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A direct coupling numerical method for solving three-dimensional interaction problems of wave and floating structures
Engineering Analysis with Boundary Elements 55 (2015) 10–27
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
A direct coupling numerical method for solving three-dimensional interaction problems of wave and floating structures Jeng-Hong Kao a,n, Jaw-Fang Lee a, Yu-Chun Chengb a b
Department of Hydraulics and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan, ROC Fifth River Management Office, Water Resources Agency, Chiayi, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2014 Accepted 6 November 2014 Available online 3 February 2015
In this article, a direct numerical method is developed to solve three-dimensional interaction problems of wave and floating structures. In the problem formulation, the diffracted wave and radiated wave by the structure are represented as one induced wave potential. The wave field is simulated using a boundary element method, and the structure motions are of six degrees of freedom. In the solution procedure, the structural motions are expressed in terms of wave potentials, which are then substituted into the wave model. The created linear algebraic system is then calculated. The present numerical model is used to simulate a floating rectangular cuboid deployed in a wave channel, and the results are compared with a theory to confirm the accuracy. The present numerical model is further used to calculate a floating structure deployed in an open sea to show the capability of the three-dimensional computation. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Coupling Interaction Three dimension Diffraction effects Boundary element method
1. Introduction Floating breakwaters are considered as alternatives to conventional fixed breakwaters deployed in coastal areas for preserving small marinas and recreational harbors from attacks of wave forces. The main advantages of floating breakwaters are (a) low-cost construction independent of water depth and seabed geological conditions, (b) easy transportation, especially for temporary facilities, and (c) ecologically advantageous since the water circulation, biological exchange, and sediment transport beneath the structure are allowed. Thus, many studies have been undertaken to investigate the performance of various types of floating breakwaters. Regarding problems of wave interaction with free floating structures, Ijima et al. [1] presented an analytic solution for wave interaction with a rectangular floating structure in two dimension, in which free floating and spring moored structures were considered. Au and Brebbia [2] used a boundary element method calculating wave forces acting on two-dimensional structures, where submerged, floating and arbitrary geometrical structures were considered. Huang [3] used a finite element method for solving interference problems between water waves and two-dimensional or three-dimensional ocean structures. The added mass, damping coefficient and wave force were discussed for different wave periods. Matsui et al. [4] solved water wave diffraction and radiation by arbitrarily shaped three-dimensional bodies using a hybrid integral equation method. The boundary element idealization was used only in an inner fluid region close to the body and local depth irregularities, while an analytical solution was employed in the outer region of constant depth extending to infinity. The two representations were matched on a fictitious vertical cylindrical surface. Lee [5] presented an analytic solution to solve the heave radiation problem of a rectangular structure. The nonhomogeneous boundary value problem was linearly decomposed into homogeneous ones, which can be readily solved. Yilmaz [6] solved the diffraction and radiation problems of a group of truncated vertical cylinders using an exact analytical method. Abul-Azm and Gesraha [7] studied wave–structure interaction problems. The structure has three degrees of freedom in motion. The research results discussed elimination effects of wave and structure motions in different configurations. Chen et al. [8] presented a complete analytic solution for wave interacting with two-dimensional floating structures by applying the analytic approach for nonhomogeneous boundary value problems proposed by Lee [5]. Diamantoulaki et al. [9] aimed at assessing the effect of pontoon spacing on the performance of floating free or moored twin pontoon breakwaters using a panel method. The results showed the
n
Corresponding author. Tel.: þ 886 6 2757575x63200; fax: þ 886 6 2741463. E-mail address: [email protected] (J.-H. Kao).
http://dx.doi.org/10.1016/j.enganabound.2014.11.013 0955-7997/& 2014 Elsevier Ltd. All rights reserved.
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
11
roll response decreased while pontoon spacing increased for both free and moored pontoon floating breakwaters. Kim et al. [10] investigated the characteristics of bending moments, shear forces and stresses at unit connections of very large floating structures (VLFS) under wave loads. The characteristics of bending moments, shear forces and stresses at unit connections were discussed at numerical examples. The three-dimensional diffraction of obliquely incident water waves by a floating structure near a wall with step-type bottom was studied using a small amplitude wave theory [11]. The wave forces and the wave elevations on the free surface were investigated for different incident wave angles and water depth ratios. For wave interaction with tension-leg or spring mooring floating structures, Lee and Lee [12] developed an analytical solution for the coupled linear problem. The fluid-induced drag on the tension legs was not considered. The only force from tension legs was the pretensioned force, which provided part of the stiffness of the floating structures. Lee [13] solved tension-leg structures interacting with linear waves. The eigen-function representations of the velocity potential and surface elevations of scattering and radiation waves were developed by Lee and Lee [12]. Analytical solutions showed that the inertia drag on tension legs was negligible compared to that due to the evanescent waves caused by the wave–structure interaction. Lee et al. [14] presented an analytical solution for the dynamic behavior of both the platform and tethers in the tension leg platform system. Elchahal et al. [15] studied wave interaction with a moored floating breakwater with a harbor boundary, and the effects of various structural parameters of the breakwater on the transmitted wave heights were discussed. Ker and Lee [16] proposed an analytical solution for the problem of waves incident on a porous tension leg platform (TLP). The permeable floating body was considered as anisotropic and homogeneous. Results showed that the drags in the porous body change the TLP behavior significantly. Lee and Ker [17] further presented an analytic solution for problems of two-dimensional tension leg structures, where the floating structure was composed of impermeable and porous regions. Floating structures with attached mooring lines can also be found in the literature. Sannasiraj et al. [18] considered moored floating structure subjected to incident waves, where the stiffness of the mooring lines was calculated using the catenary cable equation. Structural motions and mooring forces for three different mooring configurations were discussed. Diamantoulaki and Angelides [19] considered the performance of an array of floating breakwaters connected with hinge joints under the action of linear monochromatic waves in the frequency domain. The effects of the configuration of hinge joints on the response and effectiveness of the freely floating array were investigated. To date, in the solutions to interaction problems of wave and floating structures, the wave field was decomposed into a scattering problem and radiation problem. The scattering and radiation problems were solved independently; then the solutions were combined together with equations of motion of the floating structures to solve the entire problem. In this study, the direct coupling method can obtain wave fields by solving one liner algebraic system. The wave fields induced by the floating structure including scattering and radiation waves are considered as a whole induced wave potential. With a given incident wave, the induced wave potential together with motions of the floating structure is solved, where the wave problem is calculated using a boundary element method. The advantage of the proposed method is reduced computer computational time and shorter time to obtain the calculation results. The effect is more significant for the three-dimensional problem.
2. Description and solution for the three-dimensional problem The problem considered is a floating rectangular cuboid subjected to oblique incident waves, as shown in Fig. 1. A Cartesian coordinate system is adopted with positive x directed to the right, positive y into the paper, and positive z axis upward. The water depth is h, length and width of the structure are 2l and 2b, respectively, and draft is d. Incident waves propagate in the positive x direction, and the surface elevation is ηI . The center of rotation of the structure is located at Cðxc ; yc ; zc Þ and the gravity center is Oðxo ; y0 ; zo Þ. With the action of incident waves, in addition to induced waves around the floating structure, the floating structure is in six-degree of freedom motion. For steady periodic problems, the potential function of incident waves can be expressed as
ΦI ðx; y; z; tÞ ¼ ϕI ðx; y; zÞe iωt
ð1Þ
where
ϕI ðx; y; zÞ ¼ iAI
g cosh K ðz þ hÞ iKðx e cosh Kh
ω
cos α þ y sin αÞ
ð2Þ
in which AI is wave amplitude, K is wave number pffiffiffiffiffiffiffiffi (K ¼ 2π =L), L is wave length, wave period, g is gravity constant and i ¼ 1.
z
α is incident angle, ω is angular frequency (ω ¼ 2π =T), t is
y
r
l I
e
b O
C
d
x h
Fig. 1. Definition sketch of 3D waves interacting with a floating structure.
12
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2.1. The boundary value problem of the wave field The wave fields induced by the floating structure subjected to incident waves can be described by the following boundary value problem. The governing equation is the Laplace equation: ∇2 ϕðx; y; zÞ ¼ 0
ð3Þ
where ϕ is the wave potential function. The corresponding boundary conditions are [3] ∂ ϕ ω2 ¼ ϕ; ∂z g ∂ϕ ¼ 0; ∂z
z¼0
ð4Þ
z ¼ h
ð5Þ
pffiffiffiffiffiffi∂ϕ iK ϕ ¼ 0 Kr r-1 ∂r
ð6Þ
lim
in which Eqs. (4), (5) and (6) are the free surface condition, bottom condition and the radiation boundary condition, respectively. The boundary condition on the structural surface can be written as [20] I ∂ ϕþϕ ¼ iωs1 iωs5 ðz zc Þ þ iωs6 y yc ; x ¼ 7 b ð7Þ ∂x I ∂ ϕþϕ ¼ iωs2 þ iωs4 ðz zc Þ iωs6 ðx xc Þ; y ¼ 7 l ð8Þ ∂y
I ∂ ϕþϕ ∂z
¼ iωs3 iωs4 y yc þ iωs5 ðx xc Þ;
z ¼ d
ð9Þ
where sj stands for amplitude of the jth degree of freedom of the structural motions, the subscript j¼1, 2, 3, 4, 5, and 6 represents surge, sway, heave, roll, pitch, and yaw, respectively. Eqs. (7)–(9) contain unknown structural motions; therefore, equations of motion of the structure are required so that the wave field can be solved. 2.2. Equation of motion of the floating structure The matrix equation in six degrees of freedom of the three-dimensional floating structure can be expressed as [21] n o
½M ξ€ þ ½K ξ ¼ fF g in which mass matrix ½M and stiffness matrix ½K can be written, respectively, as 2 3 m y0 yc m 0 0 0 mð z 0 z c Þ 6 7 6 0 mðx0 xc Þ 7 0 m 0 mðz0 zc Þ 6 7 6 7 0 mðx0 xc Þ 0 0 m m y0 yc 6 7 6 7 b b b ½M ¼ 6 7 0 m z I ð z Þ m y y I I c 0 0 c xx yx xz 6 7 6 7 b b b 6 mðz zc Þ 7 0 mðx0 xc Þ I xy I yy I zy 0 6 7 4 5 mðx0 xc Þ m y0 yc 0 I bxz I byz I bzz 2
0 6 60 6 60 6 ½K ¼ 6 60 6 6 6 40 0
0
0
0
0
0
0
0
0
0
ρga
0
ρ
ρ
gI ay
ρ
gI ax
gI ay
ρg I ayy þ Ivz mgðz0 zc Þ
0
ρgI ax
ρgI ayx
ρgI axy a ρg I xx þ Ivz mgðz0 zc Þ
0
0
0
0
0
ð10Þ
ð11Þ
3
7 7 7 7 0 7 7 v ρgI x þ mg ðx0 xc Þ 7 7 7 7 ρgI vy þ mg y0 yc 5 0 0
ð12Þ
where I ax ¼ 4blðxc Þ 2
I axx ¼ 4bl
ð13Þ
b þxc 2 3
!
