computation Article
An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function Yingyi Liu 1, *, Ying Gou 2 , Bin Teng 2 and Shigeo Yoshida 1 1 2
*
Renewable Energy Center, Research Institute for Applied Mechanics, Kyushu University, Kasuga 8168580, Japan;
[email protected] State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China;
[email protected] (Y.G.);
[email protected] (B.T.) Correspondence:
[email protected]; Tel.: +81-80-8565-7934
Academic Editor: Qinjun Kang Received: 8 August 2016; Accepted: 12 September 2016; Published: 16 September 2016
Abstract: The present study aims to develop an efficient numerical method for computing the diffraction and radiation of water waves with horizontal long cylindrical structures, such as floating breakwaters in the coastal region, etc. A higher-order scheme is used to discretize geometry of the structure as well as the physical wave potentials. As the kernel of this method, Wehausen’s free-surface Green function is calculated by a newly-developed Gauss–Kronrod adaptive quadrature algorithm after elimination of its Cauchy-type singularities. To improve its computation efficiency, an analytical solution is derived for a fast evaluation of the Green function that needs to be implemented thousands of times. In addition, the OpenMP parallelization technique is applied to the formation of the influence coefficient matrix, significantly reducing the running CPU time. Computations are performed on wave-exciting forces and hydrodynamic coefficients for the long cylindrical structures, either floating or submerged. Comparison with other numerical and analytical methods demonstrates a good performance of the present method. Keywords: long cylindrical structure; free-surface Green function; higher-order boundary element method; multipole expansion; singularity elimination; Gauss–Kronrod; numerical quadrature; OpenMP parallelization
1. Introduction Cylindrical structures have been widely used in the rapidly-developing coastal and offshore engineering industries in recent decades, in the form of such as floating breakwaters, oscillating water columns (OWC) for power generation, etc. These devices are used to either passively avoid the large wave kinematic energy from attacking harbors or actively convert the wave energy into other kinds of energies. Cylindrical structures are important in the industries probably due to their simplicity in geometry and the relatively lower fluid forces they may experience. Extensive efforts have been made on investigation of such kind of important structures, theoretically [1–3], numerically [4–7], and experimentally [8,9]. In the numerical approaches, boundary element method should be one of the most popular tools for analysis [5–7]. However, the present work is very different from the classical research in the following aspects: (1) since the governing equation used herein is the Laplace equation instead of the Helmholtz equation, Wehausen’s free-surface Green function [10], constituted by several simple arithmetic functions, can be employed instead of Haskind’s Green function [5,11] which may require many evaluations of the modified Bessel functions; (2) since the free-surface Green function is used as the kernel instead of the Rankine Green function, meshing of the geometry could be restricted to only the body surface, such that there is no need to deal with the open boundaries, as has been Computation 2016, 4, 36; doi:10.3390/computation4030036
www.mdpi.com/journal/computation
has been accomplished by Zheng et al. [7]; (3) application of a three-node higher-order element rather 2 of 21 than a traditional constant or linear element guarantees the accuracy of geometrical/physical discretization; (4) thanks to therather exponential integral functions, the free-surface Green function 3) application of a three-nodeand higher-order element Computation 2016, 4, 36 2 of 20 could the be written form and then evaluated in a faster speed with a precise result. All of ement guarantees accuracyinofa simpler geometrical/physical nential integral the free-surface Green numerical function investigations for hydrodynamic performances of such thefunctions, above advantages facilitate en evaluated cylindrical in a faster speed with a precise result. Allapplication of accomplished by Zheng et al. [7]; (3) of a three-node higher-order element rather than a structures. investigations for hydrodynamic performances of such traditional constant or linear element guarantees the accuracy geometrical/physical discretization; These horizontal cylindrical structures are so long in itsofaxis direction that the problem to be and (4) thanks to the exponential integral functions, the free-surface Green function could be written in solved could be considered in two-dimensional. In comparison to the three-dimensional es are so long in its axis direction that theevaluated problem in to abefaster speed with a precise result. All of the above advantages a simpler form and then wave-structure interaction model within the framework of linear potential flow theory, the present imensional. In comparison to theinvestigations three-dimensional facilitate numerical for hydrodynamic performances of such cylindrical structures. simplified two-dimensional model has many fewer unknowns on the body surface, since the surface he framework of linearThese potential flow theory, the structures present are so long in its axis direction that the problem to be solved horizontal cylindrical integrations have been in substituted by lineInintegrations. In our HOBEM (higher-order boundary ny fewer unknowns on the surface, since the surface could be body considered two-dimensional. comparison to the three-dimensional wave-structure element model, for athe typical frequency