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
Figure 3.10. Substructured model and meshes for Example 2, consisting of two
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
heaving in calm water ................................................................................... 66
Figure 3.20(a). Dimensionless added mass coefficient for a rectangular structure
swaying in calm water .................................................................................. 67
Figure 3.20(b). Dimensionless damping coefficient for a rectangular structure
swaying in calm water .................................................................................. 67
Figure 3.21(a). Dimensionless added mass coefficient for a rectangular structure
rolling in calm water ..................................................................................... 68
Figure 3.21(b). Dimensionless damping coefficient for a rectangular structure
rolling in calm water ..................................................................................... 68
Figure 3.22(a). Dimensionless added mass coefficient for a twin-hull structure
heaving in calm water ................................................................................... 69
Figure 3.22(b). Dimensionless damping coefficient for a twin-hull structure
heaving in calm water ................................................................................... 69
Figure 3.23(a). Dimensionless added mass coefficient for a twin-hull structure
swaying in calm water .................................................................................. 70
Figure 3.23(b). Dimensionless damping coefficient for a twin-hull structure
swaying in calm water .................................................................................. 70
Figure 3.24(a). Dimensionless added mass coefficient for a twin-hull structure
rolling in calm water ..................................................................................... 71
Figure 3.24(b). Dimensionless damping coefficient for a twin-hull structure
rolling in calm water ..................................................................................... 71
Figure 3.25(a). Dimensionless added mass coefficient for a twin-hull structure
heaving in calm water ................................................................................... 72
Figure 3.25(b). Dimensionless damping coefficient for a twin-hull structure
heaving in calm water ................................................................................... 72
Figure 3.26(a). Dimensionless added mass coefficient for a twin-hull structure
swaying in calm water .................................................................................. 73
Figure 3.26(b). Dimensionless damping coefficient for a twin-hull structure
swaying in calm water .................................................................................. 73
Figure 3.27(a). Dimensionless added mass coefficient for a twin-hull structure
rolling in calm water ..................................................................................... 74
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
circular cylinder for Example 1 .................................................................. 103
Figure 4.6(b). Variation of total wave elevation (imaginary part) around
circular cylinder for Example 1 .................................................................. 103
Figure 4.7(a). Variation of tangential velocity (real part) around
circular cylinder for Example 1 .................................................................. 104
Figure 4.7(b). Variation of tangential velocity (imaginary part) around
circular cylinder for Example 1 .................................................................. 104
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
Figure 4.13(a). Variation of total wave elevation (real part) around
elliptical cylinder for Example 2 .................................................................. 109
XII
Figure 4.13(b). Variation of total wave elevation (imaginary part) around
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
Figure 4.16(a). Variation of total wave elevation (real part) around
elliptical cylinder for example 3 ................................................................. 111
Figure 4.16(b). Variation of total wave elevation (imaginary part) around
elliptical cylinder for example 3 ................................................................. 111
Figure 4.17. Substructured model and meshes for Example 4, consisting of two
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)
along the surface of the single square cylinder for Example 4 .................. 113
Figure 4.19(a). Computed variation of tangential velocity (real part)
along the surface of the single square cylinder for Example 4 .................. 114
Figure 4.19(b). Computed variation of tangential velocity (imaginary part)
along the surface of the single square cylinder for Example 4 .................. 114
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)
along the surface of the twin caissons for Example 5................................ 118
Figure 4.24(b). Computed variation of tangential velocity (imaginary part)
XIII
along the surface of the twin caissons for Example 5................................ 119
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
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)153
Figure 5.6. Second-order components of wave loads on the horizontal rectangular
obstacle (B/H=1.0, D/H=0.2, L/B=1.0) ...................................................... 154
Figure 5.7. Second-order components of wave loads on the horizontal rectangular
obstacle (B/H=0.2, D/H=0.4, L/B=1.0) ...................................................... 154
Figure 5.8. Second-order components of wave loads on the horizontal rectangular
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
potential, on a fixed rectangular cylinder (B/H=1, D/H=0.6).................... 156
XIV
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).................... 157
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
base angles (θ ≤ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........................ 167
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
base angles (θ ≤ 90°) for Example 2 (B/H=1.0, D/H=0.2) ........................ 168
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
Figure 5.27. Substructured model and meshes for Example 3, consisting of two
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
Figure 5.30(a). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 173
Figure 5.30(b). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 173
Figure 5.30(c). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 174
XVI
Figure 5.31(a). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 174
Figure 5.31(b). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 175
Figure 5.31(c). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) .................................................. 175
Figure 5.32(a). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 176
Figure 5.32(b). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 176
Figure 5.32(c). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 177
Figure 5.33(a). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 177
Figure 5.33(b). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 178
Figure 5.33(c). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 178
Figure 5.34(a). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 179
Figure 5.34(b). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 179