I ay ¼ 4bl yc 2
I ayy ¼ 4bl
l þ y2c 3
ð14Þ ð15Þ
! ð16Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
13
z I
r
t
e d
O
x
C
b
h
Fig. 2. Definition sketch of 2D waves interacting with a floating structure.
d I vz ¼ 4bld zc 2
ð17Þ
where m is mass of the structure, ρ is fluid density, I bxx , I byy and I bzz are moments of inertia about center of rotation in three coordinate axes; I ax and I axx are first moment and second moment of width of the structure at still water level about z axis, respectively; I ay and I ayy are first moment and second moment of length of the structure at still water level about z axis, respectively. I vz is first moment of submerged structure volume about center of rotation. The displacement function of the floating structure on the left-hand side of Eq. (10) can be written as
ξ ¼ fsge iωt ð18Þ The force expression on the right-hand side of Eq. (10) can be rewritten as
fF g ¼ f e iωt where the force amplitude is expressed in terms of the wave potentials Z
f ¼ iωρ ϕ þ ϕI fngdS
ð19Þ
ð20Þ
S
and fng ¼
n
n1
n2
n3
y yc n3 ðz zc Þn2
ðz zc Þn1 ðx xc Þn3
oT ðx xc Þn2 y yc n1
ð21Þ
in which n1, n2 and n3 are unit normal in x, y and z directions, respectively. By substituting Eqs. (18) and (19) into Eq. (10), one obtains
ω2 ½M þ ½K fsg ¼ f
ð22Þ
with the matrix inversion of Eq. (22), amplitudes of the structure motion can be obtained as h i
fsg ¼ K~ f
ð23Þ
where h i 1 K~ ¼ ω2 ½M þ ½K
ð24Þ
Eq. (22) expresses the structural motion in terms of a given incident wave and unknown induced wave potentials. In the subsequent solution methodology, Eq. (23) is substituted into boundary conditions on the structural surface in the wave problem, so that the wave field can be solved. In this study, the unknown wave field is analyzed using a boundary element method. 2.3. A boundary element model for the wave field A boundary element method is applied to solve the wave problem. Using the constant element, the boundary element equation can be expressed as [2,22–25] N X j¼1
H ij ϕj ¼
N X j¼1
Gij
∂ϕj ∂n
ð25Þ
where ϕj and ∂ϕj =∂n represent wave potential and normal derivative quantity of the jth element, respectively, and the coefficient matrices are 8 R ∂ ϕn > < 12 þ Γ j ∂n dΓ ; i ¼ j H ij ¼ ð26Þ R ϕn > : Γ j ∂∂n dΓ ; ia j
14
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 3. Variation of reflection coefficient versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 4. Variation of reflection coefficient versus relative water depth using 2D model for b/h ¼ 0.5.
Z Gij ¼
Γj
ϕn dΓ
ð27Þ
in which ϕ is the fundamental solution. For the three-dimensional Laplace equation, the fundamental solution is n
ϕn ¼
1 4π r ij
where r ij is the distance measured from the source point i to the jth element. The matrix form of Eq. (25) can be expressed as
∂ϕ ½H ϕ ¼ ½G ∂n
ð28Þ
ð29Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 5. Dimensionless amplitude for surge motion versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 6. Dimensionless amplitude for surge motion versus relative water depth using 2D model for b/h ¼ 0.5.
By substituting boundary conditions, Eqs. (4)–(9), into Eq. (29), one obtains 2
H 11 6 6 H 21 6 6 H 31 6 6 6 H 41 6 6 H 51 6 6 6 H 61 6 6 H 71 4 H 81
H 12
H 13
H 14
H 15
H 16
H 17
H 22
H 23
H 24
H 25
H 26
H 27
H 32
H 33
H 34
H 35
H 36
H 37
H 42
H 43
H 44
H 45
H 46
H 47
H 52 H 62
H 53 H 63
H 54 H 64
H 55 H 65
H 56 H 66
H 57 H 67
H 72
H 73
H 74
H 75
H 76
H 77
H 82
H 83
H 84
H 85
H 86
H 87
38 9 H 18 > > > ϕ1 > > 7> > > > H 28 7> > > > ϕ2 > 7> > > > > > ϕ H 38 7 > > > 3> 7> > > 7> < H 48 7 ϕ4 = 7 7 H 58 7> > > ϕ5 > > > > 7> > > H 68 7> > > ϕ6 > > 7> > > > 7 > > ϕ7 > H 78 5> > > > > > : ϕ ; H 88
8
15
16
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
2
G11 6 6 G21 6 6 G31 6 6 6 G41 ¼6 6 G51 6 6 6 G61 6 6 G71 4 G81
G12
G13
G14
G15
G16
G17
G22
G23
G24
G25
G26
G27
G32
G33
G34
G35
G36
G37
G42
G43
G44
G45
G46
G47
G52 G62
G53 G63
G54 G64
G55 G65
G56 G66
G57 G67
G72
G73
G74
G75
G76
G77
G82
G83
G84
G85
G86
G87
9 8 ω2 ϕ > > > 1 g 3> > > > > > G18 > > > > > iK ϕ2 > > 7> > > > G28 7> > > > 0 7> > > > I 7 > > G38 7> > ∂ ϕ4 > > þ i ω s i ω s ð z z Þ þ i ω s y y > > c 5 1 6 c ∂x > > 7< = G48 7 I 7 þ ∂ϕ5 iωs þiωs ðz z Þ iωs ðx x Þ c c 7 2 4 6 ∂y G58 7> > > > > I > > 7> > ∂ ϕ6 > > G68 7> þ i ω s þ i ω s ð z z Þ i ω s y y > > c 5 1 6 c ∂x > > 7> > > > 7 I > > G78 5> ∂ϕ7 > > þ iωs2 iωs4 ðz zc Þ þiωs6 ðx xc Þ > > > ∂y > > > > G88 > > > > ∂ϕI > ; : 8 þiωs3 þ iωs4 y y iωs5 ðx xc Þ > ∂z
c
Fig. 7. Dimensionless amplitude for heave motion versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 8. Dimensionless amplitude for heave motion versus relative water depth using 2D model for b/h ¼ 0.5.
ð30Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
17
Fig. 9. Dimensionless amplitude for pitch motion versus relative water depth using 2D model for b/h ¼1.0.
In Eq. (30) subscripts shown in the wave potentials represent the belonging boundaries, 1, 2, and 3 stand for the free surface, radiation boundary, and sea bottom, respectively. 4 and 6, 5 and 7 represents structural boundaries with normal in x and y directions, respectively. 8 represents structural bottom.
2.4. A solution procedure for wave and structure interaction problem For the interaction problem of incident waves and the floating structure, the unknown induced wave potential is expressed by the boundary element equation, Eq. (30). And the structural motion is governed by Eq. (23). It is obvious that the wave equation, Eq. (20), contains unknown structural motions, and the structure equation, Eq. (23), contains unknown wave potentials. The unknown structural motions shown on the right-hand side of Eq. (30) can be substituted using Eq. (23). Eq. (30) is then expressed in terms of wave potentials only that can be used to solve the unknown wave functions. Eq. (30) is now reformulated by moving all known variables to the right-hand side. The reformed matrix equation can be expressed as 2 2 ω G
6 H 11 g 11 6 6 6 6 6 H ω2 G 6 21 g 21 6 6 6 6 6 H 31 ω2 G31 6 g 6 6 6 6 6 H 41 ω2 G41 g 6 6 6 6 2 6 6 H 51 ωg G51 6 6 6 6 6 H ω2 G 6 61 g 61 6 6 6 6 6 H 71 ω2 G71 6 g 6 6 6 6 4 H 81 ωg2 G81
H 12 iKG12 H 22 iKG22 H 32 iKG32 H 42 iKG42
H 13 H 23 H 33 H 43
8 X
H 14 C i4 G1i i¼4 H 24
8 X
C i4 G2i
H 25
i¼4 8 X
C i4 G4i
H 35
H 53
H 54
8 X
C i4 G5i
H 45
H 63
H 64
8 X
C i4 G6i
H 55
H 82 iKG82
H 73 H 83
H 74
8 X
C i4 G7i
8 X
H 65
8 X
H 26
H 75
8 X i¼4
8 X
8 X
H 85
i¼4
C i6 G1i
H 17
8 X
H 36
8 X
C i6 G2i
H 27
H 46
8 X
H 37
H 56
8 X
C i6 G4i
H 47
H 66
8 X
C i6 G5i
H 57
H 76
8 X
C i6 G6i
H 67
H 86
8 X i¼4
C i7 G2i
8 X
C i7 G3i
8 X
C i7 G4i
8 X
C i7 G5i
i¼4
C i6 G7i
H 77
i¼4
C i5 G8i
8 X
i¼4
i¼4
C i5 G7i
C i7 G2i
i¼4
i¼4
C i5 G6i
8 X
8 X
C i7 G6i
i¼4
C i6 G8i
H 87
8 X i¼4
8 X
C i7 G7i
3 C i8 G1i
7 7 7 7 8 7 X i H 28 C 8 G2i 7 7 7 i¼4 7 7 8 X 78 9 i H 38 C 8 G3i 7 7> > > ϕ1 > > > 7> i¼4 > ϕ2 > 7> > > > > 7 > > 8 > X 7> > > ϕ i > > 3 > > H 48 C 8 G4i 7 > > > 7> < 7 ϕ4 = i¼4 7 7 8 X i > > ϕ5 > 7> > > > H 58 C 8 G5i 7> > > ϕ6 > 7> > > > i¼4 7> > > > 7> > > ϕ 8 > > 7 7 X i > > > > 7 : H 68 C 8 G6i 7 ϕ8 ; 7 i¼4 7 7 8 X 7 i H 78 C 8 G7i 7 7 7 i¼4 7 7 8 X 7 i H 88 C 8 G8i 5 H 18
i¼4
i¼4
i¼4
C i5 G5i
C i7 G1i
i¼4
C i6 G3i
i¼4
C i5 G4i
8 X i¼4
i¼4
C i5 G3i
i¼4
i¼4
H 84 C i4 G8i i¼4
C i5 G2i
i¼4
i¼4
H 72 iKG72
8 X
8 X i¼4
i¼4
i¼4
H 62 iKG62
H 16
i¼4
i¼4
H 52 iKG52
8 X
8 X
8 X
C i5 G1i
i¼4
i¼4
H 34 C i4 G3i i¼4 H 44
H 15
8 X
i¼4
18
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 10. Dimensionless amplitude for pitch motion versus relative water depth using 2D model for b/h ¼ 0.5.
Fig. 11. Definition sketch of the floating rectangular cuboid in a 3D channel and subjected to incident waves.
