical solution for a Helmholtz equation with linear

free water surface conditions using the SBFEM. This solution is applicable to solving

problems of wave diffraction by vertical cylinders and harbour oscillation problems.

The numerical results are compared with those provided by analytical methods,

experimental methods and other numerical methods such as the Infinite-Element

Method (IEM) and the BEM.

Chapter 5 applies the SBFEM to the solution of second-order monochromatic wave

diffraction by a fixed horizontal cylinder, building on the first-order solution. The

Chapter One

- 7 -

results are validated by comparison with those calculated using the method of

eigenfunction expansion.

Chapter 6 deals with the second-order bichromatic wave diffraction problem, which

is more challenging in the solution process. The wave forces with sum-frequency and

difference-frequency terms are calculated.

Chapter 7 summarizes the work in this thesis, indicating the advantages and

limitations of the SBFEM when applied to wave-structure interaction problems.

Furthermore, this chapter discusses some possible further work to overcome the

limitations.

Chapter Two

- 8 -

CHAPTER 2

LITERATURE REVIEW

2.1 General

Prediction of wave loads acting on offshore structures has been the subject of interest

for over six decades. The Morison equation, diffraction theory and the Froude-Krylov

force theory are the widely used methods for calculating wave loads in practical

designs for offshore structures. Depending on the flow regime in the vicinity of

structures, these methods have specific application scopes (Chakrabarti 1995). The

Morison equation is applicable if the flow incident on the structures separates from

the surface of the structure forming a wake (low-pressure) alternately in front of and

behind the structure (with respect to the flow direction). The diffraction theory needs

to be used if the incident wave reaching the structure experiences scattering from the

surface of the structure in the form of a reflected wave that is of the same order of

magnitude as the incident wave. For the case with neither appreciable separation from

the surface of the structure nor large evident reflection, the Froude-Krylov theory can

be employed and potential theory can he assumed to apply (Chakrabarti 1995).

Actually, one can tell relatively accurately which method is more applicable by

considering the relevant length scales in wave-body interaction (Mei 1989). The

relevant length scales include the characteristic body dimension a, the wave number

k, and the wave amplitude A. If ka≥ o(1), a structure may be regarded as large, so the

diffraction theory is applicable. When a structure is a small body (ka<<1), the

Morison equation is effective for calculating wave loads. When A/a is sufficiently

large, the local velocity gradient near the small body augments the effect of viscosity

and induces flow separation and vortex shedding (Mei 1989).

Chapter Two

- 9 -

This thesis focuses on the case of diffraction by large structures. Section 2 of this

chapter reviews briefly the mathematical developments of water wave diffraction

theory. The available analytical solutions to wave diffraction problems are listed.

Section 3 reviews the development of widely used numerical methods for wave

diffraction problems, such as the Eigenfunction Expansion Method (EEM), the FEM,

the IEM, the TFEM and the BEM (BIEM, GFM), particularly focusing on frequency-

domain solution approaches. Also, the advantages and disadvantages of these

methods are examined in this section. The last section introduces the theoretical

development of the SBFEM and its scope of application.

2.2 Mathematical development of water wave diffraction theory

2.2.1 Wave diffraction theory

In wave diffraction theory (referring to Chakrabarti 1995, Mei 1989 or other relevant

texts), it is assumed that the fluid is inviscid, incompressible and the motion

irrotational so that the fluid velocity may be expressed as the gradient of a scalar

potential Ф. The velocity potential Ф(x,y,z,t) satisfies the Laplace’s equation,

02 =∇ Φ (2-1)

within the fluid domain, where t is the time. (x,y,z) represents the co-ordinates of a

point in a rectangular Cartesian co-ordinate system, x and y are co-ordinates in the

mean free surface and z is vertically upwards. Under the assumption of potential

theory, the total velocity potential Ф can be expressed as the sum of the incident

velocity potential ФI and the scattered potential ФS, namely,

SI ΦΦΦ += (2-2)

The boundary conditions with respect to the velocity potential Ф may be expressed as

follows.

• Dynamic-boundary condition

Chapter Two

- 10 -

[ ] 021 222 =++++ zyxt ,Φ,Φ,Φg,Φ η at the free surface (2-3)

where η is the free surface elevation and g is the acceleration due to gravity.

• Kinematic-boundary condition

0=−++ zyyxxt ,Φ,,Φ,,Φ, ηηη at the free surface (2-4)

• Bottom-boundary condition

0=n,Φ at the water bottom (2-5)

where it is assumed that the bottom is impermeable and n designates the normal to

the boundary.

• Body surface-boundary condition

v,Φ n = at the structure surface (2-6)

• Radiation condition at infinity

This boundary condition at infinity requires the scattered wave must be outgoing in

a certain way. However, second-order wave radiation has been a controversial topic

and there are various view points (Drake et al. 1984).

The complete boundary-value problem defined in the preceding paragraphs is highly

nonlinear, mainly due to the free surface-boundary condition. Consequently, it is not

possible to accurately obtain the solution of the velocity potential Ф. However,

applying the perturbation technique, the original boundary-value problem can be

simplified into n approximate problems.

The velocity potential Ф may be formulated as a form of a power series with respect

to a perturbation parameter ε (wave slope),

Chapter Two

- 11 -

( )∑∞

=

=1n

nnΦΦ ε (2-7)

with

2kH

=ε (2-8)

where Ф(n) is the nth-order component of velocity potential and H is the wave height.

Correspondingly, the wave surface elevation η can be written as

( )∑∞

=

=1n

nnηεη (2-9)

The velocity potential Ф can be expanded to form of Taylor series about the mean

water surface (z = 0), hence,

( ) ( ) ( ) ( ) L+++= == 02

0 ,21,,0,,,,, zzzzz ΦΦtyxΦtyxΦ ηηη (2-10)

This thesis only deals with the problem up to the second-order, so detailed

introductions to first- and second-order problems are presented in Chapters 3-6.

2.2.2 Analytical and semi-analytical solutions

For first- and second-order problems with simple geometries, many analytical or

semi-analytical solutions have been presented in the past decades.

(a). Development of first-order (linear) solutions

For cases of infinite water depth, Havelock (1940) developed an analytical solution to

the problem of linear (first-order) diffraction by a fixed vertical circular cylinder in

deep water. Dean (1945) and Ursell (1947) presented a linear solution to wave

scattering by a vertical thin barrier. Dean & Ursell (1959) dealt with the case of a

fixed semi-immersed circular cylinder using linear wave theory. Newman (1965)

developed a linear solution to the case of an infinite step.

Chapter Two

- 12 -

However, most studies focused on cases with finite water depth. Maccamy & Fuchs

(1954) extended Havelock’s (1940) work to the case of finite depth. Twersky (1952)

constructed a solution using an iterative procedure for multiple acoustic scattering by

an arbitrary configuration of parallel cylinders. Ohkusu (1974) extended this method

to water wave problems. Miles (1967) applied the variational method to gain a

solution to wave scattering by a step shelf. Mei (1969) employed the same method to

work out the solution of surface wave scattering by rectangular obstacles. Garrett

(1971) indicated the error of Miles & Gilbert’s (1968) work and presented a solution

for calculating wave forces on a circular dock. Spring & Monkmeyer (1974) used the

direct method for the case of interaction of linear wave and multiple vertical

cylinders. Mingde & Yu (1987) studied the case of shallow water-wave diffraction of

multiple circular cylinders using the same multiple scattering techniques as Spring &

Monkmeyer (1974). Linton & Evans (1989) simplified the expression of velocity

potential in this direct solution method. Kagemoto & Yue (1986) examined a more

general case regarding arrays of axisymmetric bodies. Goo & Yoshida (1990)

extended this method to arbitrary-shaped structures numerically. Fernyhough &

Evans (1995) studied the scattering properties of an incident field upon a periodic

array of identical rectangular barriers, each extending throughout the water depth.

Maniar & Newman (1997) analysed water wave diffraction by an array of bottom-

mounted circular cylinders and investigated the near-resonant modes occurring

between adjacent cylinders at critical wave numbers. Chakrabarti (2000) describes an

analytical/numerical approach that determines the wave forces on multiple structures

in the vicinity of one another, taking into account multiple vessel interaction and

scattering in waves. It was reported that Chakrabarti’s method (Chakrabarti, 2000)

was more efficient than the direct method.

In the case of finite water depth, the study of hydrodynamic coefficients has also

attracted considerable attention in terms of predicting wave-induced loads and

motions. Ursell (1949) analysed the waves induced by a circular cylinder oscillating

on the water surface. Hulme (1982) developed a analytical solution for the case of

floating hemisphere. Havelock (1955) also investigated the case of hemisphere. Lee

(1995) applied the EEM to the solution of the heave radiation problem of a

Chapter Two

- 13 -

rectangular structure. Wu et al (1995) dealt with wave induced response of an elastic

rectangular structure in an infinite domain. Teng et al (2004) developed an analytical

solution for calculating wave radiation by a uniform cylinder in front of vertical wall,

based on the image principle. Drobyshevski (2004) presented a closed form

asymptotic formulae for all hydrodynamic coefficients for heave, sway and roll

motions. In this study, a two-dimensional rectangular profile was considered with the

under-bottom clearance assumed to be small compared with structure dimensions and

the water depth. Zheng et al (2004) employed the EEM to calculate the added mass

and damping coefficients for the buoy heave, sway and roll motions in calm water.

(b). Development of second-order solutions

Compared with the first-order problem, the second-order problem becomes much

more complex due to the effect of the nonhomogeneous free surface condition. The

second-order effect is very important in engineering problems because the high-

frequency (double-frequency and sum-frequency) component of second-order force

can cause a rapidly oscillating hydrodynamic pressure which contributes to the

fatigue of structures, while the low frequency component (difference-frequency and

zero-frequency or drift force) is the source of slowly varying exciting forces.

Analytical or semi-analytical solutions exist for some simple cases. These solutions

can be classified into three types: the indirect method, the direct method and the

approximate method.

The advantage of the indirect method lies in its ability to calculate wave loads with

ease, with no requirement for an explicit solution of second-order velocity potential.

Lighthill (1979) and Monlin (1979) developed an assistant radiation potential

approach (indirect method) for calculating second-order diffraction force on three-

dimensional bodies. Miao & Liu (1986) and Vada (1987) applied a similar method to

the solution of second-order wave forces in the case of two-dimensional infinite water

depth. Eatock Taylor & Hung (1987) calculated the second-order diffraction forces

on a vertical cylinder in regular waves using the indirect method. Williams and his

co-authors carried out a series of studies on the second-order wave loading on both

single vertical cylinders and arrays of vertical cylinders. Abul-Azm & Williams

Chapter Two

- 14 -

(1988) applied the indirect method to the case of truncated cylinders. Ghalayini &

Williams (1991) dealt with the case of vertical cylinder arrays. Moubayed &

Williams (1995a and 1995b) developed solutions for bichromatic waves.

However, the indirect method has its drawbacks, as indicated by Huang & Eatock

Taylor (1996). The indirect method can not produce the second-order free-surface

elevation and the wave run-up on the waterline, which are important quantities in the

design of a floating production system such as a tension leg platform. Furthermore,

different auxiliary radiation potentials have to be used.

The direct method is an alternative option. Isaacson (1977) applied cnoidal wave

theory to the nonlinear diffraction of a cnoidal wave around a single cylinder. This

solution is limited to the case of shallow water. Mingde & Yu (1987) used similar

methods to deal with the case of multiple cylinders. Kim & Yue (1990) worked out a

complete second-order solution for diffraction of a plane monochromatic incident

wave by an axisymmetric body. Wu & Eatock Taylor (1990) analytically solved the

problem of second order diffraction by a horizontal submerged circular cylinder in

finite water depth. Wu (1991) calculated the second-order reflection and transmission

coefficients due to wave diffraction by a submerged circular cylinder. Later, this

solution was extended to the case of wave radiation by Wu (1993a). Wu (1993b)

considered the hydrodynamic forces on a deeply submerged circular cylinder

undergoing large-amplitude motion by using a linearized free-surface condition and

the exact body boundary condition. Kriebel (1990 and 1992) and Chau & Eatock

Taylor (1992) developed a solution to the problem of second-order wave diffraction

by a bottom-seated vertical cylinder extending through the whole water depth. Sulisz

(1993) presented an analytical solution to this problem, using the EEM. Moubayed &

Williams (1994) described an eigenfunction expansion approach to the calculation of

hydrodynamic loads in regular waves. This solution is only applicable to circular

geometry. Li & Williams (1999) extended this solution to the case of a bichromatic

incident wave. Eatock Taylor & Huang (1996) presented a method for the solution of

second-order diffraction by a truncated vertical circular cylinder, using the EEM.

Eatock Taylor & Huang (1997) extended the exact theory for second-order wave

diffraction by vertical cylinder in monochromatic waves to the case of bichromatic

Chapter Two

- 15 -

incident waves. Hermans (2003) developed a new method using Green’s theorem to

calculate the interaction of free-surface waves with a floating dock. It was reported

that this method did not need to split the problem into symmetric and antisymmetric

problems, and was simpler than the EEM.

The direct method is quite complicated. This prompted some alternative approximate

methods to be developed, based on both the direct and indirect methods. Abul-Azm &

Williams (1989a and 1989b) developed a computationally efficient semi-analytical

approximate method to study second-order interference effects in structural arrays,

and presented numerical results for regular waves for arrays of two, three and four

bottom-mounted, surface-piercing and semi-immersed, truncated cylinders. The first-

order solution was based on the modified plane wave method. Newman (1990)

approximately evaluated the second-order vertical force on a horizontal rectangular

cylinder, assuming deep submergence. Second-order wave loads were obtained using

the indirect method. Williams et al (1990) compared the complete solutions with

approximate solutions for computing wave loads on arrays of bottom-mounted,

surface-piecing vertical circular cylinders in regular waves. It was found that the

approximate method was sufficient to compute hydrodynamic interference effects to

the second-order in many practical engineering situations. Sulisz & Johansson (1992)

developed an approximate approach to the problem of diffraction of second-order

monochromatic wave by a semisubmerged horizontal rectangular cylinder. Sulisz

(2002) calculated the diffraction of nonlinear waves by a horizontal rectangular

cylinder founded on a low rubble base, assuming that the pressure underneath the

cylinder was linearly dependent on the horizontal space coordinate.

Analytical and semi-analytical solutions mentioned in the preceding paragraphs are

usually able to provide good accuracy. However, these solution methods are only

applicable to simple geometries. Most practical engineering problems must employ

numerical methods.

2.3 Numerical methods for wave diffraction problems

2.3.1 Finite-element method

Chapter Two

- 16 -

Applications of the FEM to wave diffraction problems appeared in the early 1970s

(e.g. Chenault 1970, Bai 1972 & 1975). The FEM is well known for its flexibility in

handling irregular boundary problems. This is particularly attractive for wave

diffraction around offshore structures, because most offshore structures are of

irregular shape. However, one of the difficulties encountered in using the finite

element method to solve wave diffraction problems in an unbounded domain is the

implementation of the radiation boundary condition. In the finite element method, a

calculation domain of a finite size is normally used to approximate the infinite

domain on which the wave diffraction problem is defined. To satisfy the radiation

boundary condition at the outer boundary of the truncated domain, the outer boundary

has to be far away from the object investigated. The further the outer boundary, the

more nodes are needed inside the domain to maintain a certain level of accuracy of

the solution. More nodes normally implies higher computational cost. This problem

becomes more severe for three-dimensional problems. To avoid this difficulty, the so-

called hybrid-element method has been used by many researchers with moderate

success (e.g. Berkhoff 1972, Chen & Mei 1974, Bai & Yeung 1974, Krishnankutty &

Vendhan 1995, Sannasiraj et al 2000, Hsu et al. 2003 and Wu & Eatock Taylor 2003).

The hybrid-element method combines the finite element solution of the problem in a

finite domain next to the object with an analytical solution at the outer boundary of

the finite domain. It takes the advantages of both the finite element method and the

analytical method for this particular kind of problems. Zienkiewicz et al (1978)

provided a detailed review of such methods.

The IEM is a powerful technique for dealing with unbounded domain problems. Its

development can be traced back to the work by Bettess (1977). Bettess (1992) details

the development of this method and its theory. Zienkiewicz & Bettess (1976) first

applied this method to the solution of wave problems. This work was extended to

more general cases of diffraction and refraction problems. Later, the IEM was

continuously developed (Bettess 1984, Zienkiewicz & Bettess 1978, Zienkiewicz et

al 1981 and Zienkiewicz et al 1983) using the Zienkiewicz mapping. Bettess &

Bettess (1998) gave a wide overview of the application of the IEM in wave problems.

Also, Bettess & Bettess (1998) presented a new mapped infinite wave element for

Chapter Two

- 17 -

general wave diffraction problems and validated their method using the ellipse

diffraction problem. This improvement allowed the IEM to be used with elliptical and

other non-circular meshes. In conclusion, the IEM provided an effective complement

to the FEM for handling unbounded domain problems. This method works well

through coupling with the FEM.

Another popular advanced finite-element method is the Trefftz-type element method,

which is able to deal effectively with infinite fields and singularities. The idea of this

method is to seek an approximate solution of a boundary value problem using the sets

of functions that satisfy exactly the governing equation, but do not necessarily satisfy

the prescribed boundary conditions. Herrera (1984) systematically studied complete

sets of solutions of a homogenous partial differential equation and proposed a

completeness criterion called c-completeness (connectivity conditions). In the context

of the FEM, this criterion is termed T-completeness (Trefftz 1926) by Zienkiewicz.

Stein (1973) and Ruoff (1973) coupled the Trefftz-domains with finite displacement

elements using Galerkin techniques. Since then, many hybrid formulations have been

investigated (refer to Jirousek & Wroblewski 1996 for detailed survey). Cheung et al

(1991) applied this method to the solution of the Helmholtz equation for wave

diffraction problems. Jirousek & Wroblewski (1994) presented an efficient solution to

boundary-value problems based on the application of a suitable truncated T-complete

set of Trefftz functions over individual subdomains, linking the fields by a least-

squares procedure. Stojek (1998) extended this technique to solution of the Helmholtz

equation. Furthermore, Stojek et al (2000) calculated the diffraction loads on multiple

vertical cylinders with a rectangular cross section using the Trefftz-type finite

elements and demonstrated the ability of this method to deal with the unbounded

domain and singularities near sharp re-entrant corners. However, this paper also

indicated that an increase in the number of T-functions led to ill-conditioning of the

resulting system of algebraic equations, especially for the small values of wave

number. Another minor drawback of this method is that the coupling procedure needs

to be considered carefully in many cases.

2.3.2 The Green’s function method

Chapter Two

- 18 -

A Green’s function is an integral kernel that can be used to solve an inhomogeneous

differential equation with boundary conditions. Both the BIEM and BEM utilize the

Green’s function to seek the solution of boundary-value problems. The Green’s

function of simple cases can be evaluated analytically, but for most cases, numerical

procedures need to be implemented. The emphasis of this section is placed on

reviewing development of numerical techniques to calculate Green’s functions for

solving two-dimensional wave diffraction problems.

A relatively simple Green’s function method works by truncating the infinite field

using an artificial boundary (Au & Brebbia 1983, Rahman et al 1992, Carvalho &

Mesquita 1994, Drimer & Agnon 1994). In this type of approach, only the boundaries

of the solution domain are discretized spatially into elements, leading to a reduction

of the spatial dimension by one. This diminishes the effort of data preparation and

leads to fewer unknowns. However, due to the truncation of the infinite field, the

radiation condition cannot be rigorously satisfied. Furthermore, the computational

costs can be very high, since all boundaries (including the outer boundary) need to be

discretised, although the cost can be reduced for problems with system symmetries,

as suggested by Au & Brebbia (1983).

An alternative method is the BIEM, which has been widely used for wave diffraction

problems (Bai 1975, Leonard 1983, Hsu & Wu 1995, Teng 1996, Zhu et al 2000,

Miao et al 2001, Politis et al 2002, Chen 2004, etc.). The method makes use of a

singular solution that satisfies the free surface, seabed and radiation boundary

conditions. The scattered wave potential is represented by an integral of the singular

solution multiplied by a distribution function of singularities on the surface of the

object (John 1950, Murphy 1978). The strength of the singularities is then determined

by enforcing the boundary condition on the surface of the object. The BIEM is

normally very efficient, because only the surface of the object needs to be discretized,

and is widely used for calculating wave forces on offshore structures.

However the BIEM suffers from some numerical difficulties (Mikhailov 2005).

Firstly, the coefficient matrix associated with the BIEM sometimes may be ill-

conditioned (Huang & Eatock Taylor 1996). Special techniques are needed in order to

Chapter Two

- 19 -

remove the irregular frequency effect. Pien & Lee (1972) imposed an “artificial lid

boundary condition” on the interior free surface of two-hull forms when calculating

wave radiation-diffraction problems. However, it was found that the introduction of

the lid produced deviations from the results obtained using the original approach in

the frequency range below the first irregular frequency value (Wu & Price 1987).

Haraguchi & Ohmatsu (1983) and Sclavounos & Lee (1985) developed a method to

remove the irregular frequency phenomenon for the case of mono-hulls. For wave

diffraction problems involving a two-dimensional mono-hull, Ogilvie & Shin (1978)

presented two modified Green’s functions, a symmetric form and an asymmetric

form, to avoid the effect of irregular frequencies. Later, Sayer (1980) extended their

work to the case of finite water depth. Ursell (1981) presented a multiple expansion

of this modified Green’s function. Martin (1981) and Takagi (1983) attempted to use

this multiple expansion to solve realistic problems, but unfortunately the method

produced non-convergent or incorrect solutions. To develop a method to remove

irregular frequency for problems of wave diffraction-radiation by twin-hull and multi-

hulls, Wu & Price (1987) presented a new multiple Green’s function expression for

the hydrodynamic analysis of multi-hull structures in infinite water depth, building on

the work of Ogilvie & Shin (1978). Lee & Sclavounos (1989) presented a method

which could remove all irregular frequencies and eliminated undesirable effects in

numerical implementation, by selecting a purely imaginary constant of

proportionality.

The BIEM also suffers from numerical difficulties when modelling re-entrant

structure geometries or structures with sharp corners or small openings (Patel 1989,

Eatock Taylor & Teng 1993).

In conclusion, both the FEM and the BEM encounter some numerical difficulties for

solving wave diffraction problems.

2.4 Development of the scaled boundary finite-element method

Wolf (2003) presented a comprehensive historical note of the development of the

SBFEM. The SBFEM can be considered to have its roots in an algorithm presented

Chapter Two

- 20 -

by Silverster (1977) to model an unbounded domain in electrostatics and

magnetostatics. This procedure, called ballooning, may be regarded as a combination

of infinite substructuring and finite element techniques (Wolf 2003). Thatcher (1978)

and Ying (1978) independently developed similar approaches, solving an eigenvalue

problem. Dasgupta (1979 and 1982) presented a so-called cloning algorithm to model

unbounded domains in a dynamic analysis. In this computational procedure,

analogous to ballooning in statics (Silverster 1977), a finite-element cell was bounded

by two similar boundaries. Then, the dynamic-stiffness matrix of the unbounded

domain, relating to the static-stiffness and mass matrices of the cell, was derived by

solving an eigenvalue problem. This method assumed the dimensionless frequencies

of the cell were constant, so it was only applicable to statics and dynamics with a

layer of constant depth. Lysmer (1970), Waas (1972) and Kausel el al (1975)

presented the same equations, now called the thin-layer method for the consistent

boundary procedure in two dimensional cases. To improve this computational

procedure, Wolf & Weber (1982) replaced the average value of the dimensionless

frequency by two terms of a Taylor expansion. But, the analytical limit of the

infinitesimal cell width could be performed only for special cases. Later, Wolf &

Song (1994) presented the consistent infinitesimal finite element method which

allows the analytical limit of the infinitesimal width of the finite element cell to be

performed in the general case. Subsequently this method was applied to solving many

engineering problems (Wolf & Song 1996).

The key advance of this method was the derivation of fundamental equations based

on a scaled boundary coordinate transformation that leads to a system of linear

second-order ordinary differential equations in displacements with the radial

coordinate as the independent variable (Song & Wolf 1997). This computational

procedure is referred to as the SBFEM. With this theoretical breakthrough, the

solution process becomes much simpler because the procedure to seek the limit as the

cell width went to zero was not required any longer.

The SBFEM has found wide application in solving problems of elasto-statics and

elasto-dynamics, and its advantages for soil-structure interaction problems in

unbounded domains have been demonstrated (Wolf & Song 1996, Wolf 2003, Deeks

Chapter Two

- 21 -

& Wolf 2003, Deeks 2004, Doherty & Deeks 2005, Genes & Kocak 2005, Chidgzey

& Deeks 2005, Song 2004, 2005 and 2006, Hossein & Song 2006, Vu & Deeks 2006

and Yang 2006).

In the field of the fluid-structure interaction analysis, Deeks & Cheng (2003) applied

the SBFEM for potential flow around obstacles. The advantages of the SBFEM to

model potential flow problems in an unbounded domain were demonstrated through

numerical examples and through the comparisons with other numerical methods such

as the FDM. It was found that the SBFEM results in excellent agreement with

analytical solution for potential flow around a circular cylinder. It was also shown

that the scaled boundary finite-element method handles problems with velocity

singularities very efficiently and accurately.

2.5 Summary

This chapter presents a detailed review of the development of the mathematical

theory of water-wave diffraction and examined the numerical advantages and

disadvantages of the FEM and the GFM (BEM, BIEM) for solving wave diffraction

problems. The review demonstrates that neither the FEM nor the GFM (BEM, BIEM)

can very well deal with the infinite field and the singularities in the vicinity of re-

entrant corners. Furthermore, the GFM (BEM, BIEM) requires extra effort to remove

the effect of irregular frequencies.

On the other hand, in the field of solid elasto-statics and elasto-dynamics, it was

found that the SBFEM was very suitable for dealing with the unbounded domain and

singularity problems, due to the analytical nature of the solution in the radial

direction. Also, this method has been applied to calculating potential flow around

obstacles.

Until the study described in this thesis was performed, the SBFEM has not been

applied to water-wave diffraction problems, which are more challenging due to the

existence of the free water surface boundary condition. Chapters 3-6 of this thesis

Chapter Two

- 22 -

develop SBFEM solutions of wave diffraction problems and examines its numerical

benefits and difficulties.

Chapter Three

- 23 -

CHAPTER 3

LINEAR SBFEM SOLUTION OF LAPLACE EQUATION

3.1 General

A modified Scaled Boundary Finite-Element Method (SBFEM) for problems with

parallel side-faces is presented in this chapter. To overcome the inherent difficulty of

the original SBFEM for domains with parallel side-faces, a new type of local co-

ordinate system is proposed. The new local co-ordinate system allows the so-called

scaling centre of the SBFEM to move freely along an arbitrary curve and thus

eliminates the non-parallel side-face restriction in the original SBFEM. The modified

SBFEM equations are derived based on a weighted residual approach. It is found that

the modified SBFEM solution retains the analytical feature in the direction parallel to

the side-faces and satisfies the boundary conditions at infinity exactly, as in the

original SBFEM. A complete scaled boundary finite-element solution to a two-

dimensional Laplace equation with Neumann and Robin boundary conditions in a

semi-infinite domain with parallel boundaries is derived.

The modified SBFEM is then applied to solutions of two types of problems—wave

diffraction by a single and twin surface rectangular obstacles and wave radiation

induced by an oscillating mono-hull and twin-hull structures in a finite depth of

water. For wave diffraction problems, numerical results agree extremely well with the

analytical solution for the single obstacle case and other numerical results obtained

using a different approach for the twin obstacle case. For wave radiation problems,

the particular solutions to the scaled boundary finite-element equation are examined

for heave, sway and roll motions. The added mass and damping coefficients for

heave, sway and roll motions of a two-dimensional rectangular container are

Chapter Three

- 24 -

computed and the numerical results are compared with those from an independent

analytical solution and numerical solution using the boundary element method

(BEM). It is found that the SBFEM method achieves equivalent accuracy to the

conventional BEM with only a few degrees of freedom. In the last example, wave

radiation by a two-dimensional twin-hull structure is analyzed. Comparisons of the

results with those obtained using conventional Green’s Function Method (GFM)

demonstrate that the method presented in this chapter is free from the irregular

frequency problems.

3.2 Mathematical formulation

Problems of both wave diffraction by surface rectangular obstacles and wave

radiation induced by oscillating structures can be solved by seeking the solution of a

two-dimensional Laplace’s equation associated with certain conditions on the

boundary in the computational domain as shown in Figure 3.1. The entire fluid

domain is divided into N subdomains, which are denoted by NΩΩΩ ,,, 21 L . The

interface boundaries and side-faces are represented by 121

,,,−Nbbb ΓΓΓ L and

NN ssss 101101,,,, ΓΓΓΓ L , respectively.

Employing the method of the separation of variables, the velocity potential Φ may be

written as

tyxtyxΦ ωφ ie),(),,( −= (3-1)

where the ω is the angle frequency and t is the time. The velocity potential ),( yxφ is

governed by Laplace’s equation

0),(2 =∇ yxφ , in domain Ω (3-2)

The boundary conditions on the side-faces and the defining line may be specified as

φωφgy

2

=∂∂

, on the free surface of water ( 1s

Γ ) (3-3)

Chapter Three

- 25 -

0=∂∂

yφ

, on the bottom of water ( 0sΓ ) (3-4)

vn=

∂∂φ

, on the body surface (bsΓ ) (3-5)

where n represents the normal to the boundary and the over-bar denotes prescribed

values. On the boundary ∞Γ , the far-field condition needs to be satisfied, which

requires the evanescent modes of standing waves to vanish so that there only exists

propagating waves satisfying the Sommerfeld radiation condition.

3.3 Local co-ordinate system

3.3.1 Standard scaled boundary co-ordinate system for bounded domains

The standard scaled boundary co-ordinate system (Wolf & Song 2000) is shown in

Figure 3.2(a). Clearly this co-ordinate system requires rigorously that the so-called

scaling center must exist. Consequently, this local co-ordinate system can not be used

in an unbounded domain with parallel side-faces. However, it is applicable to the case

of a bounded domain. Applying this local co-ordinate system to the current

mathematical problem, a typical bounded domain 1+iΩ (see Figure 3.1) is illustrated

in Figure 3.2(b). The exterior boundary Γ (defining curve), consisting of

ibΓ , hΓ ,1+ibΓ and wΓ , is discretized in a similar manner to the conventional finite

element method. The circumferential local co-ordinate s is defined along the structure

surface and the exterior boundary, measuring the distance anticlockwise around the

boundary curve Γ . The normalized radial co-ordinate ξ is unity on the exterior

boundary curve Γ ( 1=eξ ) and zero at the scaling centre ( 0=iξ ). Thus, each value

of ξ defines a scaled version of the curve Γ . The side-faces (s=s0 and s=s1) which

coincide with the surfaces of the structure are not discretized. Therefore, the

computational subdomain 1+iΩ can be defined as 10 ≤≤ ξ and 10 sss ≤≤ .

The scaling equations relating the Cartesian co-ordinate system to the scaled

boundary co-ordinate system (Deeks & Cheng 2000) are defined as

Chapter Three

- 26 -

)(ˆ 0 sxxx ξ+= (3-6a)

)(ˆ 0 syyy ξ+= (3-6b)

where the symbols follow those used in Wolf & Song (2000). )ˆ,ˆ( yx represents the

points within the domain, ),( yx designates the points on the boundary and ),( 00 yx

the co-ordinates of the scaling centre. The derivatives in the Cartesian co-ordinate

system can be related to the standard scaled boundary co-ordinate system, namely,

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

∂∂

⎥⎦

⎤⎢⎣

⎡−

−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂∂∂

sxxyy

Jy

xss

ss

ξ

ξ1,,

,,1

ˆ

ˆ (3-7)

where the Jacobian on the curve S (ξ =1) is

ss sxsysysxJ ),()(),()( −= (3-8)

The gradient operator in the standard scaled boundary co-ordinate system can be

expressed as

ssbsb

∂∂

+∂∂

=∇ )(1)( 21 ξξ (3-9)

with

⎭⎬⎫

⎩⎨⎧−

=s

s

xy

Jsb

,,1)( 1 (3-10a)

⎭⎬⎫

⎩⎨⎧−

=xy

Jsb s,1)( 2 (3-10b)

where b1(s) and b2(s) are dependent only on the definition of boundary Γ .

3.3.2 Modified scaled boundary co-ordinate system for unbounded domains

Chapter Three

- 27 -

To eliminate the problem of locating a scaling centre in an unbounded domain with

parallel side-faces, a modified SBFEM is established on a new local co-ordinate

system, which will be termed translated boundary co-ordinate system. For the

purpose of discussion, the unbounded subdomain NΩ (Figure 3.1) is selected as the

computational domain in which the definition of the local co-ordinate system is

identical to that shown in Figure 3.3, and the representation symbols of the Nth

subdomain and its boundaries are replaced by those in Figure 3.3. If the co-ordinate

ξ is replaced by ξ− , the following resulting equations are applicable to the

unbounded subdomain 1Ω as well. Thus, the new co-ordinate system is defined in the

domain Ω enclosed by the boundary Γ ( ∞∪∪∪= ΓΓΓ10 ssS ) as shown in

Figure 3.3. The boundary S is a piece-wise smooth curve fitted on the interface with

other subdomains. To facilitate the discussion here, the boundary S is referred to as

the defining curve in this thesis. The boundary 0sΓ and

1sΓ are parallel side-faces

and the boundary ∞Γ is located at infinity. In this local co-ordinate system, the co-

ordinate s , similar to the circumferential co-ordinate in the original SBFEM co-

ordinate system, measures the distance along the defining curve. It should be noted

that the defining curve will never be a closed curve. The origin of the horizontal co-

ordinate ξ is defined on the defining curve. It is assumed that the direction towards

the interior of the domain is positive. Like the standard SBFEM, only the defining

curve needs to be discretized. This can be achieved by employing a shape function

[N(s)] in the usual finite-element manner. Using an isoparametric approach, the

Cartesian co-ordinates ),( yx of a point at position s on the defining curve may be

expressed as

)]([ xsNx = (3-11)

)]([ ysNy = (3-12)

where x and y are nodal co-ordinate vectors in the Cartesian co-ordinate system.