domain problem, 10~50 elements (or, in other ne integrations. Ininteraction ourmethod) HOBEM (higher-order boundary model within framework of linear potential flowabout theory, the present simplified words, less than 100 nodes) are sufficient to represent an arbitrary cross-sectional shape. Therefore, uency domain problem, about 10~50model elements (or, infewer other unknowns on the body surface, since the surface integrations two-dimensional has many 2 4 generally about 10 ~10 evaluations of the Green needed for each incident period, to represent an arbitrary cross-sectional shape. have been substituted by line Therefore, integrations. In ourfunction HOBEM are (higher-order boundary elementwave method) 6 Green function aremodel, needed eachOincident wave period,in forfor a typical frequency domain problem, about 10~50 elements (or,(see in other words, less than by compared to those (10 ) evaluations the three-dimensional cases [12]). Furthermore, the three-dimensional cases (see [12]). Furthermore, by 100 nodes) are sufficient to represent an arbitrary cross-sectional shape. Therefore, generally about taking advantage of the contemporary computational technologies, some special technique may be 2 ~104 some mputational technologies, special technique may be 10 evaluations of the Green function are needed for each incident wave period, compared to applied to parallelize the algorithm on multi-processor machines. 6 lti-processor machines. those (10 ) evaluations in discretization, the three-dimensional cases (see [12]). of Furthermore, by taking advantage Apartࣩ from the HOBEM efficient evaluation the free-surface Green function is n, efficient evaluation of contemporary the free-surface Green function is of the computational technologies, some special technique may be applied to parallelize another important issue in this work. Numerous studies have been performed in the field since 1980s. erous studies havethe been performed in the field since machines. 1980s. algorithm on multi-processor Noblesse and Newman have made the most important contributions for this issue [12–17]. They most important contributions for this [12–17]. They Apart from theissue HOBEM discretization, efficient evaluation of the free-surface Green function severalfrom popular methods, e.g., the local component from the far-field one and separating thedeveloped local theissue far-field onework. and separating is component another important in this Numerous studies have been performed in the field since thenthe calculate them by Newman tabulation algorithm, making the contributions singular functions hm, or making singular functions slow-varying by 1980s. Noblesse and have made the or most important for this slow-varying issue [12–17]. by subtracting some component and then approximate the resulting functions by Chebyshev approximate the resulting functions by Chebyshev They developed several popular methods, e.g., separating the local component from the far-field one These methods have been simplified in the present model since the problem simplified inapproximation. the and present model since the problem to be then calculate them by tabulation algorithm, or making the singular functions slow-varyingto be nfinite waterconsidered depth, in which ansome extremely convenient by subtracting component then approximate the resulting by Chebyshev is two-dimensional andand in infinite water depth, in whichfunctions an extremely convenient Green’s kernel in the boundary integral equation. The approximation. These methods have been simplified in the present model since the problem to be The analytical solution can be found for the Green’s kernel in the boundary integral equation. idity and efficiency of the present considered is two-dimensional and the in infinite water depth, in which numerical results inmethod. Section 3 show validity and efficiency of an theextremely present convenient method. analytical
tions
solution can be found for the Green’s kernel in the boundary integral equation. The numerical results in Section 3 show the validity and efficiency of the present method. 2. Mathematical Theory and Algorithms 2. Mathematical Theory and Algorithms
2.1. Governing Equation and Boundary Conditions
ns between linear 2.1. water waves and a longand prismatic rigid Governing Equation Boundary Conditions The problem is to consider interactions between linear water waves and a long prismatic rigid e, either floating or submerged in water of infinite depth, The problem is to consider interactions between linear or water waves and long prismatic rigid structure in arbitrary cross-sectional shape, either floating submerged inawater of infinite depth, artesian coordinate system (x, z) is defined, with its origin structure in arbitrary cross-sectional shape, either floating or submerged in water of infinite depth, as taken shown in Figure 1. TheThe right-handed l and the z-axis vertically upward. fluid domainCartesian coordinate system (x, z) is defined, with its origin as shown in Figure 1. The right-handed Cartesian system (x, z) isupward. defined, with its origin onsists of a located free surface boundary SF, anfree up-side open at the undisturbed surface level and thecoordinate z-axis taken vertically The fluid domain located at the undisturbed free surface level and the z-axis taken vertically upward. The fluid domain and a wettedissurface boundary S B on the structure. denoted by Ω, whose boundaries S consists of a free surface boundary SF, an up-side open is denoted by Ω, whose boundaries S consists of a free surface boundary SF , an up-side open boundary boundary SU, a lee-side open boundary SL, and a wetted surface boundary SB on the structure. SU , a lee-side open boundary SL , and a wetted surface boundary SB on the structure.