2
G14 6 6 G24 6 6 G34 6 6 6 G44 ¼6 6 G54 6 6 6 G64 6 6 G74 4 G84
G15
G16
G17
G25
G26
G27
G35
G36
G37
G45
G46
G47
G55
G56
G57
G65
G66
G67
G75
G76
G77
G85
G86
G87
9 8 8 X I > > > ∂ ϕ4 4 I> > > þ þ C ϕ > j j> ∂x > > > > > j¼4 3> > > > > > G18 > > > > > 8 X I > > 7> > I ∂ ϕ 5 5 > G28 7> > þ þ C ϕ j> j > > ∂y 7> > > > j ¼ 4 7 > > G38 7> > > > > > > 7> 8 X = < I G48 7 ∂ ϕ6 6 I 7 ∂x þ C j ϕj 7 G58 7> > j¼4 > > > > > 7> > > > G68 7> 8 > > X I > 7> I ∂ ϕ 7 > > 7 > > 7 þ C ϕ > > ∂y j> j G78 5> > > > > j¼4 > > > > G88 > > > 8 > > X 8 I> I > > ∂ ϕ > > 8þ > > C ϕ > j j> ∂z > > ; : j¼4
ð31Þ
where coefficients C's shown are known constants, and are expressed in Appendix A. The matrix equation, Eq. (31), can then be solved to obtain unknown wave potentials. Upon solving Eq. (31) the free surface elevation can then be calculated using Bernoulli's equation
ηðx; y; t Þ ¼
iω ϕðx; y; 0Þ Ue iωt g
ð32Þ
By substitution of wave potential ϕ and incident wave potential ϕ into Eq. (23), the structure motion can then be calculated. So far, the three-dimensional problem of a floating structure subjected to action of incident waves is completely solved. The wave field and the structural motions can all be determined. I
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
19
Fig. 12. Variation of reflection coefficient versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
Fig. 13. Dimensionless amplitude for surge motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
3. Numerical examples In order to show the accuracy and validity of the proposed method, the three potential problems are considered. Numerical examples containing 2D and 3D wave interaction with floating structure, respectively, are considered. The numerical results agreed very well with those of Ijima et al. [1]. Finally, the results show as diffraction effect on the motions of a floating structure. 3.1. Two-dimensional wave and structure interaction problem In this case, we used a two-dimensional numerical model to solve two-dimensional waves interacting with a floating structure problem, and geometrical dimensions of the problem are shown in Fig. 2. The method of creating two-dimensional model is the same as the step of Sections 2.1–2.4. Here, the amplitude of structural motion of s2, s4 and s6 is equal to zero in Eqs. (7)–(9), and we substitute 2D
20
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 14. Dimensionless amplitude for heave motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
Fig. 15. Dimensionless amplitude for pitch motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
equation of structural motion and 2D fundamental solution into Eqs. (10) and (28), respectively. For this case, the water depth is 4 m. Widths of the structure considered are b=h ¼ 1:0 and 0.5. Drafts of the structure considered are d=h ¼ 0:75, 0.5, 0.25 and 0.1. The fluid density is 1000 kg/m3. The center of rotation and gravity of the floating structure are considered to coincide. The results of reflection coefficient and three motion amplitudes versus relative water depth by using the present model are shown in Figs. 3–10. In these figures, the real and dotted lines denote the 2D theoretical results from Ijima et al. [1]; the symbols denote the present results. The comparisons present results that agree very well with the theoretical solution. Figs. 3 and 4 show reflection coefficients Kr versus relative water depth ω2 h=g for different width of structure. The reflection coefficient is larger when draft of structure increases. The reflection coefficient approaches 1 in higher relative water depth. The structural width increase can obtain better reflection effect in higher relative water depth. Figs. 5–10 show dimensionless amplitudes j s1 j =AI , j s3 j =AI and j bs5 j =AI versus relative water depth for surge, heave and pitch motions, respectively. The amplitude of surge motion versus relative water depth is shown in Figs. 5 and 6. Overall speaking, the amplitude of surge
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
21
Fig. 16. A contour plot of dimensionless wave amplitudes for a rectangular cuboid in an open sea (t¼ 4.35 s, d/h ¼ 0.75).
Fig. 17. Dimensionless amplitudes for surge motion versus relative water depth for a rectangular cuboid in an open sea.
motion is infinite when relative water depth approaches zero. Under increasing relative water depth, the amplitude of surge motion approaches zero. Furthermore, resonance frequency was obvious in b=h ¼ 1:0, and that moved towards low frequency when draft increased. For heave motion, dimensionless amplitudes j s3 j =AI versus relative water depth are shown in Figs. 7 and 8. The amplitude of heave motion is equal to the one in relative water depth equal to zero. The response resonant amplitude increases when draft increases. The resonant frequencies move toward low frequency as the width of structure increases. Dimensionless amplitudes of pitch motion j bs5 j =AI versus relative water depth are shown in Figs. 9 and 10. Overall speaking, the response amplitude became larger at resonant frequency, and the response amplitude approaches zero in other frequencies. For b=h ¼ 1:0, the resonant amplitude increases as the draft increases, but the resonant amplitude decreases as the draft increases for b=h ¼ 0:5. The resonant frequencies move toward low frequency as width of the structure increases.
22
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 18. Dimensionless amplitudes for heave motion versus relative water depth for a rectangular cuboid in an open sea.
Fig. 19. Dimensionless amplitudes for pitch motion versus relative water depth for a rectangular cuboid in an open sea.
3.2. Three-dimensional wave and structure interaction problem in the wave channel To verify the accuracy of the present model, a floating rectangular cuboid deployed in a wave channel subjected to incident waves is simulated. Geometrical dimensions of the problem are described in Fig. 11. Ideally the floating structure has the same width as the wave channel. The water depth in the wave channel is h. In the numerical model, a finite artificial truncated length of the wave channel is adopted, and on the artificial boundary a radiation boundary condition is used ∂ϕ ¼ iK ϕ; ∂n
x ¼ 7ℓa
ð33Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
23
where ℓa is the location of the artificial boundary. Furthermore, on sidewalls of the wave channel, the impermeable condition is ∂ϕ ¼ 0; ∂n
y ¼ 7l
ð34Þ
Since the structure is confined in the wave channel, the degrees of freedom of the structural motions, sway, roll, and yaw are zero. To solve the problem, the equations of motion of the structure, Eq. (23), and the boundary element equation for the wave problem, Eq. (30), are combined to obtain 2 2 ω G
6 H 11 g 11 6 6 6 6 6 H ω2 G 6 21 g 21 6 6 6 6 6 H ω2 G 6 31 g 31 6 6 6 6 6 H ω2 G 6 41 g 41 6 6 6 6 6 H 51 ω2 G51 6 g 6 6 6 6 6 H 61 ω2 G61 6 g 6 6 6 6 6 H 71 ω2 G71 g 6 6 6 6 6 4 H 81 ω2 G81 g 2
G14
6 6 G24 6 6 G34 6 6 6 G44 ¼6 6 G54 6 6 6 G64 6 6 G74 4 G84
H 12 iKG12 H 22 iKG22
H 13 H 23
8 X
H 14 C i4 G1i i ¼ 4;6 H 24
8 X
C i4 G2i
H 15
H 16
H 33
H 34
8 X
C i4 G3i
H 25
H 26
H 43
H 44
8 X
C i4 G4i
H 35
H 36
H 53
H 54
8 X
C i4 G5i
H 45
H 46
H 63
H 64
8 X
C i4 G6i
H 55
H 56
H 73
H 74
8 X
C i4 G7i
H 65
H 66
G16 G26 G36 G46 G56 G66 G76 G86
H 83
H 84
8 X
C i4 G8i
i ¼ 4;6
3 9 G18 8 8 X I > 7> > ∂ϕ4 4 I> > G28 7> > C j ϕj > þ ∂x þ > 7> > > > > j ¼ 4;6 > > G38 7 > > > 7> > > > 7> 8 X = < I G48 7 ∂ϕ6 6 I 7 ∂x þ C j ϕj 7 G58 7> > > > j ¼ 4;6 > > > 7> > > > G68 7> 8 > > X I > > ∂ϕ8 7> 8 I> > > 7 þ C ϕ j j> G78 5> ∂z > > ; : j ¼ 4;6 G88
8 X
C i6 G3i
H 37
8 X
C i6 G4i
H 47
8 X
C i6 G5i
H 57
8 X
C i6 G6i
H 67
C i6 G7i
H 77
C i6 G8i
H 87
i ¼ 4;6
H 75
H 76
i ¼ 4;6
H 82 iKG82
H 27
i ¼ 4;6
i ¼ 4;6
H 72 iKG72
C i6 G2i
i ¼ 4;6
i ¼ 4;6
H 62 iKG62
8 X
8 X i ¼ 4;6
H 85
H 86
8 X i ¼ 4;6
H 18
8 X
3 C i8 G1i
7 7 7 7 8 7 X i H 28 C 8 G2i 7 7 7 i ¼ 4;6 7 7 8 78 X i 9 H 38 C 8 G3i 7 7> > > ϕ1 > 7 > > i ¼ 4;6 7> > > ϕ2 > > 7> > > > > 8 X i 7> > > > ϕ > > 7 3 > > H 48 C 8 G4i 7> > > > = < 7 i ¼ 4;6 7 ϕ4 7 8 ϕ5 > X i 7> > > > > > > H 58 C 8 G5i 7 > 7> > > ϕ 6 > > 7 > > i ¼ 4;6 > > 7> > > ϕ 7> > > 7 8 X i > 7> > > : 7 H 68 C 8 G6i 7 ϕ8 ; 7 i ¼ 4;6 7 7 8 X 7 i H 78 C 8 G7i 7 7 7 i ¼ 4;6 7 7 8 X 7 i H 88 C 8 G8i 5 i ¼ 4;6
i ¼ 4;6
i ¼ 4;6
H 52 iKG52
H 17
i ¼ 4;6
i ¼ 4;6
H 42 iKG42
C i6 G1i
i ¼ 4;6
i ¼ 4;6
H 32 iKG32
8 X
i ¼ 4;6
ð35Þ
Notice that in Eq. (35) restrictions of the structural motions, radiation condition, Eq. (33), and side wall conditions, Eq. (34), are implemented. In the numerical computation, the water depth is 4 m, and width and length of the structure are 8 m and 1 m, respectively. Drafts of the structure considered are d=h ¼ 0:75 and 0.5. The fluid density is 1000 kg/m3. The center of rotation and gravity of the floating structure are considered to coincide. In boundary element calculation, element sizes are kept less than 1/50 of the wavelength in the higher frequency. For the lower frequency, element sizes are kept less than 1/40 of the wavelength. And the radiation boundary is located 6–12 times water depth away from the structure [26]. Since the floating structure is deployed in the channel with two sidewalls, and with normal incident waves, the simulated results are expected to have two-dimensional characteristics. Therefore, a two-dimensional theory by Ijima et al. [1] is used to compare with the results. Using the present model the reflection coefficient in front of the structure and motion amplitudes of the three degrees of freedom of the structure versus relative water depth ω2 h=g are shown in Figs. 12–15. In the figures real and dashed lines represent two-dimensional results from Ijima et al. [1], triangle or circle symbols are the present results. The comparisons indicate the present numerical results agree very well with the theory, thereby illustrating the correctness of the present model. On the other hand, in present three-dimensional results wave elevations across the wave channel are constant, also explaining that the present model is accurate. Fig. 12 shows the reflection coefficients versus relative water depth ω2 h=g. In view of tendency for high relative water depth (short wave period) the reflection coefficients approach unity. Waves are blocked in front of the structure. On the other hand, for low relative water depth waveresisting capability of the structure decreases, and most waves are transmitted behind the structure. Furthermore, resonant phenomena of the floating structure decrease wave reflections, and with increased drafts the resonant frequencies move toward low-frequency (long wave) direction. Figs. 13–15 are motion amplitudes j s1 j =AI , j s3 j =AI and j bs5 j =AI versus relative water depth for surge, heave and pitch, respectively. Overall speaking, surge motions of the structure decrease with increased water depth; in other words, surge motions decrease with the decreasing wave period, and approach zero at resonant frequency. For heave and roll motions the amplitudes decrease with increasing water depth, but resonant amplitudes increase. Furthermore, for all the three modes, resonant frequencies move toward low frequency as drafts increase. From analysis of equations of motion of the floating structure, there exists hydrostatic restoring force in heave and roll motion; therefore, response amplitudes increase at resonant frequency, and increased draft can produce more resonance amplitude. However, there is no hydrostatic restoring force in surge motion; therefore, no resonance appears.
24
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3.3. Three-dimensional wave and structure interaction problem in the open sea To show the capability of three-dimensional computation of the present numerical model, the problem of a floating structure deployed in the open sea and subjected to incident waves is considered. Geometric conditions of the floating structure are the same as the case in the wave channel mentioned above. Fig. 16 shows the contour plot of dimensionless wave amplitudes at t ¼4.35 s for structure draft d/ h¼0.75. The wave reflection shows in front of the structure, which indicates partial standing wave field. With diffracted wave passing along the structure wave focusing also appears behind the structure. Since incident waves are propagating along the major axis of the rectangular structure, the symmetric wave pattern around the structure indicates correctness of the present calculation. Fig. 17 shows amplitudes of surge motion j s1 j =AI versus the relative water depth ω2 h=g for structure draft d/h¼ 0.5 and 0.75, in which two-dimensional results by Ijima et al. [1] are also plotted to show the diffraction effects of a three-dimensional structure. For the long wave condition ω2h/g r1 where diffraction effects are not obvious, the three-dimensional results are close to the two-dimensional theory. On the other hand, for ω2h/g4 1 motion amplitudes decrease as diffraction waves transmit behind the structure. The comparison also shows three-dimensional resonant frequency moves toward higher frequency than the two-dimensional one, which implies diffraction effects increase added mass as far as equations of motion of the structure are concerned. Amplitudes of heave motion j s3 j =AI versus relative water depth ω2 h=g for structural drafts d/h¼0.5 and 0.75 are shown in Fig. 18. Comparisons of the three-dimensional and the twodimensional results indicate that resonant frequency of the structural motion increases, which also implies the added mass of the structure is increased in the three-dimensional case. Furthermore, with the increased structural draft, the added mass is further increased, which in turn increases resonant amplitude. Similar phenomena can also be observed in Fig. 19. The three-dimensional diffraction effects increase moment of inertia of the structure, which increase pitch motion of the structure. Also, the increased drafts increase pitch motions of the structure.