Figure 5.34(c). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 180
Figure 5.35(a). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 180
Figure 5.35(b). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 181
Figure 5.35(c). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.05) .................................................. 181
Figure 5.36(a). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 182
XVII
Figure 5.36(b). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 182
Figure 5.36(c). Second-order component of horizontal wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 183
Figure 5.37(a). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 183
Figure 5.37(b). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 184
Figure 5.37(c). Second-order component of vertical wave force on the obstacle B1 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 184
Figure 5.38(a). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 185
Figure 5.38(b). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 185
Figure 5.38(c). Second-order component of horizontal wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 186
Figure 5.39(a). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 186
Figure 5.39(b). Second-order component of vertical wave force on the obstacle B2 for
Example 3 (B/H=1.0, D/H=0.3, Bg=0.1) .................................................... 187
Figure 5.39(c). Second-order component of vertical wave force on the obstacle B2 for
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
the dimensionless wave number ................................................................. 217
Figure 6.3 Variation of the dimensionless double-frequency moment about (B, -D) with
the dimensionless wave number ................................................................. 218
Figure 6.4 Computed variation of the dimensionless sum-frequency wave loads with
the dimensionless wave number ................................................................. 219
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
for Example 2 (B/H=1.0, D/H=0.4, θ=30°)................................................ 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°)................................................ 227
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
for Example 2 (B/H=1.0, D/H=0.4, θ=30°)................................................ 228
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
obstacles with various base angles
for Example 2 (B/H=1.0, D/H=0.4, θ ≥ 90°).............................................. 231
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°).............................................. 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°).............................................. 232
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°).............................................. 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°).............................................. 233
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°).............................................. 234
XX
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°)............................................... 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
obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01) ........................ 235
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) ........................ 236
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) ........................ 236
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) ........................ 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) ........................ 237
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) ........................ 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) ........................ 238
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) ........................ 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) ........................ 239
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) ........................ 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) ........................ 240
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) .................. 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) .................. 241
XXI
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) .................. 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) ........................ 242
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) ........................ 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) ........................ 243
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) ........................ 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) .................. 244
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) .................. 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) .................. 245
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) ........................ 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) ........................ 246
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) ........................ 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) ........................ 247
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) ........................ 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) ........................ 248
Figure 6.40 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=0.05) ........................ 249
Figure 6.41 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=0.05) ........................ 249
XXII
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) .................. 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) ........................ 250
Figure 6.44 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=0.05) .................. 251
Figure 6.45 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=0.05) ........................ 251
Figure 6.46 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=0.1) .......................... 252
Figure 6.47 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=0.1) .......................... 252
Figure 6.48 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=0.1) .......................... 253
Figure 6.49 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=0.1) .......................... 253
Figure 6.50 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=0.1) .................... 254
Figure 6.51 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) .................... 254
Figure 6.52 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=0.1) .................... 255
Figure 6.53 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=0.1) .......................... 255
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)
with the coefficient matrices
⎥⎦
⎤⎢⎣
⎡
−−−
=−−
−−
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 -
Substituting Equation (3-71) into Equation (3-69) yields
)](][[)],([ 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)
Substituting Equation (3-75) into Equation (3-72) yields
)](][[)],([ 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.
4.2 Mathematical formulation
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 Solution process
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)
with the coefficient matrices
⎥⎦
⎤⎢⎣
⎡
−+−−
=−−
−−
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)
Substituting Equation (4-56) into Equation (4-52) yields
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.