putation domain of the problem.
d incompressible, and the motion is assumed irrotational. Figure1.1.Computation Computation domain domain of Figure ofthe theproblem. problem. monic in time with an angular frequency ω, the velocity
The fluid is assumed to be inviscid and incompressible, and the motion is assumed irrotational. For the linear small amplitude wave harmonic in time with an angular frequency ω, the velocity potential can be expressed by:
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The fluid is assumed to be inviscid and incompressible, and the motion is assumed irrotational. For the linear small amplitude wave harmonic in time with an angular frequency ω, the velocity potential can be expressed by: h i Φ ( x, z, t) = Re φ ( x, z, t) e−iωt ,
(1)
where φ is a time-independent complex velocity potential, which can be further decomposed into: 3
φ = φ0 + φ4 − iω ∑ χ j φj ,
(2)
j =1
where the three components denote incident potential, diffraction potential, and radiation potential, respectively; χj represents displacement of the body motion in each mode (sway, heave, or roll), and φj stands for the corresponding radiation potential to each motion mode. The incoming wave of amplitude A and frequency ω, propagating in the positive x direction in the water of infinite water depth, can be described by the following incident velocity potential: φ0 = −
igA Kz+iKx e , ω
(3)
where K is the infinite depth wave number defined by K = ω 2 /g. The four induced wave potentials φj (j = 1~4) must satisfy the Laplace equation:
∇2 φj = 0,
(4)
and be subjected to various boundary conditions in the fluid domain, including the free surface condition on SF : ∂φj − Kφj = 0, (5) ∂z the bottom boundary condition as z→∞:
∇φj → 0 ,
(6)
the boundary condition on the surface SB of the structure: ∂φj = ∂n
(
0 − ∂φ ( j = 4) ∂n , , nj, ( j = 1, 2, 3)
(7)
and the radiation condition in the far field boundaries SU and SL : lim
x →±∞
∂φj ± iKφj ∂x
= 0,
(8)
where n is the normal direction of the body geometry, with its three components n1 = nx , n2 = nz , n3 = (z − zc )nx − (x − xc )nz , where nx and nz are the x and z components of the unit inward normal, respectively, and (xc , zc ) is the rotation center. The subscripts j = 1, 2, 3 denote the direction of sway, heave, and roll for radiation, respectively, and j = 4 stands for the diffraction.
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2.2. Numerical Techniques According to Green’s second theorem, by employing Wehausen’s free-surface Green function as the kernel, a boundary integral equation can be obtained as: αφj (x0 ) =
Z SB
∂φj (x) ∂G (x; x0 ) φj (x) − G (x; x0 ) dS, ∂n ∂n
(9)
where α is the solid angle. The free-surface Green function is defined to be: G (x; x0 ) = ln
r −2 r1
I ∞ µ(z+ζ ) e 0
µ−K
cosµ ( x − ξ ) dµ,
(10)
where the path of the contour integral passes below the poles at µ = K; coordinate of the source point is x0 = (ξ, ζ); r is the distance between field point and source point, and r1 is the distance between field point and the image of source point with respect to the free surface. On the other aspect, if Rankine Green function were employed as the kernel, the boundary integral equation would be:
Z
αφj (x0 ) =
S F +SB +SU +S L
∂φj (x) ∂G (x; x0 ) φj (x) − G (x; x0 ) dS, ∂n ∂n
(11)
where the kernel would be simply expressed by: G (x; x0 ) = lnr.
(12)
In this paper, we denote the method based on Equations (9) and (10) as FSG_BEM, and the method based on Equations (11) and (12) as RKG_BEM. The former is applied as the present numerical method, while the latter is used as a comparison for computational efficiency. The three-node isoparametric element is selected to discretize both the geometry of body surface and the physical variables, the shape functions of which being expressed by: h1 ( η ) =
1 η ( η − 1), 2
h2 ( η ) = 1 − η 2 , h3 ( η ) =
1 η ( η + 1), 2
(13) (14) (15)
where η is the local coordinate (−1 ≤ η ≤ 1). Therefore, the velocity potential and its normal derivative on the boundary surface can be expressed straightforwardly as: 3
[ x, z] =
∑ hk (η )
h
i x k , zk ,
(16)
k =1
" # 3 ∂φ ∂φ k φ, = ∑ hk (η ) φk , . ∂n ∂n k =1
(17)
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Applying the above discretization and the body surface condition Equations (5)–(8) leads to the following discrete form of the boundary integral equation of Equation (9): NB R1 3 k ∂G (x;x0 ) αφj (x0 ) − ∑ ∑ hk (η ) φj | J (η )| dη ∂n i =1 −1 k =1 1 N R B ∂φ x ( ) 0 ∑ G (x; x0 ) ∂n | J (η )| dη, ( j = 4) , i =1 −1 = NB R1 G (x; x0 ) n j | J (η )| dη, ( j = 1, 2, 3) −∑
(18)
i =1 −1
where NB represents the number of total elements along the body surface, and J(η) the Jacobi matrix for local-global coordinate transformation, the determinant value of which is calculated by: s
| J (η )| =
2
dx dη
+
dz dη
2 .