4. Conclusions Using the direct coupling numerical solution method proposed in this study, the three-dimensional problems of floating structures subjected to incident waves can be calculated. To show the accuracy of the present model, a problem of a floating rectangular cuboid deployed in a three-dimensional wave channel and subjected to incident wave is calculated. The two-dimensional characteristics of the results compared with a two-dimensional theory show very good agreement. Floating structures deployed in an open sea subjected to incident waves are also calculated to show the three-dimensional capability of the present model. The diffraction effects of the floating structures are also discussed, that the added masses are increased to decrease surge, but to increase heave and pitch motions of the floating structures.
Appendix A From Eq. (18), the 8 9 2 s1 > > K~ 11 > > > > 6~ > > > > s > > 6 K 2 > 6 21 > > > > > < ~ s3 = 6 6 K 31 ¼6 ~ s > > 6 K 4 > 6 41 > > > > 6~ > > > > s5 > > > > 4 K 51 > > ; :s > K~ 6
61
amplitude of the structure motion can be expressed as 38 9 K~ 12 K~ 13 K~ 14 K~ 15 K~ 16 > > >f1 > 7> > > > > >f2 > K~ 22 K~ 23 K~ 24 K~ 25 K~ 26 7> > > > 7> > > < 7 K~ 32 K~ 33 K~ 34 K~ 35 K~ 36 7 f 3 = 7 K~ 42 K~ 43 K~ 44 K~ 45 K~ 46 7> > >f4 > > 7> > > >f > K~ 52 K~ 53 K~ 54 K~ 55 K~ 56 7 5> > > 5> > > > > : f6 ; K~ 62 K~ 63 K~ 64 K~ 65 K~ 66
Substituting Eq. (A1) into Eq. (25), the coefficients C of Eq. (26) can be obtained as 9 8h i ~ ~ ~ > > > > > > K 11 þ K 15 zj zc K 16 yj yc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K K z z y y þ K z z C4 ¼ ω ρ A4j c c 55 j 51 56 j c > i > > j > h > > > > ~ ~ ~ > > : þ y yc K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i ~ ~ ~ > > > > > > K 12 K 14 zj zc þ K 16 xj xc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K þ K z z x x K z z A c c c C5 ¼ ω ρ 52 54 j 56 j > 5j > j > h i > > > > > ~ ~ ~ > > : þ y yc K 62 þ K 64 zj zc K 66 xj xc ; 9 8h i ~ ~ ~ > > > > > > K 11 K 15 zj zc þ K 16 yj yc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K þ K z z y y K z z A6j c c C6 ¼ ω ρ 55 j 51 56 j c > i > > j > h > > > > ~ ~ ~ > > : þ y yc K 61 þ K 65 zj zc K 66 yj yc ;
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 12 þ K~ 14 zj zc K~ 16 xj xc > > > > > > > h i > = < X 4 ~ ~ ~ 2 ðz zc Þ K 52 K 54 zj zc þ K 56 xj xc A C7 ¼ ω ρ > 7j > j > h i > > > > > > ; : þ y y K~ 62 K~ 64 zj zc þ K~ 66 xj xc >
25
ðA5Þ
c
9 8h i > > K~ 13 K~ 14 yj yc þ K~ 15 xj xc > > > > > > > h i > = < X 4 ~ ~ ~ 2 ðz zc Þ K 53 þ K 54 yj yc K 55 xj xc A C8 ¼ ω ρ > 8j > j > h i > > > > > ~ ~ ~ > ; : þ y yc K 63 þ K 64 yj yc K 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 21 þ K 25 zj zc K 26 yj yc > > > h i > = < X 5 ~ ~ ~ 2 þ ðz zc Þ K 41 K 45 zj zc þ K 46 yj yc A4j C4 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > > : ðx xc Þ K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i ~ ~ ~ > > > > > > K 22 K 24 zj zc þ K 26 xj xc > > > h i> = < X 5 ~ ~ ~ 2 þ z z ð Þ K þ K z z x x K A c c c C5 ¼ ω ρ 42 44 j 46 j > 5j > h j > i > > > > > ~ ~ ~ > > : ðx xc Þ K 62 þ K 64 zj zc K 66 xj xc ; 9 8h i ~ ~ ~ > > > > > > K 21 K 25 zj zc þ K 26 yj yc > > > > h i = < X 5 ~ ~ ~ 2 ð Þ K þ K z z y y K þ z z A6j c c C6 ¼ ω ρ 41 45 j 46 j c > h i > > j > > > > > ~ ~ ~ > > : ðx xc Þ K 61 þ K 65 zj zc K 66 yj yc ; 9 8h i > > K~ 22 þ K~ 24 zj zc K~ 26 xj xc > > > > > > > > h i = < X þ ðz zc Þ K~ 42 K~ 44 zj zc þ K~ 46 xj xc A C 57 ¼ ω2 ρ > 7j > h j > i > > > > > > ; : ðx xc Þ K~ 62 K~ 64 zj zc þ K~ 66 xj xc > 9 8h i > > K~ 23 K~ 24 yj yc þ K~ 25 xj xc > > > > > > > h i> = < X 5 ~ ~ ~ þ ðz zc Þ K 43 þ K 44 yj yc K 45 xj xc A : C 8 ¼ ω2 ρ > 8j > h j > i > > > > > > ; : ðx xc Þ K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i > > K~ 11 K~ 15 zj zc þ K~ 16 yj yc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 51 K 55 zj zc þ K 56 yj yc A4j C4 ¼ ω ρ > i > > j > h > > > > > > ; : y y K~ 61 K~ 65 zj zc þ K~ 66 y y c
j
ðA6Þ
ðA7Þ
ðA8Þ
ðA9Þ
ðA10Þ
ðA11Þ
ðA12Þ
c
9 8h i > > K~ 12 þ K~ 14 zj zc K~ 16 xj xc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 52 þ K 54 zj zc K 56 xj xc A C5 ¼ ω ρ > 5j > j > h i > > > > > > ; : y yc K~ 62 þ K~ 64 zj zc K~ 66 xj xc > 9 8h i ~ ~ ~ > > > > > > K 11 þ K 15 zj zc K 16 yj yc > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 51 þ K 55 zj zc K 56 yj yc A6j C6 ¼ ω ρ > i > > j > h > > > > ~ ~ ~ > ; : y yc K 61 þ K 65 zj zc K 66 yj yc > 9 8h i ~ ~ ~ > > > > > > K 12 K 14 zj zc þ K 16 xj xc MK > > > h i > = < X 6 ~ ~ ~ 2 þ z z ð Þ K K z z x x þ K A c c c C7 ¼ ω ρ 52 54 j 56 j > 7j > > j > h i > > > > ~ ~ ~ > > : y yc K 62 K 64 zj zc þ K 66 xj xc ;
ðA13Þ
ðA14Þ
ðA15Þ
26
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 13 þ K~ 14 yj yc K~ 15 xj xc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 53 þ K 54 yj yc K 55 xj xc A C8 ¼ ω ρ > 8j > j > h i > > > > > > ; : y yc K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 21 K 25 zj zc þ K 26 yj yc > > > h i > = < X 7 ~ ~ ~ 2 ð Þ K K z z y y þ K z z A4j C4 ¼ ω ρ c c 41 45 j 46 j c > h i > > > j > > > > ~ ~ ~ > > : þ ðx xc Þ K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i > > K~ 22 þ K~ 24 zj zc K~ 26 xj xc > > > > > > > > h i = < X 7 ~ ~ ~ ðz zc Þ K 42 þ K 44 zj zc K 46 xj xc A C 5 ¼ ω2 ρ > 5j > h j > i > > > > > > ; : þ ðx xc Þ K~ 62 þ K~ 64 zj zc K~ 66 xj xc > 9 8h i > > K~ 21 þ K~ 25 zj zc K~ 26 yj yc > > > > > > > h i > = < X 7 ~ ~ ~ 2 ðz zc Þ K 41 þ K 45 zj zc K 46 yj yc A6j C6 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > ; : þ ðx xc Þ K 61 þ K 65 zj zc K 66 yj yc > 9 8h i ~ ~ ~ > > > > > > K 22 K 24 zj zc þ K 26 xj xc > > > > h i = < X 7 ~ ~ ~ 2 ð Þ K K z z x x þ K z z A c c c C7 ¼ ω ρ 42 44 j 46 j > 7j > h > j > i > > > > ~ ~ ~ > > : þ ðx xc Þ K 62 K 64 zj zc þ K 66 xj xc ; 9 8h i > > K~ 23 þ K~ 24 yj yc K~ 25 xj xc > > > > > > > h i> = < X 7 ~ ~ ~ 2 ðz zc Þ K 43 þ K 44 yj yc K 45 xj xc A C8 ¼ ω ρ > 8j > h j > i > > > > > > ; : þ ðx xc Þ K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 31 K 35 zj zc þ K 36 yj yc > > > h i > = < X 8 ~ ~ ~ 2 þ y yc K 41 K 45 zj zc þ K 46 yj yc A4j C4 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > > ; : ðx xc Þ K 51 K 55 zj zc þ K 56 yj yc 9 8h i ~ ~ ~ > > > > > > K 32 þ K 34 zj zc K 36 xj xc > > > > h i = < X 8 ~ ~ ~ 2 K K þ K z z x x þ y y A c c C5 ¼ ω ρ 42 44 j 46 j c > 5j > h j > i > > > > > ~ ~ ~ > > ; : ðx xc Þ K 52 þ K 54 zj zc K 56 xj xc 9 8h i > > K~ 31 þ K~ 35 zj zc K~ 36 yj yc > > > > > > > h i > = < X 8 ~ ~ ~ 2 þ y yc K 41 þ K 45 zj zc K 46 yj yc A6j C6 ¼ ω ρ > h i > > j > > > > > > > ; : ðx xc Þ K~ 51 þ K~ 55 zj zc K~ 56 y y j
ðA16Þ
ðA17Þ
ðA18Þ
ðA19Þ
ðA20Þ
ðA21Þ
ðA22Þ
ðA23Þ
ðA24Þ
c
9 8h i ~ ~ ~ > > > > > > K 32 K 34 zj zc þ K 36 xj xc > > > h i> = < X 8 ~ ~ ~ 2 K z z x x K þ K þ y y A C7 ¼ ω ρ c c 42 44 j 46 j c > 7j > h > j > i > > > > ~ ~ ~ > > ; : ðx xc Þ K 52 K 54 zj zc þ K 56 xj xc
ðA25Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 33 þ K~ 34 yj yc K~ 35 xj xc > > > > > > > h i> = < X 8 ~ ~ ~ 2 þ y yc K 43 þ K 44 yj yc K 45 xj xc A : C8 ¼ ω ρ > 8j > h j > i > > > > > > > ; : ðx xc Þ K~ 53 þ K~ 54 yj yc K~ 55 xj xc
27
ðA26Þ
where ω and ρ are angular frequency and fluid density, respectively. Aij represents jth element area on ith boundary. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
A direct coupling numerical method for solving three-dimensional interaction problems of wave and floating structures Jeng-Hong Kao a,n, Jaw-Fang Lee a, Yu-Chun Chengb a b
Department of Hydraulics and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan, ROC Fifth River Management Office, Water Resources Agency, Chiayi, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2014 Accepted 6 November 2014 Available online 3 February 2015
In this article, a direct numerical method is developed to solve three-dimensional interaction problems of wave and floating structures. In the problem formulation, the diffracted wave and radiated wave by the structure are represented as one induced wave potential. The wave field is simulated using a boundary element method, and the structure motions are of six degrees of freedom. In the solution procedure, the structural motions are expressed in terms of wave potentials, which are then substituted into the wave model. The created linear algebraic system is then calculated. The present numerical model is used to simulate a floating rectangular cuboid deployed in a wave channel, and the results are compared with a theory to confirm the accuracy. The present numerical model is further used to calculate a floating structure deployed in an open sea to show the capability of the three-dimensional computation. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Coupling Interaction Three dimension Diffraction effects Boundary element method
1. Introduction Floating breakwaters are considered as alternatives to conventional fixed breakwaters deployed in coastal areas for preserving small marinas and recreational harbors from attacks of wave forces. The main advantages of floating breakwaters are (a) low-cost construction independent of water depth and seabed geological conditions, (b) easy transportation, especially for temporary facilities, and (c) ecologically advantageous since the water circulation, biological exchange, and sediment transport beneath the structure are allowed. Thus, many studies have been undertaken to investigate the performance of various types of floating breakwaters. Regarding problems of wave interaction with free floating structures, Ijima et al. [1] presented an analytic solution for wave interaction with a rectangular floating structure in two dimension, in which free floating and spring moored structures were considered. Au and Brebbia [2] used a boundary element method calculating wave forces acting on two-dimensional structures, where submerged, floating and arbitrary geometrical structures were considered. Huang [3] used a finite element method for solving interference problems between water waves and two-dimensional or three-dimensional ocean structures. The added mass, damping coefficient and wave force were discussed for different wave periods. Matsui et al. [4] solved water wave diffraction and radiation by arbitrarily shaped three-dimensional bodies using a hybrid integral equation method. The boundary element idealization was used only in an inner fluid region close to the body and local depth irregularities, while an analytical solution was employed in the outer region of constant depth extending to infinity. The two representations were matched on a fictitious vertical cylindrical surface. Lee [5] presented an analytic solution to solve the heave radiation problem of a rectangular structure. The nonhomogeneous boundary value problem was linearly decomposed into homogeneous ones, which can be readily solved. Yilmaz [6] solved the diffraction and radiation problems of a group of truncated vertical cylinders using an exact analytical method. Abul-Azm and Gesraha [7] studied wave–structure interaction problems. The structure has three degrees of freedom in motion. The research results discussed elimination effects of wave and structure motions in different configurations. Chen et al. [8] presented a complete analytic solution for wave interacting with two-dimensional floating structures by applying the analytic approach for nonhomogeneous boundary value problems proposed by Lee [5]. Diamantoulaki et al. [9] aimed at assessing the effect of pontoon spacing on the performance of floating free or moored twin pontoon breakwaters using a panel method. The results showed the
n
Corresponding author. Tel.: þ 886 6 2757575x63200; fax: þ 886 6 2741463. E-mail address: [email protected] (J.-H. Kao).