Similarly, an interior point )ˆ,ˆ( yx in the domain can be defined as

Chapter Three

- 28 -

ξ+= )(ˆ sxx (3-13)

)(ˆ syy = (3-14)

which may be viewed as a mapping from the translated boundary co-ordinate system

to the Cartesian co-ordinate system.

Employing the mapping between the translated boundary co-ordinate system and the

Cartesian co-ordinate system, the Jacobian matrix is formulated as

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

ssss yxyxyx

sJ,ˆ,ˆ01

,ˆ,ˆ,ˆ,ˆ

)],([ ξξξ (3-15)

All spatial derivatives in the new co-ordinate system can be related to derivatives in

the Cartesian co-ordinate system using the Jacobian matrix.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂∂∂

⎥⎦

⎤⎢⎣

⎡−

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂∂∂

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂∂∂

−

sx

yJ

s

sJ

y

xs

s ξξξ1,ˆ0,ˆ1)],([

ˆ

ˆ 1 (3-16)

with the Jacobian

syJ ,ˆ= (3-17)

Thus, the gradient operator ∇ is expressed as

ssbsb

∂∂

+∂∂

=∇ )()( 21 ξ (3-18)

with

⎭⎬⎫

⎩⎨⎧−

=s

s

xy

Jsb

,ˆ,ˆ1)( 1 (3-19)

Chapter Three

- 29 -

⎭⎬⎫

⎩⎨⎧

=101)( 2 J

sb (3-20)

3.4 Scaled boundary finite-element equations

Equations (3-2)-(3-5) may be expressed in a weighted residual form

021

2 =−−∇∇ ∫∫∫ dΓvwdΓkwdΩwΓΓΩ

T φφ (3-21)

where Ω represents the part within a bounded or unbounded domain. For the case of

a bounded domain (Figure 3.2(b)), 1Γ indicates the boundary wΓ at the surface of

water while 2Γ is the side-faces at the surface of the structure. For the case of an

unbounded domain (Figure 3.3), 1Γ represents the boundary 1sΓ while 2Γ denotes

the interface bΓ . w is any weighting function and k is defined as

gk

22 ω= (3-22)

Following a similar approach to Deeks & Cheng (2003), the approximate solution to

Equation (3-21) can be formulated as

)()]([),( ξξφ asNsh = (3-23)

where shape functions [N(s)] can be defined on the defining line in the standard

finite-element manner and )( ξa is the nodal potential vector.

Applying the Galerkin approach, the weighting function can be formulated using the

same shape functions

TT sNwwsNsw )]([)()()]([),( ξξξ == (3-24)

Substituting Equation (3-23) and Equation (3-24) into Equation (3-21) results in

Chapter Three

- 30 -

0)]([)(

)()]([)]([)(

))()]([())()]([(

2

1

2

=−

−

∇∇

∫∫∫

dΓvsNw

dΓasNsNwk

dΩasNwsN

T

Γ

T

Γ

TT

Ω

T

ξ

ξξ

ξξ

(3-25)

The SBFEM applies the local co-ordinate system to Equation (3-25) to obtain the so-

called scaled boundary finite-element equation. For the convenience of discussion,

the following matrices are introduced

)]()[()]([ 11 sNsbsB = (3-26)

ssNsbsB )],()[()]([ 22 = (3-27)

Substituting Equation (3-26) and Equation (3-27) into Equation (3-25) yields

0)]([)(

)()]([)]([)(

))()]([),()](([

))()]([),()](([

2

1

2

21

21

=−

−

+

+

∫∫

∫

dΓvsNw

dΓasNsNwk

dΩasBasB

wsBwsB

Γ

TT

Γ

TT

Ω

T

ξ

ξξ

ξξ

ξξ

ξ

ξ

(3-28)

For the case of bounded domain, integrating all terms containing ξξ ),(w by parts

with respect to ξ , using Green’s identity, noting that dsdJd ξξΩ = and

introducing ξτ and sτ to transform infinitesimal lengths on the boundary sections

with constant ξ and constant s to the scaled boundary co-ordinate system, results in

dsJasBsBw

dsJasBsBw

iis

Ti

ees

TTe

ξ

ξ

ξξξ

ξξξ

),()]()][([)(

),()]([)]([)(

11

11

∫∫−

∫ ∫ +−e

i

dsdJaasBsBws

TTξ

ξ ξξξ ξξξξξ )),(),(()]([)]([)( 11

∫∫ −+s i

Tis e

TTe dsJasBsBwdsJasBsBw )()]([)]()[()()]([)]([)( 2121 ξξξξ

∫ ∫−e

i

dsdJasBsBws

TTξ

ξ ξ ξξξ ),()]([)]([)( 21

Chapter Three

- 31 -

∫ ∫

∫ ∫+

+

e

i

e

i

dsdJasBsBw

dsdJasBsBw

s

TT

s

TT

ξ

ξ

ξ

ξ ξ

ξξξ

ξ

ξξξ

)(1)]([)]([)(

),()]([)]([)(

22

12

∫− s eTT

e dsasNsNwk ξτξξ )()]([)]([)(2

∫∫ −−e

i

dsvsNwdssvsNw sTT

s eTT

e

ξ

ξ

ξ ξτξξτξξ ),()]([)(),()]([)( 00

0),()]([)(),()]([)( 11 =−− ∫∫e

i

dsvsNwdssvsNw sTT

s iTT

i

ξ

ξ

ξ ξτξξτξξ (3-29)

It is worth noting that due to different boundary conditions on the exterior boundary

( eξξ = ), the integration along the water surface is separated from the others. For

convenience, the following coefficient matrices are introduced.

∫= s

T dsJsBsBE )]([)]([][ 110 (3-30)

∫= s

T dsJsBsBE )]([)]([][ 121 (3-31)

∫= s

T dsJsBsBE )]([)]([][ 222 (3-32)

∫= s

T dssNsNM ξτ)]([)]([][ (3-33)

sTsTs svsNsvsNf τξτξξ )),(()]([)),(()]([)( 1100 −+−= (3-34)

If the solution satisfies Equation (3-29) for all sets of weighting functions )( ξw ,

the following conditions must be satisfied.

∫ −=+s

Ti

Tii dsvsNaEaE ξ

ξ τξξξ )()]([)(][),(][ 10 (3-35)

)(][)(][),(][ 210 ee

Tee aMkaEaE ξξξξ ξ =+ (3-36)

∫=+s

Te

Tee dsvsNaEaE ξ

ξ τξξξ )]([)(][),(][ 10 (3-37)

Chapter Three

- 32 -

)()(][),(])[][]([),(][ 21102

0 ξξξξξξξ ξξξ sT faEaEEEaE =−−++ (3-38)

Equation (3-35) usually satisfies the interior boundary condition for unbounded

domain problems. Since the interior boundary becomes a scaling centre for bounded

domain problems, Equation (3-35) vanishes. In fact, the solution at the scaling centre

is required to be finite, which is satisfied automatically in the final solution as

demonstrated later on. Equation (3-36) indicates the satisfaction of the boundary

condition on the water surface wΓ while the other boundary conditions on the

exterior boundary are represented in Equation (3-37). Equation (3-38) is a system of

second-order nonhomogeneous ordinary differential equations. Equation (3-38) is

termed the scaled boundary finite-element equation. This equation is weakened in the

circumferential direction in a finite element manner but remains strong in the radial

direction. Consequently, the variation of potential is analytical along the side-faces.

The nonhomogeneous term )( ξξ sf at the right hand side of Equation (3-38) is due

to the prescribed motion of the side-faces.

For the case of unbounded domain, the scaled boundary finite-element equation may

be obtained by transforming Equation (3-28) in the same way as the case of bounded

domain. However, the translated boundary co-ordinate system is employed during the

transformation. Let iξξ = at the defining curve, eξξ = at the boundary ∞Γ and

note that dsdJd ξΩ = , dsd =Γ on boundary sections with constant ξ , where the

negative sign applies on ∞Γ and ξΓ dd = on boundary sections with constant s. The

resulting scaled boundary finite-element is

∫ −=+s

Ti

Ti dsvsNaEaE )()]([)(][),(][ 10 ξξ ξ (3-39)

∫=+s

Te

Te dsvsNaEaE )]([)(][),(][ 10 ξξ ξ (3-40)

0)(])[][(),(])[]([),(][ 202

110 =−+−+ ξξξ ξξξ aEMkaEEaE T (3-41)

where [E0], [E1] and [E2] have the same definition as Equation (3-30 to 3-32) and

Chapter Three

- 33 -

)]([)]([][ 110 sNsNM T= (3-42)

Equation (3-39) and Equation (3-40) represent the relations between the nodal flow

vector and the nodal potential vector on the defining line ( iξξ = ) and the boundary

at infinity ( eξξ = ), respectively. Equation (3-41) is an n-dimensional system of

homogeneous second-order ordinary differential equations with constant coefficients.

3.5 Solution process

3.5.1 Bounded domain solution

To be consistent with previous work (Deeks & Cheng 2003), a matrix solution of

Equation (3-38) is first sought in a general form and then the specific boundary

conditions are incorporated into the solution. Initially the homogeneous solution of

Equation (3-38) is considered. To transform Equation (3-38) into a simpler ordinary

differential equation, defining

)(][),(][)( 10 ξξξξ ξ aEaEq T+= (3-43)

⎭⎬⎫

⎩⎨⎧

=)()(

)(ξξ

ξqa

X (3-44)

and then substituting Equation (3-43) and Equation (3-44) into the homogeneous part

of Equation (3-38) yields

)(][),( ξξξ ξ XZX = (3-45)

with the coefficient matrices

⎥⎦

⎤⎢⎣

⎡

−−

=−−

−−

1011

1012

101

10

]][[][]][[][][][][

][EEEEEE

EEEZ T

T

(3-46)

Introducing constant integration vector cb, X(ξ) can be expressed as

Chapter Three

- 34 -

)]([)( bcXX ξξ = (3-47)

Substituting Equation (3-47) into Equation (3-45) yields

)](][[)],([ ξξξ ξ XZX = (3-48)

To obtain the solution of Equation (3-48), Jordan decomposition of the matrix [Z] is

performed,

]][[]][[ ΛTTZ = (3-49)

with

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=Λ

−

+

][0010

][

][

j

j

λ

λ

, 1,,2,1 −= nj L (3-50)

in which the transform matrix [T] is invertible, [λj+] and [λj

-] are the eigenvalues of the

matrix [Z], Re(λj+)>0 and Re(λj

-)<0 . Using the transform matrix [T], the matrix

[X(ξ)] can be expressed as

)](][[)]([ ξξ YTX = (3-51)

where the matrix [Y(ξ)] satisfies

)](][[)],([ ξξξ ξ YΛY = (3-52)

The solution of Equation (3-52) may be expressed as

][)]([ ΛY ξξ = (3-53)

Substituting Equation (3-52) into Equation (3-51) first and then substituting Equation

(3-51) into Equation (3-47) yields

][)( ][ bΛ cΦX ξξ = (3-54)

Chapter Three

- 35 -

Partitioning all matrices and vectors in Equation (3-54) into block forms results in

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

−

0][][][][

)(2

1][

][][

2221

1211b

bP

cc

TTTT

Xλ

λ

ξξξ

ξ (3-55)

To obtain a finite solution at the scaling centre, 2bc must be equal to zero, leading

to

)]([)( 1bb cAa ξξ = (3-56a)

)]([)( 1bb cQq ξξ = (3-56b)

with

][11 ][)]([

+

= λξξ TAb (3-57a)

][21 ][)]([

+

= λξξ TQb (3-57b)

Eliminating the constant vector 1bc in Equation (3-51) leads to

)()(][ ξξ qaH b = (3-58)

with

1)]()][([][ −= ξξ bbb AQH (3-59)

Noting that the solution of Equation (3-58) is for the homogeneous part of Equation

(3-38), the subscript h is introduced to represent the homogeneous solution. In the

same manner, s is introduced to represent the particular solution and e for the entire

solution. As the entire solution on the exterior boundary (ξ =1) is unknown, Equation

(3-58) can be expressed as

][][ bs

bbs

bee

b aHqqaH +−= (3-60)

Chapter Three

- 36 -

For convenience, separating the nodal flow vector beq into two parts yields

21 be

be

be qqq += (3-61)

with

⎭⎬⎫

⎩⎨⎧

=0

1

bweb

eq

q (3-62a)

⎭⎬⎫

⎩⎨⎧

=

2bbe

be q

q0

(3-62b)

where bweq is the nodal flow vector on the boundary at the water surface, while the

nodal flow vector on the other boundaries is denoted by bbeq . Substituting Equation

(3-36) into Equation (3-43) yields

][ 2 bwe

bwe aMkq = (3-63)

where bwea is the nodal potential vector corresponding to bw

eq . Substituting

Equation (3-63) into Equation (3-62a), Equation (3-62a) may be written as

][ 1e

bwbe aHq = (3-64)

where

⎥⎦

⎤⎢⎣

⎡=

000][

][2 Mk

H bw (3-65)

In fact, an expression of the form of Equation (3-65) can always be obtained by

arranging the position of nodes. Substituting Equation (3-64) into Equation (3-60)

yields

][])[]([ 2 bs

bbs

bee

bwb aHqqaHH +−=− (3-66)

Chapter Three

- 37 -

Considering Equation (3-37) and Equation (3-43), bbeq may be determined at the

boundary 1=ξ as

∫= s

Tbbe dsvsNq )]([ (3-67)

Apparently the dimensions of the vector bweq and bb

eq depend on the number of

elements on the corresponding boundaries.

The form of the particular solution consisting of bsa and b

sq depends on the

concrete boundary condition at the side-faces, which is discussed later. Once the

nodal potentials are determined through Equation (3-66), the entire potential field can

be found by Equation (3-23).

3.5.2 Unbounded domain solution

In terms of Equation (3-39), the nodal flow vector is defined as

)(][),(][)( 10 ξξξ ξ aEaEq T+= (3-68)

Using Equation (3-39) and Equation (3-68), the scaled boundary finite-element

equation (Equation (3-41)) may be transformed into the following form,

)(][),( ξξ ξ XZX = (3-69)


⎥⎦

⎤⎢⎣

⎡

−−−

=−−

−−

1010

21

1012

101

10

]][[][][]][[][][][][

][EEMkEEEE

EEEZ T

T

(3-70)

To express Equation (3-69) in the form of a matrix equation, introducing the

integration constant vector ∞c , Equation (3-44) may be written as

)]([)( ∞= cXX ξξ (3-71)

Chapter Three

- 38 -


)](][[)],([ ξξ ξ XZX = (3-72)

For the Hamiltonian matrix [Z], the eigenvalues consist of two groups with opposite

signs

⎥⎥⎦

⎤

⎢⎢⎣

⎡= −

+

][00][

][j

jΛλ

λ, nj ,,2,1 L= (3-73)

with Re(λj+)≥0 and Re(λj

-)≤0. Furthermore, it is assumed that the positive imaginary

number is placed in the block ][ −jλ . The corresponding eigenvalue problem can be

formulated

]][[]][[ ΛΦΦZ = (3-74)

where [Φ] denotes the eigenvector matrix. Due to the physical meaning of the

eigenvalues (which will be discussed later on), the eigenvector matrix [Φ] is always

invertible. Hence, to solve Equation (3-72), the matrix [X(ξ)] is expressed as

)](][[)]([ ξξ YΦX = (3-75)


)](][[)],([ ξξ ξ YΛY = (3-76)

The solution of this matrix equation is formulated as

ξξ ][)]([ ΛeY = (3-77)

Substituting the solution into Equation (3-75) and then substituting resulting matrix

[X(ξ)] back into Equation (3-71) yields

][)( ][ ∞Φ= ceX Λ ξξ (3-78)

Chapter Three

- 39 -

Partitioning [Φ] and constant vector ∞c in Equation (3-78) into block matrix with

nn× dimension blocks and block vector with 1×n dimension vectors respectively

results in

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

∞

∞

−

+

)(2

1][

][

2221

1211

cc

ee

ΦΦΦΦ

Xj

j

ξλ

ξλ

ξ (3-79)

The physical meaning of the eigenvalues will be examined before the far-field

boundary condition is applied to determine the integration constants. For problems of

wave diffraction and wave radiation, a complex eigenvalue corresponds to a mode of

the scattered wave when nodal potentials of the scattered wave are taken as

unknowns, eigenvalues with a zero real part represent propagating wave modes, and

eigenvalues with a negative real part correspond to the evanescent modes of the

standing wave. To satisfy the far-field boundary condition, the evanescent modes of

the standing wave must vanish at infinity and the propagating wave is outgoing,

which indicates that 1∞c must be zero so that the term with ][ +

jλ vanishes.

Consequently, Equation (3-75) becomes

)]([)( 2∞∞= cAa ξξ (3-80a)

)]([)( 2∞∞= cQq ξξ (3-80b)

with

ξλξ ][12 ][)]([ jeΦA −∞ = (3-81a)

ξλξ ][22 ][)]([ jeΦQ −∞ = (3-81b)

Eliminating the constant vector 2∞c in Equation (3-80) leads to

)(][)( ξξ aHq ∞= (3-82)

with

Chapter Three

- 40 -

1)]()][([][ −∞∞∞ = ξξ AQH (3-83)

When ξ=0, Equation (3-82) may be written as

][ ∞∞∞ = aHq (3-84)

Once the nodal potential vector a(ξ) is obtained, the integration constant vector

2∞c can be determined as

][ 12

∞−∞∞ = aAc (3-85)

With the known integration constant vector 2∞c , the entire potential field can be

found by substituting Equation (3-85) into Equation (3-80a), then using Equation (3-

23).

3.5.3 Coupling solutions of bounded and unbounded domains

At the interfaces between subdomains, the matching conditions with respect to the

nodal potential and nodal flow need to be satisfied. Hence, at a typical interface biΓ ,

the nodal potential vector ia and the nodal flow vector iq at the boundary of the

ith subdomain are related to those at the same boundary of the (i+1)th subdomain by

1 += ii aa (3-86a)

1 +−= ii qq (3-86b)

For problems of wave diffraction and radiation, the scattered wave potential is taken

to be unknowns both in an unbounded domain and a bounded domain, for

conveniences of establishing a global equation. Consequently, assembling the

solution of all subdomains into a global equation yields

][ qaH S = (3-87)

with

Chapter Three

- 41 -

][][][][ ∞+−= HHHH bwb (3-88)

and

][ 2 bs

bbs

be aHqqq +−= (3-89)

where aS represents the scattered potential vector whose dimension is equal to the

number of all nodes. Due to the match condition of scattered nodal flow

vector(Equation (3-86b)), the values in 2beq are zero at the nodes of interfaces

between bounded domains and are determined by the incident velocity at the nodes of

interfaces between bounded and unbounded domains.

3.6 Results and discussions

3.6.1 Wave reflection and transmission

In this problem, a structural system subjected to incident wave travelling towards the

positive x-axis is considered, as shown in Figure 3.4. It is assumed that the fluid is

inviscid, irrotational and incompressible in the present analysis. The total velocity

potential φ consists of the incident velocity potential Iφ and the scattered velocity

potential Sφ . In linear wave theory, the incident wave potential Iφ may be formulated

as

kxI kH

HzkgA iecosh

)(coshi +−=ω

φ (3-90)

where k is the wave number, A is the wave amplitude, g is the acceleration due to

gravity, ω is angular frequency and H is the depth of water. The wave elevation η at

any point on the free surface can be found by

φωηgi

= (3-91)

The reflection coefficient Kr and the transmission coefficient Kt are defined as

Chapter Three

- 42 -

I

rrK

ηη

= (3-92)

I

ttK

ηη

= (3-93)

where rη , tη and Iη represent the elevation amplitudes of the reflected wave, the

transmitted wave and the incident wave respectively. Since no energy loss is

assumed, the reflected coefficient and the transmission coefficient satisfy

122 =+ tr KK (3-94)

It can be seen that the reflected wave elevation can be computed from the unbounded

subdomain 1Ω and the transmitted wave elevation from the unbounded subdomain

2Ω by letting ξ = ∞, since the evanescent modes of the standing wave vanish at ξ =

∞.

For this problem, since the structural system is considered to be fixed, the velocity

normal to the side-faces in bounded domains is zero, leading to a homogenous scaled

boundary finite-element equation (Equation (3-33)).

Two numerical examples are presented to demonstrate the accuracy of the present

method and its ability to model problems with parallel side-faces. The first example

involves the scattering of plane waves around a floating rectangular cylinder in water

of a finite depth. The numerical solution using the modified SBFEM is compared

with the analytical solution by Mei & Black (1969) for this problem. The other

practical example considers the reflection and transmission of plane waves by twin

floating rectangular obstacles with a small gap. The problem is computed numerically

using both the modified SBFEM and a simple Green’s function method. Comparison

of the numerical results demonstrates the validity of the modified SBFEM for wave

reflection/transmission around structures.

3.6.1.1 Wave scattering by a single surface obstacle (Example 1)

Chapter Three

- 43 -

To analyse the reflection and transmission of plane waves by a single floating

obstacle, the entire fluid domain is divided into two unbounded subdomains and two

bounded subdomains. The boundaries are discretized with three-node quadratic

elements. The mesh is illustrated in Figure 3.4. Three meshes with different densities

are employed. The coarse mesh consists of 2 elements for the discretization of the

defining curve in each of the unbounded subdomains and 8 scaled boundary finite-

elements for the boundary discretization in each of the bounded domains. The

medium mesh is composed of 3 elements for the defining curve discretization and 12

scaled boundary finite-elements for boundary discretization in each of the bounded

domains. The fine mesh contains twice as many elements as in the coarse mesh case.

The Cartesian co-ordinate system (x, z) is defined as shown in Figure 3.4. The width

of the block is denoted by B, the draft by T and the depth by H. Wave loads on this

block and the reflection and transmission coefficient are computed with the

assumption of B=H.

The numerical results obtained using the modified SBFEM with different mesh

densities for Example 1 are presented in Figure 3.5 to Figure 3.9, together with those

obtained using the analytical solution by Mei & Black (1969). Figure 3.5 and Figure

3.6 show the variations of the normalized horizontal and vertical wave force

amplitudes with the relative width of the obstacle for different relative draft to water

depth ratios respectively. Figure 3.7 shows the variation of the normalized moment

around y-axis with the relative width for different relative draft to water depth ratios.

Figure 3.8 and Figure 3.9 plot the corresponding reflection and the transmission

coefficients. It can be seen from Figure 3.5 to Figure 3.9 that the numerical results

compare extremely well with the analytical solution by Mei & Black (1969) even

with the coarse mesh.

3.6.1.2 Wave scattering by twin surface obstacles (Example 2)

In this example, the reflection and transmission of plane waves by twin surface

obstacles with a small gap are analyzed. The entire fluid domain is divided into two

unbounded subdomains and four bounded subdomains, as shown in Figure 3.10. As

with the first example, three meshes with different density are used. In the coarse

mesh, the defining curves in the unbounded domains 1Ω and 6Ω are discretized with

Chapter Three

- 44 -

2 elements, the boundary of bounded subdomains 2Ω and 5Ω with 8 scaled

boundary finite-elements and 3Ω and 4Ω with 6 scaled boundary finite-elements.

The medium mesh consists of 4 elements for the discretization of the defining curves

in the unbounded domains, 16 scaled boundary finite-elements for the discretization

of the boundary of bounded subdomains 2Ω and 5Ω , and 15 scaled boundary finite-

elements for that of 3Ω and 4Ω . The fine mesh is composed of 6 elements for the

discretization of the defining lines and 24 scaled boundary finite-elements for the

boundary discretization in bounded subdomain 2Ω and 5Ω and 22 scaled boundary

finite-elements for 3Ω and 4Ω . The width of the blocks is represented by B and the

gap Bg between the blocks is taken to be 0.01B. Assuming B=H, the ratio of the draft

T to the depth H of water is 0.3.

This example is also computed by a simple Green’s Function Method (GFM) for the

purpose of comparison. In this GFM, the Green’s function is defined in terms of a

simple (Rankine) source. Since the evanescent modes have negligible contributions to

the potential when |x| tends to infinity, the disturbance potential in the far field is only

related to the propagating modes. Thus, for the current problem, the computation

domain is truncated at x = ± 7B. Three-node quadratic elements are employed for the

discretization of boundaries. The sources are located at all nodes and central points of

both boundaries parallel to the z-axis. Three meshes, namely the coarse, medium and

fine meshes, are used in the calculation. The coarse, medium and fine meshes are

composed of 175, 347 and 691 elements respectively.

Figure 3.11(a) shows the computed variation of the amplitude of the normalized

horizontal wave force on the block B1 with normalized wave number. It can be seen

from Figure 3.11(a) that both methods predicted almost identical horizontal wave

forces except near the resonance frequency kB=π. Figure 3.11(b) illustrates an

enlarged portion of Figure 3.11(a) near the resonance frequency. It can be seen that

the results predicted by the modified SBFEM converged on the medium mesh. The

predicted resonance frequency is kB=3.14, very close to the theoretical value of kB=π

suggested in Miao et al (2000). In contrast, the simple Green’s function method

showed considerable mesh dependence. The calculations converged rather slowly as

Chapter Three

- 45 -

can be observed from Figure 3.11(b). The predicted resonance frequency is about

kB=3.11, slightly lower than the theoretical value of kB=π.

The trends observed in Figures 3.11 are also applicable to the predicted horizontal

forces on Block 2 (Figure 3.12) and vertical forces on Block 1 (Figure 3.13) and

Block 2 (Figure 3.14). The modified SBFEM produced mesh independent results on

the medium mesh while the simple Green’s function method shows some mesh

dependence even on the fine mesh. Figure 3.15 and 3.16 plot the variation of squared

reflection and transmission coefficients calculated using both methods. The trend in

the reflection and transmission coefficients is very similar to those observed for wave

forces. Figure 3.17 shows the variations of the sum of squared reflection and

transmission coefficients calculated using both methods on different meshes. It

should be noted that the sum of squared reflection and transmission coefficients

should be unity if the calculations have converged. It can be seen that the predicted

sum of squared reflection and transmission coefficients by the modified SBFEM

departed slightly from unity near the resonance frequency on the coarse mesh but

converged to unity for all frequencies on the medium and fine meshes. In contrast, the

predicted sum of squared reflection and transmission coefficients by the simple

Green’s function method failed to converge to unity near the resonance frequency

even on the fine mesh. The comparison presented here suggests the modified SBFEM

is more accurate near the resonance frequency and is less mesh dependent than the

simple Green’s function method. This may be attributed to the fact that the modified

SBFEM does not require the discretization on the surface of bodies and truncated

boundaries and satisfies the boundary conditions on the surface of bodies and at

infinity exactly.

3.6.2 Wave radiation

Wave radiation by a two-dimensional rectangular structure in calm water of finite

depth is considered here. As with the wave diffraction problems, potential flow

theory can be applied. Based on the Cartesian co-ordinate system defined in Figure

3.4, the displacement function μ that represents harmonic motions of the structure can

be expressed as

Chapter Three

- 46 -

tAe ωμ i−= (3-95)

where A is the amplitude of the motion, ω is the motion frequency and t is the time.

The velocity of the structural surface can be found by deriving Equation (3-95) with

respect to the time t. Particular solutions bsa of scaled boundary finite-element

equation in the bounded domains (Equation (38)) can be determined by these

boundary conditions associated with the periodic motion of the floating structure.

For the heave motion, the boundary condition on the body surface is formulated as

0=∂∂

xφ

, 2Bx = and 0≤≤− zT (3-96a)

Az

ωφ i−=∂∂

, 2Bx ≤ and Tz −= (3-96b)

A particular solution is found of the following form

)( bs

bs aa ξξ = (3-97)

where ξ is the dimensionless radial co-ordinate. Substituting this particular solution

into Equation (38) yields

])[][][]([ 12110 s

Tbs fEEEEa −−−+= (3-98)

in which matrices [E0], [E1] and [E2] and the vector fs are defined as above. The

nodal flow bsq vector corresponding to the nodal potential b

sa can be found by

][,][ 10bs

Tbs

bs aEaEq += ξξ (3-99)

Likewise, for the sway motion, the boundary condition on the body surface can be

expressed as

Ax

ωφ i−=∂∂

, 2Bx = and 0≤≤− zT (3-100a)

Chapter Three

- 47 -

0=∂∂

zφ

, 2Bx ≤ and Tz −= (3-100b)

The particular solution is found to be of the same form as that for the heave motion.

For the roll motion around a rolling centre (xc, zc), the boundary condition on the body

surface can be expressed as

)(i czzAx

−−=∂∂ ωφ

, 2Bx = and 0≤≤− zT (3-101a)

)(i cxxAz

−=∂∂ ωφ

, 2Bx ≤ and Tz −= (3-101b)

Due to the occurrence of variables x and z in the Equation (3-101a) and Equation (3-

101b), the form of the particular solution is different from that in the heave and sway

motions. A particular solution can be found in the following form

)( 2 bs

bs aa ξξ = (3-102)

Substituting this particular solution into Equation (38) yields

])[][2][2][4( 12110 s

Tbs fEEEEa −−−+= (3-103)

It is noted that the solution process presented here is based on the fact that the rolling

center (xc, zc) coincides with the origin of the Cartesian co-ordinate system. If this not

the case, another particular solution of the same form as Equation (3-97) needs to be

included in Equation (3-103) to cancel the term induced by the translation of the

rolling centre.

To demonstrate the accuracy and efficiency of the SBFEM, two numerical examples

are considered in this section. In the first example (Example 3), the added mass and

damping coefficients for heave, sway and roll motions of a two-dimensional

rectangular container are computed and the numerical results are compared with the

analytical solution and the numerical results of an independent study using the

boundary element method (Zheng et al 2004). In the other example (Example 4),

Chapter Three

- 48 -

wave radiation by a two-dimensional twin-hull structure is analyzed. Numerical

results are compared with the numerical results of an independent study using

Green’s function method (Wu & Price 1987).

3.6.2.1 Wave radiation by an oscillating rectangular structure (Example 3)

The rectangular structure modelled in Figure 3.1 is investigated for the case H/T=3.0

and B/T=1.0, using the same three meshes described in Section 3.5.1.1. The following

quantities are defined for comparison purposes. For the heave and sway motions,

dimensionless added mass Ca and radiation damping Cd are defined as

BTAfCa ρω2

)Re(= (3-104)

BTAfCd ρω2

)Im(−= (3-105)

where f represents the vertical wave force in the heave motion or the horizontal wave

force in the sway motion. For the roll motion, dimensionless added mass and

radiation damping are defined as

TABmC c

a 22

)Re(2ρω

= (3-106)

TABmC c

d 23

)Im(2ρω

−= (3-107)

where mc denotes the moment about the rolling centre which is located at the origin

of the Cartesian co-ordinate system.

Figures 3.19-21 plot the variation of the added mass and damping coefficients with

the dimensionless wave number kH. The same example is also computed by Lee

(1995) and Zheng et al (2004). The analytical solutions plotted in Figures 3.19-3.21

are extracted from Zheng et al (2004). The results of the BEM in the Figure 3.21 are

obtained from Lee (1995) while those in Figure 3.20 and Figure 3.21 are obtained

from Zheng et al (2004). It can be seen from Figure 3.19 to Figure 3.21 that the

Chapter Three

- 49 -

numerical results obtained using SBFEM agree extremely well with the analytical

solution even with the coarse mesh. The SBFEM outperforms the BEM in predicting

the added mass coefficient of the roll motion as shown in Figure 3.21(a).

3.6.2.2 Wave radiation by an oscillating twin-hull structure (Example 4)

The problem considered in this example is identical to that addressed by Wu & Price

(1987). Wu & Price (1987) proposed the multiple Green’s function method (MGFM)

to remove the first irregular frequency that appeared when the problem was solved

using Green function method (GFM). The present numerical results are compared

with those obtained using both GFM and MGFM by Wu & Price (1987).

The substructured model and medium mesh in Figure 3.10 are used in this example.

For the purpose of comparisons with the results of Wu & Price (1987), the

configuration of computational model is defined as described in Wu & Price (1987).

The gap Bg between structures is equal to B. The definitions of the dimensionless

added mass and damping coefficients are consistent with those presented in Wu &

Price (1987). For the heave and sway motions, the dimensionless added mass Ca and

the radiation damping Cd are defined in the same way as in Equation (3-106) and

Equation (3-107) respectively. For the roll motion, dimensionless added mass and

radiation damping are defined as

TABmC c

a 22

)Re(ρω

= (3-108)

TABmC c

d 23

)Im(ρω

−= (3-109)

Wu & Price (1987) carried out the calculations in deep water regime with no specific

information on the water depth used. To be comparable with the calculations carried

out by Wu & Price (1987), the case of H/T=20 is employed in this study. Figures

3.22-3.24 plot the variation of the dimensionless added mass and damping

coefficients with ω2B/2g, along with the results from the GFM and MGFM. It can be

seen that the SBFEM does not suffer from the irregular frequency problem that the

GFM encountered at frequency of ω2B/2g = 1.715 (Wu & Price (1987)). Due to the

Chapter Three

- 50 -

appearance of the resonant wave in this practical problem, it was almost impossible

for the GFM to distinguish a resonant frequency from an irregular frequency. It is

also clear that the present numerical results agree very well with those obtained using

MGFM by Wu & Price (1987).

To investigate the influence of water depth on the hydrodynamic coefficients, the

present method is also applied to simulate the cases in the intermediate water

(H/T=10 and H/T=4) and in the shallow water (H/T=1.5) regimes. Figures 3.25-3.27

plot the variation of the added mass and damping coefficients with ω2B/2g. It is

observed that the change in water depth has significant effect on the hydrodynamic

characteristics of the floating structures, especially at low wave frequencies. The

heave and roll motion added mass coefficients are strongly affected by the existence

of the seabed while the sway motion added mass coefficient is not very sensitive to

the change of water depth. The damping coefficients are sensitive to the change of

water depth at low wave frequencies but are rarely affected at high wave frequencies.