(19)
By employing a collocation process for Equation (18) that the source point is arranged to be put on each grid node on the immersed body surface mesh, a linear algebraic system could be obtained in closed form: [ A ] N × N φ j N × N = [ B ] N ×4 , where N is the total number of nodes. Solution of the above linear system is sensitive to the diagonal terms of the left-hand side influence matrix [A] which, therefore, needs to be evaluated precisely with caution. However, direct calculation of these diagonal terms is usually inaccurate and troublesome, due to the high singularity of the Green function in the case when the field point and the source point coincides with each other. Fortunately, this weakness can be avoided by considering a constant flux across the fluid (φ = 1); thereafter we obtain: N
Aii = −
∑
(i = 1, . . . , N ).
Aij ,
(20)
j=1,j6=i
In calculation of each influence coefficient Aij (j 6= i), OpenMP parallelization technique is employed to distribute the computation burden on multiple processors of a single computer. The parallelization works well since calculation of the influence coefficient on one element is independent from that on another element. After that, the Gauss elimination algorithm is used to solve the linear system, which is extremely robust regardless of arbitrary shape of the structure. Given solution for the linear system, we can get the wave exciting force, added mass and added damping by directly integrating the corresponding hydrodynamic pressure over the immersed body surface, respectively, i.e.: Z f j = iρω
(φ0 + φ4 ) n j dS,
(21)
SB
and: aij +
Z ibij = ρ φi n j dS. ω
(22)
SB
2.3. Direct Calculation of Free-Surface Green’s Function As pointed out in the introduction, accurate calculation of the free-surface Green function is of great importance to the final solution of the problem. At a preliminary step, we may apply the function
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decomposition method which was proposed by Newman [17]. Equation (10) can be decomposed into a simplified form: r (23) G = ln − 2F1 ( x, z) − 2πieK (z+ζ ) cosK ( x − ξ ), r1 where i denotes the imaginary unit. The singular part of Equation (23) is: F1 ( x, z) = PV
Z ∞ µ(z+ζ ) e 0
µ−K
cosµ ( x − ξ ) dµ,
(24)
where PV denotes the Cauchy principle value of the integral. The principle task here is to evaluate the real function F1 (x, z) for all relevant values of input parameters (x, z) of possible physical interest [17]. Using the identity: Z 2K dµ PV = 0, µ −K 0 Equation (24) can be written [18] in a more convenient form from the view point of numerical evaluation: F1 ( x, z) =
Z 2K f 1 (µ) − f 1 (K )
µ−K
0
dµ +
Z ∞ f 1 (µ) 2K
µ−K
dµ,
(25)
where: f 1 (µ) = eµ(z+ζ ) cosµ ( x − ξ ).
(26)
In the neighborhood of µ = K, linear approximation may be applied such that: f 1 (µ) − f 1 (K ) = f 10 (K ) = eK(z+ζ ) [(z + ζ ) cosK ( x − ξ ) − ( x − ξ ) sinK ( x − ξ )]. µ−K
(27)
Hence, Equation (25) can be evaluated accurately by an adaptive Gauss–Kronrod-type quadrature algorithm (see Appendix A). Following a similar procedure, the derivatives of the Green function with respect to x and z can be normalized as: ! 1 1 Gx = ( x − ξ ) − 2 + 2KF2 ( x, z) + 2πiKeK(z+ζ ) sinK ( x − ξ ), (28) r2 r1 Gz =
(z − ζ ) (z + ζ ) − − 2KF3 ( x, z) − 2πiKeK(z+ζ ) cosK ( x − ξ ), r2 r12
(29)
where their singular parts are: F2 ( x, z) = PV
F3 ( x, z) = PV
Z ∞ µeµ(z+ζ ) 0
µ−K
Z ∞ µeµ(z+ζ ) 0
µ−K
sinµ ( x − ξ ) dµ,
(30)
cosµ ( x − ξ ) dµ.