http://dx.doi.org/10.1016/j.enganabound.2014.11.013 0955-7997/& 2014 Elsevier Ltd. All rights reserved.
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
11
roll response decreased while pontoon spacing increased for both free and moored pontoon floating breakwaters. Kim et al. [10] investigated the characteristics of bending moments, shear forces and stresses at unit connections of very large floating structures (VLFS) under wave loads. The characteristics of bending moments, shear forces and stresses at unit connections were discussed at numerical examples. The three-dimensional diffraction of obliquely incident water waves by a floating structure near a wall with step-type bottom was studied using a small amplitude wave theory [11]. The wave forces and the wave elevations on the free surface were investigated for different incident wave angles and water depth ratios. For wave interaction with tension-leg or spring mooring floating structures, Lee and Lee [12] developed an analytical solution for the coupled linear problem. The fluid-induced drag on the tension legs was not considered. The only force from tension legs was the pretensioned force, which provided part of the stiffness of the floating structures. Lee [13] solved tension-leg structures interacting with linear waves. The eigen-function representations of the velocity potential and surface elevations of scattering and radiation waves were developed by Lee and Lee [12]. Analytical solutions showed that the inertia drag on tension legs was negligible compared to that due to the evanescent waves caused by the wave–structure interaction. Lee et al. [14] presented an analytical solution for the dynamic behavior of both the platform and tethers in the tension leg platform system. Elchahal et al. [15] studied wave interaction with a moored floating breakwater with a harbor boundary, and the effects of various structural parameters of the breakwater on the transmitted wave heights were discussed. Ker and Lee [16] proposed an analytical solution for the problem of waves incident on a porous tension leg platform (TLP). The permeable floating body was considered as anisotropic and homogeneous. Results showed that the drags in the porous body change the TLP behavior significantly. Lee and Ker [17] further presented an analytic solution for problems of two-dimensional tension leg structures, where the floating structure was composed of impermeable and porous regions. Floating structures with attached mooring lines can also be found in the literature. Sannasiraj et al. [18] considered moored floating structure subjected to incident waves, where the stiffness of the mooring lines was calculated using the catenary cable equation. Structural motions and mooring forces for three different mooring configurations were discussed. Diamantoulaki and Angelides [19] considered the performance of an array of floating breakwaters connected with hinge joints under the action of linear monochromatic waves in the frequency domain. The effects of the configuration of hinge joints on the response and effectiveness of the freely floating array were investigated. To date, in the solutions to interaction problems of wave and floating structures, the wave field was decomposed into a scattering problem and radiation problem. The scattering and radiation problems were solved independently; then the solutions were combined together with equations of motion of the floating structures to solve the entire problem. In this study, the direct coupling method can obtain wave fields by solving one liner algebraic system. The wave fields induced by the floating structure including scattering and radiation waves are considered as a whole induced wave potential. With a given incident wave, the induced wave potential together with motions of the floating structure is solved, where the wave problem is calculated using a boundary element method. The advantage of the proposed method is reduced computer computational time and shorter time to obtain the calculation results. The effect is more significant for the three-dimensional problem.
2. Description and solution for the three-dimensional problem The problem considered is a floating rectangular cuboid subjected to oblique incident waves, as shown in Fig. 1. A Cartesian coordinate system is adopted with positive x directed to the right, positive y into the paper, and positive z axis upward. The water depth is h, length and width of the structure are 2l and 2b, respectively, and draft is d. Incident waves propagate in the positive x direction, and the surface elevation is ηI . The center of rotation of the structure is located at Cðxc ; yc ; zc Þ and the gravity center is Oðxo ; y0 ; zo Þ. With the action of incident waves, in addition to induced waves around the floating structure, the floating structure is in six-degree of freedom motion. For steady periodic problems, the potential function of incident waves can be expressed as
ΦI ðx; y; z; tÞ ¼ ϕI ðx; y; zÞe iωt
ð1Þ
where
ϕI ðx; y; zÞ ¼ iAI
g cosh K ðz þ hÞ iKðx e cosh Kh
ω
cos α þ y sin αÞ
ð2Þ
in which AI is wave amplitude, K is wave number pffiffiffiffiffiffiffiffi (K ¼ 2π =L), L is wave length, wave period, g is gravity constant and i ¼ 1.
z
α is incident angle, ω is angular frequency (ω ¼ 2π =T), t is
y
r
l I
e
b O
C
d
x h
Fig. 1. Definition sketch of 3D waves interacting with a floating structure.
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J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
2.1. The boundary value problem of the wave field The wave fields induced by the floating structure subjected to incident waves can be described by the following boundary value problem. The governing equation is the Laplace equation: ∇2 ϕðx; y; zÞ ¼ 0
ð3Þ
where ϕ is the wave potential function. The corresponding boundary conditions are [3] ∂ ϕ ω2 ¼ ϕ; ∂z g ∂ϕ ¼ 0; ∂z
z¼0
ð4Þ
z ¼ h
ð5Þ
pffiffiffiffiffiffi∂ϕ iK ϕ ¼ 0 Kr r-1 ∂r
ð6Þ
lim
in which Eqs. (4), (5) and (6) are the free surface condition, bottom condition and the radiation boundary condition, respectively. The boundary condition on the structural surface can be written as [20] I ∂ ϕþϕ ¼ iωs1 iωs5 ðz zc Þ þ iωs6 y yc ; x ¼ 7 b ð7Þ ∂x I ∂ ϕþϕ ¼ iωs2 þ iωs4 ðz zc Þ iωs6 ðx xc Þ; y ¼ 7 l ð8Þ ∂y
I ∂ ϕþϕ ∂z
¼ iωs3 iωs4 y yc þ iωs5 ðx xc Þ;
z ¼ d
ð9Þ
where sj stands for amplitude of the jth degree of freedom of the structural motions, the subscript j¼1, 2, 3, 4, 5, and 6 represents surge, sway, heave, roll, pitch, and yaw, respectively. Eqs. (7)–(9) contain unknown structural motions; therefore, equations of motion of the structure are required so that the wave field can be solved. 2.2. Equation of motion of the floating structure The matrix equation in six degrees of freedom of the three-dimensional floating structure can be expressed as [21] n o
½M ξ€ þ ½K ξ ¼ fF g in which mass matrix ½M and stiffness matrix ½K can be written, respectively, as 2 3 m y0 yc m 0 0 0 mð z 0 z c Þ 6 7 6 0 mðx0 xc Þ 7 0 m 0 mðz0 zc Þ 6 7 6 7 0 mðx0 xc Þ 0 0 m m y0 yc 6 7 6 7 b b b ½M ¼ 6 7 0 m z I ð z Þ m y y I I c 0 0 c xx yx xz 6 7 6 7 b b b 6 mðz zc Þ 7 0 mðx0 xc Þ I xy I yy I zy 0 6 7 4 5 mðx0 xc Þ m y0 yc 0 I bxz I byz I bzz 2
0 6 60 6 60 6 ½K ¼ 6 60 6 6 6 40 0
0
0
0
0
0
0
0
0
0
ρga
0
ρ
ρ
gI ay
ρ
gI ax
gI ay
ρg I ayy þ Ivz mgðz0 zc Þ
0
ρgI ax
ρgI ayx
ρgI axy a ρg I xx þ Ivz mgðz0 zc Þ
0
0
0
0
0
ð10Þ
ð11Þ
3
7 7 7 7 0 7 7 v ρgI x þ mg ðx0 xc Þ 7 7 7 7 ρgI vy þ mg y0 yc 5 0 0
ð12Þ
where I ax ¼ 4blðxc Þ 2
I axx ¼ 4bl
ð13Þ
b þxc 2 3
!