This is probably because the wave energy for low frequency waves is more uniformly

distributed in water column than that for high frequency waves. Therefore the

radiation of low frequency waves around the structures is strongly affected by the gap

between the structure and the seabed. For the sway and roll motions, wave resonance

is found to take place at ω2B/2g = 1.595 for H/T=20, H/T=10 and H/T=4 while

resonance occurs at ω2B/2g = 1.558 for H/T=1.5. No resonance is observed for the

heave motion, regardless of water depth.

3.7 Summary

A modified SBFEM is developed to handle a class of wave force problems with

parallel side-faces. To overcome the inherent difficulty of the original SBFEM in

solving problems with parallel side-faces, a new type of local co-ordinate system is

proposed. The new local co-ordinate system is based on the translation of the defining

curve and eliminates the non-parallel side-face restriction in the original SBFEM. The

modified SBFEM equations are derived based a weighted residual approach. It is

found that the modified SBFEM solution retains the analytical feature in the direction

Chapter Three

- 51 -

parallel to side-faces and satisfies the boundary conditions at infinity exactly, as in

the original SBFEM.

The application of the modified SBFEM to solving wave diffraction problems by

surface fixed structures and wave radiation problems by oscillating structures in water

of finite depth in the present study demonstrates:

• The scaled boundary finite-element method is accurate and robust even with a

small number of freedoms;

• The scaled boundary finite-element method is free of the irregular frequency

problems;

• The scaled boundary finite-element method can be an excellent complementary

method to the existing numerical methods for some specific problems.

Chapter Three

- 52 -

1Ω 2Ω 3Ω iΩ 1+iΩ 2−NΩ 1−NΩ

Ns0Γ

Ns1Γ

1−NbΓ

NΩ

1bΓ

Figure 3.1. Configuration of the mathematical problem.

ibsΓ

ξ

s

scaling centre side-face

side-face

defining curve S

(a)

Figure 3.2(a). The original boundary co-ordinate system.

Chapter Three

- 53 -

x

y

),( yx)ˆ,ˆ( yx

s

ξ

side-face

Figure 3.3. The translated boundary co-ordinate system.

defining curve ∞ ΩS

1sΓ

0sΓ

∞Γ

side-face

Figure 3.2(b). The original boundary co-ordinate system.

ξ - axis

s - axis

typical scaled boundary finite-element

scaling centre

side-face

s = s0

s = s1 exterior boundary ξe =1

(b)

wΓ

ibΓ

1+ibΓ

hΓ

Chapter Three

- 54 -

(a) coarse mesh

(b) medium mesh (c) fine mesh

B

T

H defining curve

scaling centre

Figure 3.4. Substructured model and meshes for Example 1, consisting of two unbounded domains and two bounded domains.

z

x

Incident wave

Reflected wave

Transmitted wave

2Ω1Ω

Chapter Three

- 55 -

f z /ρg

HA

Figure 3.6. Vertical wave force (amplitude) for Example 1.

kB

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Coarse

Medium

Fine

Exact

T/H=0.25

T/H=0.5

T/H=0.75

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Coarse

Medium

Fine

Exact

f x /ρ

gHA

Figure 3.5. Horizontal wave force (amplitude) for Example 1.

kB

T/H=0.25

T/H=0.5

T/H=0.75

Chapter Three

- 56 -

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Coarse

Medium

Fine

Exact

K r

Figure 3.8. Reflection coefficient for Example 1.

kB

T/H=0.25

T/H=0.75

T/H=0.5

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5

Coarse

Medium

Fine

Exact

my /ρ

gHBA

Figure 3.7. Moment around y-axis (amplitude) for Example 1.

kB

T/H=0.25T/H=0.5

T/H=0.75

Chapter Three

- 57 -

K t

Figure 3.9. Transmission coefficient for Example 1.

kB

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Coarse

Medium

Fine

Exact

T/H=0.75

T/H=0.25

T/H=0.5

Chapter Three

- 58 -

(a) coarse mesh

(b) medium mesh

(c) fine mesh

B

T

H defining curve

scaling centre

z

x

Bg

B1 B2

1Ω

2Ω 3Ω 4Ω 5Ω 6Ω

B

Figure 3.10. Substructured model and meshes for Example 2, consisting of two unbounded domains and four bounded domains.

Chapter Three

- 59 -

f x /ρ

gHA

Figure 3.11(b). Horizontal wave forces (amplitude) on the block B1.

kB

-3

0

3

6

9

12

15

18

21

24

27

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse(B1)

Medium(B1)

Fine(B1)

BEM(CoarseB1)

BEM(MediumB1)

BEM(FineB1)

f x /ρ

gHA

Figure 3.11(a). Horizontal wave forces (amplitude) on the block B1.

kB

-5

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse(B1)

Medium(B1)

Fine(B1)

BEM(CoarseB1)

BEM(FineB1)

Chapter Three

- 60 -

f x /ρ

gHA

Figure 3.12(b). Horizontal wave forces (amplitude) on the block B2.

kB

-3

0

3

6

9

12

15

18

21

24

27

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse(B2)

Medium(B2)Fine(B2)

BEM(CoarseB2)BEM(MediumB2)

BEM(FineB2)

f x /ρ

gHA

Figure 3.12(a). Horizontal wave forces (amplitude) on the block B2.

kB

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse(B2)

Medium(B2)

Fine(B2)

BEM(CoarseB2)

BEM(FineB2)

Chapter Three

- 61 -

f z /ρg

BA

Figure 3.13(b). Vertical wave forces (amplitude) on the block B1.

kB

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse(B1)

Medium(B1)

Fine(B1)

BEM(CoarseB1)

BEM(MediumB1)

BEM(FineB1)

f z /ρg

BA

Figure 3.13(a). Vertical wave forces (amplitude) on the block B1.

kB

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse(B1)

Medium(B1)

Fine(B1)

BEM(CoarseB1)

BEM(FineB1)

Chapter Three

- 62 -

f z /ρg

BA

Figure 3.14(b). Vertical wave forces (amplitude) on the block B2.

kB

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse(B2)

Medium(B2)

Fine(B2)BEM(CoarseB2)

BEM(MediumB2)

BEM(FineB2)

f z /ρg

BA

Figure 3.14(a). Vertical wave forces (amplitude) on the block B2.

kB

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse(B2)

Medium(B2)

Fine(B2)

BEM(CoarseB2)

BEM(FineB2)

Chapter Three

- 63 -

K r2

Figure 3.15(b). Squared reflection coefficient for Example 2.

kB

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse

Medium

Fine

BEM(Coarse)

BEM(Medium)

BEM(Fine)

K r2

Figure 3.15(a). Squared reflection coefficient for Example 2.

kB

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse

Medium

Fine

BEM(Coarse)

BEM(Fine)

Chapter Three

- 64 -

K t2

Figure 3.16(b). Squared transmission coefficient for Example 2.

kB

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse

Medium

Fine

BEM(Coarse)

BEM(Medium)

BEM(Fine)

K t2

Figure 3.16(a). Squared transmission coefficient for Example 2.

kB

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse

Medium

Fine

BEM(Coarse)

BEM(Fine)

Chapter Three

- 65 -

Figure 3.18. Cartesian co-ordinate system and substructured computational domain.

z

x o

1Ω 2Ω 3Ω 4Ω

1bΓ

2bΓ

3bΓ

1wΓ 2wΓ

1hΓ 2hΓ

side-face

side-face

side-face

side-face

side-face H

B/2

T

B/2

K r2 +

Kt2

Figure 3.17. Summation of squared reflection and transmission coefficients for Example 2.

kB

0.90

0.95

1.00

1.05

1.10

1.15

1.20

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30

Coarse

Medium

Fine

BEM(Coarse)

BEM(Medium)

BEM(Fine)

Chapter Three

- 66 -

C d

kH

Figure 3.19(b). Dimensionless damping coefficient for a rectangular structure heaving in calm water.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

C a

kH

Figure 3.19(a). Dimensionless added mass coefficient for a rectangular structure heaving in calm water.

Chapter Three

- 67 -

C d

kH

Figure 3.20(b). Dimensionless damping coefficient for a rectangular structure swaying in calm water.

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

C a

kH

Figure 3.20(a). Dimensionless added mass coefficient for a rectangular structure swaying in calm water.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

Chapter Three

- 68 -

kH

Figure 3.21(b). Dimensionless damping coefficient for a rectangular structure rolling in calm water.

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

C d

C a

kH

Figure 3.21(a). Dimensionless added mass coefficient for a rectangular structure rolling in calm water.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14

Coarse

M edium

Fine

Analytical solution

BEM

Chapter Three

- 69 -

C d

ω2B/2g

Figure 3.22(b). Dimensionless damping coefficient for a twin-hull structure heaving in calm water.

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

C a

ω2B/2g

Figure 3.22(a). Dimensionless added mass coefficient for a twin-hull structure heaving in calm water.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

Chapter Three

- 70 -

C d

ω2B/2g

Figure 3.23(b). Dimensionless damping coefficient for a twin-hull structure swaying in calm water.

0.0

2.0

4.0

6.0

8.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

C a

ω2B/2g

Figure 3.23(a). Dimensionless added mass coefficient for a twin-hull structure swaying in calm water.

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

Chapter Three

- 71 -

C d

ω2B/2g

Figure 3.24(b). Dimensionless damping coefficient for a twin-hull structure rolling in calm water.

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

C a

ω2B/2g

Figure 3.24(a). Dimensionless added mass coefficient for a twin-hull structure rolling in calm water.

-2.0

0.0

2.0

4.0

6.0

8.0

0.0 0.5 1.0 1.5 2.0 2.5

SBFEM (H/T=20)

GFM

M GFM

Chapter Three

- 72 -

C d

ω2B/2g

Figure 3.25(b). Dimensionless damping coefficient for a twin-hull structure heaving in calm water.

0.0

4.0

8.0

12.0

16.0

20.0

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

C a

ω2B/2g

Figure 3.25(a). Dimensionless added mass coefficient for a twin-hull structure heaving in calm water.

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

Chapter Three

- 73 -

C d

ω2B/2gFigure 3.26(b). Dimensionless damping coefficient for a

twin-hull structure swaying in calm water.

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

C a

ω2B/2g

Figure 3.26(a). Dimensionless added mass coefficient for a twin-hull structure swaying in calm water.

-10.0

0.0

10.0

20.0

30.0

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

Chapter Three

- 74 -

C d

ω2B/2g

Figure 3.27(b). Dimensionless damping coefficient for a twin-hull structure rolling in calm water.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

C a

ω2B/2g

Figure 3.27(a). Dimensionless added mass coefficient for a twin-hull structure rolling in calm water.

-5.0

0.0

5.0

10.0

15.0

0.0 0.5 1.0 1.5 2.0 2.5

H/T=20

H/T=10

H/T=4

H/T=1.5

Chapter Four

- 75 -

CHAPTER 4

LINEAR SOLUTION OF THE HELMHOLTZ EQUATION

4.1 General

This chapter attempts to extend the Scaled Boundary Finite-Element Method

(SBFEM) to solve the two-dimensional Helmholtz equation. The solution for a

bounded domain is similar to that of a Laplace’s equation. However, the solution

process for an unbounded domain becomes more challenging. The reason is that the

resulting scaled boundary finite-element equation is a system of non-homogeneous

second-order ordinary differential equations. In this chapter, this system of equations

is first transformed into a system of first-order ordinary differential equations with an

irregular singularity (Wasow 1965) at infinity. The first-order ordinary differential

equation is then solved by using an asymptotic expansion method. In the asymptotic

expansion, the Sommerfeld radiation condition at infinity is enforced as rigorously as

the scalar Sommerfeld radiation condition is satisfied by the Hankel function.

This approach is applicable to two-dimensional computational domains of any shape

including unbounded domains. The accuracy and efficiency of this method are

illustrated by numerical examples of wave diffraction around vertical cylinders and

harbour oscillation problems. Computation results are compared with those obtained

using analytical methods, other conventional numerical methods and physical

experiments. It is found that the present method is completely free from the irregular

frequency difficulty that the conventional Green’s Function Method (GFM) often

encounters. It is also found that the present method does not suffer from

computational stability problems at sharp corners, is able to resolve velocity

singularities analytically at sharp structure corners by choosing the structure surfaces

Chapter Four

- 76 -

as side-faces and produces more accurate solutions than conventional numerical

methods with far less number of degrees of freedom. With these attractive attributes,

the scaled boundary finite-element method is an excellent complement to

conventional numerical methods for solving the two dimensional Helmholtz equation.


A typical wave diffraction problem shown in Figure 4.1 is considered here. Applying

the assumption of linearized wave theory, the total velocity potential ΦT may be

expressed as the summation of the incident wave potential ΦI and the scattered wave

potential ΦS. Employing the method of the separation of variables, the velocity

potential may be written as

tzZyxtzyxΦ ωφ ie)(),(),,,( −= (4-1)

where Φ denotes any one of ΦT , ΦI and ΦS , Z(z) is the corresponding water depth

function, ω is the angular frequency, t is the time and i is the imaginary unit. The

velocity potential ),( yxφ is governed by the two dimensional Helmholtz equation

022 =+∇ φφ k in domain Ω (4-2)

where k is the wave number. On the boundaries (wetted body boundary cΓ and

infinity boundary ∞Γ ) of the domain Ω , Neumann boundary condition and

Sommerfeld radiation condition may be expressed as

0=∂∂

nTφ at the body surface cΓ (4-3)

0)i(lim =−∂∂

∞→ SS

rk

rr φφ

at infinity ∞Γ (4-4)

in which n designates the unit normal to the boundary and r is the radial coordinate. It

should be noted that different unknowns are employed deliberately on boundary cΓ

and ∞Γ in Equation (4-3) and Equation (4-4). The boundary condition specified by

Chapter Four

- 77 -

Equation (4-3) will result in a homogeneous scaled boundary finite-element equation

that is convenient to solve and the boundary condition specified by Equation (4-4)

will facilitate a precise satisfaction of the radiation condition at infinity.

The scaled boundary finite-element solution of wave diffraction problems around

multiple obstacles in an unbounded domain can be constructed in a way similar to

that illustrated in Figure 4.2(a). The entire computational domain is divided into an

unbounded domain and a bounded domain with a common interface of Γb. The scaled

boundary finite-element solution of the two dimensional Helmholtz equation is

modified to seek solutions of Equation (4-2) in the unbounded domain with relevant

boundary conditions on Γb and Γ∞ and in the bounded domain with appropriate

boundary conditions on Γb and Γc. The boundary condition on Γb can be eliminated

by matching the unbounded and bounded solutions on Γb.

4.3 Scaled boundary finite-element equation

As illustrated in Figure 4.2 (a), the computational domain can be divided into a

number of subdomains, denoted as )1,,3,2,1( += NiΩi L . A typical subdomain

( iΩ in Figure 4.2(a)) has a scaling centre (solid point in Figure 4.2(b)) and is

bounded by two side faces (s0 and s1 in Figure 4.2(b)) or Γci, Γbi, Γi and Γi+1. The

scaled boundary finite-element solution in the bounded domain can be obtained by

matching the solutions of Equation (4-2) on internal boundaries Γi (i=1 to N). The

scaled boundary finite-element solution in a typical subdomain iΩ is presented here

and the solutions in the other subdomains can be obtained using the same approach.

Suppose iΩΓ is comprised of Γbi , Γi and Γi+1. The total velocity potential on

iΩΓ

satisfies

nT vn

−=∂∂φ

on iΩΓ (4-5)

Chapter Four

- 78 -

Where nv is the velocity component outward normal to the boundary iΩΓ . Applying

the weighted residual approach to Equation (4-2) and Equation (4-3) and employing

Green’s identity results in

02 =−−∇∇ ∫∫∫iii

dvwdwkdw nTTT

ΩΓΩΩΓΩφΩφ (4-6)

where w is a weighting function.

To apply the SBFEM to the solution to Equation (4-6), a so-called scaled boundary

co-ordinate system needs to be introduced. A typical scaled boundary co-ordinate

system is shown in Figure 4.2(b). For the case of bounded domain, the scaled radial

coordinate ξ is 1 on ξ=ξe ( 1+∪∪ iibi ΓΓΓ ) and zero at the scaling centre.

Consequently, each value of ξ defines a scaled version of the curve S. The

circumferential co-ordinate s measures the distance anticlockwise around a defining

curve S. The bounded domain iΩ is defined as the region enclosed by 0≤ξ≤1 and

s0≤s≤s1. For the case of unbounded domain, the scaled radial coordinate ξ equals 1 at

the boundary 1+∪∪ iibi ΓΓΓ and ξe tends to infinity (See Deeks & Cheng 2003 for

more detail). The defining curve S can be discretized by shape functions [N(s)] in the

classical finite-element manner. Thus, an approximate solution ),( sh ξφ to Equation

(4-6) is sought in the form

)()]([),( ξξφ asNsh = (4-7)

where vector )( ξa represents radial nodal functions analogous to nodal values in

the standard finite element method. At each node i the function )(ξia designates the

variation of the total potential in the radial direction. Since no discretization is carried

out in the radial direction, the scaled boundary finite-element method keeps the radial

solution )( ξa analytical. Using the scaled boundary transformation detailed by

Deeks & Cheng (2003) [6], the Laplace operator ∇ can be expressed as

ssbsb

∂∂

+∂∂

=∇ )(1)( 21 ξξ (4-8)

Chapter Four

- 79 -

where )( 1 sb and )( 2 sb are vectors that are only dependent on the definition of

the curve S. The approximate velocity vector ),( svh ξ can be formulated as

)()]([1),()]([),( 21 ξξ

ξξ ξ asBasBsvh += (4-9)

where, for convenience,

)]()[()]([ 11 sNsbsB = (4-10)

ssNsbsB )],()[()]([ 22 = (4-11)

Applying the Galerkin approach, the weighting function w can be formulated

employing the same shape function as the approximation for the potential (Equation

(4-7)).

TT sNwwsNsw )]([)()()]([),( ξξξ == (4-12)

Substituting Equation (4-7), Equation (4-8) and Equation (4-12) into (4-6),

integrating all terms containing ξξ ),(w by parts with respect to ξ , using Green’s

identity, introducing ξτ to transform infinitesimal lengths on the boundary sections

with constant ξ to the scaled boundary coordinate system, introducing the following

coefficient matrices for convenience

∫= s

T dsJsBsBE )]([)]([][ 110 (4-13)

∫= s

T dsJsBsBE )]([)]([][ 121 (4-14)

∫= s

T dsJsBsBE )]([)]([][ 222 (4-15)

∫= s

T dsJsNsNM )]([)]([][ 0 (4-16)

and collecting common terms yield

Chapter Four

- 80 -

0))(][)(1][

)(])[][]([)(]([)(

))]([)(][)(]([)(

))]([)(][)(]([)(

02

2

1100

10

10

=+−

−++−

++−

−+

∫∫∫

ξξξξξ

ξξξξ

τξξξξ

τξξξξ

ξ

ξ ξξξ

ξξ

ξξ

daMkaE

,aEEE,aEw

dsvsNaE,aEw

dsvsNaE,aEw

e

i

TT

s nT

iT

iiT

i

s nT

eT

eeT

e

(4-17)

The following conditions hold if Equation (4-17) is satisfied for all sets of weighting

functions )( ξw .

∫ −=+s n

Ti

Tii dsvsNaEaE ξ

ξ τξξξ ))(()(][),(][ 10 (4-18)

∫=+s n

Te

Tee dsvsNaEaE ξ

ξ τξξξ )]([)(][),(][ 10 (4-19)

0)(][

)(][),(])[][]([),(][

022

21102

0

=+

−−++

ξξ

ξξξξξ ξξξ

aMk

aEaEEEaE T

(4-20)

Equation (4-18) and Equation (4-19) indicate the relationships between the nodal

potential and the integrated nodal flow on the boundary with constant radial co-

ordinate ξ. For the case of bounded domain considered in Figure 4.2(b), ξe represents

the sum of Γbi Γi and Γi+1 and ξi is actually the scaling centre with ξi=0. For the case

of unbounded domain, ξi represents the sum of Γbi Γi and Γi+1 and ξe tends to infinity.

Equation (4-20) is the scaled boundary finite-element equation. It can be seen that

Equation (4-20) is a homogeneous second-order ordinary differential equation, due to

the use of the total velocity potential as an unknown in the bounded domain.


4.4.1 Bounded domain solution

Song and Wolf [23] presented the solution to Equation (4-20) for elasto-dynamic

problems in a bounded domain. However, this solution is based on a specific property

of rigid body movements in solid mechanics that is not available for the wave

Chapter Four

- 81 -

diffraction problem considered in this study. Therefore the application of this solution

to the problem investigated here is not straightforward. An improved method has

been proposed in this study to obtain an analytical solution to Equation (4-20).

To solve Equation (4-20), a transformation of Equation (4-20) to a set of first-order

ordinary differential equations of order 2n can be performed (Song & Wolf 1998).

Considering the satisfaction of Equation (4-18) and Equation (4-19), no generality is

lost by assuming

)(][),(][)( 10 ξξξξ ξ aEaEq T+= (4-21)

Combining Equation (4-20) with Equation (4-21) and introducing the independent

variables

ξξ k= (4-22)

and

)]([)()(

)( cXqa

X ξξξ

ξ =⎭⎬⎫

⎩⎨⎧

= (4-23)

result in

)](][[)](][[)],([ 2 ξξξξξ ξ XMXZX −−= (4-24)


⎥⎦

⎤⎢⎣

⎡

−+−−

=−−

−−

1011

1012

101

10

]][[][]][[][][][][

][EEEEEE

EEEZ T

T

(4-25)

and

⎥⎦

⎤⎢⎣

⎡=

0][00

][0M

M (4-26)

Chapter Four

- 82 -

For the Hamiltonian matrix [Z], the eigenvalues consist of two groups of values with

opposite signs

⎥⎦

⎤⎢⎣

⎡−

=][0

0][][

j

j

λλ

Λ , nj ,,2,1 L= (4-27)

where Re(λj)≥0. The corresponding eigenvalue problem can be formulated as

]][[]][[ ΛΦΦZ = (4-28)

where [Φ] denotes the eigenvector matrix.

Based on the physical meaning of Equation (4-27) and Equation (4-28), zero

eigenvalues appear in pairs in Equation (4-27). A zero eigenvalue leads to a constant

potential in the entire fluid domain. This means that there is no flow in the fluid

domain. Thus the eigenvectors of zero eigenvalues are not linearly independent any

more, leading to a singular matrix [Φ]. The method proposed by Song and Wolf

(1998) to solve Equation (4-24) is not applicable here, because the eigenvectors of

zero eigenvalue are linearly independent for the solid mechanics problems

investigated. Therefore, to obtain the solution for wave problems, Jordan

decomposition of the matrix [Z] is performed, namely,

]][[]][[ ΛTTZ = (4-29)

with

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⎥⎦

⎤⎢⎣

⎡=

][0010

][

][

j

j

λ

λ

Λ , 1,,2,1 −= nj L (4-30)

in which the transform matrix [T] is invertible. Although programming of Jordan

decomposition for general matrix could be tedious and unstable, Jordan

decomposition of the matrix [Z] does not reduce the efficiency of the computer

program too much when advantage is taken of the simple structural form of the

Chapter Four

- 83 -

matrix [Λ] and completely known eigenvalues. In fact, based on the eigenvector

matrix [Φ], the matrix [Z] can be easily formed using the characteristic of the Jordan

chain.

Following the procedures provided by Gantmacher (1959), through a series of matrix

transformations and the solution of a system of recursion equations, the analytical

solution of Equation (4-24) may be expressed as

][][)](][[)]([ URTX ξξξξ Λ= (4-31)

where the matrix )]([ ξR is formulated as a power series in ξ with a leading identity

matrix [I]

LL +++++= ][][][][)]([ 22

41

20 m

m RRRRR ξξξξ (4-32)

with

][][ 0 IR = (4-33)

and the matrix [Λ] is an upper triangular matrix with eigenvalues being on the

diagonal entries and [U] is also an upper triangular matrix with zero diagonal entries.

For convenience, the matrix function ][Uξ may be written as

)]([][ ξξ YU = (4-34)

Partitioning all the matrices on the right hand side of Equation (4-31) and constant

vector c into block matrix with nn× dimensions and block vector with

1×n dimensions respectively, then substituting Equation (4-31) and Equation (4-34)

into Equation (4-23) yields

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡×

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

−

)]([)]([)]([)]([

0

)]([)]([)]([)]([

][][][][

)(

2

1

2221

1211][

][][

2221

1211

2221

1211

cc

YYYY

RRRR

TTTT

X

P

ξξξξ

ξξξ

ξξξξ

ξ

λ

λ (4-35)

Chapter Four

- 84 -

To obtain a finite solution for 0=ξ , 2c must be zero. For brevity, introducing

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

)]([)]([)]([)]([

][][][][

)]([)]([)]([)]([

2221

1211

2221

1211

2221

1211

ξξξξ

ξξξξ

RRRR

TTTT

KKKK

(4-36)

into Equation (4-35) yields

)]([)( 1cAa ξξ = (4-37)

)]([)( 1cQq ξξ = (4-38)

with

)]([)]([)]([ 111][

11 cYKA ξξξξ λ= (4-39)

)]([)]([)]([ 111][

21 cYKQ ξξξξ λ= (4-40)

Eliminating the constant vector 1c in Equation (4-37) and Equation (4-38) leads to

)()]([)( ξξξ aHq b= (4-41)

with

1)]()][([)]([ −= ξξξ AQH b (4-42)

Once the nodal potential vector )( ξa is obtained, the integration constant vector

1c can be determined as

)()]([ 11 ξξ aAc −= (4-43)

With the known integration constant vector 1c , the entire potential field can be

found by substituting Equation (4-37) into Equation (4-7). The velocity field can be

determined by substituting the integration constant vector 1c into Equation (4-38)

to determine the nodal flow vector )( ξq , then employing Equation (4-21) to derive

Chapter Four

- 85 -

the derivative of the nodal potential vector )( ξa with respect to the radial co-

ordinate ξ and substituting the derivative into Equation (4-9).

4.4.2 Unbounded domain solution

For an unbounded domain ( ∞≤≤ ξ1 ), Equation (4-24) is more difficult to solve

because it has an irregular singular point at ∞=ξ , and all eigenvalues of the matrix

][M are zero. In solid mechanics problems, a high-frequency asymptotic expansion

of the dynamic stiffness can be constructed using the radiation condition (Wolf 2003).

The displacements obtained from the high-frequency asymptotic expansion method

outlined do not satisfy the Sommerfeld radiation condition automatically in the same

way as the Hankel function does for the scalar case. Therefore, the solution procedure

for an unbounded domain provided by Song & Wolf (1998) can not be applied

directly to the problem at hand.

In the work discussed here, an asymptotic expansion for the scatted wave potential as

∞→ξ is obtained using procedures suggested by Wasow (1965). The resulting

solution is able to satisfy the Sommerfeld radiation condition automatically. This is

one of the major contributions of the current work. Due to its generality, the

asymptotic expansion presented here is also valid in the solid mechanics problems.

Since the derivation of the procedure is rather long, only a summary is presented

here. For full details the reader should refer to the text by Wasow (1965).

Equation (4-24) can be re-written as

)](]][[][[)],([ 21 ξξξξ ξ XZMX −− −−= (4-44)

Since the eigenvalues of the square matrix ][M are all zero, it is not possible to

obtain an asymptotic expansion for )]([ ξX directly. Instead a transformation is

introduced so that )]([ ξX is replaced by )]([ ξG , where

)]()][(][[)]([ ξξξ GPTX = (4-45)

Chapter Four

- 86 -

with

∑∞

=

−+=1

][][)]([m

mmPIP ξξ (4-46)

The coefficient matrices ][ mP and ][T are selected so that the transformed

differential equation (4-44) becomes

)]()][([)],([1 ξξξξ ξ GBG =− (4-47)

where the matrix function )]([ ξB may be expressed as

∑∞

=

−=0

][)]([m

mmBB ξξ (4-48)

and each of the coefficient matrices ][ mB in this expansion are block diagonal.

Insertion of Equation (4-46) and Equation (4-48) and rearrangement, according to the

same power of ξ , leads to recursion formulae that allow determination of these

coefficient matrices.

With a view to obtaining a solution that can be interpreted as propagating waves,

analogous to the Hankel functions for the scalar case, a shearing transformation of the

form

)]()][([)]([ ξξξ USG = (4-49)

is introduced, where

)]([)]([)]([)]([)]([ 10 ξξξξξ nj SSSSS ⊕⊕⊕⊕= LL (4-50)

with

⎥⎦

⎤⎢⎣

⎡=

−1001

)]([ξ

ξjS (4-51)

Chapter Four

- 87 -

When Equation (4-49) is substituted into Equation (4-47) the system of differential

equations becomes

)]()][([)],([ ξξξ ξ UCU = (4-52)

where )]([ ξC has an asymptotic expansion of the form

∑∞

=

−=0

][)]([m

mmCC ξξ (4-53)

The Hamiltonian form of the matrix ][Z ensures that ][ 0C can be expressed as

]][[][][ 10 QQC Λ= − (4-54)

with the diagonal eigenvalue matrix ][Λ and the eigenvector matrix ][Q . The

eigenvalues of ][Λ consist of two groups with opposite signs

⎥⎦

⎤⎢⎣

⎡−

=][

][][

j

jΛλ

λ, nj ,,2,1 L= (4-55)

with 0)Re( =jλ and 0)Im( >jλ , and the eigenvalues with the same sign in the

matrix ][Λ satisfy λλλλ ==== nL21 , which has been rigorously testified by

Wasow (1965).

This suggests a further transformation of the form

)]()][(][[)]([ ξξξ YPQU = (4-56)

where

∑∞

=

−+=1

][][)]([m

mmPIP ξξ (4-57)


Chapter Four

- 88 -

)]()][([)],([ ξξξ ξ YBY = (4-58)

The matrix )]([ ξB can be expressed as

∑∞

=

−+=1

][][)]([m

mmBΛB ξξ (4-59)

Substituting Equations (4-56), (4-57) and (4-59) into Equation (4-58) again leads to a

recursive system through which the coefficient matrices ][ mP and ][ mB , which are

block diagonal, can be computed. Equation (4-58) can be solved. The approach is

employed first by Sibuya (1958) for asymptotic expansions in terms of a parameter

and is outlined by Wasow (1965). A particular solution of the final matrix equation

(Equation (4-58)) can be expressed as

ξΛξξξ ][][ e)]([)]([ 1BZY = (4-60)

with

)]([)]([ ξξ GeZ = and ∑∞

=

−

−=

2

1

])[1

()]([m

m

m

Bm

G ξξ (4-61)

Substituting Equations (4-45), (4-49), (4-56) and (4-60) into Equation (4-23) results

in

e)]([)( ][][ 1 cYX B ξΛξξξ = (4-62)

with

)]()][(][)][()][(][[)]([ ξξξξξ ZPQSPTY = (4-63)

Partitioning the matrices )]([ ξY and ][ 1B in Equation (4-62) into square submatrices

of order nn × in the same way as the eigenvalues in Equation (4-55) are partitioned

results in

Chapter Four

- 89 -

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

−

−

e)]([e)]([

e)]([e)]([)()(

2

1

][][22][][21

][][12][][11

221

111

221

111

cc

YY

YYqa

ii

ii

BB

BB

ξλξλ

ξλξλ

ξξξξ

ξξξξξξ

(4-64)

with the integration constants partitioned similarly as

⎭⎬⎫

⎩⎨⎧

=

2

1

cc

c (4-65)

Equation (4-64) is the general solution. To determine the constants c , the radiation

condition and the body boundary condition must be considered.

For the radiation condition, it is required that there is no return of the scatted waves

from infinity. For convenience, the solution of the velocity potential is written as

)]()][([)]()][([)()]([ 2211 cAsNcAsNasN ξξξφ +== (4-66)

with

ξλξξξ ][][111 e)]([)]([

111 iBYA = (4-67)

ξλξξξ ][][122 e)]([)]([

221 iBYA −= (4-68)

Inspection of the exponential terms of )]([ 1 ξA and )]([ 2 ξA indicates that

)]()][([ 11 cAsN ξ represents the outgoing wave potential while

)]()][([ 22 cAsN ξ represents the incoming wave potential. Since the diffracted

waves must satisfy the Sommerfeld radiation condition, the returning wave potential

must vanish, and so 0 2 =c . In fact )]([ 1 ξA and )]([ 2 ξA in the matrix case are

similar to the Hankel functions of the first kind and the second kind respectively in

the structure of the series, and )]([ 1 ξA satisfies the Sommerfeld radiation condition

automatically in the same way as the Hankel function does in the scalar form.

Chapter Four

- 90 -

Since 0 2 =c , the top section of Equation (4-64) determines the set of integration

constants 1c corresponding to any set of nodal potentials as

)()]([ 111 ii aAc ξξ −= (4-69)

while the bottom section of Equation (4-64) then relates the integrated nodal flows to

the nodal potentials as

)()]([)()]()][([)( 111 iiiiii aHaAQq ξξξξξξ ∞− == (4-70)

4.5 Assembly of subdomains

From this point onwards, variables associated with the bounded domain and the

unbounded domain solutions are identified with subscript b and ∞ respectively. The

assembly of the nodal potential and the nodal flow of bounded domains into a global

equation results in

)()())()()](([ ξξξξξ bS

bI

bS

bI

b qqaaH +=+ (4-71)

where the total nodal potential is the sum of incident and scattered potentials and

nodal flow is expressed as the sum of nodal flows induced by the incident and

scattered waves respectively, with subscript I indicating the incident wave and

subscript S the scattered wave. At the interfaces between two adjacent subdomains,

the nodal flows are equal in magnitude but of opposite signs. Therefore the

superposed nodal flow at all interfaces of subdomains is zero.