4.6 Results and discussions
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
Angle round cylinders in degrees
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
Angle round cylinders in degrees
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
Angle round cylinders in degrees
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.
Angle round cylinders in degrees
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.
Angle round cylinders in degrees
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=1.0)SBFEM (h/a=0.5)
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=2.0)SBFEM (h/a=1.0)
Exact (h/a=1.0)SBFEM (h/a=0.5)
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.
Angle round cylinders in degrees
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
Angle round cylinders in degrees
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.
Angle round cylinders in degrees
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.
Angle round cylinders in degrees
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.
Angle round cylinder in degrees
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.
Angle round cylinder in degrees
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.
Angle round cylinder in degrees
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 -
Angle round cylinder in degrees
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
Angle round cylinder in degrees
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
Leading caisson (SBFEM )Trailing caisson (SBFEM )Leading caisson (BEM )Trailing caisson (BEM )
Chapter Four
- 118 -
Angle round cylinder in degrees
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 -
Angle round cylinder in degrees
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
Leading caisson (SBFEM )Trailing caisson (SBFEM )Leading caisson (BEM )Trailing caisson (BEM )
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 -
5.2 Mathematical formulation
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.
5.4 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)
Substituting Equation (5-51) into Equation (5-44) yields
)(][),( 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)
Substituting Equation (5-55) and Equation (5-56) into Equation (5-51) yields
⎭⎬⎫
⎩⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
−
+
)()(
)()(
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.
5.5 Results and discussions
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
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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
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ads
fH
fV
m
kH
Chapter Five
- 154 -
Figure 5.7. Second-order components of wave loads on the horizontal rectangular obstacle (B/H=0.2, D/H=0.4, L/B=1.0).
Dim
ensio
nles
s wav
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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)
Figure 5.6. Second-order components of wave loads on the horizontal rectangular obstacle (B/H=1.0, D/H=0.2, L/B=1.0).
Dim
ensio
nles
s wav
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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)
Figure 5.8. Second-order components of wave loads on the horizontal rectangular obstacle (B/H=0.2, D/H=0.2, L/B=1.0).
Dim
ensio
nles
s wav
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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
Bounded domain Bounded domain
Scaling center Scaling center
D H
B B
Incident wave
Unbounded domain Unbounded
domain
Bounded domain Bounded 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
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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
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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
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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
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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
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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
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ads
kH
Figure 5.30(b). Second-order component of horizontal wave force on the obstacle B2 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 5.30(a). Second-order component of horizontal wave force on the obstacle B2 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
- 174 -
Dim
ensio
nles
s wav
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ads
kH
Figure 5.31(a). Second-order component of vertical wave force on the obstacle B2 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 5.30(c). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.01).
-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
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ads
kH
Figure 5.31(c). Second-order component of vertical wave force on the obstacle B2 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 5.31(b). Second-order component of vertical wave force on the obstacle B2 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
- 176 -
Dim
ensio
nles
s wav
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ads
kH
Figure 5.32(b). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
Figure 5.32(a). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.33(a). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
Figure 5.32(c). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.33(c). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.33(b). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.34(b). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
Figure 5.34(a). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.35(a). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
Figure 5.34(c). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.35(c). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
Figure 5.35(b). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.05).
-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
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ads
kH
Figure 5.36(b). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.36(a). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
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ads
kH
Figure 5.37(a). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
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ads
kH
Figure 5.36(c). Second-order component of horizontal wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.37(c). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.37(b). Second-order component of vertical wave force on the obstacle B1 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.38(b). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.38(a). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.39(a). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.38(c). Second-order component of horizontal wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.39(c). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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
Figure 5.39(b). Second-order component of vertical wave force on the obstacle B2 for Example 3 (B/H=1.0, D/H=0.3, Bg=0.1).