I ay ¼ 4bl yc 2
I ayy ¼ 4bl
l þ y2c 3
ð14Þ ð15Þ
! ð16Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
13
z I
r
t
e d
O
x
C
b
h
Fig. 2. Definition sketch of 2D waves interacting with a floating structure.
d I vz ¼ 4bld zc 2
ð17Þ
where m is mass of the structure, ρ is fluid density, I bxx , I byy and I bzz are moments of inertia about center of rotation in three coordinate axes; I ax and I axx are first moment and second moment of width of the structure at still water level about z axis, respectively; I ay and I ayy are first moment and second moment of length of the structure at still water level about z axis, respectively. I vz is first moment of submerged structure volume about center of rotation. The displacement function of the floating structure on the left-hand side of Eq. (10) can be written as
ξ ¼ fsge iωt ð18Þ The force expression on the right-hand side of Eq. (10) can be rewritten as
fF g ¼ f e iωt where the force amplitude is expressed in terms of the wave potentials Z
f ¼ iωρ ϕ þ ϕI fngdS
ð19Þ
ð20Þ
S
and fng ¼
n
n1
n2
n3
y yc n3 ðz zc Þn2
ðz zc Þn1 ðx xc Þn3
oT ðx xc Þn2 y yc n1
ð21Þ
in which n1, n2 and n3 are unit normal in x, y and z directions, respectively. By substituting Eqs. (18) and (19) into Eq. (10), one obtains
ω2 ½M þ ½K fsg ¼ f
ð22Þ
with the matrix inversion of Eq. (22), amplitudes of the structure motion can be obtained as h i
fsg ¼ K~ f
ð23Þ
where h i 1 K~ ¼ ω2 ½M þ ½K
ð24Þ
Eq. (22) expresses the structural motion in terms of a given incident wave and unknown induced wave potentials. In the subsequent solution methodology, Eq. (23) is substituted into boundary conditions on the structural surface in the wave problem, so that the wave field can be solved. In this study, the unknown wave field is analyzed using a boundary element method. 2.3. A boundary element model for the wave field A boundary element method is applied to solve the wave problem. Using the constant element, the boundary element equation can be expressed as [2,22–25] N X j¼1
H ij ϕj ¼
N X j¼1
Gij
∂ϕj ∂n
ð25Þ
where ϕj and ∂ϕj =∂n represent wave potential and normal derivative quantity of the jth element, respectively, and the coefficient matrices are 8 R ∂ ϕn > < 12 þ Γ j ∂n dΓ ; i ¼ j H ij ¼ ð26Þ R ϕn > : Γ j ∂∂n dΓ ; ia j
14
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 3. Variation of reflection coefficient versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 4. Variation of reflection coefficient versus relative water depth using 2D model for b/h ¼ 0.5.
Z Gij ¼
Γj
ϕn dΓ
ð27Þ
in which ϕ is the fundamental solution. For the three-dimensional Laplace equation, the fundamental solution is n
ϕn ¼
1 4π r ij
where r ij is the distance measured from the source point i to the jth element. The matrix form of Eq. (25) can be expressed as
∂ϕ ½H ϕ ¼ ½G ∂n
ð28Þ
ð29Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 5. Dimensionless amplitude for surge motion versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 6. Dimensionless amplitude for surge motion versus relative water depth using 2D model for b/h ¼ 0.5.
By substituting boundary conditions, Eqs. (4)–(9), into Eq. (29), one obtains 2
H 11 6 6 H 21 6 6 H 31 6 6 6 H 41 6 6 H 51 6 6 6 H 61 6 6 H 71 4 H 81
H 12
H 13
H 14
H 15
H 16
H 17
H 22
H 23
H 24
H 25
H 26
H 27
H 32
H 33
H 34
H 35
H 36
H 37
H 42
H 43
H 44
H 45
H 46
H 47
H 52 H 62
H 53 H 63
H 54 H 64
H 55 H 65
H 56 H 66
H 57 H 67
H 72
H 73
H 74
H 75
H 76
H 77
H 82
H 83
H 84
H 85
H 86
H 87
38 9 H 18 > > > ϕ1 > > 7> > > > H 28 7> > > > ϕ2 > 7> > > > > > ϕ H 38 7 > > > 3> 7> > > 7> < H 48 7 ϕ4 = 7 7 H 58 7> > > ϕ5 > > > > 7> > > H 68 7> > > ϕ6 > > 7> > > > 7 > > ϕ7 > H 78 5> > > > > > : ϕ ; H 88
8
15
16
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
2
G11 6 6 G21 6 6 G31 6 6 6 G41 ¼6 6 G51 6 6 6 G61 6 6 G71 4 G81
G12
G13
G14
G15
G16
G17
G22
G23
G24
G25
G26
G27
G32
G33
G34
G35
G36
G37
G42
G43
G44
G45
G46
G47
G52 G62
G53 G63
G54 G64
G55 G65
G56 G66
G57 G67
G72
G73
G74
G75
G76
G77
G82
G83
G84
G85
G86
G87
9 8 ω2 ϕ > > > 1 g 3> > > > > > G18 > > > > > iK ϕ2 > > 7> > > > G28 7> > > > 0 7> > > > I 7 > > G38 7> > ∂ ϕ4 > > þ i ω s i ω s ð z z Þ þ i ω s y y > > c 5 1 6 c ∂x > > 7< = G48 7 I 7 þ ∂ϕ5 iωs þiωs ðz z Þ iωs ðx x Þ c c 7 2 4 6 ∂y G58 7> > > > > I > > 7> > ∂ ϕ6 > > G68 7> þ i ω s þ i ω s ð z z Þ i ω s y y > > c 5 1 6 c ∂x > > 7> > > > 7 I > > G78 5> ∂ϕ7 > > þ iωs2 iωs4 ðz zc Þ þiωs6 ðx xc Þ > > > ∂y > > > > G88 > > > > ∂ϕI > ; : 8 þiωs3 þ iωs4 y y iωs5 ðx xc Þ > ∂z
c
Fig. 7. Dimensionless amplitude for heave motion versus relative water depth using 2D model for b/h ¼ 1.0.
Fig. 8. Dimensionless amplitude for heave motion versus relative water depth using 2D model for b/h ¼ 0.5.
ð30Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
17
Fig. 9. Dimensionless amplitude for pitch motion versus relative water depth using 2D model for b/h ¼1.0.
In Eq. (30) subscripts shown in the wave potentials represent the belonging boundaries, 1, 2, and 3 stand for the free surface, radiation boundary, and sea bottom, respectively. 4 and 6, 5 and 7 represents structural boundaries with normal in x and y directions, respectively. 8 represents structural bottom.
2.4. A solution procedure for wave and structure interaction problem For the interaction problem of incident waves and the floating structure, the unknown induced wave potential is expressed by the boundary element equation, Eq. (30). And the structural motion is governed by Eq. (23). It is obvious that the wave equation, Eq. (20), contains unknown structural motions, and the structure equation, Eq. (23), contains unknown wave potentials. The unknown structural motions shown on the right-hand side of Eq. (30) can be substituted using Eq. (23). Eq. (30) is then expressed in terms of wave potentials only that can be used to solve the unknown wave functions. Eq. (30) is now reformulated by moving all known variables to the right-hand side. The reformed matrix equation can be expressed as 2 2 ω G
6 H 11 g 11 6 6 6 6 6 H ω2 G 6 21 g 21 6 6 6 6 6 H 31 ω2 G31 6 g 6 6 6 6 6 H 41 ω2 G41 g 6 6 6 6 2 6 6 H 51 ωg G51 6 6 6 6 6 H ω2 G 6 61 g 61 6 6 6 6 6 H 71 ω2 G71 6 g 6 6 6 6 4 H 81 ωg2 G81
H 12 iKG12 H 22 iKG22 H 32 iKG32 H 42 iKG42
H 13 H 23 H 33 H 43
8 X
H 14 C i4 G1i i¼4 H 24
8 X
C i4 G2i
H 25
i¼4 8 X
C i4 G4i
H 35
H 53
H 54
8 X
C i4 G5i
H 45
H 63
H 64
8 X
C i4 G6i
H 55
H 82 iKG82
H 73 H 83
H 74
8 X
C i4 G7i
8 X
H 65
8 X
H 26
H 75
8 X i¼4
8 X
8 X
H 85
i¼4
C i6 G1i
H 17
8 X
H 36
8 X
C i6 G2i
H 27
H 46
8 X
H 37
H 56
8 X
C i6 G4i
H 47
H 66
8 X
C i6 G5i
H 57
H 76
8 X
C i6 G6i
H 67
H 86
8 X i¼4
C i7 G2i
8 X
C i7 G3i
8 X
C i7 G4i
8 X
C i7 G5i
i¼4
C i6 G7i
H 77
i¼4
C i5 G8i
8 X
i¼4
i¼4
C i5 G7i
C i7 G2i
i¼4
i¼4
C i5 G6i
8 X
8 X
C i7 G6i
i¼4
C i6 G8i
H 87
8 X i¼4
8 X
C i7 G7i
3 C i8 G1i
7 7 7 7 8 7 X i H 28 C 8 G2i 7 7 7 i¼4 7 7 8 X 78 9 i H 38 C 8 G3i 7 7> > > ϕ1 > > > 7> i¼4 > ϕ2 > 7> > > > > 7 > > 8 > X 7> > > ϕ i > > 3 > > H 48 C 8 G4i 7 > > > 7> < 7 ϕ4 = i¼4 7 7 8 X i > > ϕ5 > 7> > > > H 58 C 8 G5i 7> > > ϕ6 > 7> > > > i¼4 7> > > > 7> > > ϕ 8 > > 7 7 X i > > > > 7 : H 68 C 8 G6i 7 ϕ8 ; 7 i¼4 7 7 8 X 7 i H 78 C 8 G7i 7 7 7 i¼4 7 7 8 X 7 i H 88 C 8 G8i 5 H 18
i¼4
i¼4
i¼4
C i5 G5i
C i7 G1i
i¼4
C i6 G3i
i¼4
C i5 G4i
8 X i¼4
i¼4
C i5 G3i
i¼4
i¼4
H 84 C i4 G8i i¼4
C i5 G2i
i¼4
i¼4
H 72 iKG72
8 X
8 X i¼4
i¼4
i¼4
H 62 iKG62
H 16
i¼4
i¼4
H 52 iKG52
8 X
8 X
8 X
C i5 G1i
i¼4
i¼4
H 34 C i4 G3i i¼4 H 44
H 15
8 X
i¼4
18
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 10. Dimensionless amplitude for pitch motion versus relative water depth using 2D model for b/h ¼ 0.5.
Fig. 11. Definition sketch of the floating rectangular cuboid in a 3D channel and subjected to incident waves.
2
G14 6 6 G24 6 6 G34 6 6 6 G44 ¼6 6 G54 6 6 6 G64 6 6 G74 4 G84
G15
G16
G17
G25
G26
G27
G35
G36
G37
G45
G46
G47
G55
G56
G57
G65
G66
G67
G75
G76
G77
G85
G86
G87
9 8 8 X I > > > ∂ ϕ4 4 I> > > þ þ C ϕ > j j> ∂x > > > > > j¼4 3> > > > > > G18 > > > > > 8 X I > > 7> > I ∂ ϕ 5 5 > G28 7> > þ þ C ϕ j> j > > ∂y 7> > > > j ¼ 4 7 > > G38 7> > > > > > > 7> 8 X = < I G48 7 ∂ ϕ6 6 I 7 ∂x þ C j ϕj 7 G58 7> > j¼4 > > > > > 7> > > > G68 7> 8 > > X I > 7> I ∂ ϕ 7 > > 7 > > 7 þ C ϕ > > ∂y j> j G78 5> > > > > j¼4 > > > > G88 > > > 8 > > X 8 I> I > > ∂ ϕ > > 8þ > > C ϕ > j j> ∂z > > ; : j¼4
ð31Þ
where coefficients C's shown are known constants, and are expressed in Appendix A. The matrix equation, Eq. (31), can then be solved to obtain unknown wave potentials. Upon solving Eq. (31) the free surface elevation can then be calculated using Bernoulli's equation
ηðx; y; t Þ ¼
iω ϕðx; y; 0Þ Ue iωt g
ð32Þ
By substitution of wave potential ϕ and incident wave potential ϕ into Eq. (23), the structure motion can then be calculated. So far, the three-dimensional problem of a floating structure subjected to action of incident waves is completely solved. The wave field and the structural motions can all be determined. I
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
19
Fig. 12. Variation of reflection coefficient versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
Fig. 13. Dimensionless amplitude for surge motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
3. Numerical examples In order to show the accuracy and validity of the proposed method, the three potential problems are considered. Numerical examples containing 2D and 3D wave interaction with floating structure, respectively, are considered. The numerical results agreed very well with those of Ijima et al. [1]. Finally, the results show as diffraction effect on the motions of a floating structure. 3.1. Two-dimensional wave and structure interaction problem In this case, we used a two-dimensional numerical model to solve two-dimensional waves interacting with a floating structure problem, and geometrical dimensions of the problem are shown in Fig. 2. The method of creating two-dimensional model is the same as the step of Sections 2.1–2.4. Here, the amplitude of structural motion of s2, s4 and s6 is equal to zero in Eqs. (7)–(9), and we substitute 2D
20
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 14. Dimensionless amplitude for heave motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
Fig. 15. Dimensionless amplitude for pitch motion versus relative water depth for a rectangular cuboid in a wave channel (d/h ¼ 0.5 and 0.75).
equation of structural motion and 2D fundamental solution into Eqs. (10) and (28), respectively. For this case, the water depth is 4 m. Widths of the structure considered are b=h ¼ 1:0 and 0.5. Drafts of the structure considered are d=h ¼ 0:75, 0.5, 0.25 and 0.1. The fluid density is 1000 kg/m3. The center of rotation and gravity of the floating structure are considered to coincide. The results of reflection coefficient and three motion amplitudes versus relative water depth by using the present model are shown in Figs. 3–10. In these figures, the real and dotted lines denote the 2D theoretical results from Ijima et al. [1]; the symbols denote the present results. The comparisons present results that agree very well with the theoretical solution. Figs. 3 and 4 show reflection coefficients Kr versus relative water depth ω2 h=g for different width of structure. The reflection coefficient is larger when draft of structure increases. The reflection coefficient approaches 1 in higher relative water depth. The structural width increase can obtain better reflection effect in higher relative water depth. Figs. 5–10 show dimensionless amplitudes j s1 j =AI , j s3 j =AI and j bs5 j =AI versus relative water depth for surge, heave and pitch motions, respectively. The amplitude of surge motion versus relative water depth is shown in Figs. 5 and 6. Overall speaking, the amplitude of surge
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
21
Fig. 16. A contour plot of dimensionless wave amplitudes for a rectangular cuboid in an open sea (t¼ 4.35 s, d/h ¼ 0.75).