For the unbounded domain, the resulting equation may be expressed as

)()()]([ ξξξ ∞∞∞ = SS qaH (4-72)

Since the exterior boundary of the bounded domain is the interior boundary of the

unbounded domain, the nodal flow at the interface between the bounded domain and

the unbounded domain are equal in magnitude but opposite in sign. Matching the

Chapter Four

- 91 -

nodal potential of the scattered wave at the interface and assembling Equation (4-71)

and Equation (4-72) into a common global equation yields

)()]([)()()])([)](([ ξξξξξξ bI

bbI

bS

b aHqaHH −=− ∞ (4-73)

which is a linear system of equations in terms of the only unknown )( ξbSa . Once

the nodal potential vector )( ξbSa of the scattered wave is obtained through Equation

(4-73), the total nodal potential vector for the bounded domain is found. Substituting

the resulting total nodal potential vector into Equation (4-43) to obtain the integral

constant vector and using Equation (4-7) and Equation (4-9), the entire potential field

and velocity field can be determined.


To validate the present method, some typical numerical examples regarding wave

diffraction by objects of various geometric cross sections and harbour oscillations

excited by water waves are calculated in this section. The first example is related to

wave diffraction by a vertical circular cylinder. This is a classical problem for which

an analytical solution is available. The SBFEM results are compared with this

analytical solution, the finite/infinite-element solution and boundary-element solution

to examine its accuracy and convergence. A second example is then is addressed to

demonstrate the ability of the method to solve problems of wave diffraction by an

elliptical cylinder. The third and fourth examples are concerned with wave diffraction

by a single square cylinder and twin caissons with a small gap respectively. The last

two examples deal with harbour oscillation problems.

4.6.1 Wave diffraction by piercing-surface structures

4.6.1.1 Wave diffraction by a circular cylinder (Example 1)

To demonstrate the accuracy of the method, as the first example, the analytical

solution of wave diffraction by a circular cylinder in water of a constant depth,

developed by MacCamy & Fuchs (1954), is compared with the scaled boundary finite

element solution presented in this paper. If the incident wave is taken to come in from

Chapter Four

- 92 -

left to right, the analytical solution of the velocity potential of the scattered wave can

be expressed as

∑∞

=

+−=

0

cos)(icosh

)(coshi),,(m

mmm

ms mkrHAkh

hzkgAzr θεω

θφ (4-74)

with

)()(

kaHkaJA

m

mm ′

′−= (4-75)

and

⎩⎨⎧

≥=

=)1(2)0(1

mm

mε (4-76)

where a is the radius of the circular cylinder, mJ is the Bessel function of the first

kind and mH is the Hankel function of the first kind with order m. Correspondingly,

the total wave elevation ( 0=z ) can be calculated as

∑∞

= ′′

−=0

cos)]()()(

)([i),(m

mm

mm

mm mkrH

kaHkaJ

krJAr θεθη (4-77)

Taking into account the symmetry of the problem, a half of the circular cylinder is

illustrated in Figure 4.3. The boundary of the cylinder is discretized with three-noded

quadratic elements. In order to illustrate the dependence of the method on the number

of elements used in the calculation, a half of the cylinder is discretized with three

meshes. The coarse mesh is composed of 8 elements, the intermediate mesh consists

of 12 elements and the fine mesh has 16 elements. The three meshes are as shown in

Figure 4.4. Scaling centre is placed on the origin of the coordinate system.

In this example, waves of unit amplitude are considered incident at the angle of 0° to

the x-axis and the dimensionless frequency of the wave (ka) is taken to be 2. The

number of terms of the analytical series solution (Equation (4-74) and Equation (4-

Chapter Four

- 93 -

77)) computed here is 513. The number of the terms of the series solution addressed

in this paper is 15.

Along with the analytical solution, the real and the imaginary parts of the scattered

wave elevation are plotted in Figures 4.5(a) and 4.5(b) respectively and those of the

total wave elevation are illustrated in Figures 4.6(a) and 4.6(b), respectively. It can be

seen from the figures that the scaled boundary finite-element solution shows excellent

agreement with the analytical solution even on the coarse mesh of eight elements.

The finite element results using 16 finite and infinite elements for the whole cylinder

(8 quadrilateral elements for a half of the cylinder) presented by Bettess &

Zienkiewicz (1977) for the same problem are also given in Figure 4.5 and Figure 4.6

correspondingly. It can be seen from Figure 4.5 and Figure 4.6 that the scaled

boundary finite element method gives better prediction than the finite element method

with the same number of elements.

The boundary element results for the same problem given by Au & Brebbia (1984)

are reproduced in Figures 4.6(a) and 4.6(b) for the purpose of comparison. In the

boundary element solution, 40 elements were employed in terms of symmetry (a half

of the cylinder was used). It can be seen from Figures 4.6(a) and 4.6(b) that the

present method gives much better results than the boundary element method with

fewer elements.

Figures 4.7(a) and 4.7(b) show the distributions of the tangential velocity (real and

imaginary parts) along the surface of the cylinder, together with the corresponding

analytical solutions. It can be seen that the numerical solutions by the scaled

boundary finite element method again agree very well with the analytical solution.

Figures 4.8(a) and 4.8(b) give the contour plots of wave elevation in the deep water

for a dimensionless wave number of 4. The coarse mesh is used here to generate the

contour plots. It can be seen from Figures 4.8(a) and 4.8(b) that the effect of scattered

wave decreases with the increasing distance from the surface of cylinder. The vector

plots of velocity at the water surface are given in Figures 4.9(a) and 4.9(b).

Table 1 lists the dimensionless total horizontal wave force and total wave moment on

the whole circular cylinder computed by the scaled boundary finite-element method

Chapter Four

- 94 -

and those obtained using the analytical solution. It can be seen from the Table 1 that

the scaled boundary finite element solutions converge towards the analytical solution

as the mesh is refined and compare very well with the analytical solution even with

the coarse mesh. The variations of the amplitudes of dimensionless wave force and

wave moment with the increasing wave number and water depth are plotted in Figure

4.10(a) and Figure 4.10(b) respectively. Again, the results agree very well with the

analytical solution for the range of frequency and water depth considered. No sign of

irregular frequency problems in the scaled boundary finite element solutions

presented here.

4.6.1.2 Wave diffraction by an elliptical cylinder with an aspect ratio of 2:1 (Example

2)

Wave diffraction by an elliptical cylinder with an aspect ration of 2:1 is also modelled

in this study. The incident wave is 30o to the principle axis of the elliptical cylinder as

shown in Figure 4.11. The boundary of the cylinder is still discretized with three-

noded quadratic scaled boundary finite elements. The coarse mesh consists of 8

elements, the intermediate mesh 16 elements and the fine mesh 24 elements, as

shown in Figure 4.12.

Figures 4.13(a) and 4.13(b) plot the results of the real and the imaginary components

of total wave elevation around the cylinder, together with the infinite element method

results by Bettess & Bettess (1998). Bettess & Bettess (1998) employed a mapped

infinite wave element method. The computational mesh used by Bettess & Bettess

(1998) was composed of two concentric circles, containing 2 radial elements and 24

circumferential elements. It can be seen from Figures 4.13(a) and 4.13(b) that the

scaled boundary finite-element method with the intermediate mesh (12 elements)

produced identical results to those by the infinite element method with a total of 48

elements (Bettess & Bettess 1998).

4.6.1.3 Wave diffraction by an elliptical cylinder with an aspect ratio of 4:1 (Example

3)

The third example involves an elliptical cylinder of an aspect ratio of 4:1. The scaled

boundary finite element solutions are again compared with the results using the

Chapter Four

- 95 -

infinite element method reported by Bettess & Bettess (1998). The meshes and other

setups used in this example are the same as in the previous example for both the

present study and the study by Bettess & Bettess (1998) except that aspect ratio is 4:1

in this example as shown in Figure 4.14. The scaled boundary finite element meshes

used in this example are as shown in Figure 4.15.

Figures 4.16(a) and 4.16(b) illustrate the results of the real and the imaginary

components of total wave elevation around the cylinder, compared with the results of

the infinite element method (Bettess & Bettess 1998). Again the results produced by

the scaled boundary finite element method on the intermediate mesh agree very well

with the results from the infinite element method, although some noticeable

difference exists between the results obtained with the fine mesh and the intermediate

mesh in the scaled boundary element method. This difference is mainly because the

size of the scaled boundary finite elements increases as the aspect ratio increases.

4.6.1.4 Wave diffraction by a square cylinder (Example 4)

Wave diffraction by a square cylinder is now considered. It is assumed that the

incident wave travels in the same direction of the x-axis (Figure 4.17(a)). The

symmetry of the problem allows a half of the cylinder with a unit width to be

modelled. The entire fluid domain is divided into one unbounded subdomain and two

bounded subdomains as illustrated in Figure 4.17(a). The scaling centres for the

bounded domains are placed at the two corners and the scaling centre for the

unbounded domain is located at the centre of the cylinder. The side-faces for the

bounded and unbounded domains are identified in Figure 4.17(a) and the side faces

are not discretized in the solution process. Other boundaries are discretized with

three-node quadratic scaled boundary finite elements. Three different meshes are

employed, as shown in Figure 4.17. In the coarse mesh for this example, 8 elements

are used to discretize the unbounded domain and 6 elements in each of the bounded

domains. The medium mesh consists of 12 elements for the unbounded domain and

10 elements for each of the bounded domains. The fine mesh is composed of 16

elements for the unbounded domain and 14 elements for each of the bounded

domains. A semicircle as shown in Figure 4.17 is taken as the inner boundary of the

unbounded domain.

Chapter Four

- 96 -

In the conventional boundary-element method, the boundary of the computational

field is discretized by two meshes, comprising 16 and 800 three-node quadratic

elements respectively. Through a number of numerical tests, it is found that 16

elements is essential for the boundary element method to provide convergent results

and 800 elements is needed to produce results of comparable accuracy with the

present scaled boundary finite-element method.

Figures 4.18(a) and 4.18(b) plot the real part and imaginary part of variations of wave

elevation along the surface of the single cylinder correspondingly, together with the

results obtained using the boundary-element method (16 elements). It can be seen

from Figures 4.18(a) and 4.18(b) that both methods predicted identical wave

elevations on the cylinder surface. The numerical results on surface elevation are

insensitive to the number of elements used in the calculations. Both methods achieved

mesh independent results with only few elements. The real part and imaginary part of

tangential velocity are plotted in Figures 4.19(a) and 4.19(b) correspondingly. It can

be seen from Figures 4.19(a) and 4.19(b) that tangential velocity is singular at the two

corners of the cylinder (45o and 135o). As expected, the present method predicts this

singularity extremely well. Even on the coarse mesh, the present approach provides

an excellent approximation to the singular velocity at the corners. In contrast, the

conventional boundary element method failed to predict the singular velocity at the

corners, even with an extremely dense mesh up to 800 elements.

Figure 4.20 plots the variation of normalized amplitude of horizontal wave force on

the single square cylinder with the normalized wave number kL. The conventional

boundary-element method, in which 16 elements are used, provides a good agreement

with the scaled boundary finite-element method (medium mesh), except near kL=

π5 where a sharp force increase is observed. It is understood that kL= π5

corresponds to the well-known irregular frequency for the conventional boundary

element method.

4.6.1.5 Wave diffraction by twin caissons with a small gap (Example 5)

The fifth example considered in this study involves wave diffraction by a pair of

caissons with a small gap. The incident wave considered in this example is identical

Chapter Four

- 97 -

to that in Example 4. The configuration of the caissons is shown in Figure 4.21. The

breath B is a constant of 2. The width of the gap (d) is 0.01B and the length of gap (L)

is taken as 2 in the analysis of the potential and velocity field under a fixed wave

number (k=2). Noting the symmetry of problem, only a half of the caissons is

modeled. The entire computational domain is combined with one unbounded

subdomain and four bounded subdomains as illustrated in Figure 4.22. A relatively

coarse mesh with 8 three-node quadratic elements for the discretization of unbounded

subdomain and 5 three-node quadratic elements for one bounded subdomain is used

in the simulations. Two meshes with 32 elements and 1600 elements are used to

model this problem using the conventional boundary element method.

The real part and imaginary part of the computed variation of wave elevation along

the surface of the twin caissons are plotted in Figures 4.23(a) and 4.23(b),

respectively. The boundary-elements method with 32 elements produces an excellent

agreement with the scaled boundary finite-element in computing wave elevations.

Figures 4.24(a) and 4.24(b) plot the real part and imaginary part of computed

variation of tangential velocity along the surface of the twin caissons respectively.

Again, the scaled boundary finite-element method demonstrates its ability to model

singularities and discontinuities.

Predicted normalized horizontal wave force on the caissons is given in Figure 4.25. It

can be seen from Figure 4.25 that there is a sharp increase of wave force at around kL

= 3.1. This sharp increase in the wave force was found to be due to a resonant

phenomenon (Miao et al 2001). It can be seen that both scaled boundary finite-

element method and the conventional boundary element method predicted the sharp

force increase at around kL = 3.1 quite well. However the boundary element method

also predicted two sharp increases in the wave force at around kL = π2 and π5

while the scaled boundary finite-element method did not predict these two peaks at

all. Further investigations indicate that the peaks predicted by the boundary element

method at kL = π2 and π5 are not physical and are purely due to the irregular

frequency problems of the boundary element method. Based on the approach

suggested by Lee & Sclavounos (1989), it was predicted that irregular frequencies

will appear at kL = π2 and π5 for the problem considered in this study using the

Chapter Four

- 98 -

boundary element method. It is interesting to see that the magnitude of wave force

increase caused by the irregular frequency at kL = π5 is in a similar magnitude to

the force increase by resonance at kL = 3.1. It would have been difficult to

distinguish physical and unphysical sharp wave force increases using the boundary

element method for more complicated problems.

4.6.2 Harbour oscillations

4.6.2.1 Wave diffraction by a rectangular narrow bay (Example 6)

Problems of harbour oscillations excited by incident waves are also examined in this

study. As shown in Figure 4.26, a rectangular narrow bay with constant depth of

water is subjected to an incident wave train travelling in the direction of the positive

x-axis. Due to the symmetry of the problem, only a half of the bay is computed here.

The width of the bay is 6m, the length is 31m and the depth of water is 25.73m. Point

C is the centre of the back wall. This problem has been calculated by many numerical

methods (Ippen & Goda 1963, Madsen & Larsen 1987, Zhao & Teng 2004) and these

numerical solutions were compared with experimental data (Ippen & Goda 1963, Lee

1971). Zhao & Teng (2004) combined the finite-element solution for inner of the bay

with the boundary-element solution for outer region of the bay to compute this

problem. The computational domain in the inner bay is discretized by 186 four-node

rectangular finite elements. In contrast, the scaled boundary finite-element method

only needs solve a one-dimensional problem. The scaled boundary finite-element

method uses 12 three-node quadratic elements for the discretization of boundary and

the computational domain is divided into one bounded domain and one unbounded

domain, as shown in Figure 4.27. Figure 4.28 plots the variation of dimensionless

wave elevation at the point C with the dimensionless wave number, along with the

other numerical and experimental results. As is apparent, the scaled boundary finite-

element solution compares well with other results, even though very few elements are

employed.

4.6.2.2 Wave diffraction by a square harbour with two straight breakwaters

(Example 7)

Chapter Four

- 99 -

A more complicated example for a harbour oscillation problem is also illustrated.

Wave diffraction into a square harbour with two straight breakwaters as shown in

Figure 4.29 is simulated using the present scaled boundary finite-element method.

The constant depth of water is taken to be a/10. The dimensionless wave number ka

is set at 30. Zhao & Teng (2004) used finite-element methods to calculate this

problem and provided boundary-element solutions for comparisons. The domain in

the harbour was discretized by 10000 four-node rectangular finite elements in Zhao &

Teng (2004). In the scaled boundary finite-element method, only 112 3-node

quadratic elements are used (Figure 4.30). The computational domain is substructured

into ten bounded subdomains (from Ω1 to Ω10 ) and one unbounded subdomain (Ω11).

Thus, only 24 elements are employed for the discretization of boundary of every

bounded subdomain and 16 elements for that of unbounded subdomain. It is worth

noting that no discretization is required on the wall and the centre line (x-axis) as the

side-faces coincide with these boundaries. Therefore, the solutions along the wall and

the centerline remain semi-analytical. Figures 4.31(a), (b) and (c) plot the distribution

of wave elevation along the wall and the centerline. It is found the scaled boundary

finite-element solution agrees extremely well with the solutions obtained by the FEM

and the BEM.

4.7 Summary

A scaled boundary finite element solution is developed for the two-dimensional

Helmholtz equation. The scaled boundary finite element formulation for the two-

dimensional Helmholtz equation is derived based on a weighted residual approach. A

complex solution procedure for a system of second order ordinary differential

equations is proposed and implemented. Also, this newly developed semi-analytical

technique is applied to the solution of wave diffraction by structures with various

geometric cross sections and harbour oscillations excited by water waves. The

comparisons of the SBFEM with other solution methods demonstrate that the SBFEM

is accurate. The method does not suffer from the difficulties often encountered by

BEM/GFM such as irregular frequency and sharp corner problems.

Chapter Four

- 100 -

Scaling centre for bounded domain

Scaling centre for unbounded domain

Unbounded domain

Typical bounded domain

Side-faces

s-axis

ξ-axis Typical scaled boundary finite element

S

O

Figure 4.2. Substructuring configuration and scaled boundary co-ordinate definition.

ξ-axis

cΓ

ξ=ξe

s=s0

s=s1

iΩ

(a) Subdomains (b) Scaled boundary co-ordinates

1Ω 2Ω …

…

NΩ 1−NΩ

1bΓ 2bΓ

biΓ biΓ

iΓ

1+iΓ

iΓ

1+iΓ

1+ΩN

Incident wave

y

x

x

z

o η

h

Figure 4.1. Definition sketch of wave diffraction around obstacles.

o

Ω∞Γ

cΓ

cΓ

Chapter Four

- 101 -

(a) Coarse mesh (b) Intermediate mesh (c) Fine mesh

Figure 4.4. Scaled boundary finite-element meshes for a circular cylinder.

x

y

r

o

Incident wave α= 0°

θ

Figure 4.3. Circular cylinder (ka=2).

Chapter Four

- 102 -

Figure 4.5(b). Variation of scattered wave elevation (imaginary part) around circular cylinder for Eample 1.

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0 20 40 60 80 100 120 140 160 180

Coarse

Intermediate

Fine

Exact

8 Infinite elements

Angle round cylinders in degrees

Scat

ted

wav

e el

evat

ion

Figure 4.5(a). Variation of scattered wave elevation (real part) around circular cylinder for Example 1.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0 20 40 60 80 100 120 140 160 180

Coarse

Intermediate

Fine

Exact

8 Infinite elements

Scat

ted

wav

e el

evat

ion


Chapter Four

- 103 -

Figure 4.6(b). Variation of total wave elevation (imaginary part) around circular cylinder for Example 1.

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

0 20 40 60 80 100 120 140 160 180

CoarseIntermediateFineExactBEM


Tota

l wav

e el

evat

ion

Figure 4.6(a). Variation of total wave elevation (real part) around circular cylinder for Example 1.

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

0 20 40 60 80 100 120 140 160 180

CoarseIntermediateFineExactBEM


Tota

l wav

e el

evat

ion

Chapter Four

- 104 -

Figure 4.7(b). Variation of tangential velocity (imaginary part) around circular cylinder for Example 1.


Tota

l vel

ocity

-6

-4

-2

0

2

4

6

0 20 40 60 80 100 120 140 160 180

Coarse

Intermediate

Fine

Exact

Figure 4.7(a). Variation of tangential velocity (real part) around circular cylinder for Example 1.


Tota

l vel

ocity

-8

-6

-4

-2

0

2

4

6

0 20 40 60 80 100 120 140 160 180

Coarse

Intermediate

FineExact

Chapter Four

- 105 -

Figure 4.8 (b). Contour plots of wave elevation (imaginary part, ka=4) for Example 1.

Cylinder

-1.0

00

-1.000

-0.500

-0.5

00

-0.5

00

-0.500.50

0

-0.5

00

-0.5

00

-0.500

-0. 500

-0.500

0.00

0

0.00

0

0.0 0

00.

000

0.00

0

0.00

0

0.00

0

0.000

0.000

0.00

0

0.00

0

0.00

0

0.000

0.5000.500

0.50

0

0.500

0.50

0 0.50

0

0.50

0

0.50 0

1.000

1.000

Incident wave

-1.000

-1.0

00

- 1. 000

-0.500-0.500

-0.5

00

-0. 50 0

-0.5

00

-0.5

00

0.00

0

0.000

0.000

0.00

0

0.000

0.000

0.00

00.

000

0.00

00.

000 0.

500

0.500

0.500

0.500

0.500

0.50

0

1.000

1.000

1.00

0

1.000

1.000

Figure 4.8 (a). Contour plots of wave elevation (real part, ka=4) for Example 1.

Cylinder

Incident wave

Chapter Four

- 106 -

Figure 4.9(b). Vector plots of velocity at the water surface (imaginary part, ka=4) for Example 1.

Cylinder

Incident wave

Figure 4.9(a). Vector plots of velocity at the water surface (real part, ka=4) for Example 1.

Cylinder

Incident wave

Chapter Four

- 107 -

Figure 4.10(b). Wave moment on circular cylinder for Example 1.

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

SBFEM (h/a=2.0)

Exact (h/a=2.0)SBFEM (h/a=1.0)


Exact (h/a=0.5)

ka

my / ρ

gAha

2 f x

/ρgA

a2

-0.8

0.0

0.8

1.6

2.4

3.2

4.0

4.8

5.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

SBFEM (h/a=2.0)



Exact (h/a=0.5)

Figure 4.10(a). Horizontal wave force on circular cylinder for Example 1.

ka

Chapter Four

- 108 -

Table 4.1. Variation in dimensionless wave force and wave moment computed with the meshes of increasing density.

Wave force ( 2gAaf x ρ ) Wave moment ( 2gAham y ρ )

Real part Imaginary part Real part Imaginary

part coarse -0.1880 -1.6668 0.0716 0.6347 Intermediate -0.1907 -1.6783 0.0726 0.6391 SBFEM fine -0.1917 -1.6823 0.0730 0.6406

Analytical solution -0.1929 -1.6875 0.0735 0.6426

(a) Coarse mesh (b) Medium mesh (c) Fine mesh

Figure 4.12. Scaled boundary finite-element meshes for an elliptical cylinder for Example 2.

x

y

a 2a

o

Incident wave

α= 30°

Figure 4.11. Elliptical cylinder of aspect ratio 2:1 (ka=4) for Example 2.

Chapter Four

- 109 -

-2.4

-1.6

-0.8

0.0

0.8

1.6

2.4

3.2

0 30 60 90 120 150 180 210 240 270 300 330 360

CoarseIntermediateFineInfinite element

Figure 4.13(b). Variation of total wave elevation (imaginary part) around elliptical cylinder for Example 2.


Tota

l wav

e el

evat

ion

Figure 4.13(a). Variation of total wave elevation (real part) around elliptical cylinder for Example 2.

-2.4

-1.6

-0.8

0.0

0.8

1.6

2.4

0 30 60 90 120 150 180 210 240 270 300 330 360

Coarse

Intermediate

Fine

Infinite element


Tota

l wav

e el

evat

ion

Chapter Four

- 110 -

Figure 4.15. Scaled boundary finite-element meshes for an elliptical cylinder for Example 3.

(a) Coarse mesh (b) Medium mesh (c) Fine mesh

y

x

Incident wave α= 30°

Figure 4.14. Elliptical cylinder of aspect ratio 4:1 (ka=4) for Example 3.

a 4a o

Chapter Four

- 111 -

Figure 4.16(b). Variation of total wave elevation (imaginary part) around elliptical cylinder for example 3.


Tota

l wav

e el

evat

ion

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 30 60 90 120 150 180 210 240 270 300 330 360

CoarseIntermediateFineInfinite element

Figure 4.16(a). Variation of total wave elevation (real part) around elliptical cylinder for example 3.


Tota

l wav

e el

evat

ion

-2.4

-1.6

-0.8

0.0

0.8

1.6

2.4

0 30 60 90 120 150 180 210 240 270 300 330 360

Coarse

Intermediate

Fine

Infinite element

Chapter Four

- 112 -

Figure 4.17. Substructured model and meshes for Example 4, consisting of two bounded subdomains and one unbounded subdomain.

(b) Medium mesh (c) Fine mesh

(a) Coarse mesh

Side-face for unbounded domain

Bounded domain

x Scaling centre for unbounded domain

y Scaling centre for bounded domain

Unbounded domain

Side-face for bounded domain

Chapter Four

- 113 -

Figure 4.18(b). Computed variation of wave elevation (imaginary part) along the surface of the single square cylinder for Example 4.

Angle round cylinder in degrees

Wav

e el

evat

ion

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

0 20 40 60 80 100 120 140 160 180

Coarse

Medium

Fine

BEM

Figure 4.18(a). Computed variation of wave elevation (real part) along the surface of the single square cylinder for Example 4.


Wav

e el

evat

ion

-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2

0 20 40 60 80 100 120 140 160 180

Coarse

Medium

Fine

BEM

Chapter Four

- 114 -

Figure 4.19(b). Computed variation of tangential velocity (imaginary part) along the surface of the single square cylinder for Example 4.


Tang

entia

l Vel

ocity

-12.0

-8.0

-4.0

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

0 20 40 60 80 100 120 140 160 180

Coarse

M edium

Fine

BEM (16 elements)

BEM (800 elements)

Figure 4.19(a). Computed variation of tangential velocity (real part) along the surface of the single square cylinder for Example 4.


Tang

entia

l Vel

ocity

-12.0

-8.0

-4.0

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

0 20 40 60 80 100 120 140 160 180

Coarse

M edium

Fine

BEM (16 elements)

BEM (800 elements)

Chapter Four

- 115 -

Incident wave L

B B d

Figure 4.21. Configuration of wave diffraction by twin rectangular caissons for Example 5.

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

0 1 2 3 4 5 6 7 8

SBFEM

BEM

Figure 4.20. Horizontal wave force (amplitude) on the single square cylinder for Example 4.

kL

f x/2ρ

gAhL

(tanh

kh/k

h)

Chapter Four

- 116 -


Wav

e el

evat

ion

Figure 4.23(a). Computed variation of wave elevation (real part) along the surface of the twin caissons for Example 5.

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 20 40 60 80 100 120 140 160 180

Leading caisson (SBFEM )Trailing caisson (SBFEM )Leading caisson (BEM )Trailing caisson (BEM )

x

y

Unbounded domain

Bounded domain Bounded domain

Scaling centre for unbounded domain

Scaling centre for bounded domain

Side-face for unbounded domain

Side-face for bounded domain

Figure 4.22. Substructured model and mesh for Example 5, consisting of four bounded subdomains and one unbounded subdomain.

Chapter Four

- 117 -

Wav

e el

evat

ion


Figure 4.23(b). Computed variation of wave elevation (imaginary part) along the surface of the twin caissons for Example 5.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100 120 140 160 180


Chapter Four

- 118 -


Tang

entia

l vel

ocity

Figure 4.24(a). Computed variation of tangential velocity (real part) along the surface of the twin caissons for Example 5.

-9

-6

-3

0

3

6

9

12

15

18

21

24

27

30

0 20 40 60 80 100 120 140 160 180

Leading cassion (SBFEM )

Trailing cassion (SBFEM )

Leading cassion (32 BEM elements)Trailing cassion (32 BEM elements)

Leading cassion (1600 BEM elements)

Trailing cassion (1600 BEM elements)

Chapter Four

- 119 -


Tang

entia

l vel

ocity

Figure 4.24(b). Computed variation of tangential velocity (imaginary part) along the surface of the twin caissons for Example 5.

-18

-15

-12

-9

-6

-3

0

3

6

9

12

15

18

21

24

27

30

0 20 40 60 80 100 120 140 160 180

Leading cassion (SBFEM )Trailing cassion (SBFEM )Leading cassion (32 BEM elements)Trailing cassion (32 BEM elements)Leading cassion (1600 BEM elements)Trailing cassion (1600 BEM elements)

Chapter Four

- 120 -

f x/2ρ

gAhL

(tanh

kh/k

h)

Figure 4.25. Horizontal wave force (amplitude) on the twin caissons for Example 5.

kL

0

2

4

6

8

10

12

14

16

18

20

22

24

0 1 2 3 4 5 6 7 8


Chapter Four

- 121 -

Figure 4.27. Mesh and substructure definition for Example 6.

Unbounded domain

Bounded domain

scaling centre for bounded domain

scaling centre for unbounded domain

C

L=31m

6m C x

y

Figure 4.26. Configuration of the rectangular narrow bay for Example 6.

Incident wave

Chapter Four

- 122 -

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5

SBFEM Zhao et al. 2004Lee 1971 (Theory)Lee 1971 (Experiment)Madsen & Larsen 1987Ippen & Goda 1963

η/A

kL

Figure 4.28. Variation of dimensionless wave elevation (amplitude) at the point C with dimensionless wave number for Example 6.

Chapter Four

- 123 -

Unbounded domain

Ω1

Ω2

Ω3

Ω4

Ω5

Ω6

Ω7

Ω8

Ω9

Ω10

Ω11

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10 C11

Figure 4.30. Mesh for Example 7, consisting of ten bounded domains and one unbounded domain.

C1-C10 are the scaling centers for the bounded domains and C11is the scaling center for the unbounded domain

x

y

a/4

a/4

a/2

a/2

a

Figure 4.29. Configuration of the square harbor with straight breakwaters for Example 7.

Incident wave

Chapter Four

- 124 -

Wav

e el

evat

ion

x/a

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1

SBFEM

FEM

BEM

Figure 4.31(b). Variation of wave elevation (amplitude) at y/a=0.5 for Example 7.

x/a

Wav

e el

evat

ion

0

0.5

1

1.5

2

2.5

3

3.5

0 0.25 0.5 0.75 1

SBFEM

FEM

BEM

Figure 4.31(a). Variation of wave elevation (amplitude) at y/a=0 for Example 7.

Chapter Four

- 125 -

Wav

e el

evat

ion

y/a

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5

SBFEM

FEM

BEM

Figure 4.31(c). Variation of wave elevation (amplitude) at x/a=1 for Example 7.

Chapter Five

- 126 -

CHAPTER 5

SECOND-ORDER SOLUTION TO MONOCHROMATIC WAVE DIFFRACTION PROBLEMS

5.1 General

The previous two chapters develop scaled boundary finite-element solutions to linear

wave diffraction problems. These studies demonstrate that the SBFEM is a novel

semi-analytical approach to linear wave diffraction problems, particularly suited to

handling the radiation condition at infinity and singularities in the field near sharp re-

entrant corners. It is well known that the second-order solution of wave diffraction

problems can be derived from the first-order solution with minor modifications. It is

therefore expected that the SBFEM can be employed to solve the second-order

problems, retaining the salient features demonstrated in linear problems.

Based on the work presented in the Chapter 3, the SBFEM solution of second-order

monochromatic wave diffraction by two-dimensional rectangular structures in water

of finite depth is developed in this chapter. The second-order wave loading

component induced by the first-order dynamic pressure is evaluated based on the

first-order velocity potential. Since both the first and second-order velocity potential

fields are determined explicitly in the context of the scaled boundary finite-element

solution, the second-order component of wave loading can be calculated directly.

In the following sections, the derivation and solution process of the scaled boundary

finite-element equations for second-order wave scattering problems are presented in

detail. The numerical results are compared with other semi-analytical and

experimental solutions. The accuracy and efficiency of the method are investigated.

Chapter Five

- 127 -


As shown in Figure 5.1, a fixed semi-submerged rectangular obstacle with a draft of

D is subjected to a train of monochromatic incident wave travelling in the direction of

positive x-axis. Considering the Stokes nonlinear wave theory up to the second-order,

the governing equation of second-order problem is the Laplace equation in terms of

the second-order velocity potential Φ(2)(x,z,t),

0)2(2 =∇ Φ , within the fluid domain Ω . (5-1)

where for convenience of applying the SBFEM at a later stage, the fluid domain is

divided into four subdomains Ω1, Ω2, Ω3 and Ω4, such that

4321 ΩΩΩΩΩ ∪∪∪= .

The boundary condition on the water surface may be expressed as

)2()2()2( ,, Qg ttz =+ΦΦ (5-2)

with

tttzzzt gQ ,2),1,(, )1()1()1()1()1()2( ΦΦΦΦΦ ∇⋅∇−+= (5-3)

and

)e),(Re( i)1()1( tzx ωφΦ −= (5-4)

),()e),(Re( )2(i2)2()2( zxzx t φφΦ ω += − (5-5)

where Φ(1)(x,z,t) is the first-order velocity potential, Re denotes the real part of

complex quantity, i represents the unit of imaginary number, ω is angular frequency,

g is the acceleration due to gravity and t is the time. Since ),()2( zxφ does not

contribute to the second-order wave forces when it is substituted into the Bernoulli

equation, it is sufficient to consider only the time-dependent term in the current

Chapter Five

- 128 -

problem. Discarding the term ),()2( zxφ and then substituting Equations (5-5) into

Equation (5-3) yields

e),(Re i2)2()2( tzxQ ωΞ −= (5-6)

with

)1()1()1()1(2

)1()2( i),,(2i φφωφφωφωΞ ∇⋅∇+−=

ggg zzz (5-7)

Assuming the bottom boundary and the body surface to be impermeable and

enforcing Neumann boundary condition at the interfaces of subdomains, the

boundary-value problem in any subdomain may be defined as,

0)2(2 =∇ φ , within the subdomain iΩ , i=1,2,3,4. (5-8)

)2()2(2

)2( 4, Ξφωφ =−gz , at the free surface of water. (5-9)

0,)2( =nφ , at the bottom of water. (5-10)

vn =,)2(φ , at the surface of body. (5-11)

where n is the unit normal vector pointing inside of fluid domain and v is the

prescribed velocity. Only outgoing wave exist at infinity. The radiation condition at

infinity will be discussed further when the solution process of scaled boundary finite-

element equations is addressed.