-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 Mathematical formulation
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)
Substituting Equation (6-10) into Equation (6-12) yields
+++ =+− 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)
Substituting Equation (6-18) into Equation (6-13) yields
1)(, 2 =+− ZgZ jiz ωω (6-19)
The solution of Equation (6-19) can be expressed as
)])(cosh[( hzkkDZ ji ++= (6-20)
Substituting Equation (6-20) into Equation (6-18) yields
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
φφφφφφωωωω
+++
++−
+++
=
∗∗
∗∗∗
∗∗∗−
(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.
6.4 Solution process
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.
Substituting Equation (6-63) into Equation (6-61) yields
)(][),( 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)
Substituting Equation (6-69) and Equation (6-70) into Equation (6-63) yields
⎭⎬⎫
⎩⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
±
±
±±
±±
±
±
±−
±+
)()(
)()(
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)
6.5 Results and discussions
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
φφφφφφωωωω
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Chapter Six
- 222 -
Figure 6.11 Sum-frequency second-order transmission coefficients (kjH=1.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Figure 6.10 Sum-frequency second-order reflection coefficients (kjH=1.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Chapter Six
- 223 -
Figure 6.13 Difference-frequency second-order transmission coefficients (kjH=1.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Figure 6.12 Difference-frequency second-order reflection coefficients (kjH=1.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Chapter Six
- 224 -
Figure 6.15 Sum-frequency second-order transmission coefficients (kjH=2.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Figure 6.14 Sum-frequency second-order reflection coefficients (kjH=2.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Chapter Six
- 225 -
Figure 6.17 Difference-frequency second-order transmission coefficients (kjH=2.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
Figure 6.16 Difference-frequency second-order reflection coefficients (kjH=2.0).
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
D=0.4H(Coarse)D=0.4H(Medium)D=0.4H(Fine)D=0.6HD=0.8H
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
30 degree45 degree60 degree90 degree
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
90 degree120 degree135 degree150 degree
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
30 degree45 degree60 degree90 degree
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
90 degree120 degree135 degree150 degree
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
30 degree45 degree60 degree90 degree
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
90 degree120 degree135 degree150 degree
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
30 degree45 degree60 degree90 degree
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
90 degree120 degree135 degree150 degree
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
30 degree45 degree60 degree90 degree
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
90 degree120 degree135 degree150 degree
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
30 degree45 degree60 degree90 degree
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
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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
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s w
ave
load
s
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1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45
Coarse mesh
Medium mesh
Fine mesh
Chapter Six
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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
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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
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Coarse mesh
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Chapter Six
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Figure 6.41 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=0.05)
kiH
Dim
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s w
ave
load
s
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Coarse mesh
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Fine mesh
Figure 6.40 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=0.05)
kiH
Dim
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s w
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Fine mesh
Chapter Six
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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
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s w
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load
s
0
4
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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
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s w
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Chapter Six
- 251 -
Figure 6.45 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=0.05)
kiH
Dim
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s w
ave
load
s
048
12162024283236
0 1 2 3 4
Coarse mesh
Medium mesh
Fine mesh
Figure 6.44 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=0.05)
kiH
Dim
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Chapter Six
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Figure 6.47 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=0.1)
kiH
Dim
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s w
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load
s
0123456789
0 1 2 3 4
Coarse mesh
Medium mesh
Fine mesh
Figure 6.46 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=0.1)
kiH
Dim
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Chapter Six
- 253 -
Figure 6.49 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=0.1)
kiH
Dim
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s
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3.5
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0 1 2 3 4
Coarse mesh
Medium mesh
Fine mesh
Figure 6.48 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=0.1)
kiH
Dim
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s w
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1.5
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Chapter Six
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Figure 6.51 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)
kiH
Dim
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s w
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s
0
4
8
12
16
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0 1 2 3 4
Coarse mesh
Medium mesh
Fine mesh
Figure 6.50 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=0.1)
kiH
Dim
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s w
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25
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0 1 2 3 4
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Fine mesh
Chapter Six
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Figure 6.53 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=0.1)
kiH
Dim
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nles
s w
ave
load
s
0
4
8
12
16
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0 1 2 3 4
Coarse mesh
Medium mesh
Fine mesh
Figure 6.52 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=0.1)
kiH
Dim
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Chapter Seven
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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
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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
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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
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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
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