Fig. 17. Dimensionless amplitudes for surge motion versus relative water depth for a rectangular cuboid in an open sea.
motion is infinite when relative water depth approaches zero. Under increasing relative water depth, the amplitude of surge motion approaches zero. Furthermore, resonance frequency was obvious in b=h ¼ 1:0, and that moved towards low frequency when draft increased. For heave motion, dimensionless amplitudes j s3 j =AI versus relative water depth are shown in Figs. 7 and 8. The amplitude of heave motion is equal to the one in relative water depth equal to zero. The response resonant amplitude increases when draft increases. The resonant frequencies move toward low frequency as the width of structure increases. Dimensionless amplitudes of pitch motion j bs5 j =AI versus relative water depth are shown in Figs. 9 and 10. Overall speaking, the response amplitude became larger at resonant frequency, and the response amplitude approaches zero in other frequencies. For b=h ¼ 1:0, the resonant amplitude increases as the draft increases, but the resonant amplitude decreases as the draft increases for b=h ¼ 0:5. The resonant frequencies move toward low frequency as width of the structure increases.
22
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
Fig. 18. Dimensionless amplitudes for heave motion versus relative water depth for a rectangular cuboid in an open sea.
Fig. 19. Dimensionless amplitudes for pitch motion versus relative water depth for a rectangular cuboid in an open sea.
3.2. Three-dimensional wave and structure interaction problem in the wave channel To verify the accuracy of the present model, a floating rectangular cuboid deployed in a wave channel subjected to incident waves is simulated. Geometrical dimensions of the problem are described in Fig. 11. Ideally the floating structure has the same width as the wave channel. The water depth in the wave channel is h. In the numerical model, a finite artificial truncated length of the wave channel is adopted, and on the artificial boundary a radiation boundary condition is used ∂ϕ ¼ iK ϕ; ∂n
x ¼ 7ℓa
ð33Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
23
where ℓa is the location of the artificial boundary. Furthermore, on sidewalls of the wave channel, the impermeable condition is ∂ϕ ¼ 0; ∂n
y ¼ 7l
ð34Þ
Since the structure is confined in the wave channel, the degrees of freedom of the structural motions, sway, roll, and yaw are zero. To solve the problem, the equations of motion of the structure, Eq. (23), and the boundary element equation for the wave problem, Eq. (30), are combined to obtain 2 2 ω G
6 H 11 g 11 6 6 6 6 6 H ω2 G 6 21 g 21 6 6 6 6 6 H ω2 G 6 31 g 31 6 6 6 6 6 H ω2 G 6 41 g 41 6 6 6 6 6 H 51 ω2 G51 6 g 6 6 6 6 6 H 61 ω2 G61 6 g 6 6 6 6 6 H 71 ω2 G71 g 6 6 6 6 6 4 H 81 ω2 G81 g 2
G14
6 6 G24 6 6 G34 6 6 6 G44 ¼6 6 G54 6 6 6 G64 6 6 G74 4 G84
H 12 iKG12 H 22 iKG22
H 13 H 23
8 X
H 14 C i4 G1i i ¼ 4;6 H 24
8 X
C i4 G2i
H 15
H 16
H 33
H 34
8 X
C i4 G3i
H 25
H 26
H 43
H 44
8 X
C i4 G4i
H 35
H 36
H 53
H 54
8 X
C i4 G5i
H 45
H 46
H 63
H 64
8 X
C i4 G6i
H 55
H 56
H 73
H 74
8 X
C i4 G7i
H 65
H 66
G16 G26 G36 G46 G56 G66 G76 G86
H 83
H 84
8 X
C i4 G8i
i ¼ 4;6
3 9 G18 8 8 X I > 7> > ∂ϕ4 4 I> > G28 7> > C j ϕj > þ ∂x þ > 7> > > > > j ¼ 4;6 > > G38 7 > > > 7> > > > 7> 8 X = < I G48 7 ∂ϕ6 6 I 7 ∂x þ C j ϕj 7 G58 7> > > > j ¼ 4;6 > > > 7> > > > G68 7> 8 > > X I > > ∂ϕ8 7> 8 I> > > 7 þ C ϕ j j> G78 5> ∂z > > ; : j ¼ 4;6 G88
8 X
C i6 G3i
H 37
8 X
C i6 G4i
H 47
8 X
C i6 G5i
H 57
8 X
C i6 G6i
H 67
C i6 G7i
H 77
C i6 G8i
H 87
i ¼ 4;6
H 75
H 76
i ¼ 4;6
H 82 iKG82
H 27
i ¼ 4;6
i ¼ 4;6
H 72 iKG72
C i6 G2i
i ¼ 4;6
i ¼ 4;6
H 62 iKG62
8 X
8 X i ¼ 4;6
H 85
H 86
8 X i ¼ 4;6
H 18
8 X
3 C i8 G1i
7 7 7 7 8 7 X i H 28 C 8 G2i 7 7 7 i ¼ 4;6 7 7 8 78 X i 9 H 38 C 8 G3i 7 7> > > ϕ1 > 7 > > i ¼ 4;6 7> > > ϕ2 > > 7> > > > > 8 X i 7> > > > ϕ > > 7 3 > > H 48 C 8 G4i 7> > > > = < 7 i ¼ 4;6 7 ϕ4 7 8 ϕ5 > X i 7> > > > > > > H 58 C 8 G5i 7 > 7> > > ϕ 6 > > 7 > > i ¼ 4;6 > > 7> > > ϕ 7> > > 7 8 X i > 7> > > : 7 H 68 C 8 G6i 7 ϕ8 ; 7 i ¼ 4;6 7 7 8 X 7 i H 78 C 8 G7i 7 7 7 i ¼ 4;6 7 7 8 X 7 i H 88 C 8 G8i 5 i ¼ 4;6
i ¼ 4;6
i ¼ 4;6
H 52 iKG52
H 17
i ¼ 4;6
i ¼ 4;6
H 42 iKG42
C i6 G1i
i ¼ 4;6
i ¼ 4;6
H 32 iKG32
8 X
i ¼ 4;6
ð35Þ
Notice that in Eq. (35) restrictions of the structural motions, radiation condition, Eq. (33), and side wall conditions, Eq. (34), are implemented. In the numerical computation, the water depth is 4 m, and width and length of the structure are 8 m and 1 m, respectively. Drafts of the structure considered are d=h ¼ 0:75 and 0.5. The fluid density is 1000 kg/m3. The center of rotation and gravity of the floating structure are considered to coincide. In boundary element calculation, element sizes are kept less than 1/50 of the wavelength in the higher frequency. For the lower frequency, element sizes are kept less than 1/40 of the wavelength. And the radiation boundary is located 6–12 times water depth away from the structure [26]. Since the floating structure is deployed in the channel with two sidewalls, and with normal incident waves, the simulated results are expected to have two-dimensional characteristics. Therefore, a two-dimensional theory by Ijima et al. [1] is used to compare with the results. Using the present model the reflection coefficient in front of the structure and motion amplitudes of the three degrees of freedom of the structure versus relative water depth ω2 h=g are shown in Figs. 12–15. In the figures real and dashed lines represent two-dimensional results from Ijima et al. [1], triangle or circle symbols are the present results. The comparisons indicate the present numerical results agree very well with the theory, thereby illustrating the correctness of the present model. On the other hand, in present three-dimensional results wave elevations across the wave channel are constant, also explaining that the present model is accurate. Fig. 12 shows the reflection coefficients versus relative water depth ω2 h=g. In view of tendency for high relative water depth (short wave period) the reflection coefficients approach unity. Waves are blocked in front of the structure. On the other hand, for low relative water depth waveresisting capability of the structure decreases, and most waves are transmitted behind the structure. Furthermore, resonant phenomena of the floating structure decrease wave reflections, and with increased drafts the resonant frequencies move toward low-frequency (long wave) direction. Figs. 13–15 are motion amplitudes j s1 j =AI , j s3 j =AI and j bs5 j =AI versus relative water depth for surge, heave and pitch, respectively. Overall speaking, surge motions of the structure decrease with increased water depth; in other words, surge motions decrease with the decreasing wave period, and approach zero at resonant frequency. For heave and roll motions the amplitudes decrease with increasing water depth, but resonant amplitudes increase. Furthermore, for all the three modes, resonant frequencies move toward low frequency as drafts increase. From analysis of equations of motion of the floating structure, there exists hydrostatic restoring force in heave and roll motion; therefore, response amplitudes increase at resonant frequency, and increased draft can produce more resonance amplitude. However, there is no hydrostatic restoring force in surge motion; therefore, no resonance appears.
24
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
3.3. Three-dimensional wave and structure interaction problem in the open sea To show the capability of three-dimensional computation of the present numerical model, the problem of a floating structure deployed in the open sea and subjected to incident waves is considered. Geometric conditions of the floating structure are the same as the case in the wave channel mentioned above. Fig. 16 shows the contour plot of dimensionless wave amplitudes at t ¼4.35 s for structure draft d/ h¼0.75. The wave reflection shows in front of the structure, which indicates partial standing wave field. With diffracted wave passing along the structure wave focusing also appears behind the structure. Since incident waves are propagating along the major axis of the rectangular structure, the symmetric wave pattern around the structure indicates correctness of the present calculation. Fig. 17 shows amplitudes of surge motion j s1 j =AI versus the relative water depth ω2 h=g for structure draft d/h¼ 0.5 and 0.75, in which two-dimensional results by Ijima et al. [1] are also plotted to show the diffraction effects of a three-dimensional structure. For the long wave condition ω2h/g r1 where diffraction effects are not obvious, the three-dimensional results are close to the two-dimensional theory. On the other hand, for ω2h/g4 1 motion amplitudes decrease as diffraction waves transmit behind the structure. The comparison also shows three-dimensional resonant frequency moves toward higher frequency than the two-dimensional one, which implies diffraction effects increase added mass as far as equations of motion of the structure are concerned. Amplitudes of heave motion j s3 j =AI versus relative water depth ω2 h=g for structural drafts d/h¼0.5 and 0.75 are shown in Fig. 18. Comparisons of the three-dimensional and the twodimensional results indicate that resonant frequency of the structural motion increases, which also implies the added mass of the structure is increased in the three-dimensional case. Furthermore, with the increased structural draft, the added mass is further increased, which in turn increases resonant amplitude. Similar phenomena can also be observed in Fig. 19. The three-dimensional diffraction effects increase moment of inertia of the structure, which increase pitch motion of the structure. Also, the increased drafts increase pitch motions of the structure.