5.3 Scaled boundary finite-element equations

To apply the SBFEM to modelling the singularities of the velocity field at the sharp

corners of a rectangular obstacle and the boundary condition at infinity, the

computational domain is divided into several bounded and unbounded subdomains as

shown in Figure 5.1. In this section, a typical bounded subdomain 3Ω and

Chapter Five

- 129 -

unbounded subdomain 4Ω are used to derive the scaled boundary finite-element

equations and the corresponding solutions for the two types of subdomains.

5.3.1 Scaled boundary finite-element equations for a bounded subdomain

The system of the partial differential equations of the boundary-value problem may

be formulated in weighted residual form in terms of a weighting function w

0)( )2()2(2)2( =−+−∇∇ ∫∫∫ ΓΓΞφκΩφΓΓΩ

dvwdwdwbsw

T (5-12)

with

g

22 4ωκ = (5-13)

where Ω denotes of the appropriate subdomain ( 2Ω or 3Ω ), wΓ represents the

boundary of free surface of water and bsΓ represents other boundaries with Neumann

boundary conditions.

To discretize Equation (5-12) using the scaled boundary co-ordinate system, an

approximate solution to the second-order velocity potential is proposed in the form

)()]([)2( ξφ asN= (5-14)

in which [N(s)] are standard finite element shape functions discretising the defining

curve and )( ξa is a nodal potential vector. Since )2(φ∇=v , the velocity may be

written as

)()]([1),()]([ 21 ξξ

ξ ξ asBasBv += (5-15)

where, for convenience,

)]()[()]([ 11 sNsbsB = (5-16a)

Chapter Five

- 130 -

ssNsbsB )],()[()]([ 22 = (5-16b)

Applying the Galerkin approach, the weighting function w is approximated by the

same shape functions, leading to

)()]([ ξwsNw = (5-17)

Then substituting Equations (5-14) and (5-17) into Equation (5-13) yields

0)]([)(

))()()]([()]([)(

))()]([1)()](([

))()]([1)()](([

)2(2

21

21

=−

+−

+

+

∫∫

∫

dΓvsNw

dΓs,ΞasNsNw

dΩasB,asB

wsB,wsB

bs

w

Γ

TT

Γ

TT

Ω

T

ξ

ξξκξ

ξξ

ξ

ξξ

ξ

ξ

ξ

(5-18)

All terms containing ξξ ),(w are integrated by parts with respect to ξ , using

Green’s identity. Noting that dsdJd ξξΩ = , introducing ξτ and sτ to transform

infinitesimal lengths on the boundary sections with constant ξ and constant s to the

scaled boundary co-ordinate system, and including the boundary conditions, Equation

(5-18) becomes

∫ −=+s

Ti

Tii dsvsNaEaE ξ

ξ τξξξ )()]([)(][),(][ 10 (5-19)

Πξκξξξ ξ +=+ )(][)(][),(][ 210 ee

Tee aMaEaE (5-20)

∫=+s n

Te

Tee dsvsNaEaE ξ

ξ τξξξ )]([)(][),(][ 10 (5-21)

)()(][),(])[][]([),(][ 21102

0 ξξξξξξξ ξξξ sT faEaEEEaE =−−++ (5-22)

where the coefficient matrices

∫= s

T dsJsBsBE )]([)]([][ 110 (5-23)

Chapter Five

- 131 -

∫= s

T dsJsBsBE )]([)]([][ 121 (5-24)

∫= s

T dsJsBsBE )]([)]([][ 222 (5-25)

∫= s

T dssNsNM ξτ)]([)]([][ (5-26)

sTsTs svsNsvsNf τξτξξ )),(()]([)),(()]([)( 1100 −+−= (5-27)

∫= s eT dsssN ξτξΞΠ ),()]([ )2( (5-28)

have been introduced to simplify the resulting integral equation (referring to Chapter

3 for more detail).

Since the surface of obstacle is impermeable and the obstacle is fixed, the

nonhomogeneous term )( ξsf in the scaled boundary finite-element equation

(Equation (5-22)) vanishes. Hence,

0)(][),(])[][]([),(][ 21102

0 =−−++ ξξξξξ ξξξ aEaEEEaE T (5-29)

Equation (5-29) is termed the homogenous scaled boundary finite-element equation

in the SBFEM. Observing Equation (5-29), it can be found that the original governing

equation is weakened in the circumferential direction but remains strong in the radial

direction. Equations (5-19)-(5-22) are the scaled boundary finite-element forms of

boundary conditions, relating the nodal velocity potential and nodal flow. Equation

(5-19) vanishes when ξi is equal to zero. In other words, the inner boundary of the

domain degenerates into a point – the scaling centre. The boundary condition at the

scaling centre may be replaced by the requirement that the solution remains finite.

Equations (5-20) and (5-21) are satisfied on the boundary wΓ and bsΓ respectively.

The analytical matrix solution of Equation (5-29) associated with boundary

conditions (Equations (5-20) and (5-21)) can be found in Chapter 3.

5.3.2 Scaled boundary finite-element equations for an unbounded subdomain

Chapter Five

- 132 -

In the previous derivation of scaled boundary finite-element equations for a bounded

subdomain, the equations are written in terms of the total second-order wave

potential. However, for an unbounded subdomain, the boundary condition at infinity

can be applied more conveniently if the scattered second-order wave potential is

taken as the unknown quantity. The total wave potential )2(φ may be expressed as the

summation of the incident wave potential )2(Iφ and the scattered wave potential )2(

Sφ ,

namely,

)2()2()2(SI φφφ += (5-30)

Substituting Equation (5-30) into Equations (5-8)-(5-11) yields

0)2(2 =∇ Sφ , within the domain. (5-31)

)2()2(2

)2( 4, SSzS gΞφωφ =− , at the free surface of water. (5-32)

0,)2( =nSφ , at the bottom of water. (5-33)

SnS v=,)2(φ , at the oriented line. (5-34)

with

)1()1()1()1(2

)1()2(2

)2()2( i),,(2i)4,( φφωφφωφωφωφΞ ∇⋅∇+−+−−=

gggg zzzIzIS (5-35)

where the first-order and second-order incident potentials can be written as

kx)(I kh

)hz(kA i1 ecosh

coshig +−=ω

φ (5-36a)

kxI kh

hzkA i24

2)2( e

sinh)(2cosh

8i3 +

−=ωφ (5-36b)

Chapter Five

- 133 -

A weighted residual equation in terms of the scattered wave potential may be written

as

0)( )2()2(2)2( =−+−∇∇ ∫∫∫ ΓΓΞφκΩφΓΓΩ

dvwdwdwbsw

SSSST (5-37)

Then, following the same approach to the derivation of the scaled boundary finite-

element equations as applied above for the bounded subdomain, weakening Equation

(5-37) in the direction of local co-ordinate s results in

∫ −=+s n

Ti

Ti dsvsNaEaE ξ

ξ τξξ )()]([)(][),(][ 10 (5-38)

∫=+s n

Te

Te dsvsNaEaE ξ

ξ τξξ )]([)(][),(][ 10 (5-39)

)()(])[][()(])[]([)(][ 202

110 ξξκξξ ξξξ tT paEM,aEE,aE −=−+−+ (5-40)

with

sT sNsNM τ)]([)]([][ 110 = (5-41)

)2(1 )]([)( S

sTt sNp Ξτξ = (5-42)

Equation (5-40) is the scaled boundary finite-element equation for the unbounded

subdomain. It is a system of nonhomogeneous second-order ordinary differential

equations with constant coefficients. Equation (5-38) relates to the nodal potential

)( ξa of the scattered wave and the nodal flow at the oriented line (ξi=0). Equation

(5-39) represents the external boundary condition, but when ξe tends to infinity, it is

replaced by the boundary condition at infinity. In the physical sense, at infinity,

scattered waves must be outgoing. This boundary condition is imposed in the solution

process.


Chapter Five

- 134 -

The analytical matrix solution of scaled boundary finite-element equations in a

bounded subdomain was introduced in Chapter 3. In this section, emphasis is placed

on the solution of the scaled boundary finite-element equations in an unbounded

subdomain. Song & Wolf (1999) applied the method of variation of parameters to the

solution of a nonhomogeneous scaled boundary finite-element equation for elasto-

static problems with body loads. A similar approach is developed here to the solution

of Equation (5-40) associated with the boundary conditions for second-order wave

scattering problems.

5.4.1 General solution

Using the standard technique to simplify the scaled boundary finite-element equation,

introducing

)(][),(][)( 10 ξξξ ξ aEaEq T+= (5-43)

then combining Equation (5-43) with Equation (5-40) results in a matrix equation

)()(][,)( ξξξ ξ FXZX += (5-44)

with

⎥⎦

⎤⎢⎣

⎡

−−−

=−−

−−

1010

21

1012

101

10

]][[][][]][[][][][][

][EEMEEEE

EEEZ T

T

κ (5-45)

⎭⎬⎫

⎩⎨⎧

=)()(

)(ξξ

ξqa

X (5-46)

⎭⎬⎫

⎩⎨⎧−

=)(

0)(

ξξ

tpF (5-47)

The homogeneous solution of equation (5-44) may be expressed as

][)( ][ CeX ξΛΦξ = (5-48)

Chapter Five

- 135 -

where matrices ][Λ and ][Φ are the eigenvalue matrix and eigenvector matrix, that

is,

]][[]][[ ΛΦΦ =Z (5-49)

Due to the properties of matrix [Z] (Wolf, 2003), the eigenvalues consist of two

groups with opposite signs

⎥⎦

⎤⎢⎣

⎡=

−

+

][][

][m

m

λλ

Λ , .,,3,2,1 nm L= (5-50)

where +mλ represents the eigenvalues with positive real part or a negative pure

imaginary number while −mλ denotes correspondingly those with negative real part or

a positive pure imaginary number.

Applying the method of variation of parameters to the resulting solution of the

homogenous equation, the general solution of Equation (5-48) may be expressed as

)(][)( ][ ξΦξ ξΛ CeX = (5-51)


)(][),( 1][ ξΦξ ξΛξ FeC −−= (5-52)

Thus, C(ξ) can be determined from the solution of Equation (5-52), namely,

)(][)(0

][ cduuFAeC u += ∫ −ξ Λξ (5-53)

with

1][][ −= ΦA (5-54)

Partitioning the vector C(ξ) and the matrix ][Φ into the same block forms as the

vector X(ξ) and the matrix ][Λ results in

Chapter Five

- 136 -

⎭⎬⎫

⎩⎨⎧

=)()(

)(2

1

ξξ

ξCC

C (5-55)

⎥⎦

⎤⎢⎣

⎡=

2221

1211][ΦΦΦΦ

Φ (5-56)

The corresponding matrix [A] is also written in block form as

⎥⎦

⎤⎢⎣

⎡=

2221

1211][AAAA

A (5-57)


⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

−

+

)()(

)()(

2

1][

][

2221

1211

ξξ

ΦΦΦΦ

ξξ

ξλ

ξλ

CC

ee

qa

m

m

(5-58)

or

)(][)(][)( 2][

121][

11 ξΦξΦξ ξλξλ CeCea mm−+

+= (5-59a)

)(][)(][)( 2][

221][

21 ξΦξΦξ ξλξλ CeCeq mm−+

+= (5-59b)

Thus, Equation (5-53) becomes,

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧−⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

∫ −

+

−

−

2

1

02221

1211][

][

2

1

)(0

)()(

cc

duupAA

AA

ee

CC

u

u

m

mξ

λ

λ

ξξ

(5-60)

or

)(][)( 10 12][

1 cduupAeC tum +−= ∫

+−ξ λξ (5-61a)

)(][)( 20 22][

2 cduupAeC tum +−= ∫

−−ξ λξ (5-62b)

Chapter Five

- 137 -

where c1 and c2 are the blocks of the integration constant vector c.

Consequently, the general solution of scaled boundary finite element equation in

terms of the nodal potential vector is determined by Equations (5-59a).

5.4.2 Boundary condition at infinity

The solution at infinity should be finite, hence,

0)( 1 =∞=ξC (5-63)

It can be seen from Equation (5-59a) that a positive pure imaginary eigenvalue

corresponds to an outgoing propagating mode of the scattered wave at infinity while a

negative pure imaginary eigenvalue corresponds to a returning propagating mode of

the scattered wave at infinity. Equation (5–59a) also suggests that an eigenvalue with

a negative real part corresponds to an evanescent mode, vanishing at infinity.

Therefore, at infinity, the solution (Equation (5-59a)) properly describes the

properties of the far-field velocity potential of scattered waves.

Substituting Equation (5-63) into Equation (5-61a) yields

∫∞ − +

=0 12

][1 )(][ duupAec t

umλ (5-64)

Inserting Equation (5-64) into Equation (5-61a) results in

∫∞ − +

=ξ

λξ duupAeC tum )(][)( 12

][1 (5-65)

Details of the explicit expression of )( 1 ξC are provided in Appendix 5-A.

Eliminating )( 2 ξC in Equations (5-59a) and (5-59b) leads to

)(])[]][[]([)()(]][[ 1][

111

1222211

1222 ξΦΦΦΦξξΦΦ ξλ Ceqa m+−− −−= (5-66)

The standard form of the finite element equation can be expressed as

)()()(][ ξξξ RqaH += (5-67)

Chapter Five

- 138 -

with

11222 ]][[][ −= ΦΦH (5-68)

and

)(])[]][[]([)( 1][

111

122221 ξΦΦΦΦξ ξλ CeR m+−−−= (5-69)

Taking into account

][]][[][][ 111

1222211

12 ΦΦΦΦ −− −=A (5-70)

and substituting Equation (5-70) and Equation (5-A-12) into Equation (5-69) yields

TIkmi

kxiniii

Imji

n

jjinjnijji

n

i

sNAeIk

eckAkAgA

eI

ccggg

AR

i

s

ji

)](][][])[])[i((

))(2

2i3[(

]])[])[((

)2ii

23i([][)(

112]))[i((1)1(

i2

22)1()1(2

4

]))[((1)1()1(

122

2)1()1()1(3

5

1

112

)1(

)1()1(

ξλ

ξλλ

λλ

γλλω

λλλ

γγλωλλωωξ

+−+−

−−

+−+−−

=

−−−

=

−

−

−−

−+

−+++

−+

++= ∑∑

(5-71)

The final solution for all nodal potentials can be obtained by following the technique

employed in Chapter 3, assembling the equations from all subdomains into a global

equation and solving this global equation for the complex field.

5.4.3 Determination of wave forces

Since the first-order and the second-order velocity potentials have been determined,

the wave force can be calculated using classical wave theory (Mei 1989). The total

hydrodynamic pressure p and wave forces F(t) may be expressed as

2ΦΦρΦρρ ∇⋅∇

−∂∂

−−=t

gzp (5-72)

∫∫=S

dsnptF v)( (5-73)

Chapter Five

- 139 -

where s is the wetted surface of the obstacle. Introducing the expressions of the

perturbation of the total potential Φ and the wave loads F into Equation (5-73) yields

the time-dependent (double-frequency) component of the second-order wave forces

f(2),

∫∫∫ −∇⋅∇−=CS

dng

dsnfb

Γφρωφφωφρ vv 2)1(2

)1()1()2()2(

4]

41i2[ (5-74)

where Sb is the equilibrium surface of the body up to the still water level and C

indicates the waterline contour (or waterline points if the problem is two-

dimensional). The horizontal and vertical components (fH and fV) of the wave forces

f(2) can be determined using Equation (5-74). The moment m can be found in a similar

manner.


This section attempts to apply the SBFEM for calculating the second-order wave

diffraction problems. Firstly, the problem of second-order wave diffraction by a two-

dimensional semi-submerged rectangular obstacle is studied to investigate the

accuracy and efficiency of the computational procedure presented in this Chapter.

The computed results of the second-order wave force and hydrodynamic pressure

using the SBFEM are compared to the semi-analytical solution presented (Sulisz

1993), the direct and indirect GFM solution (Drimer & Agnon 1994), the hybrid

BEM solution (Drimer & Agnon 1994) and physical experimental results (Sulisz &

Johansson 1992). Secondly, the SBFEM is applied to solving problem of the second-

order wave diffraction by an obstacle with a trapezoidal cross-section. The effect of

geometric parameters of the cross section on the second-order wave force is studied.

Finally, the resonance phenomenon in the second-order wave force induced by wave

diffraction by twin obstacles with a small gap is illustrated.

5.5.1 Wave diffraction by a rectangular obstacle (Example 1)

In the first example the second-order components of wave loads on a semisubmerged

horizontal rectangular obstacle are calculated using the SBFEM. The accuracy of the

Chapter Five

- 140 -

method is investigated through comparison with semi-analytical results published by

Sulisz (1993). Four cases with various dimension ratios are investigated, as indicated

in Table 5.1.

Table 5.1. Four cases with different dimension ratios

B/H D/H

Case 1 1.0 0.4

Case 2 1.0 0.2

Case 3 0.2 0.4

Case 4 0.2 0.2

For Case 1 and Case 2, the meshes for computation are shown in (a-1), (b-1) and (c-

1) of Figure 5.3. The coarse mesh consists of 18 elements. The numbers of elements

in the medium mesh and the fine mesh are 36 and 72 respectively. Likewise, three

types of meshes, coarse mesh (12 elements), medium mesh (24 elements) and fine

mesh (48 elements) are employed for Case 3 and Case 4, as shown in (a-2), (b-2) and

(c-2) of Figure 3. Three-node quadratic elements are used for the discretization of the

boundaries of both the bounded and unbounded subdomains. For all the cases, the

amplitudes of horizontal forces and vertical forces are nondimensionalized by the

factor ρgA2. The moment is taken about (B,-D) and its amplitude is normalized by the

factor ρgA2H.

Table 5.2. Error estimation of SBFEM solutions σ

Mesh fH fV m

coarse 0.20589 0.02017 0.03033

medium 0.02974 0.01696 0.02557

fine 0.02972 0.01647 0.02449

Chapter Five

- 141 -

Figure 5.4 shows the variations of the amplitudes of dimensionless horizontal force

fH, vertical force fV and moment m with wave frequency for Case 1. To illustrate the

accuracy of SBFEM solutions, an error estimator σ is defined as

∑=

−=n

icisi

ci

pppn 1

2)(1σ (5-75)

where pci represents the digitalized results of Sulisz (1993), cip is the average value

of pci and psi is the corresponding SBFEM solution, n denotes the number of

digitalized points and equals 11 herein. As shown in Table 5.2, the error of SBFEM

solutions decreases with the increase of mesh density. It can be seen that the results

obtained using the medium mesh and fine mesh agree well with the semi-analytical

results of Sulisz (1993).

Table 5.3. Error estimation of SBFEM solutions σ

L/B fH fV m

1.0 0.20589 0.02017 0.03033

0.6 0.04051 0.01762 0.02598

0.2 0.03427 0.01743 0.02525

The effect of the size of bounded domain on the results is illustrated in Figure 5.5.

Three sizes of bounded domain are analysed, L=B, L=0.6B and L=0.2B. The size of

the elements is identical for all three domain sizes. This means that the smaller

bounded domain was, the fewer elements were used. The Table 5.3 shows the error

variation with the decrease of the ratio L/B. The results show that despite this, the

smaller the bounded domain size is, the more accurate are the results. This outcome is

not surprising due to the unique features of the SBFEM. There is no discretization of

the boundary of water surface in the unbounded subdomain. For a larger unbounded

domain, this means less overall discretization errors and therefore more accurate

results. This demonstrates that the scaled boundary finite-element method is able to

Chapter Five

- 142 -

obtain results with good accuracy even using few elements. Figures 5.6-5.8 plot the

second-order components of the wave loads for the other cases. In each case the

results agree very well with the semi-analytical results.

In this example, the accuracy and efficiency of the SBFEM is also compared to the

GFM. The same semisubmerged horizontal rectangular obstacle is used, with the

geometry B/H=1.0 and with two cases of different draft, D/H=0.8 and D/H=0.6. Only

the component of the second-order force contributed by the second-order potential,

dsnFbS

v∫∫= )2()2( i2 ωφρ (5-76)

is presented. The SBFEM analysis employs three meshes of various refinement

levels, a coarse mesh (12 elements), a medium mesh (24 elements) and a fine mesh

(48 elements). The L/B is taken to be 0.2. Figures 5.9-5.12 show the scaled boundary

finite-element solutions converge quickly, and even the coarse mesh can produce

accurate results. Drimer & Agnon (1994) used a hybrid boundary-element method to

solve the same example. This hybrid method used the simple Rankine resource as a

Green’s function in the interior region and matched the analytical solution of the

exterior region by ensuring velocity and pressure continuity on the interface between

the two regions. This direct method can calculate the second-order wave force by

solving explicitly for the second-order potential. Drimer & Agnon (1994) employed

60 elements in their calculations. It can be seen from Figures 5.9-5.12 that the results

obtained from the SBFEM agree well with those from the direct method. The results

of the indirect method provided by Drimer & Agnon are also plotted in Figures 5.9-

5.12 for the purpose of comparison. The indirect method used the ‘assisting potential’

to calculate the second-order wave force, without explicitly solving the second-order

potential. Though this method can obtain the good accuracy for the second-order

wave force, it cannot provide information about the hydrodynamic pressure.

The hydrodynamic pressure is calculated using the SBFEM in this example. For the

purpose of comparisons, the geometry set is B=0.305m, D=0.3m, H=0.4m and

A=0.041m. The wave number is taken to be 7.5. As before, three levels of mesh

refinement are used, a coarse mesh (12 elements), a medium mesh (24 elements) and

Chapter Five

- 143 -

a fine mesh (48 elements). The results are compared with the experimental and

approximate theoretical solutions reported by Sulisz & Johansson (1992) in Figure

5.13. The first-order component of hydrodynamic pressure obtained from the SBFEM

agrees well with the experimental results, but the second-order component of

hydrodynamic pressure is slightly lower than the experimental results. Drimer &

Agnon (1994) also calculated this example using a hybrid GFM with 60 elements.

Figure 5.13 shows that the present method has similar accuracy to that of Drimer &

Agnon (1994), although only 12 elements were employed.

5.5.2 Wave diffraction by a trapezoidal obstacle (Example 2)

This section applies the SBFEM to solving problems of second-order wave

diffraction by a trapezoidal obstacle. The influence of base angle θ (Figure 5.14) on

the second-order wave forces is investigated.

The substructured model for this example is shown in Figure 5.14. The computational

domain consists of two bounded domains and two unbounded domains. In the first

case of this example, the geometry set is B/H=1.0, D/H=0.4, θ=30°. 3-node quadratic

elements are employed to discretize the boundary of subdomains. The assignment of

scaled boundary elements is shown in Figure 5.14.

In this case the sharp base angle results in a singular first-order velocity field near

sharp corners. Since the second-order wave force is related to the first-order velocity

field (see Equation 5-74), the accuracy and convergence of the first-order velocity

plays an important role in calculating the second-order components of wave loads.

Since the SBFEM does not need to discretize the surface of obstacles, the solution to

the first-order velocity is analytical in the radial direction. The SBFEM employs an

explicit and analytical expression (Equation 5-65) for the integral along the water

surface in the unbounded domains, so it is expected to be able to accurately predict

the second-order wave loads in a fluid field with sharp corners. In this case, three

types of mesh with 24, 48 and 96 elements respectively are used to study the accuracy

and convergence of the SBFEM when solving such problems. As with Example 1, the

amplitudes of horizontal forces and vertical forces are nondimensionalized by the

Chapter Five

- 144 -

factor ρgA2. The moment is taken about (B,-D) and its amplitude is normalized by the

factor ρgA2H.

Figures 5.15-5.17 plot the second-order components of horizontal wave force, vertical

wave force and moment about (B,-D) on the obstacle. It can be seen that the SBFEM

solution converges well with the increase of mesh density and no irregular frequency

is found. The computed results demonstrate that the SBFEM has a satisfactory rate of

convergence when calculating the second-order components of wave loads. In Figure

5.15-5.17, it can be seen that the effects of second-order wave forces in the region

with smaller wave numbers (kH<1.0) are more significant.

To examine the influence of base angle size on the second-order components of wave

loads, cases with θ = 45°, 60°, 90°, 120°, 135°, 150° are calculated as well. The mesh

with 48 elements (Figure 5.14) is employed for calculating these cases.

The results are compared in Figures 5.18-5.20. For clarity, the cases of θ ≥ 90° and θ

≤ 90° are illustrated in different figures ((a)s and (b)s in Figures 5.18-5.20). The

differences between the second-order horizontal wave forces in the various cases are

not significant, particularly when θ ≥ 90°, as shown in Figures 5.18(a) and 5.18(b).

However, the variation of second-order vertical wave forces with the change of base

angle is significant when θ ≤ 90°, although the differences are still small in the case

of θ ≥ 90°. It is can be seen that the second-order component of vertical wave forces

has a rapid rise with the increase of base angle until θ is 90° (Figure 5.19(a)). Again,

the change of the second-order component of vertical wave force becomes very small

when θ ≥ 90° (Figure 5.19(b)). The moment has a similar trend to the vertical wave

force, as shown in Figure 5.20(a) and 5.20(b). In conclusion, the results computed

with the SBFEM show that the corner sharpness has a significant effect on the

second-order component of wave loads.

Another case, with the geometry set to B/H=1.0, D/H=0.2, is calculated to study the

effect of sharp corners on the second-order component of wave loads in deeper water

than the first case (D/H=0.4). Again, the case with θ = 30° is calculated with three

meshes to investigate the convergence of the SBFEM. Like the first case, 24, 48 and

Chapter Five

- 145 -

96 quadratic elements are used in the three meshes respectively to discretize the

boundary of the subdomains.

Figures 5.21-5.23 plot the variation of dimensionless second-order component of

horizontal, vertical wave forces and moment about (B,-D) with the dimensionless

wave number (kH). As shown in Figures 5.21-5.23, the SBFEM solutions converge

quickly with the increase of mesh density. Comparing with the results of the first case

(D/H=0.4), the noticeable differences between the computed second-order wave loads

occur when kH<2.0.

As with the shallow water, the cases with θ = 45°, 60°, 90°, 120°, 135°, 150° are

calculated using the mesh with 48 elements (Figure 5.14) to examine the effect of the

base angle size of obstacle on the second-order component of wave loads. The

variations of the dimensionless second-order wave loads with the dimensionless wave

number are plotted in Figures 5.24-5.26. It turns out that the effect of base angle size

of obstacle on the second-order component of wave loads in the case of D/H=0.2 is

similar to that of the case D/H=0.4. When θ ≤ 90°, the change of the second-order

wave loads becomes more and more significant as the base angle decreases, while the

change is not noticeable when θ ≥ 90°. This is because a sharp corner in the fluid

domain makes the first-order velocity change dramatically, and the sharper the corner

is, the greater is effect on the first-order velocity field, and the second-order wave

loads change considerably. When dealing with this class of problem, the SBFEM

does not need to discretize the boundary at the surface of structure, so the first-order

velocity field can be expressed analytically in the radial direction. Thus, the SBFEM

is able to calculate the second-order wave loads for various geometries of fluid

domain very well.

In summary, it can been seen the SBFEM can predict the second-order wave loads

with a rapid rate of convergence, even if there exist sharper corners in the fluid

domain.

5.5.3 Wave diffraction by twin rectangular obstacles (Example 3)

The first-order scaled boundary finite-element solutions to wave diffraction by twin

rectangular obstacle were examined in Section 3.6.1.2. This section applies the

Chapter Five

- 146 -

SBFEM to study the second-order effect for the same problem. The geometry set

(Figure 5.27) of numerical example is the same as the example in Section 3.6.1.2,

namely, B/H=1.0, D/H=0.3 and Bg=0.01. The computational domain is substructured

into four bounded domains and two unbounded domains. This example uses the same

meshes as Section 3.6.1.2 for calculating the second-order wave diffraction problem

(refer to Section 3.6.1.2 for details on meshes). The amplitudes of second-order wave

forces are nondimensionalized by the factor ρgA2.

Figures 5.28(a), (b) and (c) show the variation of the dimensionless second-order

horizontal wave force on the obstacle B1 with the normalized wave number. As

shown in Figure 5.28(a), the second-order effect is weak in most frequency bands.

However, two resonant frequency bands are found. For clarity, the portions of Figure

5.28(a) near the resonant frequencies are plotted enlarged in Figures 5.28(b) and

5.28(c). It is found that the numerical approximations of the resonant frequencies

converge to kH=1.02 and kH=3.14, respectively. In Section 3.6.2.1, the theoretical

(Miao et al 2000) and computed results shows the resonant frequencies of the first-

order problem are at approximately kH=nπ, n=1,2…. Since the wave frequency of

second-order incident wave is two times of that of the first-order, resonances for the

second-order problem are expected to occur at kH=nπ/2, n=1,2…. As shown in

Figures 5.28(a), (b) and (c), the computed resonant frequencies by SBFEM is 1.02

and 3.14. Clearly the SBFEM produces excellent solutions when predicting the

resonant phenomena of second-order wave diffraction by twin obstacles with a

narrow gap.

The variation of the second-order vertical wave force on the obstacle B1 with the

normalized wave number is illustrated in Figure 5.29(a). Figures 5.29(b) and 5.29(c)

are the enlarged portions near the resonant frequencies. As before, the resonant

phenomena is found at kH=1.02 and kH=3.14. The computed results for the second-

order wave forces on the obstacle B2 are plotted in Figures 5.30 and 5.31. The

resonant frequencies occur at kH=1.02 and kH=3.14 again. Also, it can be seen that

the amplitudes of second-order wave forces on the obstacles B1 and B2 have a

similar trend.

Chapter Five

- 147 -

To investigate how the size of the gap affects the resonant phenomena of the second-

order wave diffraction problems, two more cases (Bg=0.05 and Bg=0.1) are

calculated using the SBFEM. For the case of Bg=0.05, Figures 5.32(a) and 5.33(a)

show the variation of the second-order horizontal and vertical wave forces on

obstacle B1, while Figures 5.34(a) and 5.35(a) show the results computed for obstacle

B2. To make clear the detail of convergence of the results, the solutions near the

resonant frequencies bands are enlarged and plotted in (b) and (c) of Figures 5.32-

5.35. The results show the resonance becomes weaker due to the increase of the gap

between two obstacles. Also, the increase of the gap distance results in a change of

the resonant frequencies. The resonant frequencies take place at kH=0.90 and

kH=2.59 in this case. When the gap distance increases to Bg=0.1, the computed

results of the second-order wave forces, as shown in Figures 5.36-5.39, are similar to

those when Bg=0.05 in overall trend, although the resonant frequencies become

lower. The first resonant frequency decreases from 0.90 to 0.81 and the second one

falls from 2.59 to 2.21.

This example demonstrates that the SBFEM has good ability to deal with second-

order wave diffraction by two obstacles. Particularly, it is found that the SBFEM is

able not only to calculate the first-order resonant frequencies induced a narrow gap

between two obstacles, as discussed in Chapter 3, but also to predict very well the

second-order resonant frequencies of such problems.

5.6 Summary

In this chapter the scaled boundary finite-element method is extended to the solution

of Laplace equation associated with nonlinear boundary condition at water surface.

The scaled boundary finite-element solution up to the second-order is presented for

problems of monochromatic wave diffraction by semisubmerged horizontal obstacles

in a finite depth of water. This approach does not require an assisting radiation

potential to be calculated, and is able to obtain the solution of the second-order

velocity potential directly. The computed results show this technique does not

encounter numerical difficulties such as the irregular frequency problem or the

evaluation of singular integrals, such as occur in the Green’s function method or the

Chapter Five

- 148 -

BEM, so it can be used for calculating the resonant phenomena of second-order wave

diffraction problems. Due to the reduction of space dimensions, fewer elements are

needed in the SBFEM than are required in the standard FEM. Because the SBFEM

does not need to discretize the boundary at the surface of structures and the first-order

velocity field is analytical in the radial direction, the scaled boundary finite-element

solution to the second-order wave loads converges quickly with the increase of mesh

density.

Appendix 5-A Derivation of variation parameter C1(ξ).

To derive an explicit expression of the vector C1(ξ), Equation (42) may be

simplified first by using the free surface condition of the second-order potential of

incident waves and the first-order potential of scattered waves, namely,

)1()1()1()1(2

)1()2(2

)2( i),,(2i4, IIzzIzIIIzI gggg

φφωφφωφωφωφ ∇⋅∇+−=− (5-A-1)

04, )1(2

)1( =− SzS gφωφ (5-A-2)

where the total first-order wave potential is written as the summation of potentials of

incident waves and scattered waves

)1()1()1(SI φφφ += (5-A-3)

Substituting Equations (5-A-1)-(5-A-3) and the governing equation of the first-order

wave potential 0)1(2 =∇ φ into Equation (42) yields

),,(2i,),2(i)2(

23i

,2i),(i

23i

)1()1()1()1()1()1()1()1(3

5

)1()1(2)1(2)1(3

5)2(

xxISxxSIxSxISI

xxSSxSSS

ggg

ggg

φφφφωφφωφφω

φφωφωφωΞ

++++

++= (5-A-4)

Chapter Five

- 149 -

For conveniences, Equation (49) is expressed as

)()]([)( 1 ξΘξ Tt sNp = (5-A-5)

where

)2()( SsΞτξΘ = (5-A-6)

In the context of the modified scaled boundary co-ordinate system (Equation (34a)),

for the subdomain shown in Figure 5.2(b),

1=sτ (5-A-7)

The shape function [N(s1)] is specified as

]1,0,,0,0[)]([ 1 L=sN (5-A-8)

Substituting Equations (5-A-4)-(5-A-7) into Equation (70) yields

)](][[))(2i

)2(i)2(23i(

)2i)(i

23i(

)](][[)()(

112)1()1()1()1(

)1()1()1()1(3

5][

)1()1(2)1(2)1(3

5][

112][

1

sNAdu,,g

,,gg

e

du,g

,gg

e

sNAduΘeC

xxISxxSI

xSxISIu

xxSSxSSu

u

m

m

j

φφφφω

φφωφφω

φφωφωφω

ξξ

ξ

λ

ξ

λ

ξ

λ

++

++

++=

=

∫

∫

∫

∞ −

∞ −

∞ −

+

+

(5-A-9)

The first-order scaled boundary finite-element solution (Chapter 3) of scattered waves

at the free surface of water can be expressed as

])][([ )1(2

][)1(121

)1( )1(

cesN iS

ξλΦφ−

= , .,,3,2,1 ni L= (5-A-10)

where ][ )1(12Φ is the first-order eigenvector matrix associated with the first-order

eigenvalue matrix ][ )1( −iλ and n denotes the number of nodes. Substituting Equation

(5-A-8) into Equation (5-A-10) yields

Chapter Five

- 150 -

∑=

−

=n

iiniS ce i

12

)1( )1(

)( ξλγξφ (5-A-11)

where niγ is the element at the ith row and the nth column in the eigenvector matrix

][ )1(12Φ , −)1(

iλ and c2i are the eigenvalue and the integration constant at the ith row,

respectively. Substituting Equation (5-A-11) and Equation (43a) into Equation (5-A-

9) leads to

TIkmi

kxiniii

Imji

n

jjinjnijji

n

i

sNAeIk

eckAkAgA

eI

ccggg

C

mi

s

mji

)](][][])[])[i((

))(2

2i3[(

]])[])[((

)2ii

23i([)(

112])[])[i((1)1(

i2

22)1()1(2

4

])[])[((1)1()1(

122

2)1()1()1(3

5

11

)1(

)1()1(

ξλλ

ξλλλ

λλ

γλλω

λλλ

γγλωλλωωξ

+−

+−−

−+−+−

−−

−+−+−−

=

−−−

=

−+

−+++

−+

++−= ∑∑

(5-A-12)

where [I] represents the identity matrix.

Chapter Five

- 151 -

Figure 5.2. Local co-ordinate systems in a bounded and an unbounded subdomain.

(a) bounded subdomain (b) unbounded subdomain

s=s0

s=s1

s=s0

s=s1

ξi=0

ξe=1

ξ-axis

s-axis

ξ-axis

s-axis scaling center

Defining line

(xs,ys) (x,y)

(x0,y0)

(xs,ys) (x,y)

B B L L

H

z

x

Incident wave

1Ω 2Ω 3Ω 4Ω

Figure 5.1. Definition of the boundary-value problem.

D

Chapter Five

- 152 -

Figure 5.3. Substructure models and computational mesh.

(a-1) Coarse mesh (B/H=1.0) (a-2) Coarse mesh (B/H=0.2)

(b-1) Medium mesh (B/H=1.0) (b-2) Medium mesh (B/H=0.2)

(c-1) Fine mesh (B/H=1.0) (c-2) Fine mesh (B/H=0.2)

Chapter Five

- 153 -

Figure 5.5. Second-order components of wave loads on the horizontal rectangular obstacle with different sizes of the

bounded domain (B/H=1.0, D/H=0.4).

Dim

ensio

nles

s wav

e lo

ads

0

5

10

15

20

0 1 2 3 4 5

L=B

L=0.6B

L=0.2BAnalytic(Horizontal force)

Analytic(Vertical force)

Analytic(Moment)

fH

fV

m

kH

0

5

10

15

20

0 1 2 3 4 5

CoarseMediumFineAnalytic(Horizontal force)

Analytic(Vertical force)Analytic(Moment)

Figure 5.4. Second-order components of wave loads on the horizontal rectangular obstacle (B/H=1.0, D/H=0.4, L/B=1.0).

Dim

ensio

nles

s wav

e lo

ads

fH

fV

m

kH

Chapter Five

- 154 -


Dim

ensio

nles

s wav

e lo

ads

0

5

10

15

20

0 1 2 3 4 5

CoarseMedium

FineAnalytic(Horizontal Force)Analytic(Vertical Force)

Analytic(Moment)

fH

fV

m

kH

0

5

10

15

20

0 1 2 3 4 5

Coarse

Medium

FineAnalytic(Horizontal Force)

Analytic(Vertical Force)

Analytic(Moment)


Dim

ensio

nles

s wav

e lo

ads

fH

fV

m

kH

Chapter Five

- 155 -

0

5

10

15

20

0 1 2 3 4 5

CoarseMedium

FineAnalytic(Horizontal force)Analytic(Vertical force)

Analytic(Moment)


Dim

ensio

nles

s wav

e lo

ads

fH

fV

m

kH

Chapter Five

- 156 -

Figure 5.10. Vertical second-order wave loads, related to second-order velocity potential, on a fixed rectangular cylinder (B/H=1, D/H=0.8).

0

3

6

9

12

15

0 2 4 6 8 10

Coarse

Medium

Fine

Indirect method

Direct method

wave length / depth

|FV (2

) | / ρ

gA2

Figure 5.9. Horizontal second-order wave loads, related to second-order velocity potential, on a fixed rectangular cylinder (B/H=1, D/H=0.8).

0

2

4

6

8

10

12

0 2 4 6 8 10

Coarse

Medium

Fine

Indirect method

Direct method

wave length / depth

|FH (2

) | / ρ

gA2

Chapter Five

- 157 -

Figure 5.12. Vertical second-order wave loads, related to second-order velocity potential, on a fixed rectangular cylinder (B/H=1, D/H=0.6).

0

3

6

9

12

15

0 2 4 6 8 10

Coarse

Medium

Fine

Indirect method

Direct method

wave length / depth

|FV (2

) | / ρ

gA2

Figure 5.11. Horizontal second-order wave loads, related to second-order velocity potential, on a fixed rectangular cylinder (B/H=1, D/H=0.6).

0

2

4

6

8

10

12

0 2 4 6 8 10

Coarse

Medium

Fine

Indirect method

Direct method

wave length / depth

|FH (2

) | / ρ

gA2

Chapter Five

- 158 -

Figure 5.13. Pressure on the bottom of a semi-submerged horizontal cylinder of rectangular cross-section

(B=0.305m, D=0.3m, H=0.4m, A=0.014m, kH=3).

0

5

10

15

20

25

30

-0.4 -0.2 0 0.2 0.4

Coarse

Medium

Fine

Experiments

Sulisz&Johansson(1992)

Drimer&Agnon(1994)

Mod

ulus

of s

econ

d-or

der p

ress

ure

x (m)

(b)

(a)

0

5

10

15

20

25

30

35

-0.4 -0.2 0 0.2 0.4

Coarse

Medium

Fine

Experiments

Sulisz&Johansson(1992)

Drimer&Agnon(1994)

Mod

ulus

of f

irst-o

rder

pre

ssur

e

x (m)

Chapter Five

- 159 -

(a) o90<θ

Incident wave

B B

HD

Unbounded domain

Unbounded domain


Scaling center Scaling center

D H

B B

Incident wave

Unbounded domain Unbounded

domain


Scaling center Scaling center

(b) o90>θ

θ θ

θ θ

Figure 5.14. Substructured model for problems of second-order wave diffraction by a trapezoidal obstacle.

Chapter Five

- 160 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.15. Second-order component of horizontal wave forces on the obstacle for Example 2 (B/H=1.0, D/H=0.4, θ =30°).

0

3

6

9

12

15

0 1 2 3 4 5

24 elements

48 elements

96 elements

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.16. Second-order component of vertical wave forces on the obstacle for Example 2 (B/H=1.0, D/H=0.4, θ =30°).

0

4

8

12

16

20

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Five

- 161 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.18(a). Second-order component of horizontal wave force on obstacles with various base angles (θ ≤ 90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.17. Second-order component of moment about (B,-D) for Example 2 (B/H=1.0, D/H=0.4, θ =30°).

0

5

10

15

20

25

30

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Five

- 162 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.19(a). Second-order component of vertical wave force on obstacles with various base angles (θ ≤90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.18(b). Second-order component of horizontal wave force on obstacles with various base angles (θ ≥90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Chapter Five

- 163 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.20(a). Second-order component of moment about (B,-D) on obstacles with various base angles (θ ≤90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.19(b). Second-order component of vertical wave force on obstacles with various base angles (θ ≥90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Chapter Five

- 164 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.21. Second-order component of horizontal wave forces on the obstacle for Example 2 (B/H=1.0, D/H=0.2, θ =30°).

0

3

6

9

12

0 1 2 3 4 5

24 elements

48 elements

96 elements

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.20(b). Second-order component of moment about (B,-D) on obstacles with various base angles (θ ≥90°) for Example 2

(B/H=1.0, D/H=0.4).

0

5

10

15

20

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Chapter Five

- 165 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.23. Second-order component of moment about (B,-D) for Example 2 (B/H=1.0, D/H=0.2, θ =30°).

0

5

10

15

20

25

30

35

0 1 2 3 4 5

24 elements

48 elements

96 elements

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.22. Second-order component of vertical wave forces on the obstacle for Example 2 (B/H=1.0, D/H=0.2, θ =30°).

0

3

6

9

12

15

18

21

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Five

- 166 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.24(b). Second-order component of horizontal wave force on obstacles with various base angles (θ ≥ 90°) for Example 2

(B/H=1.0, D/H=0.2).

0

2

4

6

8

10

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.24(a). Second-order component of horizontal wave force on obstacles with various base angles (θ ≤ 90°) for Example 2

(B/H=1.0, D/H=0.2).

0

2

4

6

8

10

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Chapter Five

- 167 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.25(b). Second-order component of vertical wave force on obstacles with various base angles (θ ≥ 90°) for Example 2

(B/H=1.0, D/H=0.2).

0

5

10

15

20

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.25(a). Second-order component of vertical wave force on obstacles with various base angles (θ ≤ 90°) for Example 2

(B/H=1.0, D/H=0.2).

0

5

10

15

20

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Chapter Five

- 168 -

Dim

ensio

nles

s wav

e lo

ads

kH

0

5

10

15

20

25

0 1 2 3 4 5

30 degree

45 degree

60 degree

90 degree

Figure 5.26(a). Second-order component of moment about (B,-D) on obstacles with various base angles (θ ≤ 90°) for Example 2

(B/H=1.0, D/H=0.2).

Dim

ensio

nles

s wav

e lo

ads

kH

0

5

10

15

20

25

0 1 2 3 4 5

90 degree

120 degree

135 degree

150 degree

Figure 5.26(b). Second-order component of moment about (B,-D) on obstacles with various base angles (θ ≥ 90°) for Example 2

(B/H=1.0, D/H=0.2).

Chapter Five

- 169 -

(a) coarse mesh

(b) medium mesh

(c) fine mesh

B

D

H

Scaling centre

Bg

B1 B2

B

Figure 5.27. Substructured model and meshes for Example 3, consisting of two unbounded domains and four bounded domains.

Incident wave

Bounded domainUnbounded domain Unbounded domain

Chapter Five

- 170 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.28(b). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

-10

0

10

20

30

40

50

60

0.90 0.94 0.98 1.02 1.06 1.10

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.28(a). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

-10

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 171 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.29(a). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

-20

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.28(c). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

-10

0

10

20

30

40

50

60

70

3.00 3.04 3.08 3.12 3.16 3.20

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 172 -

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.29(c). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

0

200

400

600

800

1000

1200

3 3.04 3.08 3.12 3.16 3.2

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH

Figure 5.29(b). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).

-1

0

1

2

3

4

5

6

7

8

0.9 0.94 0.98 1.02 1.06 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 173 -

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

0.90 0.94 0.98 1.02 1.06 1.10

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 174 -

Dim

ensio

nles

s wav

e lo

ads

kH


-20

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

70

3 3.04 3.08 3.12 3.16 3.2

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 175 -

Dim

ensio

nles

s wav

e lo

ads

kH


0

200

400

600

800

1000

1200

3 3.04 3.08 3.12 3.16 3.2

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-1

0

1

2

3

4

5

6

7

8

0.9 0.94 0.98 1.02 1.06 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 176 -

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 177 -

Dim

ensio

nles

s wav

e lo

ads

kH


-200

20

406080

100

120140160180

200220

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

2 2.2 2.4 2.6 2.8 3

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 178 -

Dim

ensio

nles

s wav

e lo

ads

kH


-200

204060

80100120

140160180

200220

2 2.2 2.4 2.6 2.8 3

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 179 -

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 180 -

Dim

ensio

nles

s wav

e lo

ads

kH


-200

204060

80100120

140160180

200220

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

2 2.2 2.4 2.6 2.8 3

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 181 -

Dim

ensio

nles

s wav

e lo

ads

kH


-200

2040

6080

100

120140

160180

200220

2 2.2 2.4 2.6 2.8 3

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 182 -

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

12

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 183 -

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 184 -

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 185 -

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

12

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 186 -

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5 4

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-5

0

5

10

15

20

25

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

Coarse mesh

Medium mesh

Fine mesh

Chapter Five

- 187 -

Dim

ensio

nles

s wav

e lo

ads

kH


-10

0

10

20

30

40

50

60

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Coarse mesh

Medium mesh

Fine mesh

Dim

ensio

nles

s wav

e lo

ads

kH


-2

0

2

4

6

8

10

0.5 0.6 0.7 0.8 0.9 1 1.1

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 188 -

CHAPTER 6

SECOND-ORDER SOLUTION TO BICHROMATIC WAVE DIFFRACTION PROBLEMS

6.1 General

In Chapter 5, the SBFEM solution for monochromatic wave diffraction problems was

developed. In the context of a continuous spectrum, a monochromatic wave can be

regarded as the double-frequency component of two linear waves with identical

angular frequencies. However, the interaction between two linear waves with

different angular frequencies is of significance in the analysis and design of offshore

structures. Although the magnitudes of these nonlinear effects are in general second-

order, they may be of primary concern when such excitations are near the natural

frequencies of an offshore structure. For example, the sum-frequency wave load may

excite the response of ‘springing’ vibrations of ship hulls and tension leg platforms,

while the difference-frequency wave load is the excitation source of horizontal-plane

motions of moored ships (Kim & Yue 1990, Eatock Taylor & Huang 1997, Chen

2004).

Due to the presence of interaction between linear waves with different angular

frequencies, the nonhomogeneous term in the free water-surface boundary condition

becomes much more complicated than in the monochromatic wave diffraction

problem. This chapter attempts to generalize the scaled boundary finite-element

solution of regular wave problems to irregular wave cases and examines the ability of

this method to handle complex boundary conditions.

Chapter Six

- 189 -


6.2.1 Second-order incident potential of bichromatic wave

The physical model considered here is the same as that described in Chapter 5, except

that the incident monochromatic wave is a bichromatic wave. For clarity and

completeness, the incident velocity potential of a bichromatic wave is first derived.

Under the assumption of bichromatic wave theory, the first-order velocity potential

Ф(1) may be expressed as

]Re[ i)1(i)1()1( tj

ti

ji eeΦ ωω φφ −− += (6-1)

with

xk

l

l

l

ll

lehk

hzkA i)1(

)cosh()](cosh[

)ig

(+−

=ω

φ , l = i,j. (6-2)

where Al, ωl, kl, are the wave amplitude, angle frequency and wave number of the lth

linear wave, respectively.

To commence, it is noted that two arbitrary complex quantities always satisfy

)Re(21)Re(

21)Re()Re( *ababba += (6-3)

where a and b denote any two complex numbers, and * represents the conjugate of

the complex number.

Applying Equation (6-3) to the nonhomogeneous term of free water surface-boundary

condition (Equation (5-3)), Q(2) can be expressed as

tziixiizziiizii

i ie,,,,g

Q ωφωφωφφωφφω i222

3(2) )22Re[i(

21 −++−=

t)zjzijxjxijzzjiizji

ji jie,,,,,,g

ωωφφωφφωφφωφφωω +−++−+ i(

2

)22i(

Chapter Six

- 190 -

tzjziixjxiijzzijjzi

ji ije,,,,,,g

)i(2

)22i( ωωφφωφφωφφωφφωω +−++−+

])22i( i2223

tzjjxjjzzjjjzjj

j je,,,,g

ωφωφωφφωφφω −++−+

)22Re[i(21 3

*zizii

*xixii

*zziii

*zii

i ,,,,,,g

φφωφφωφφωφφω

++−+

t*zjzij

*xjxij

*zzjii

*zji

ji jie,,,,,,g

)i(2

)22i( ωωφφωφφωφφωφφωω −−−−−+

tzj

*ziixj

*xiij

*zzijj

*zi

ji ije,,,,,,g

)i(2

)22i( ωωφφωφφωφφωφφωω −−−−−+

)]22i(3

*zjzjj

*xjxjj

*zzjjj

*zjj

j ,,,,,,g

φφωφφωφφωφφω

++−+ (6-4)

The second-order incident velocity potential Ф(2) may be expressed as

Ctee

eeeeΦ

jjiit)(

jit)(

ij

t)(ji

t)(ij

tjj

tii

)(

ijji

ijjiji

+++++

+++=−−−−−−−−

+−++−+−+−+

]

Re[ii

ii2i2i2

φφφφ

φφφφωωωω

ωωωωωω

(6-5)

where +iiφ and +

jjφ represent the double-frequency potentials induced by waves with

frequencies iω and jω , +ijφ and +

jiφ represent the sum-frequency potentials by

interaction components of linear waves, −ijφ and −

jiφ represent the difference-

frequency potentials by interaction components of linear waves, −iiφ and −

jjφ represent

the zero-frequency potential induced by waves with frequency iω and jω

respectively. C is a constant (Stoker 1957). The last term of Equation (6-5) is

spatially independent, and only contributes to the vertical force on the structure

(Moubayed & Williams 1995). Consequently the loading induced by this term can be

Chapter Six

- 191 -

calculated separately. Taking into account that the double-frequency and zero-

frequency terms are the special cases (ωi=ωj) results in

]Re[Φ )(i)(i)(i)(i)2( tji

tij

tji

tij

ijjiijji eeee ωωωωωωωω φφφφ −−−−−−+−++−+ +++= (6-6)

Furthermore, with the symmetry conditions

]Re[]Re[ ii t)(ji

t)(ij

ijji ee ωωωω φφ +−++−+ = (6-7a)

]Re[]Re[ ii t)(*ji

t)(ij

jiji ee ωωωω φφ −−−−−− = (6-7b)

Equation (6-6) can be expressed as

]Re[ )i()i()2( tt jiji eeΦ ωωωω φφ −−−+−+ += (6-8)

with

+++ += jiij φφφ (6-9a)

*jiij−−− += φφφ (6-9b)

For simplicity and with no loss of generality, only the solution of velocity potential of

incident waves associated with the term tjie )i( ωω +− is discussed in detail here.

The term associated with tjie )i( ωω +− in Equation (6-4) may be expressed as

tzjzijxjxijzzjiizji

ji jie,,,,,,g

Q )i(2

)22(2i ωωφφωφφωφφωφφ

ωω +−++−= (6-10)

Correspondingly, the governing equation (Equation (5-1)) and the free water surface-

boundary condition (Equation (5-2)) may be reduced to

02 =∇ +ijφ (6-11)

Chapter Six

- 192 -

Qeg tiijjizij

ji =+− +−++ )(2 ])(,[ ωωφωωφ (6-12)


+++ =+− ijijjizij qg φωωφ 2)(, (6-13)

with

zjzijxjxijzzjiizjiji

ij ,,,,,,g

q φφωφφωφφωφφωω

ii2i

2i 2

++−=+ (6-14)

Noting that

0)1(2 =∇ lφ (6-15)

)1(2)1(llzl ,g φωφ = , l=i,j (6-16)

and substituting Equation (6-15) and Equation (6-16) into Equation (6-14) results in

xkkijjjijiji

ji

jiij

jiekgkkgAA

q )(i222432 ]22[2

i ++ −−+−

= ωωωωωωωω

(6-17)

Applying the method of separation of variables, +ijφ may be expressed as

)(zZqijij++ =φ (6-18)


1)(, 2 =+− ZgZ jiz ωω (6-19)

The solution of Equation (6-19) can be expressed as

)])(cosh[( hzkkDZ ji ++= (6-20)


Chapter Six

- 193 -

])cosh[()(])sinh[()(1

2 hkkhkkkkgD

jijijiji ++−++=

ωω (6-21)

Consequently the solution for the sum-frequency ( ji ωω + ) velocity potential is

determined as

2)(])tanh[()(])cosh[(/)])(cosh[(

jijiji

jijiijij hkkkkg

hkkhzkkq

ωωφ

+−++

+++= ++ (6-22)

In the same way, the solution for the sum-frequency ( ij ωω + ) velocity potential can

be written as

2)(])tanh[()(])cosh[(/)])(cosh[(

jijiji

jijijiji hkkkkg

hkkhzkkq

ωωφ

+−++

+++= ++ (6-23)

Combining Equation (6-22) and Equation (6-23) yields

2)(])tanh[()(])cosh[(/)])(cosh[(

jijiji

jiji

hkkkkghkkhzkk

qωω

φ+−++

+++= ++ (6-24)

where

xkk

jj

j

ii

i

jiji

jijiji

x)kk(jiiijijij

ji

ji

x)kk(ijjjijiji

ji

ji

jiij

ji

ji

ji

ehk

khk

k

hkhkkkgAA

ekgkkgAA

ekgkkgAA

qqq

)i(22

2

i222432

i222432

)])cosh()cosh(

(21

)1))tanh()(tanh(([i

]22[2

i

]22[2

i

+

+

+

+++

+−

−+

−=

−−+−

+

−−+−

=

+=

ωω

ωωωω

ωωωωωωωω

ωωωωωωωω

(6-25)

Chapter Six

- 194 -

In the limit of a single regular wave, ij ωω → , the incident velocity potential with

sum-frequency term ji ωω + (Equation 6-22) reduces to the well-known second-

order uniform Stokes wave (5-36b). It is apparent that Equation (6-23) can produce

the same result for this limiting case.

Replacing ),( jjk ω in Equation (6-22) with ),( jjk ω−− leads to the solution of

difference-frequency ( ji ωω − ) incident velocity potential,

2)(])tanh[()(])cosh[(/)])(cosh[(

jijiji

jijiijij hkkkkg

hkkhzkkq

ωωφ

−−−−

−+−= −− (6-26)

with

xkkijjjijiji

ji

*ji

ijjiekgkkg

AAq )i(222432 ]22[

2i −− −−+−= ωωωωωω

ωω (6-27)

Likewise, the solution of difference-frequency ( ij ωω − ) incident velocity potential

can be expressed as

2)(])tanh[()(])cosh[(/)])(cosh[(

jijiji

jijijiji hkkkkg

hkkhzkkq

ωωφ

−−−−

−+−= −− (6-28)

with

xkkjiijiijij

ji

j*

iji

jiekgkkgAA

q )i(-222432 ]22[2i −− −−+−= ωωωωωω

ωω (6-29)

Substituting Equation (6-26) and Equation (6-28) into Equation (6-9b) yields,

2)(]))tanh[((]))]/cosh[()(cosh[(

jijiji

jiji

hkkkkghkkhzkk

qωω

φ−−−−

−+−= −− (6-30)

where

Chapter Six

- 195 -

xkk

jj

j

ii

i

jiji

jiji

*ji

xkkjiijiijij

ji

*ji

xkkijjjijiji

ji

*ji

*jiij

ji

ji

ji

ehk

khk

k

hkhkkkgAA

ekgkkgAA

ekgkkgAA

qqq

)i(22

2

)i(-222432

)i(222432

)])cosh()cosh(

(21

)1))tanh()(tanh(([i

]22[2i

]22[2i

)(

−

−

−

−−

−+

+−

=

−−+−+

−−+−=

+=−

ωω

ωωωω

ωωωωωωωω

ωωωωωωωω

(6-31)

6.2.2 Second-order wave force

Like the expression for the second-order velocity potential of a bichromatic wave, the

second-order wave force consists of sum-difference and difference-frequency

components. The wave pressure p in terms of complete velocity potential can be

expressed as

22 /Φ,Φgzp t ∇−−−= ρρρ (6-32)

Hence, the wave force )(tFv

acting on a body may be expressed as

∫∫=S

dSnptF rv)( (6-33)

in which S denotes the wetted surface of body. Considering the two-dimensional case

(Figure 5.1) and substituting Equation (2-7) into Equation (6-33), the second-order

component of the wave force can be written as

WC FFtFrrv

+=)()2( (6-34)

with

dSnΦ

,ΦFD t

)(C

rr)

2(-

2)1(0

-

2∇

+= ∫ρ

ρ (6-35a)

Chapter Six

- 196 -

dSnΦgzF tWrr

),( )1(

0 2 ερ

ερη

+−= ∫ (6-35b)

where D is the draft (See Figure 5.1).

Substituting Equation (6-1) and Equation (6-8) into Equation (6-35a) yields

tC

tCC

jiji efefF )i()i( )()( ωωωω −−−+−+ +=vvr

(6-36)

with

dSnf jijiDCrv

]21)([i )1()1(0

-φφφωωρ ∇∇−+= ++ ∫ (6-37a)

dSnf jijiDCrv

]21)([(i *)1()1(0

-φφφωωρ ∇∇−−= −− ∫ (6-37b)

Substituting Equation (2-9) and Equation (6-1) into Equation (6-35b) results in

tW

tWW

jiji efefF )i()i( )()( ωωωω −−−+−+ +=vvr

(6-38)

with

)1()1(

2 jijiW gf φφωωρ

∇∇−=+v (6-39a)

*)1()1(

2 jijiW gf φφωωρ

∇∇=−v (6-39b)

Thus, taking into account Equations (6-37a and b) and Equations (6-39a and b), the

second-order wave force also can be written as the summation of sum-frequency

(high-frequency) and difference-frequency (low-frequency) wave loads

−+ += FFtFrrv

)()2( (6-40)

where

Chapter Six

- 197 -

tWC

jieffF )i()( ωω +−+++ +=vvr

(6-41a)

tWC

jieffF )i()( ωω −−−−− +=vvr

(6-41b)

6.2.3 Second-order wave surface elevation

The second-order wave surface elevation )2(η may be expressed as

]21[1 )1()1()1()1()2()2( ΦΦΦΦ

g tzt ∇⋅∇++−= ηη (6-42)

Substituting Equation (6-1), Equation (6-8) into Equation (6-42) yields

)Re( )(i)(i)2( tt jiji ee ωωωω ηηη −−−+−+ += (6-43)

with

)1()1()1()1()1()1(2 2

1),,(2

)(ijijzizji

jiji

gggφφφφφφ

ωωφ

ωωη ∇⋅∇−+−

+= ++ (6-44a)

*)1()1(*)1()1(*)1()1(2 2

1),,(2

)(ijijzizji

jiji

gggφφφφφφ

ωωφ

ωωη ∇⋅∇−++

−= −− (6-44b)

where +η and −η are the sum-frequency and difference-frequency components of

the second-order wave surface elevation, respectively.

6.3 Scaled boundary finite-element equation

Following the procedure described in Chapter 5, for a bounded domain, the scaled

boundary finite-element equations may be expressed as

ΠaMaE,aE eeT

ee +=+ ±±± )(][)(][)(][ 210 ξκξξξ ξ (6-45)

∫ ±±± =+s

Te

Tee dsvsNaE,aE ξ

ξ τξξξ )]([)(][)(][ 10 (6-46)

Chapter Six

- 198 -

0)(][)(])[][]([)(][ 21102

0 =−−++ ±±± ξξξξξ ξξξ aE,aEEE,aE T (6-47)

with

gji

22 )( ωω

κ±

=± (6-48)

∫ ±=s e

T dssΞsNΠ ξτξ ),()]([ (6-49)

where the positive sign represents the high-frequency term and the negative sign

indicates the low-frequency term. The expressions of +Ξ and −Ξ are

+++ += jiij ΞΞΞ (6-50a)

*jiij ΞΞΞ )( −−− += (6-50b)

with

xxjii

xjxij

jijiji

ij

,g

,,g

gΞ

)1()1()1()1(

)1()1(3

432

2ii

2)i(2

φφω

φφω

φφωωωω

++

+=+

(6-51a)

xx*

jii

xjxij

jijiji

ij

gg

gΞ

,)(2i

,)(,i

)(2

)i(-2

)1()1()1()1(

)1()1(3

432

φφω

φφω

φφωωωω

+−

+=

∗

∗−

(6-51b)

For an unbounded domain, to clear the radiation condition at infinity to be enforced

conveniently (refer to Chapter 3), the scattered nodal potential vector is again selected

as the dependent variable of the equations. Consequently, the scaled boundary finite-

element equations are

∫ ±±± −=+s

TiS

TiS dsvsNaEaE ξ

ξ τξξ )()]([)(][),(][ 10 (6-52)

Chapter Six

- 199 -

∫ ±±± −=+s

TeS

TeS dsvsNaEaE ξ

ξ τξξ )()]([)(][),(][ 10 (6-53)

)()(])[][(

),(])[]([),(][

202

110

ξξκ

ξξ ξξξ

±±

±±

−=−+

−+

tS

ST

S

paEM

aEEaE (6-54)

with

sT sNsNM τ)]([)]([][ 110 = (6-55)

±± = SsT

t ΞsNp τξ )]([)( 1 (6-56)

where ±SΞ has a different form from that in the equations for bounded domains,

formulated as

+++ += Sji

SijS ΞΞΞ (6-57a)

*Sji

SijS ΞΞΞ )( −−− += (6-57b)

with

),,,(2

)i(

),,,,,,()i(

)(2

)i(2

)1()1()1()1()1()1(

)1()1()1()1()1()1(

)1()1()1()1()1()1(3

432

xxS

jS

ixxS

jI

ixxI

jS

iji

xS

jxS

ixS

jxI

ixI

jxS

iji

Sj

Si

Sj

Ii

Ij

Si

jijiSij

g

g

gΞ

φφφφφφωω

φφφφφφωω

φφφφφφωωωω

+++

+

+++

+

+++

=+

(6-58a)

),)(,)(,)((2i

),)(,,)(,,)(,(i

))()()((2

)i(-2

)1()1()1()1()1()1(

)1()1()1()1()1()1(

)1()1()1()1()1()1(3

432

xx*S

jS

ixxS

jI

ixxI

jS

ii

xS

jxS

ixS

jxI

ixI

jxS

ij

Sj

Si

Sj

Ii

Ij

Si

jijiSij

g

g

gΞ

φφφφφφω

φφφφφφω


+++

++−

+++

=

∗∗

∗∗∗

∗∗∗−

(6-58b)

Chapter Six

- 200 -

Comparing Equation (6-58) with Equation (6-51), it can be seen that Equation (6-58)

contains only the effects of the first-order interactions of the incident and scattered

waves and two first-order scattered wave interactions, cancelling the effect of two

first-order incident wave interactions.

Due to the first-order component interactions, the scaled boundary finite-element

equation for bichromatic wave diffraction problems becomes much more complicated

than that for monochromatic wave cases. Fortunately, the forms of differential

equations for the two types of problems are identical. Consequently the major task of

solving the resulting scaled boundary finite-element equation is to determine the

integration constants analytically.


The second-order solution of bichromatic wave diffraction problems is based on the

first-order solutions of two linearized wave with different frequencies. Since the

solution method of linear problems was introduced in Chapter 3, the first-order

solutions are regarded as known qualities here. The bounded domain solution

procedure for bichromatic wave diffraction problems is very similar to that of

monochromatic wave diffraction problems, except that the incident wave potential

should be replaced with Equation (6-24) and Equation (6-30) respectively for the

sum-frequency and the difference-frequency cases. Consequently this section focuses

on the derivation of the unbounded domain solution only.

6.4.1 General solution

Using the technique of the standard scaled boundary finite-element method and

introducing

⎭⎬⎫

⎩⎨⎧

=±

±±

)()(

)(ξξ

ξqa

X (6-59)

⎭⎬⎫

⎩⎨⎧−

= ±±

)(0

)(ξ

ξp

F (6-60)

Chapter Six

- 201 -

Equation (6-52) and Equation (6-54) can be transformed into the following forms

)()(][),( ξξξ ξ±±±± += FXZX (6-61)

with

⎥⎦

⎤⎢⎣

⎡

−−−

=−±−

−−±

1010

21

1012

101

10

]][[][][]][[][][][][

][EEMEEEE

EEEZ T

T

κ (6-62)

Following the same notation conventions used in the proceeding paragraphs, the

positive and negative signs in superscripts denote the sum-frequency and difference-

frequency cases respectively.

The general solution of Equation (6-61) can be written as

)(][)( ][ ξξ ξ ±±± ±

= CeΦX Λ (6-63)

where the matrices ][ ±Λ and ][ ±Φ are the eigenvalue matrix and eigenvector matrix

respectively, satisfying,

]][[]][[ ±±±± = ΛΦΦZ (6-64)

As noted previously, the eigenvalues consist of two groups with opposite signs

⎥⎦

⎤⎢⎣

⎡=

±−

±+±

][][

][m

mλΛλ

, .,,3,2,1 nm L= (6-65)

where +mλ represents the eigenvalues with a positive real part or a negative pure

imaginary number, while −mλ denotes correspondingly those with negative real part or

a positive pure imaginary number.


)(][),( 1][ ξξ ξξ

±−±−± ±

= FΦeC Λ (6-66)

Chapter Six

- 202 -

The solution of Equation (6-66) can be expressed as

)(][)(0

][ ±±±−± += ∫±

cduuFAeC uΛξξ (6-67)

with

1][][ −±± = ΦA (6-68)

Partitioning the vector )( ξ±C and the matrix ][ ±Φ into the same block forms as

the vector )( ξ±X and the matrix ±][Λ results in

⎭⎬⎫

⎩⎨⎧

= ±

±±

)()()(

2

1

ξξξ

CCC (6-69)

⎥⎦

⎤⎢⎣

⎡= ±±

±±±

2221

1211][ΦΦΦΦΦ (6-70)

In the same way, the matrix ][ ±A is written in block form as

⎥⎦

⎤⎢⎣

⎡= ±±

±±±

2221

1211][AAAAA (6-71)


⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

±

±

±±

±±

±

±

±−

±+

)()(

)()(

2

1][

][

2221

1211

ξξ

ξξ

ξλ

ξλ

CC

ee

ΦΦΦΦ

qa

m

m

(6-72)

or

)(][)(][)( 2][

121][

11 ξξξ ξλξλ ±±±±± ±−±+

+= CeΦCeΦa mm (6-73a)

)(][)(][)( 2][

221][

21 ξξξ ξλξλ ±±±±± ±−±+

+= CeΦCeΦq mm (6-73b)

Equation (6-67) becomes

Chapter Six

- 203 -

)(][)( 10 12][

1±±±−± +−= ∫

±+

cduupAeC tum

ξ λξ (6-74a)

)(][)( 20 22][

2±±±−± +−= ∫

±+

cduupAeC tum

ξ λξ (6-74b)

where 1±c and 2

±c are the blocks in the integration constant vector ±c .