4. Conclusions Using the direct coupling numerical solution method proposed in this study, the three-dimensional problems of floating structures subjected to incident waves can be calculated. To show the accuracy of the present model, a problem of a floating rectangular cuboid deployed in a three-dimensional wave channel and subjected to incident wave is calculated. The two-dimensional characteristics of the results compared with a two-dimensional theory show very good agreement. Floating structures deployed in an open sea subjected to incident waves are also calculated to show the three-dimensional capability of the present model. The diffraction effects of the floating structures are also discussed, that the added masses are increased to decrease surge, but to increase heave and pitch motions of the floating structures.
Appendix A From Eq. (18), the 8 9 2 s1 > > K~ 11 > > > > 6~ > > > > s > > 6 K 2 > 6 21 > > > > > < ~ s3 = 6 6 K 31 ¼6 ~ s > > 6 K 4 > 6 41 > > > > 6~ > > > > s5 > > > > 4 K 51 > > ; :s > K~ 6
61
amplitude of the structure motion can be expressed as 38 9 K~ 12 K~ 13 K~ 14 K~ 15 K~ 16 > > >f1 > 7> > > > > >f2 > K~ 22 K~ 23 K~ 24 K~ 25 K~ 26 7> > > > 7> > > < 7 K~ 32 K~ 33 K~ 34 K~ 35 K~ 36 7 f 3 = 7 K~ 42 K~ 43 K~ 44 K~ 45 K~ 46 7> > >f4 > > 7> > > >f > K~ 52 K~ 53 K~ 54 K~ 55 K~ 56 7 5> > > 5> > > > > : f6 ; K~ 62 K~ 63 K~ 64 K~ 65 K~ 66
Substituting Eq. (A1) into Eq. (25), the coefficients C of Eq. (26) can be obtained as 9 8h i ~ ~ ~ > > > > > > K 11 þ K 15 zj zc K 16 yj yc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K K z z y y þ K z z C4 ¼ ω ρ A4j c c 55 j 51 56 j c > i > > j > h > > > > ~ ~ ~ > > : þ y yc K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i ~ ~ ~ > > > > > > K 12 K 14 zj zc þ K 16 xj xc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K þ K z z x x K z z A c c c C5 ¼ ω ρ 52 54 j 56 j > 5j > j > h i > > > > > ~ ~ ~ > > : þ y yc K 62 þ K 64 zj zc K 66 xj xc ; 9 8h i ~ ~ ~ > > > > > > K 11 K 15 zj zc þ K 16 yj yc > > > h i > = < X 4 ~ ~ ~ 2 ð Þ K þ K z z y y K z z A6j c c C6 ¼ ω ρ 55 j 51 56 j c > i > > j > h > > > > ~ ~ ~ > > : þ y yc K 61 þ K 65 zj zc K 66 yj yc ;
ðA1Þ
ðA2Þ
ðA3Þ
ðA4Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 12 þ K~ 14 zj zc K~ 16 xj xc > > > > > > > h i > = < X 4 ~ ~ ~ 2 ðz zc Þ K 52 K 54 zj zc þ K 56 xj xc A C7 ¼ ω ρ > 7j > j > h i > > > > > > ; : þ y y K~ 62 K~ 64 zj zc þ K~ 66 xj xc >
25
ðA5Þ
c
9 8h i > > K~ 13 K~ 14 yj yc þ K~ 15 xj xc > > > > > > > h i > = < X 4 ~ ~ ~ 2 ðz zc Þ K 53 þ K 54 yj yc K 55 xj xc A C8 ¼ ω ρ > 8j > j > h i > > > > > ~ ~ ~ > ; : þ y yc K 63 þ K 64 yj yc K 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 21 þ K 25 zj zc K 26 yj yc > > > h i > = < X 5 ~ ~ ~ 2 þ ðz zc Þ K 41 K 45 zj zc þ K 46 yj yc A4j C4 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > > : ðx xc Þ K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i ~ ~ ~ > > > > > > K 22 K 24 zj zc þ K 26 xj xc > > > h i> = < X 5 ~ ~ ~ 2 þ z z ð Þ K þ K z z x x K A c c c C5 ¼ ω ρ 42 44 j 46 j > 5j > h j > i > > > > > ~ ~ ~ > > : ðx xc Þ K 62 þ K 64 zj zc K 66 xj xc ; 9 8h i ~ ~ ~ > > > > > > K 21 K 25 zj zc þ K 26 yj yc > > > > h i = < X 5 ~ ~ ~ 2 ð Þ K þ K z z y y K þ z z A6j c c C6 ¼ ω ρ 41 45 j 46 j c > h i > > j > > > > > ~ ~ ~ > > : ðx xc Þ K 61 þ K 65 zj zc K 66 yj yc ; 9 8h i > > K~ 22 þ K~ 24 zj zc K~ 26 xj xc > > > > > > > > h i = < X þ ðz zc Þ K~ 42 K~ 44 zj zc þ K~ 46 xj xc A C 57 ¼ ω2 ρ > 7j > h j > i > > > > > > ; : ðx xc Þ K~ 62 K~ 64 zj zc þ K~ 66 xj xc > 9 8h i > > K~ 23 K~ 24 yj yc þ K~ 25 xj xc > > > > > > > h i> = < X 5 ~ ~ ~ þ ðz zc Þ K 43 þ K 44 yj yc K 45 xj xc A : C 8 ¼ ω2 ρ > 8j > h j > i > > > > > > ; : ðx xc Þ K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i > > K~ 11 K~ 15 zj zc þ K~ 16 yj yc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 51 K 55 zj zc þ K 56 yj yc A4j C4 ¼ ω ρ > i > > j > h > > > > > > ; : y y K~ 61 K~ 65 zj zc þ K~ 66 y y c
j
ðA6Þ
ðA7Þ
ðA8Þ
ðA9Þ
ðA10Þ
ðA11Þ
ðA12Þ
c
9 8h i > > K~ 12 þ K~ 14 zj zc K~ 16 xj xc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 52 þ K 54 zj zc K 56 xj xc A C5 ¼ ω ρ > 5j > j > h i > > > > > > ; : y yc K~ 62 þ K~ 64 zj zc K~ 66 xj xc > 9 8h i ~ ~ ~ > > > > > > K 11 þ K 15 zj zc K 16 yj yc > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 51 þ K 55 zj zc K 56 yj yc A6j C6 ¼ ω ρ > i > > j > h > > > > ~ ~ ~ > ; : y yc K 61 þ K 65 zj zc K 66 yj yc > 9 8h i ~ ~ ~ > > > > > > K 12 K 14 zj zc þ K 16 xj xc MK > > > h i > = < X 6 ~ ~ ~ 2 þ z z ð Þ K K z z x x þ K A c c c C7 ¼ ω ρ 52 54 j 56 j > 7j > > j > h i > > > > ~ ~ ~ > > : y yc K 62 K 64 zj zc þ K 66 xj xc ;
ðA13Þ
ðA14Þ
ðA15Þ
26
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 13 þ K~ 14 yj yc K~ 15 xj xc > > > > > > > h i > = < X 6 ~ ~ ~ 2 þ ðz zc Þ K 53 þ K 54 yj yc K 55 xj xc A C8 ¼ ω ρ > 8j > j > h i > > > > > > ; : y yc K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 21 K 25 zj zc þ K 26 yj yc > > > h i > = < X 7 ~ ~ ~ 2 ð Þ K K z z y y þ K z z A4j C4 ¼ ω ρ c c 41 45 j 46 j c > h i > > > j > > > > ~ ~ ~ > > : þ ðx xc Þ K 61 K 65 zj zc þ K 66 yj yc ; 9 8h i > > K~ 22 þ K~ 24 zj zc K~ 26 xj xc > > > > > > > > h i = < X 7 ~ ~ ~ ðz zc Þ K 42 þ K 44 zj zc K 46 xj xc A C 5 ¼ ω2 ρ > 5j > h j > i > > > > > > ; : þ ðx xc Þ K~ 62 þ K~ 64 zj zc K~ 66 xj xc > 9 8h i > > K~ 21 þ K~ 25 zj zc K~ 26 yj yc > > > > > > > h i > = < X 7 ~ ~ ~ 2 ðz zc Þ K 41 þ K 45 zj zc K 46 yj yc A6j C6 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > ; : þ ðx xc Þ K 61 þ K 65 zj zc K 66 yj yc > 9 8h i ~ ~ ~ > > > > > > K 22 K 24 zj zc þ K 26 xj xc > > > > h i = < X 7 ~ ~ ~ 2 ð Þ K K z z x x þ K z z A c c c C7 ¼ ω ρ 42 44 j 46 j > 7j > h > j > i > > > > ~ ~ ~ > > : þ ðx xc Þ K 62 K 64 zj zc þ K 66 xj xc ; 9 8h i > > K~ 23 þ K~ 24 yj yc K~ 25 xj xc > > > > > > > h i> = < X 7 ~ ~ ~ 2 ðz zc Þ K 43 þ K 44 yj yc K 45 xj xc A C8 ¼ ω ρ > 8j > h j > i > > > > > > ; : þ ðx xc Þ K~ 63 þ K~ 64 yj yc K~ 65 xj xc > 9 8h i ~ ~ ~ > > > > > > K 31 K 35 zj zc þ K 36 yj yc > > > h i > = < X 8 ~ ~ ~ 2 þ y yc K 41 K 45 zj zc þ K 46 yj yc A4j C4 ¼ ω ρ > h i > > j > > > > > ~ ~ ~ > > ; : ðx xc Þ K 51 K 55 zj zc þ K 56 yj yc 9 8h i ~ ~ ~ > > > > > > K 32 þ K 34 zj zc K 36 xj xc > > > > h i = < X 8 ~ ~ ~ 2 K K þ K z z x x þ y y A c c C5 ¼ ω ρ 42 44 j 46 j c > 5j > h j > i > > > > > ~ ~ ~ > > ; : ðx xc Þ K 52 þ K 54 zj zc K 56 xj xc 9 8h i > > K~ 31 þ K~ 35 zj zc K~ 36 yj yc > > > > > > > h i > = < X 8 ~ ~ ~ 2 þ y yc K 41 þ K 45 zj zc K 46 yj yc A6j C6 ¼ ω ρ > h i > > j > > > > > > > ; : ðx xc Þ K~ 51 þ K~ 55 zj zc K~ 56 y y j
ðA16Þ
ðA17Þ
ðA18Þ
ðA19Þ
ðA20Þ
ðA21Þ
ðA22Þ
ðA23Þ
ðA24Þ
c
9 8h i ~ ~ ~ > > > > > > K 32 K 34 zj zc þ K 36 xj xc > > > h i> = < X 8 ~ ~ ~ 2 K z z x x K þ K þ y y A C7 ¼ ω ρ c c 42 44 j 46 j c > 7j > h > j > i > > > > ~ ~ ~ > > ; : ðx xc Þ K 52 K 54 zj zc þ K 56 xj xc
ðA25Þ
J.-H. Kao et al. / Engineering Analysis with Boundary Elements 55 (2015) 10–27
9 8h i > > K~ 33 þ K~ 34 yj yc K~ 35 xj xc > > > > > > > h i> = < X 8 ~ ~ ~ 2 þ y yc K 43 þ K 44 yj yc K 45 xj xc A : C8 ¼ ω ρ > 8j > h j > i > > > > > > > ; : ðx xc Þ K~ 53 þ K~ 54 yj yc K~ 55 xj xc
27
ðA26Þ
where ω and ρ are angular frequency and fluid density, respectively. Aij represents jth element area on ith boundary. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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