6.4.2 Determination of integration constants

The solution at infinity should be finite, hence, the )( 1 ξ±C must satisfy

0)( 1 =∞→± ξC (6-75)

Equation (6-73a) describes the properties of the far-field velocity potential of the

scattered wave at infinity. It is noted that a pure imaginary eigenvalue corresponds to

a propagating mode of the scattered wave. A positive quantity indicates an outgoing

wave, satisfying with the boundary condition at infinity, while a negative quantity

represents a returning wave. An eigenvalue with a negative real part corresponds to

an evanescent mode, vanishing at infinity.

Substituting Equation (6-74a) into Equation (6-75) yields

∫∞ ±±−± ±+

=0 12

][1 )(][ duupAec t

umλ (6-76)

Inserting Equation (6-76) back into Equation (6-74a) results in

∫∞ ±±−± ±+

=ξ

λξ duupAeC tum )(][)( 12

][1 (6-77)

The explicit expression of )( 1 ξ±C is provided in Appendix 6-A.

Applying the equation

])[]][[]([][ 111

1222211

12±−±±±−± −= ΦΦΦΦA (6-78)

Chapter Six

- 204 -

and cancelling )( 2 ξ±C in Equations (6-73a) and (6-73b) results in the standard

form of the finite element equation

)()()(][ ξξξ ±±±± += RqaH (6-79)

with

11222 ]][[][ −±±± = ΦΦH (6-80)

and

)(][)( 1][1

12 ξξ ξλ ±−±± ±+

−= CeAR m (6-81)

Matching the boundary conditions of the bounded and unbounded domain, Equation

(6-79) can be solved with respect to the nodal potential vector ±a . Then,

substituting ±a into Equation (6-73a) yields

][][ 212111±±±±± += cΦcΦa (6-82)

and the integration constants 2±c can be expressed as

][][ 111-1

122±±±±± −= cΦaΦc (6-83)


Three numerical examples regarding bichromatic wave diffraction are computed in

this section using the SBFEM. The accuracy and convergence of the SBFEM for

solving the bichromatic wave diffraction problems are validated through these

numerical examples. In the first example, the scaled boundary finite-element

solutions to the sum-frequency and difference-frequency wave loads on a single

horizontal rectangular are illustrated. Also, the computed second-order wave

reflection and transmission coefficients are discussed. Then in the second example

the effect of sharp corners in the fluid domain on the sum-frequency and difference-

Chapter Six

- 205 -

frequency wave loads are studied. The last example demonstrates the ability of the

SBFEM to solve problems of bichromatic wave diffraction by twin rectangular

obstacles. For the purpose of comparison, the same set of problem geometries used in

the examples discussed in Chapter 5 is employed again, but this time with

bichromatic waves.

6.5.1 Wave diffraction by a rectangular obstacle (Example 1)

6.5.1.1 Second-order wave forces

As is well known, when ωi is identical to ωj, the sum-frequency wave diffraction

problem reduces to a monochromatic wave diffraction problem. Sulisz (1993)

developed an analytical solution for such a problem. To validate the present method,

the double-frequency (ωi=ωj) component of wave loads is computed initially using

the computational procedure described in this Chapter. In this numerical example, the

configuration of the computational domain is taken to be L=0.2B, B/H=1.0 and

D/H=0.4 (Figure 5.1). Three meshes of differing levels of refinement (Figures 5.1(a-

2, b-2 and c-2)) are used to study the convergence of the numerical results. The

coarse mesh, the medium mesh and the fine mesh consist of 12, 24 and 48 elements

respectively.

Figures 6.1-6.3 plot the computed variations of dimensionless wave loads, along with

the analytical solution presented by Sulisz (1993). It can be seen that the present

numerical results agree well with the analytical solutions for the sum frequency

problem and converge quickly.

The general sum-frequency and difference-frequency components of wave loads are

also computed. The second-order sum-frequency and difference-frequency horizontal

and vertical wave forces are normalized by the factor ρgAiAj and the moment about

(B, -D) by the factor ρgAiAjH. The range of the dimensionless wave number kiH is

taken to be identical to the double frequency case (Figures 6.1-6.3). In the present

numerical calculation, it is assumed that ωi+ωj=7.0. Figure 6.4 and Figure 6.5 plot the

computed variations of second-order components of sum-frequency and difference-

frequency wave loads (horizontal wave force, vertical wave force and moment about

Chapter Six

- 206 -

(B,-D) respectively. No numerical difficulties (such as irregular frequency problems)

were encountered by the SBFEM for this example.

6.5.1.2 Second-order wave reflection and transmission coefficients

Using the SBFEM, the second-order wave reflection and transmission coefficients in

the sum-frequency and difference-frequency problems are investigated in this section.

When −∞→x , the evanescent modes in the reflected waves vanish and there exists

only the propagating mode of the second-order reflected waves (refer to the Section

6.4.2). Consequently the second-order wave reflection coefficient ±)2(

rK may be

expressed as

||||)2(

±

±±=

I

rrK

ηη

(6-84)

where the subscripts ± represent the cases of sum-frequency and difference-

frequency, ηr and ηI indicate the second-order reflected wave surface elevation and

the second-order incident wave surface elevation, respectively.

Likewise, when +∞→x , the second-order component of transmission waves is

composed of the second order incident wave and the propagating mode of the second-

order transmission wave. Correspondingly, the second-order transmission wave

coefficient ±)2(

tK can be expressed as

||||)2(

±

±±=

I

ttK

ηη

(6-85)

where ±tη denotes the second-order transmission wave surface elevation.

The example discussed in the Section 6.5.1 is now extended to the water depths

D/H=0.4, D/H=0.6 and D/H=0.8. The second-order reflection and transmission

coefficients with respect to various wave frequency combinations are computed for

the sum-frequency and difference-frequency problems.

Chapter Six

- 207 -

Figures 6.6-6.9 compare the second-order wave reflection and transmission

coefficients for the sum-frequency and difference-frequency problems. The

dimensionless wave number kiH ranges from 0.01 to 3.0 and the other dimensionless

wave number kjH is taken to be 0.01. It can be clearly seen that the overall trends of

variations of the second order wave reflection and transmission coefficients are

similar to those of first-order problem (refer to Chapter 3). This is because the

influence of the linear wave with the very small wave number kjH on the other linear

wave is so weak that the second order effect of wave interaction can be ignored. The

second-order wave reflection coefficients for both sum-frequency and difference-

frequency problems increase with the increasing dimensionless draft, while the wave

transmission coefficients decrease.

For the case of kjH=1.0, as shown in Figures 6.10, the sum-frequency second-order

wave reflection coefficient dramatically increases with the increasing dimensionless

wave number kiH from 0 to around 2.0 but reaches to equilibrium for kiH >2.0. When

the draft increases from 0.4H to 0.8H, it can be seen that the second-order wave

reflection coefficient slightly decreases. Figure 6.11 plots variations of the sum-

frequency second-order wave transmission coefficients. In contrast to the case of

kjH=0.01, the sum-frequency second-order wave transmission coefficient shows a

different trend. It decreases initially and then increases when kiH is greater than a

critical value and less than 1.25 and approaches to equilibrium when kiH is greater

than 1.25. From Figure 6.10 and Figure 6.11, it can be seen that the second-order

effect in the case of kjH=1.0 becomes more noticeable than when kjH=0.01. This is

probably because the increase of the dimensionless wave number kjH leads to a

higher sum-frequency ( ji ωω + ). Consequently, the effect of two linear waves

interaction becomes significant. For the difference-frequency second-order wave

reflection coefficients illustrated in Figures 6.12 and 6.13, it can be seen that there are

dramatic increases of both reflected and transmitted wave surface elevations

approximately between kiH =0.7 and kiH =1.5. The increases become more

considerable with the increasing draft. In this wave frequency range, the maximum

wave reflection and transmission coefficient happens where kiH =1.0. This point

corresponds to the case of zero-frequency. The ratio of the draft to water depth has no

Chapter Six

- 208 -

noticeable effect on the difference-frequency wave reflection and transmission

coefficients.

For the case of kjH=2.0, compared with the cases of kjH=0.01 and kjH=1.0, the sum-

frequency wave reflection and transmission coefficients become higher. They

gradually increase when kiH is less than 2.0 and then tend to be stable, as shown in

Figures 6.14 and 6.15. Like the case of kjH=1.0, the sum-frequency second-order

wave reflection coefficient slightly declines with increasing draft, while the variation

of draft does not lead to a dramatic change in the transmission coefficients. It is worth

noting in Figures 6.16 and 6.17 that significant increases of difference-frequency

wave reflection and transmission coefficients are found when the dimensionless wave

number kiH is between about 1.5 and 2.5. The maximum difference-frequency wave

reflection and transmission coefficients again occur in the case of zero-frequency (kiH

= kjH=2.0) and higher ratio of the draft to the water depth leads to larger difference-

frequency second-order wave reflection and transmission coefficients.

6.5.2 Wave diffraction by a trapezoidal obstacle (Example 2)

One of advantages of the SBFEM is that it is able to deal with the singularity of the

velocity field near sharp corners with ease. This is because the SBFEM does not need

to discretize the boundary at the surface of structures and the solutions in the radial

direction are analytical. As discussed in section 6.2.2, the solutions to sum-frequency

and difference-frequency wave loads in problems of bichromatic wave diffraction are

related to the first-order solution of velocity field, so the SBFEM may be a very good

numerical approach for predicting of the sum-frequency and difference-frequency

wave loads, due to its advantage in modelling the first-order velocity field around

sharp comers.

Section 5.5.2 in Chapter 5 investigated the accuracy and convergence of the SBFEM

in handling problems with sharp corners, and studied the effect of the sharp corners in

the fluid domain on the monochromatic second-order wave forces. This section will

discuss the case of bichromatic waves. To allow comparison, both the geometry and

the assignment of elements for numerical examples in section 5.5.2 are employed

again for the bichromatic wave case (See Figure 5.14). The second-order sum-

Chapter Six

- 209 -

frequency and difference-frequency horizontal and vertical wave forces are

normalized by the factor ρgAiAj, andthe moment about (B, -D) by the factor ρgAiAjH.

In all cases, ji ωω + is taken to be 7.

Firstly, the case with the base angle θ=30° is calculated. Three meshes are employed,

with 24, 48 and 96 elements respectively. The computed variation of the

dimensionless sum-frequency horizontal wave force with the dimensionless wave

number (kiH) is plotted in Figure 6.18. It can be seen that the scaled boundary finite-

element solutions converge very quickly with the increase of mesh density. The

results of the sum-frequency vertical wave force and moment about (B,-D) are shown

in Figures 6.19 and 6.20. The solutions also converge very well. As discussed in the

proceeding paragraphs, the accuracy of the computed second-order wave forces

depends on the accuracy of the solutions to the first-order velocity field. The SBFEM

can calculate the first-order velocity field near sharp corners accurately, so the sum-

frequency solutions to wave loads from the SBFEM converge quickly. Figures 6.21-

6.23 show the computed variation of the dimensionless difference-frequency

horizontal wave force, vertical wave force and moment about (B,-D), respectively.

The difference-frequency results demonstrate the same good convergence of the

SBFEM.

In order to investigate the influence of base angle (θ) on the sum-frequency and

difference-frequency wave loads, the cases with θ=45°, 60°, 120°, 135° and 150° are

calculated with the 48-element mesh. These results, along with the solution for the

cases of θ=30° and θ=90° (discussed in the section 6.5.1) are shown in Figures 6.24-

6.29. It is found that the changes of sum-frequency horizontal wave force (Figures

6.24(a) and 6.24(b) induced by the variation of base angle are not considerable.

However, the sum-frequency vertical wave force and moment about (B,-D) change

significantly when θ≤90° (Figures 6.25(a) and 6.26(a)), but not noticeably in the

cases where θ≥90° (Figures 6.25(b) and 6.26(b)). The calculated difference-frequency

wave loads show that the difference-frequency horizontal wave force, vertical wave

force and moment about (B, -D) change significantly when the base angle becomes

sharper (Figures 6.27(a), 6.27(b) and 6.27(c)). In contrast, with an increase of the

base angle, the effect of changes of base angle on the difference-frequency horizontal

Chapter Six

- 210 -

wave force, vertical wave force and moment about (B,-D) becomes smaller and

smaller.

In conclusion, the sharper the corners in the fluid domain, the greater the effect on the

sum-frequency and difference-frequency wave loads.

6.5.3 Wave diffraction by twin rectangular obstacles (Example 3)

The geometries of the numerical examples discussed in the section 5.5.3 are

employed again in this section to study the problem of bichromatic wave diffraction

by twin rectangular obstacles with a narrow gap. Again, in all cases ji ωω + is taken

to be 7 and the amplitudes of second-order wave forces are normalized by the factor

ρgAiAj. The three types of meshes used in the section 5.5.3, the coarse mesh, medium

mesh and fine mesh, are used again for the case of bichromatic waves in this section.

In the first case of this example, the gap distance is considered to be 0.01. The first-

order solution and the double-frequency solution for this case have already been

discussed in Chapter 3 and Chapter 5. This section applies the SBFEM to calculate its

sum-frequency and difference-frequency solutions. The computed variation of the

sum-frequency component of the horizontal wave force on obstacle B1 with the

normalized wave number kiH is plotted in Figure 6.30(a). The resonant phenomena

can be found at the frequency bands around kiH=0.48 and kiH=3.14 where the

normalized wave number kjH of the other first-order wave equals 3.14 and 0.48

correspondingly. That means the resonant phenomena take place when either of

normalized first-order wave numbers kiH and kjH in bichromatic waves is close to the

resonant frequencies 0.48 or 3.14. For clarity, the enlarged portions of Figure 6.30(a)

at the resonant frequency bands are shown in Figures 6.30(b) and 6.30(c). The results

of the sum-frequency component of second-order vertical wave force on obstacle B1

are illustrated in Figures 6.31(a), (b) and (c), where (b) and (c) are enlarged portions

at the resonant frequency bands. The same resonant phenomena are found in

computed results with respect to the sum-frequency vertical wave forces on obstacle

B1. It can be seen that the resonant frequencies for the horizontal wave force and the

vertical wave force are consistent. For the other obstacle (B2), Figures 6.32 and 6.33

show the computed variation of the sum-frequency horizontal and vertical wave

Chapter Six

- 211 -

forces with the normalized wave number kiH. It is found that the sum-frequency

horizontal wave force and the sum-frequency vertical wave force acting on the two

obstacles are very similar.

The difference-frequency solutions for this case are illustrated in Figures 6.34-6.37.

Figures 6.34(a) plot the variation of difference-frequency component of second-order

wave force on obstacle B1. It is found that there are four resonant frequency bands

for this problem. They occur at around kiH=0.24, 0.48, 3.14 and 3.99 when kjH equals

3.99, 3.14, 0.48 and 0.24 respectively. Figures 6.34(b) and 6.34(c) show the

convergence of the difference-frequency solutions obtained from the SBFEM when

the normalized wave numbers are close to the resonant frequencies with enlarged

diagrams. The computed results for the difference-frequency component of the

vertical wave force on obstacle B1 are plotted in Figures 6.35(a). At the same

frequency bands (kiH=0.24, 0.48, 3.14 and 3.99), resonant frequencies are found. The

enlarged portions of Figure 6.35(a) near these frequencies are shown in Figures

6.35(b) and 6.35(d). In Figure 6.35(a), it seems that a abrupt change takes place at

around kiH=1.41. For clarity, the diagram from kiH=1.38 to kiH=1.45 is enlarged and

plotted in Figure 6.35(c). It can be seen clearly that this abrupt change results from

the mesh density. With an increase of mesh density, the solution becomes smoother

and the abrupt change vanishes. The results regarding the difference-frequency

horizontal and vertical wave forces on obstacle B2 are shown in Figures 6.36 and

6.37, respectively. Compared with the results of difference-frequency wave forces on

the obstacle B1, the resonant phenomena occurs at the same frequency bands.

In this section another two cases (Bg equals 0.05 and 0.1) are used to investigate the

effect of the gap distance between two obstacles on the resonant phenomena of

bichromatic waves. Figures 6.38-6.41 give the computed sum-frequency wave forces

on obstacle B1. The resonant frequencies are found to be kiH=0.69 and 2.57. The kjH

equals 2.57 and 0.69 respectively. In the difference-frequency solutions of wave

forces on obstacle B1, as shown in Figures 6.42 and 6.43, two more resonant

frequencies, kiH=0.32 and 3.69, are found besides kiH=0.69 and 2.57. Corresponding

to kiH=0.32 and 3.69, kjH equals 3.69 and 0.32 when the resonant phenomena

happen. The same resonant phenomena appear in the computed results of difference-

frequency wave forces on obstacle B2, as shown in Figures 6.44-6.45. The numerical

Chapter Six

- 212 -

results obtained from the SBFEM for the case Bg=0.1 are shown in Figure 6.46-6.53.

The resonant frequencies for the sum-frequency solutions are kiH=0.85 and 2.21

where kjH equals 0.85 and 2.21 respectively. For the difference-frequency solutions,

the resonant frequencies are found to be kiH=0.39, 0.85, 2.21 and 3.44 where kjH is

equal to 3.44, 2.21, 0.85 and 0.39.

6.6 Summary

This Chapter extends the SBFEM to the second-order solution of bichromatic wave

diffraction problems. This class of problem involves both the sum-frequency and the

difference-frequency cases, due to the interaction of first-order terms in the

inhomogeneous free surface-boundary condition, leading to a more complicated

system of ordinary differential equations than that of the monochromatic wave

diffraction problems. To seek the solution of the resulting scaled boundary finite-

element equation, an analytical integration procedure is implemented, employing the

first-order scaled boundary finite-element solutions with different wave numbers. The

computational procedure is implemented and numerical results of the second-order

components of wave loads are validated by examining an extreme case (double-

frequency) where an analytical solution is available. The numerical results indicate

that the SBFEM is able to retain an analytical solution in the radial direction for the

bichromatic wave diffraction problems, so the unbounded domain and singularities

can be handled with ease. No numerical difficulties such as irregular frequency

problems are encountered in the examples examined in this chapter.

Chapter Six

- 213 -

Appendix 6-A Derivation of variation parameter C1±(ξ).

The first-order scaled boundary finite-element solution of incident and scattered wave

potentials at the mean water surface can be expressed as (See Chapter 2 for more

details)

)(i)1( i)( uxk

l

lIl

clegA +−=ω

ξφ l=i,j. (6-A-1)

∑=

−

=n

llnS

l ce l

1

)1(2

)1( )(α

αξλ

ααγξφ l=i,j. (6-A-2)

where ωl and kl denote the angular frequency and wave number of the l-th incident

linearized wave, and Al is the corresponding wave amplitude. The symbol αγ ln

represents the element at the joint of the n-th row and the α-th column of the

eigenvector matrix while −αλl is the eigenvalue of the α-th eigenvector. )1(

2αlc is the α-

th element of the first-order integration vector.

Firstly, the case associated with the sum-frequency term is considered. For the

convenience define

)()]([)( 1 ξξ ++ = ΘsNp Tt (6-A-3)

where

++ = SsΞΘ τξ )( (6-A-4)

In the context of the modified scaled boundary co-ordinate system, for the subdomain

shown in Figure 5.2(b),

1=sτ (6-A-5)

The shape function [N(s1)] is specified as

]1,0,,0,0[)]([ 1 L=sN (6-A-6)

Chapter Six

- 214 -

The variation parameter C1±(ξ) can be expressed as

)()()( 111 ξξξ +++ += jiij CCC (6-A-7)

For the purpose of generality, the derivation procedure of )( 1 ξ+ijC is introduced

here.

Substituting Equations (6-A-3)-(6-A-6) into Equation (6-75) yields

)](][[

),,,(2

)i(

),,,,,,()i(

)(2

)i(2

)](][[)()(

112

)1()1()1()1()1()1(

)1()1()1()1()1()1(

)1()1()1()1()1()1(3

432

][

112][

1

sNAdu

g

g

g

e

sNAduΘeC

ij

xxS

jS

ixxS

jI

ixxI

jS

iji

xS

jxS

ixS

jxI

ixI

jxS

iji

Sj

Si

Sj

Ii

Ij

Si

jiji

u

ijiju

ij

ijm

ijm

+

∞ −

+∞ +−+

+++

+

+++

+

+++

=

=

∫

∫++

++

φφφφφφωω

φφφφφφωω


ξξ

ξ

λ

ξ

λ

(6-A-8)

Then substituting Equation (6-A-1) and Equation (6-A-2) into Equation (6-A-8)

results in

Chapter Six

- 215 -

T112

)i)((i

2

22

432

)i)((i

2

2

2

432

))((22

1 1

23

432

1

)](][][i)(

))(21i

2)(2

[(

]i)(

)2

i2

)(2[(

])(

))(2ii

2)i(2

([

)(

sNAek

eAc

kg

ek

eAc

kk

g

ecc

ggg

C

ijk

iijmj

xkijjn

ji

jij

i

jiji

k

jijmi

xkjiin

j

ijij

j

jiji

ijmji

jjiijnin

n n

ji

jijjiji

ij

iijmj

ci

jijmi

cj

ijmji

+±−

++−

−−

±−

++−

−

−+

++−−

′′

= =

−−−

+

++−

++−

++−−

±−

+±+

+

±−

−±+

+

−+

+++

−

=

∑ ∑

ξλλ

α

αα

αα

ξλλ

α

αα

α

ξλλλ

βα

βα

α βββα

α

α

βα

λλγ

λωλω

ωωωωω

λλγ

ωω

λω

ωωωω

λλλγγ

λω

λλωωωωω

ξ

(6-A-9)

For the difference-frequency case, the following equation is introduced.

*111 )()()( ξξξ −−− += jiij CCC (6-A-7)

Replacing ωj , kj with –ωj and –kj , and all other terms with subscript j with the

corresponding complex conjugates will lead to the expression for )( 1 ξ−ijC .

Chapter Six

- 216 -

T112

)i)()((*

i*2

*

2*

2

432

)i)((i

2

2

2

432

))()((*

22*

1 1

2**3

432

1

)](][][i)()(

)()(

))(21)(i

2)(-2

[(

]i)(

)2

i2

)(-2[(-

])()()()(

)))((2i

)(i

2)i(-2

([

)(

*

*

sNAek

eAc

kg

ek

eAc

kk

g

ecc

ggg

C

ijk

iijmj

xkijjn

ji

jij

i

jiji

k

jijmi

xk*jiin

j

ijij

j

jiji

ijmji

*jjiijnin

n n

ji

jijjiji

ij

iijmj

ci

jijmi

cj

ijmji

−±−

−+−

−−

±−

−+−

−

−+

++−−

′′

= =

−−−

−

−+−

−+−

−+−−

±−

++

+

±−

++

+

−+

+−+

−

=

∑ ∑

ξλλ

α

αα

αα

ξλλ

α

αα

α

ξλλλ

βα

βα

α βββα

α

α

βα

λλγ

λωλω

ωωωωω

λλγ

ωω

λω

ωωωω

λλλγγ

λω

λλωωωωω

ξ

m

m

(6-A-10)

Exchanging the position of subscripts i and j in all terms of Equation (6-A-9) and

Equation (6-A-10) yields the expressions of )( 1 ξ+jiC and )( 1 ξ−

jiC .

Chapter Six

- 217 -

Figure 6.2 Variation of the dimensionless double-frequency vertical wave force with the dimensionless wave number.

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

0 1 2 3 4 5

Coarse meshMedium meshFine meshSulisz (1993)

Figure 6.1 Variation of the dimensionless double-frequency horizontal wave force with the dimensionless wave number.

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

0 1 2 3 4 5

Coarse meshMedium meshFine MeshSulisz (1993)

Chapter Six

- 218 -

Figure 6.3 Variation of the dimensionless double-frequency moment about (B, -D) with the dimensionless wave number.

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

0 1 2 3 4 5

Coarse meshMedium meshFine meshSulisz (1993)

Chapter Six

- 219 -

Figure 6.5 Computed variation of the dimensionless difference-frequency wave loads with the dimensionless wave number.

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

0 1 2 3 4 5

Horizontal force

Vertical force

Moment

Figure 6.4 Computed variation of the dimensionless sum-frequency wave loads with the dimensionless wave number.

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

0 1 2 3 4 5

Horizontal force

Vertical force

Moment

Chapter Six

- 220 -

Figure 6.7 Sum-frequency second-order transmission coefficients (kjH=0.01).

kiH

K t(2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H

Figure 6.6 Sum-frequency second-order reflection coefficients (kjH=0.01).

kiH

K r(2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Chapter Six

- 221 -

Figure 6.9 Difference-frequency second-order transmission coefficients (kjH=0.01).

kiH

K t(2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Figure 6.8 Difference-frequency second-order reflection coefficients (kjH=0.01).

kiH

K r(2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Chapter Six

- 222 -


kiH

K t(2

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0



kiH

K r(2

)

0.0

0.8

1.6

2.4

3.2

4.0

4.8

5.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Chapter Six

- 223 -


kiH

K t(2

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0



kiH

K r(2

)

0.0

0.6

1.2

1.8

2.4

3.0

3.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Chapter Six

- 224 -


kiH

K t(2

)

0.0

0.4

0.8

1.2

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0



kiH

K r(2

)

0.0

1.6

3.2

4.8

6.4

8.0

9.6

0 0.5 1 1.5 2 2.5 3


Chapter Six

- 225 -


kiH

K t(2

)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0



kiH

K r(2

)

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0


Chapter Six

- 226 -

Figure 6.19 Computed variation of the dimensionless sum-frequency vertical wave loads with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5

24 elements

48 elements

96 elements

Figure 6.18 Computed variation of the dimensionless sum-frequency horizontal wave loads with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Six

- 227 -

Figure 6.21 Computed variation of the dimensionless difference-frequency horizontal wave force with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

0 1 2 3 4 5

24 elements

48 elements

96 elements

Figure 6.20 Computed variation of the dimensionless sum-frequency moment about (B,-D) with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

70

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Six

- 228 -

Figure 6.23 Computed variation of the dimensionless difference-frequency moment about (B,-D) with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5

24 elements

48 elements

96 elements

Figure 6.22 Computed variation of the dimensionless difference-frequency vertical wave force with the dimensionless wave number for Example 2.

(B/H=1.0, D/H=0.4, θ=30°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

24 elements

48 elements

96 elements

Chapter Six

- 229 -

Figure 6.24(b) Sum-frequency component of second-order horizontal wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5

90 degree120 degree135 degree150 degree

Figure 6.24(a) Sum-frequency component of second-order horizontal wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5


Chapter Six

- 230 -

Figure 6.25(b) Sum-frequency component of second-order vertical wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

140

0 1 2 3 4 5


Figure 6.25(a) Sum-frequency component of second-order vertical wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5


Chapter Six

- 231 -

Figure 6.26(b) Sum-frequency component of second-order moment about (B,-D) on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

140

0 1 2 3 4 5


Figure 6.26(a) Sum-frequency component of second-order moment about (B,-D) on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

70

0 1 2 3 4 5


Chapter Six

- 232 -

Figure 6.27(b) Difference-frequency component of second-order horizontal wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5


Figure 6.27(a) Difference-frequency component of second-order horizontal wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5


Chapter Six

- 233 -

Figure 6.28(b) Difference-frequency component of second-order vertical wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

140

0 1 2 3 4 5


Figure 6.28(a) Difference-frequency component of second-order vertical wave force on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

0 1 2 3 4 5


Chapter Six

- 234 -

Figure 6.29(b) Difference-frequency component of second-order moment about (B,-D) on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≥90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

140

0 1 2 3 4 5


Figure 6.29(a) Difference-frequency component of second-order moment about (B,-D) on obstacles with various base angles for Example 2.

(B/H=1.0, D/H=0.4, θ ≤ 90°)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5


Chapter Six

- 235 -

Figure 6.30(b) Sum-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Coarse mesh

Medium mesh

Fine mesh

Figure 6.30(a) Sum-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 236 -

Figure 6.30(c) Sum-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50

Coarse mesh

Medium mesh

Fine mesh

Figure 6.31(a) Sum-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

02468

1012141618

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 237 -

Figure 6.31(c) Sum-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

02468

1012141618

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50

Coarse mesh

Medium mesh

Fine mesh

Figure 6.31(b) Sum-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

02468

1012141618

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 238 -

Figure 6.32(b) Sum-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Coarse mesh

Medium mesh

Fine mesh

Figure 6.32(a) Sum-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 239 -

Figure 6.33(a) Sum-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.32(c) Sum-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 240 -

Figure 6.33(c) Sum-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50

Coarse mesh

Medium mesh

Fine mesh

Figure 6.33(b) Sum-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 241 -

Figure 6.34(b) Difference-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

0.1 0.2 0.3 0.4 0.5 0.6

Coarse mesh

Medium mesh

Fine mesh

Figure 6.34(a) Difference-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 242 -

Figure 6.35(a) Difference-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.34(c) Difference-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 243 -

Figure 6.35(c) Difference-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45

Coarse mesh

Medium mesh

Fine mesh

Figure 6.35(b) Difference-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

0.1 0.2 0.3 0.4 0.5 0.6

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 244 -

Figure 6.36(a) Difference-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.35(d) Difference-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 245 -

Figure 6.36(c) Difference-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

3 3.2 3.4 3.6 3.8 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.36(b) Difference-frequency component of second-order horizontal wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

20

40

60

80

100

120

0.1 0.2 0.3 0.4 0.5 0.6

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 246 -

Figure 6.37(b) Difference-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

16

0.1 0.2 0.3 0.4 0.5 0.6

Coarse mesh

Medium mesh

Fine mesh

Figure 6.37(a) Difference-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

16

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 247 -

Figure 6.37(d) Difference-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

16

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.37(c) Difference-frequency component of second-order vertical wave force on the obstacle B2 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.01)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

16

1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 248 -

Figure 6.39 Sum-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

14

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.38 Sum-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 249 -


(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

1

2

3

4

5

6

7

8

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 250 -

Figure 6.43 Difference-frequency component of second-order vertical wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

24

28

32

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Figure 6.42 Difference-frequency component of second-order horizontal wave force on the obstacle B1 for Example 3.

(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

70

80

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 251 -


(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

048

12162024283236

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.05)

kiH

Dim

ensio

nles

s w

ave

load

s

0

10

20

30

40

50

60

70

80

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 252 -


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0123456789

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

2

4

6

8

10

12

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 253 -


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

0.5

1

1.5

2

2.5

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 254 -


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

35

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Six

- 255 -


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

4

8

12

16

20

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh


(B/H=1.0, D/H=0.3, Bg=0.1)

kiH

Dim

ensio

nles

s w

ave

load

s

0

5

10

15

20

25

30

35

0 1 2 3 4

Coarse mesh

Medium mesh

Fine mesh

Chapter Seven

- 256 -

CHAPTER 7

CONCLUSIONS

7.1 Summary

This thesis developed scaled boundary finite-element solutions to first-order and

second-order wave diffraction problems. The solution methods described in this

thesis are concerned with the solutions of those boundary-value problems which are

governed either by the Laplace equation or the Helmholtz equation. The boundary

conditions of wave diffraction problems are usually complex, due to the effect of free

water surface and body motions. Accordingly, the work reported in this thesis also

presented detailed solution procedures for cases with Neumann boundary conditions

and Robin boundary conditions. In applying these solution methods to boundary-

value problems, many practical numerical examples were computed to investigate the

advantages and disadvantages of the scaled boundary finite-element method. The

following paragraphs summarize the major work and findings contained in the

preceding chapters.

This thesis commenced with an introduction to the motivation of carrying out this

study. The benefits and limitations of the existing numerical methods for solving

wave diffraction problems were reviewed. Most numerical methods encounter

difficulties of certain degrees when attempting to rigorously satisfy the radiation

condition and handle any singularities in the velocity field. A newly developed

numerical method, the scaled boundary finite-element method, has been found to be

particularly good at dealing with unbounded domains and singularities in problems of

elastic-statics and elastic-dynamics. Extending the scaled boundary finite-element

method to the area of ocean wave research and investigating the performance of this

method in various cases became the main aims of this thesis.

Chapter Seven

- 257 -

A detailed review of the development of both the mathematical theory and numerical

methods for wave diffraction problems followed. It was observed that it is impossible

to solve the complete boundary-value problem of wave diffraction, due to the

presence of highly nonlinear free water surface boundary conditions. Consequently

this type of boundary-value problem is usually simplified into individual nth-order

boundary-value problems using a perturbation technique (the perturbation parameter

is wave slope), theoretically up to any order. This thesis limits discussions only up to

the second-order. The history and significant theoretical advances in the scaled

boundary finite-element method were also discussed in Chapter 2.

The first issue addressed by this study was the solution of the two-dimensional

Laplace’s equation associated with a linearized water surface boundary condition

using the scaled boundary finite-element method. The standard scaled boundary

finite-element method needs the existence of a so-called scaling centre to define the

computational domain. This requirement, to some extent, prevents the standard scaled

boundary finite-element method from being applied to problems bounded by a semi-

infinite unbounded domain with parallel side-faces. This thesis developed a new local

coordinate system to overcome this limitation. Since the solution procedure of the

standard scaled boundary finite-element method still holds for the new local

coordinate system, the semi-analytical nature of the scaled boundary finite-element

method is retained in the new local coordinate system. The new local coordinate

approach is different from the existing approximation approaches in solid mechanics.

The present method has a concise derivation and is conceptually consistent with the

standard scaled boundary finite-element method.

Generally speaking, the scaled boundary finite-element method transforms the

resulting equations into an eigenvalue problem. The current work found that the

behaviour of the zero eigenvalues in the fluid area is different from those in solid

mechanics. The two eigenvectors corresponding to the zero eigenvalues are linearly

dependent for wave diffraction problems while they are linearly independent for

problems in solid mechanics. The linearly dependent eigenvectors made analytical

solution of the resulting matrix differential equations a challenge. The advanced

solution process presented in this thesis employs the Jordan chain to seek the solution

of matrix differential equation in the complex number space. The entire solution

Chapter Seven

- 258 -

procedure was implemented and used to calculate several practical numerical

examples, including wave reflection and transmission by a single and multiple

structures and wave radiation induced by oscillating bodies with a certain draft.

Numerical results were compared with available analytical solutions and results

obtained using other numerical methods. Comparisons with the analytical solutions

demonstrated that the scaled boundary finite-element method has very high accuracy

and that the results converge rapidly. Furthermore, it was found that the scaled

boundary finite-element method predicts the resonance phenomenon resulting from a

narrow gap between structures very well, and is free of irregular frequency problems.

In addition to examining the ability of the scaled boundary finite-element method to

solve Laplace’s equation, this thesis explored its applicability to boundary-value

problems governed by the Helmholtz equation. The challenge for solving the the

Helmholtz equation lies in the “irregular singular” property of the resulting scaled

boundary finite-element equation for an unbounded domain. Wolf (2003) presented

an approximate numerical approach to obtain the solution of such a system of

differential equations. The work reported in Chapter 4 employs the method developed

by Wasow (1965) to formulate a high-frequency asymptotic expansion for the nodal

potential satisfying the radiation condition directly. Using a substructuring technique,

the unbounded and bounded domain solutions are matched at the interface of a

bounded domain and an unbounded domain to generate the entire solution of

problems. The solution procedure of the Helmholtz equation in the context of the

scaled boundary finite-element method was applied to problems of wave diffraction

by a single vertical cylinder or cylinder group and harbour oscillation induced by

linear incident waves. The computed wave elevation, velocity distributions and wave

forces for the case with a circular cross section were compared with the analytical

solution and numerical results of conventional numerical methods. Excellent

accuracy of the new method was demonstrated again in the problems involving the

Helmholtz equation. Cases with elliptical, rectangular cross sections were also

computed. The scaled boundary finite-element method modeled the singularity of the

velocity field near sharp corners extremely well. For the case of two cylinders, the

method calculated the resonant phenomenon accurately because it does not suffer

from the effect of irregular frequency. The numerical examples addressed in this

Chapter Seven

- 259 -

thesis included the case of harbour oscillation induced by linear incident waves. The

computed results compared well with those available in literature.

Based on the linear scaled boundary finite-element solutions, Chapter 5 established a

second-order solution for monochromatic wave diffraction problems. The second-

order problem arises from the presence of the nonhomogeneous water surface

boundary condition. The second-order scaled boundary finite-element equation is still

solved analytically, making use of the analytical expression of the first-order solution

in the radial direction. In general, the second-order wave loads are the engineers’

concern. The formulation of the second-order wave force involves the terms of the

first-order velocity. Consequently the second-order wave forces can be predicted

accurately only if the first-order velocity field is modelled very well. However, when

sharp re-entrant corners appear in the domain, most numerical methods can not

calculate such first-order velocity fields easily. Nevertheless, the scaled boundary

finite-element method provided accurate results without requiring any special

treatment in this case. As a result, the second-order wave forces were accurately

calculated by the scaled boundary finite-element method. The numerical example

presented in Chapter 5 clearly demonstrated this.

A more general second-order problem is the case of bichromatic incident waves. In

this case, the interaction of the two first-order incident waves with different

frequencies induces the terms with the sum-frequency and difference-frequency in the

boundary-value problem. This type of problem is usually of significance in practical

engineering analysis and design. Chapter 6 of this thesis described in detail how to

establish the scaled boundary finite-element equation and solve the resulting

equations in this case. The major effort required is to explicitly determine the

integration constants. From the programming point of view, this procedure became

more complicated than that in Chapter 5. In the numerical examples, the extreme case

of the double-frequency wave loads was calculated first for the purpose of

comparison. The results agreed well with the analytical solution for the

monochromatic wave diffraction problem. The sum-frequency and difference-

frequency results were also given. It appeared that the scaled boundary finite-element

method performed well even for the bichromatic wave diffraction problems.

Chapter Seven

- 260 -

Finally, for clarity, the major advantages of the scaled boundary finite-element

method are summarized as follows:

• Rigorously satisfy the radiation condition.

• Clearly describe the far-field wave property in the physical sense.

• Model the singularity of velocity field near sharp re-entrant corners with ease.

• Remain analytical attribute in the radial direction.

• Only boundaries of computational domains are discretized so as to reduce the

spatial dimension by one.

• Free of irregular frequency.

• No fundamental solution required.

• No singular integral required.

• No need to couple with other numerical methods.

7.2 Future work

Like other numerical methods, there is still room for SBFEM to be improved. Firstly,

the coefficient matrix of the global equation is fully populated, and an entire

eigenvalue value equation needs to be numerically solved. Consequently, to some

extent, this cancels out some of the advantages and efficiency brought by the reduced

dimensions of physical problems. Secondly, it is worthwhile to continue exploring the

ability of the scaled boundary finite-element method to address boundary-value

problems with complex boundary conditions. For instance, the problem of

bichromatic wave diffraction by freely-floating bodies will be a challenge because the

free water surface boundary condition and the body surface boundary condition

become very complicated due to the interaction of the first-order qualities.

Consequently it will be very difficult to find the analytical solution of scaled

boundary finite-element equations. Finally, the scaled boundary finite-element

Chapter Seven

- 261 -

method may face significant challenges for handling problems of wave diffraction by

a three-dimensional rectangular caisson in water of finite depth.

In order to improve the scaled boundary finite-element method to tackle the

difficulties mentioned above, a few potential methods are probably feasible. First of

all, the scaled boundary finite-element method could be applied by dividing the entire

computational domain into several subdomains, then computing the solution of every

subdomain one by one and finally matching these solutions at the interface between

subdomains. Parallel computing techniques could be developed to compute the

solution of subdomains simultaneously so that the computational time can be

reduced. The development of new PC processor types, like Dual-core and Multi-core,

make it possible to explore parallel computing techniques for the scaled boundary

finite-element method. As far as the analytical solution method for the scaled

boundary finite-element method goes, a particular local coordinate system might give

some assistance, as was done in Chapter 2. Likewise, a proper coordinate mapping

would be valuable for addressing problems of three-dimensional wave diffraction by

a rectangular caisson in finite water depth.

7.3 Concluding remarks

Taking into account all aspects summarized in this chapter, the conclusion can be

made that the scaled boundary finite-element method, combining many of advantages

of the finite-element method and boundary-element method with features of its own,

does provide powerful assistance to the existing numerical methods for solving wave

diffraction problems, particularly for dealing with the radiation condition and

singularity of velocity field near sharp corners. This new method can be a competitive

alternative to traditional numerical methods in the future, but, there is still room for

further improvement at this stage.

References

- 262 -

REFERENCES

[1] Abul-Azm AG, Williams AN. (1988) Second-order diffraction loads on truncated cylinders. J. Waterway, Port, Coastal Ocean Dic. ASCE 114, 436-454.

[2] Abul-Azm AG, Williams AN. (1989a) Approximation of second-order diffraction loads on arrays of vertical circular cylinders. J.Fluids Struct. 3, 17-36.

[3] Abul-Azm AG, Williams AN. (1989b) Second-order diffraction loads on arrays of semi-immersed circular cylinder. J.Fluids Struct. 3, 365-388.

[4] Au MC, Brebbia CA. (1983) Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling. 7, 106-114.

[5] Bai, KJ. (1972) A variational method in potential flow with a free surface. PhD thesis. Univ.Calif., Berkeley. 137pp.

[6] Bai, KJ. (1975) Diffraction of oblique waves by an infinite cylinder. J.Fluid Mech. 68, 513-535.

[7] Bai, KJ, Yeung, R. (1974) Numerical solutions of free-surface flow problems. Proc.10th Symp.Naval Hydrodyn., Cambridge, Mass. 609-647.

[8] Berkhoff, JCW. (1972) Computation of combined refraction-diffraction. Proc.13th Coastal Eng.Conf., Vancouver. 1, 471-490.

[9] Bettess P. (1977) Infinite elements. Int.J.Numer.Methods Engng. 11, 53-64.

[10] Bettess P. (1992) Infinite elements. Sunderland: Penshaw Press.

[11] Bettess P, Emson CRI, Chiam TC. (1984) A new mapped infinite element for exterior wave problems. Chapter 17 of Numer. Methods in Coupled Systems, John Wiley, Chichester.

[12] Bettess JA, Bettess P. (1998) A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem. Comput. Methods Appl. Mech. Engrg. 164,17-48.

[13] Bettess P, Zienkiewicz OC. (1977) Diffraction and refraction of surface waves using finite and infinite elements. International Journal for numerical methods in Engineering. 11, 1271-1290.

[14] Bishop RED, Price WG, Wu Y. (1986) A general linear hydroelasticity theory of floating bodies in a seaway. Phil.Trans.Roy.Soc. A316: 375.

References

- 263 -

[15] Black JL, Mei CC, Bray MCG. (1971) Radiation and scattering of water waves by rigid bodies. J. Fluid Mech. 46: 151-164.

[16] Brebbia CA. (1984) Topics in Boundary element research (Volume 1 Basic priciples and applications). Springer-Verlag Berlin, Heidelberg. 97-122.

[17] Carvalho ER, Mesquita E. (1994) Numerical evaluation of hydrodynamic forces and stattionary response of floating bodies using the boundary element method. Proceedings of the Fourth International Offshore and Polar Engineeering Conference. Vol. III, 237-243.

[18] Genes MC, Kocak S. (2005) Dynamic soil-structure interaction analysis of layered unbounded media via a coupled finite element/boundary element/scaled boundary finite element model. International Journal for Numerical Methods in Engineering. 62(6), 798-823.

[19] Chakrabarti SK. (1995) Hydrodynamics of offshore structures. Southampton.

[20] Chakrabarti SK. (2000) Hydrodynamic interaction forces on multi-moduled structures. Ocean Engineering. 27, 1037-1063.

[21] Chau FP, Eatock Taylor R. (1992) Second-order wave diffraction by a vertical cylinder. J.Fluid Mech. 240, 571-599.

[22] Chen HS, Mei CC. (1974) Oscillations and wave forces in a man-made harbor in the open sea. Proc.10th Symp. Naval Hydrodyn., Cambridge, Mass. 573-596.

[23] Chen JT, Lin TW, Chen IL, Lee YJ. (2005) Fictitious frequency for the exterior Helmholtz equation subject to the mixed-type boundary condition using BEMs. Mechanics Research Communications. 32, 75-92.

[24] Chen JT, Lin JH, Kuo SO, Chyuan SW. (2001) Boundary element analysis for the Helmhotz eigenvalue problems with a multiply connected domain. Proc.R.Soc.Lond.A. 457, 2521-2546.

[25] Chen X. (2004) Hydrodynamics in Offshore and Naval Applicatoins – Part I. The 6th International Conference on HydroDynamics. 1-28.

[26] Chenault, DW. (1970) Motion of a ship at the free surface. M.S. thesis. Naval Post Grad. Sch., Monterey, Calif.

[27] Cheung YK, Jin WG, Zienkiewicz OC. (1991) Solution of Helmholtz equation by Trefftz method. International Journal for Numerical Methods in Engineering. 32, 63-78.

[28] Chidgzey SR, Deeks AJ. (2005) Determination of coefficients of crack tip asymptotic fields using the scaled boundary finite element method. Engineering Fracture Mechanic.72(13), 2019-2036.

References

- 264 -

[29] Dasgupta G. (1979) Impedance matrices for unbounded media. Proceedings Seventh Canadian Congress of Applied Mechanics, Sherbrooke, 891-892.

[30] Dasgupta G. (1982) A finite element formulation for unbounded homogeneous continua. Journal of Applied Mechanics, 49, 136-140.

[31] Dean WR. (1945) On the reflexion of surface waves by a submerged plane barrier. Prof. Camb. Phil. Soc. 41, 231.

[32] Deeks AJ. (2004) Prescribed side-face displacements in the scaled boundary finite-element method. Computers & Structures. 82(15-16), 1153-1165.

[33] Deeks AJ, Cheng L. (2003) Potential flow around obstacles using the scaled boundary finite-element method. International journal for numerical methods in fluids. 41,721-741.

[34] Deeks AJ, Wolf JP. (2003) Semi-analytical solution of Laplace’s equation in non-equilibrating unbounded problems. Computers & Structure. 81(15), 1525-1537.

[35] Doherty JP, Deeks AJ (2005) Adaptive coupling of the finite-element and scaled boundary finite-element methods for non-linear analysis of unbounded media. Computers and Geotechnics. 32(6), 436-444.

[36] Drake KR, Eatock Taylor R, Hung SM. (1984) Comments on ‘Wave diffraction by large offshore structures: an exact second-order theory by M. Rahman. Applied Ocean Research. 6(2), 90-100.

[37] Drimer N, Agnon Y. (1994) A hybrid boundary element method for second-order wave-body interaction. Applied Ocean Research. 16, 27-45.

[38] Eatock Taylor R, Hung SM. (1987) Second-order diffraction forces on a vertical cylinder in regular waves. Applied Ocean Research. 9, 19-30.

[39] Eatock Taylor R, Huang JB. (1997) Semi-analytical formulation for second-order diffraction by vertical cylinder in bichromatic waves. J. Fluids Struct. 11, 465-484.

[40] Eatock Taylor R, Teng B. (1993) The effect of corners on diffraction/radiation forces and wave drift damping. The 25th Annual Offshore Technology Conference, Houston, Texas, USA. 571-581.

[41] Fernyhough M, Evans DV. (1995) Scattering by a periodic array of rectangular blocks. J.Fluid.Mech. 305, 263-279.

[42] Gantmacher FR. (1959) The theory of matrices. Chelsea publishing company, New York, N.Y. Vol.I..

[43] Garrett CJR. (1971) Wave forces on a circular dock. J.Fluid Mech. 46(1), 129-139.

References

- 265 -

[44] Ghalayini SA, Williams AN. (1991) Nonlinear wave forces on vertical cylinder arrays. J.Fluids Struct. 5, 1-32.

[45] Givoli D, Patlashenko I, Keller JB. (1997) High-order boundary conditions and finite elements for infinite domains. Comput. Methods Appl. Mech. Engng. 143: 13-39.

[46] Goo J-S, Yoshida K. (1990) A numerical method for huge semi-submersible response in waves. SNAME Transactions 98, 365-387.

[47] Haraguchi T, Ohmatsu S. (1983) On an improved solution of the oscillation problem on non-wall sided floating bodies and a new method for eliminating the irregular frequencies. Trans. West-Japan Soc. Nav. Archi. 66, 9. (in Japanese).

[48] Sclavounos, PD and Lee CH. Topics on boundary-element solutions of wave radiation-diffraction problems. 4th Numerical Hryrodynamics Conf. Washington, DC, September 1985.

[49] Harari I., Barbone PE, Slavutin M, Shalom R. (1998) Boundary infinite elements for the Helmholtz equation in exterior domains. Int. J. Numer. Meth. Engng. 41: 1105-31.

[50] Havelock TH. (1940) Pressure of water waves on a fixed obstacle, Proc. Royal Society, London, Vol. A175, 409-421.

[51] Havelock TH. (1955) Waves due to a floating hemisphere making periodic heaving oscillations. Prof.R.Soc.Lond.A 231, 1-7.

[52] Hossein BM, Song Ch. (2006) Time-harmonic response of non-homogeneous elastic unbounded domains using the scaled boundary finite-element method. Earthquake Engineering and Structural Dynamics. 35(3), 357-383.

[53] Hsu H. and Wu Y., The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall. Ocean Engineering 1997; 24(2): 177-199.

[54] Hsu TW, Lan YJ, Tsay TK, Lin KP. (2003) Second-order radiation boundary condition for water wave simulation with large angle incidence. Journal of Engineering Mechanics. 129(12), 1429-1438.

[55] Hermans AJ. (2003) Interaction of free-surface waves with a floating dock. Journal of Engineering Mathematics. 45, 39-53, 2003.

[56] Herrera I. (1984) Boundary Methods. An Algebraic Theory. Pitman Publishing Ltd, London, 1984.

[57] Hulme A. (1982) The wave forces on a floating hemisphere undergoing forced periodic oscillations. J.Fluid Mech. 121, 443-463.

References

- 266 -

[58] Huang JB, Eatock Taylor R. (1996) Semi-analytical solution for second-order wave diffraction by a truncated circular cylinder in monochromatic waves. J.Fluid Mech. 319, 171-196.

[59] Isaacson M de St Q. (1977) Shallow wave diffraction around large cylinder. Journal of Waterway, Port, Coastal and Ocean Division, ASCE(Proc.paper 12756). 103(WW1), 69.

[60] Jirousek J, Wroblewski A. (1996) T-elements: State of the art and future trends. Arch.Comp.Meth.Engng. 3, 323-434.

[61] John F. (1950) On the motion of floating bodies II, Communications in Pure and Applied Mathematics. 3, 45-101

[62] Kagemoto H, Yue DKP. (1986) Interactions among multiple three-dimensional bodies in water waves: an exact algebraic method. J. Fluid Mech. 166, 189-209.

[63] Kausel E, Roësset JM, Waas G. (1975) Dynamic analysis of footings on layered media. Journal of Engineering Mechanics, 101, 679-693.

[64] Kim M-H, Yue DKP. (1990) The complete second-order diffraction solution for an axisymmetric body. Part 2. Bichromatic incident waves and body motions. J. Fluid Mech. 211, 557-593.

[65] Kriebel DL. (1990) Nonlinear wave interaction with a vertical circulare cylinder, part 1: diffraction theory. Ocean Engng. 17, 345-377.

[66] Krieble DL. (1992) Nonlinear wave interaction with a vertical circulare cylinder, part 1: diffraction theory. Ocean Engng. 19, 75-99.

[67] Krishnankutty P, Vendhan CP. (1995) Three-dimensional finite element analysis of the diffraction-radiation problem of hydrodynamically compact structures. Marine Structures. 8, 525-542.

[68] Ippen AT, Goda Y. (1963) Wave-induced oscillation in harbours: the solution a rectangular harbour connected to the open sea. Hydrodynamic Lab, Massachusetts Institute of Technology, Report 59.

[69] John, F. (1950) On the motion of floating bodies Ⅱ, Communications in Pure and Applied Mathematics, 3,45-101.

[70] Jirousek J, Wroblewski A. (1994) Least-squares T-elements: Equivalent FE and BE forms of a substructure-oriented boundary solution approach. Communications in Numerical Methods in Engineering. 10, 21-32.

[71] Lee C-H, Sclavounos PD. (1989) Removing the irregular frequencies from integral equations in wave-body interactions. J.Fluid Mech. 207, 393-418.

[72] Lee CK. (1988) A direct boundary-element method for three-D wave diffraction and radiation problems. Ocean Engng. 15, 431-455.

References

- 267 -

[73] Lee J-F. (1995) On the heave radiation of a rectangular structure. Ocean Engineering. 22, 19-34.

[74] Lee JJ. (1971) Wave induced oscillations in harbours of arbitrary geometry. J.Fluid Mech. 45, 375-394

[75] Leonard JW, Huang MC, Hudspeth RT. (1983) Hydrodynamic interference between floating cylinders in oblique seas. Applied Ocean Research. 5, 158-166.

[76] Li W, Williams AN. (1999) The diffraction of second-order bichromatic waves by a semi-immersed horizontal rectangular cylinder. J. Fluids and Structures. 13, 381-397.

[77] Lighthill MJ. (1979) Waves and hydrodynamic loading. Proceedings Behavior of Offshore Structures (BOSS), London, U.K.. 1-40.

[78] Linton CM, Evans DV. (1990) The interaction jof waves with arrays of vertical circular cylinders. J.Fluid.Mech. 215, 549-569.

[79] Lysmer, J. (1970) Lumped mass method of r Rayleigh waves. Bulletin of the Seismological Society of America. 60, 89-104.

[80] Maccamy RC, Fuchs RA. (1954) Wave forces on piles: a diffraction theory. Tech. Memo 69. US Army Corps of Engineers Beach Erosion Board, Washington DC.

[81] Madsen PA, Larsen J. (1987) An efficient finite-difference approach to the mild-slop equation. Costal Engineering. 11, 329-351.

[82] Maniar HD, Newman JN. (1997) Wave diffraction by a long array of cylinders. J.Fluid Mech. 339, 309-330.

[83] Martin PA. (1981) On the null-field equations for water-wave radiation problem. J.Fluid Mech. 113, 315.

[84] Martin PA, Rizzo FJ. (1995) Partitioning, boundary integral equations, and exact Green’s functions. International Journal for Numerical Methods in Engineering. 38, 3483-3495.

[85] Mei CC. (1989) The applied dynamics of ocean surface waves. Singapore.

[86] Mei CC, Black JL. (1969) Scattering of surface waves by rectangular obstacles in waters of finite depth. J.Fluid.Mech. 38(3), 499-511.

[87] Molin B. (1979) Second-order diffraction loads on three-dimensional bodies. Applied Ocean Research. 1, 197-202.

[88] Moubayed WI, Williams AN. (1994) The second-order diffraction loads and associated motions of a freely floating cylindrical body in regular waves: an eigenfunction expansion approach. J.Fluids Struct. 8, 417-451.

References

- 268 -

[89] Moubayed WI, Williams AN. (1995a) Second-order hydrodynamic interactions in an array of vertical cylinders in bichromatic waves. J.Fluids Struct. 9, 61-98.

[90] Moubayed WI, Williams AN. (1995b) The sum- and difference-frequency wave loads and motions of a freely floating circular cylinder. J.Fluids Struct. 9, 323-355.

[91] Miao GP, Liu YZA. (1986) A theoretical study of the second-order wave forces for two dimensional bodies. Proc.of the Fifth Int. Offshore Mech. And Artic Eng. Symp.. 2, 330-336.

[92] Miao GP, Ishida H, Saitoh T. (2000) Influence of gaps between multiple floating bodies on wave forces. China Ocean Engineering. 14, 407-422.

[93] Miao G, Saitoh T, Ishida H. (2001) Water wave interaction of twin large scale caissons with a small gap between. Coastal Engineering Journal. 43(1), 39-58.

[94] Mikhailov SE. (2005) Will the boundary (-domain) integral-equation methods survive? Preface to the special issue on non-traditional boundary (-domain) intergral-equation methods. Journal of Engineering Mathematics. 51, 197-198.

[95] Miles JW. (1967) Surface-wave scattering matrix for a shelf. J.Fluid.Mech. 28, 755.

[96] Miles JW, Gilbert JF. (1968) Scattering of gravity waves by a circular dock. J.Fluid Mech. 34, 783.

[97] Mingde S, Yu P. (1987) Shallow water diffraction around multiple large cylinders. Appl.Ocean Res. 9, 31-36.

[98] Murphy JE. (1978) Integral equation failure in wave calculations. Journal of Waterways, Port, Coastal and Ocean Division. 104(WW4), 330-334.

[99] Newman JN. (1965) Propagation of water waves over an infinite step. J.Fluid.Mech. 32, 549.

[100] Newman JN. (1990) Second-harmonic wave diffraction at large depths. J.Fluid Mech. 213, 59-70.

[101] Ogilvie F, Shin YS. (1978) Integral-equation solutions for time-dependent free-surface problems. J. Soc. Nav. Archi. Japan. 143, 41.

[102] Ohkusu M. (1974) Hydrodynamic forces on multiple cylinders in waves. Proc.Intl Symp. On Dynamics of Marine Vehicles and Structures in Waves, University College, London. 107-112.

[103] Patel MH. (1989) Dynamics of offshore structures. Butterworths, London.

[104] Pien PC, Lee CM. (1972) Motion and resistance of a low waterplane catamaran. 9th Symp. Nacal Hydrodynamics, 463-539.

References

- 269 -

[105] Politis CG, Papalexandris MV, Athanassoulis GA. (2002) A boundary integral equation method for oblique water-wave scattering by cylinders governed by the modifed Helmholtz equation. Applied Ocean Research. 24, 215-233.

[106] Rahman M. (1984) Wave diffraction by large offshore structures: an exact second-order theory. Applied Ocean Research. 6(2), 90-100.

[107] Rahman M, Statish MG., Xiang Y. (1992) Wave diffraction due to large offshore structures: a boundary element analysis. Ocean Engng. 19, 271-287.

[108] Ruoff G. (1973) Die praktiche Berechnung der Kopplungsmatrizen bei der Kombination der Trefftzschen Methode und der Methode der finiten Elmente bei fiachen Schalen. Finite Elemente Congress in der Static. W. Ernst & Sohn, Berlin, 242-159.

[109] Sannasiraj SA, Sundaravadivelu R, Sundar V. (2000) Diffraction-radiation of multiple floating structures in directional waves. Ocean Engng. 28, 201-234.

[110] Sayer P. (1980) An integral-equation method for determining the fluid motion due to a cylinder heaving on water of finite due to a cylinder heaving on water of finite depth. Proc.R.Soc.Lond.A. 372, 93-110.

[111] Silverster PP, Lowther DA, Carpenter CJ, Wyatt EA. (1977) Exterior finite elements for 2-dimensional field problems with open boundaries. Proceedings of the Institution of Electrical Engineerings. 124, 1267-1270.

[112] Song Ch. (2004) A matrix function solution for the scaled boundary finite-element equation in statics. Computer Methods in Applied Mechanics and Engineering, 193(23-26), 2325-2356.

[113] Song Ch. (2005) Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners. Engineering Fracture Mechanics. 72(10), 1498-1530.

[114] Song Ch. (2006) Dynamic analysis of unbounded domains by a reduced set of base functions. Computer Methods in Applied Mechanics and Engineering. 195(33-36), 4075-4094.

[115] Song Ch, Wolf JP. The scaled boundary finite element method – alias consistent infinitesimal finite element cell method – for elastodynamics. Computer Methods in Applied Mechanics and Engineering, 1997, 147, 329-355.

[116] Song Ch, Wolf JP. (1998) The scaled boundary finite element method: analytical solution in frequency domain. Computer Methods in Applied Mechanics and Engineering. 164, 249-264.

[117] Sibuya Y. (1958) Second order linear ordinary differential equations containing a large parameter. Proc.Japan.Acad. 34, 229-234.

References

- 270 -

[118] Spring BH, Monkmeyer PL. (1974) Interaction of plane waves with vertical cylinders. Proc.14th Intl. Conf. on Coastal Engineering, Chap.107, 1828-1845.

[119] Stein E. (1973) Die Kombination des modifizierten Trefftzschen Vertahrens mit der Methods der finiten Elemente. Finite Elemente Congr.in der Static. 172-185.

[120] Stoker JJ. (1957) Water waves: The Mathematical Theory with Application. New York, Wiley Interscience.

[121] Stojek M. (1998) Least-squares Trefftz-type elements for the Helmholtz equation. International Journal for Numerical Methods in Engineering. 41, 831-849.

[122] Stojek M, Markiewicz M, Mahrenholtz O. (2000) Diffraction loads on multiple vertical cylinders with rectangular cross section by Trefftz-type finite elements. Computers and Structures. 75: 335-345.

[123] Sulisz W. (1993) Diffraction of second-order surface waves by semi-submerged horizontal rectangular cylinder. ASCE Journal of Waterway, Port, Costal and Ocean Division. 119, 160-171.

[124] Sulisz W, Johansson M. (1992) Second-order wave loading on a horizontal rectangular cylinder of substantial draught. Applied Ocean Research. 14, 333-340.

[125] Sulisz W. (2002) Diffraction of nonlinear waves by horizontal rectangular cylinder founded on low rubble base. Applied Ocean Research. 24, 235-245.

[126] Takagi M, Furukawa H, Takagi K. (1983) On the precision of various hydrodynamic solutions of two-dimensional oscillating bodies. Int. Workshop on Ship and Platform Motions, Berkeley. 450-466.

[127] Teng B, Li Y. (1996) A unique solvable higher order BEM for wave diffraction and radiation. China Ocean Engineering. 10(3), 333-342.

[128] Teng B, Cheng L, Liu S, Li F. (2002) Re-examination of interaction of water waves with a semi-infinite elastic plate. Applied Ocean Research. 23, 357-368.

[129] Teng B, Ning DZ, Zhang XT. (2003) Wave radiation by a uniform cylinder in front of a vertical wall. Ocean Engineering. 31(2), 201-224.

[130] Thatcher RW. (1978) On the finite element method for unbounded regions. SIMAM Journal on Numerical Analysis. 15, 466-477.

[131] Trefftz E. (1926) Ein Gegenstück zum Ritz’schen Verfahren. Pro. 2nd Int.Congr.Appl. Mech.Zuich, 131-137.

[132] Twersky V. (1952) Multiple scattering of radiation by an arbitrary configuration of parallel cylinders. J. Acoust. Soc. Am. 24, 42-46.

References

- 271 -

[133] Ursell F. (1947) The effect of a fixed vertical barrier on surface waves in deep water. Proc.Camb. Phil. Soc. 43, 374.

[134] Ursell F. (1949) On the heaving motion of a circular cylinder on the surface of a fluid. Q.J.Mech.Appl.Math. 2, 218-231.

[135] Ursell F. (1981) Irregular frequencies and the motion of floating bodies. J.Fluid Mech. 105, 143.

[136] Vada T. (1987) A numerical solution of the second-order wave diffraction problem for a submerged cylinder of arbitrary shape. Journal of Fluid Mechanics. 174, 23-37.

[137] Vu TH, Deeks AJ. (2006) Use of higher-order shape functions in the scaled boundary finite element method. International Journal for Numerical Methods in Engineering. 65(10), 1714-1733.

[138] Waas, G. (1972) Linear two-dimensional analysis of soil dynamics problems in Semi-infinite layered media. PhD dissertation, University of California, Berkeley, CA.

[139] Wasow W. (1965) Asymptotic expansions for ordinary differential equations. John Wiley and Sons, New York.

[140] Williams AN, Abul-Azm AG, Ghalayini SA. (1990) A comparison of complete and approximate solutions of second order diffraction loads on arrays of vertical cylinders. Ocean Engng. 17, 427-446.

[141] Wu C, Watanabe E, Utsunomiya T. (1995) An eigenfunction expansion-matching method for analyzing the wave-induced responses of an elastic floating plate. Applied Ocean Research. 17, 301-310.

[142] Wu GX. (1991) On the second-order reflection and transmission by a horizontal cylinder. Applied Ocean Research. 13, 58-62.

[143] Wu GX. (1993a) Second-order wave radiation by a submerged horizontal circular cylinder. Applied Ocean Research. 15, 293-303.

[144] Wu GX. (1993b) Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude motion. Journal of Fluid Mechanics. 254, 41-58.

[145] Wu GX, Eatock Taylor R.(1990) The second order diffraction force on a horizontal cylinder in finite water depth. Applied Ocean Research. 12(3), 106-111.

[146] Wu GX, Eatock Taylor R. (2003) The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies. Ocean Engineering. 30, 387-400.

References

- 272 -

[147] Wu X, Price WG. (1987) A multiple Green’s function expression for the hydrodynamic analysis of multi-hull structures. Applied Ocean Research. 9(2), 58-66).

[148] Wolf JP. (2003) The scaled boundary finite element method, John Wiley and Sons, Chichester.

[149] Wolf JP, Song Ch. (1994) Dynamic-stiffness matrix of unbounded soil by finite-element multi-cell cloning. Earthquake Engineering and Structural Dynamics. 23, 233-250.

[150] Wolf JP, Song Ch. (1996) Finite-element modelling of unbounded Media, John Wiley and Sons, Chichester.

[151] Wolf JP, Song Ch. (2000) The scaled boundary finite-element method – a primer: derivations. Computers and structures. 78, 191-210.

[152] Wolf JP, Weber B. (1982) On calculating the dynamic-stiffness matrix of the unbounded soil by cloning. International Symposium on Numerical Models in Geotechnics. Zürich. 486-494.

[153] Yang Z. (2006) Fully automatic modelling of mixed-mode crack propagation using scaled boundary finite element method. Engineering Fracture Mechanics. 73(12),1711-1731

[154] Ying L. (1978) The infinite similar element method for calculating stress intensity factors. Scientia Sinica. 21, 19-43. (also presented in Chinese at the National Conference on Fracture Mechanics, Nanning, 1974)

[155] Zhao M, Teng B. (2004) A composite numerical model for wave diffraction in a harbour with varying water depth. Acta Oceanologica Sinica, 2004, 23, 367-375.

[156] Zheng YH, You YG, Shen YM. (2004) On the radiation and diffraction of water waves by a rectangular buoy. Ocean Engineering. 31, 1063-1082.

[157] Zhu SP, Liu HW, Chen K. (2000) A general DRBEM model for wave refraction and diffraction. Engineering Analysis with Boundary Elements. 24, 377-390.

[158] Zienkiewicz OC, Bettess P. (1976) Infinite elements in the study of fluid structure interaction problems. Proc. 2nd Int.Symp.on Comp.Methods Appl.Sci., Versailles 1975; also published in : J. Ehlers et al., eds., Lecture Notes in Physics, Springer-Verlag, Berlin, Vol.58.

[159] Zienkiewicz OC, Bettess P. (1978) Fluid structure interaction. Numer. Methods in Offshore Engineering, Chapter 5, John Wiley.

[160] Zienkiewicz OC, Bettess P, Chiam TC, Emson CRI. (1981) Numer. Methods for Unbounded Field Problems and a New Infinite Element Formulation, ASME, AMD, New York, 46, 115-148.

References

- 273 -

[161] Zienkiewicz OC, Bettess P, Kelly DW. (1978) The finite element method of determining fluid loading on rigid structures—two and three dimensional formulations. Numerical Methods in Offshore Engineering, John Wiley, Chichester, England. 141-183.

[162] Zienkiewicz OC, Emson CRI, Bettess P. (1983) A novel boundary infinite element. Int. J.Numer.Methods Engrg. 19, 393-404.

extending the scaled boundary finite-element method … · extending the scaled boundary...

Documents