(31)
Equations (30) and (31) can be formulated as the same form as Equation (25), where: f 2 (µ) = µeµ(z+ζ ) sinµ ( x − ξ ),
(32)
f 3 (µ) = µeµ(z+ζ ) cosµ ( x − ξ ),
(33)
f 20 (K ) = eK (z+ζ ) [sinK ( x − ξ ) + K (z + ζ ) sinK ( x − ξ ) + K ( x − ξ ) cosK ( x − ξ )],
(34)
f 30 (K ) = eK (z+ζ ) [cosK ( x − ξ ) + K (z + ζ ) cosK ( x − ξ ) − K ( x − ξ ) sinK ( x − ξ )].
(35)
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2.4. Fast Evaluation by the Analytical Method Although calculation of the free-surface Green function becomes applicable following the method described in Section 2.3, a large amount of computation time would be consumed due to the direct integration by the meticulous adaptive numerical quadrature method. The reason for that is the effort of dealing with singularity in the denominator, as well as the oscillating inherence of the integrand. A possible way for its improvement is to derive an alternative analytical expression which can automatically remove the troublesome singularity, as described below. Based on McIver [19], we can obtain the following representation for the principle value of the singular integral without too much of difficulty:
PV
Z ∞ µ(z+ζ )+iµX e 0
µ−K
K (z+ζ )+iKX (− πi + E ( K ( z + ζ ) + iKX )) , 1 e dµ = −eK(z+ζ ) Ei (−K (z + ζ )) , eK (z+ζ )+iKX ( πi + E (K (z + ζ ) + iKX )) , 1
X < 0, (z + ζ ) ≤ 0 X = 0, (z + ζ ) ≤ 0 , (36) X > 0, (z + ζ ) ≤ 0
where the exponential integrals are defined as: Ei ( x ) =
E1 ( Z ) =
Z x t e −∞
Z ∞ −t e Z
t
t
dt,
dt,
( x > 0),
(37)
( arg | Z | < π ).
(38)
Let Z = K (z + ζ ) + iKX, the real part of Equation (10) can then be written as: Re e Z (−πi + E1 ( Z )) , r Re { G } = ln − 2 Re −e Z Ei (−Kz) , r1 Re e Z ( πi + E ( Z )) , 1
X < 0, (z + ζ ) ≤ 0 X = 0, (z + ζ ) ≤ 0 , X > 0, (z + ζ ) ≤ 0
(39)
while the imaginary part is obtained by applying the residue theorem, after which we find: Im { G } = −2iπRe e Z .
(40)
Based on the following identities of the exponential integral, i.e.,: e− Z dE1 ( Z ) =− , dZ Z
(41)
dEi (z) e−z = , dz z
(42)
It is possible to calculate derivatives of the Green function with respect to x and z, in which their real parts are corresponding to:
Re { Gx } = ( x − ξ )
1 1 − 2 2 r r1
!
n o Z (− πi + E ( Z )) − iK , Re iKe 1 Z −2 0, n o Re iKe Z ( πi + E ( Z )) − iK , 1
Z
n o Z (− πi + E ( Z )) − K , Re Ke 1 n oZ (z − ζ ) (z + ζ ) 1 z Re { Gz } = − −2 Re −Ke Ei ( −Kz) + KZ , n o r2 r12 Re Ke Z ( πi + E1 ( Z )) − K , Z
X < 0, (z + ζ ) ≤ 0 X = 0, (z + ζ ) ≤ 0 , (43) X > 0, (z + ζ ) ≤ 0 X < 0, (z + ζ ) ≤ 0 X = 0, (z + ζ ) ≤ 0 , X > 0, (z + ζ ) ≤ 0
(44)
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respectively. Their imaginary parts are easily obtained by applying the residue theorem as: 2iKπIm e Z ,
(45)
Im { Gz } = −2iKπRe e Z .
(46)
Im { Gx } =
Through this method, we are able to calculate the free-surface Green function in a fast manner, since Equations (39), (40), and (43)–(46) are all in analytical form, which just simply consists of the exponential functions and the trigonometric functions. In terms of the following two coordinates: X = K ( x − ξ ) , Y = K | z + ζ |.
(47)
The three singular functions in Equations (24), (30) and (31) can be expressed as F1 (X, Y), F2 (X, Y), and F3 (X, Y). Figures 2–4 show a comparison between plots of the three singular functions calculated by the direct integration method and the analytical solution method. In general, the two methods get almost same results which are hard to be distinguished from each other. It is obviously to see that F1 (X, Y) and F3 (X, Y) are even functions in symmetric with respect to the Y axis, while F2 (X, Y) is an odd function which is anti-symmetric about the Y axis. Remarkable variations with a period of π in parallel to the X axis can be observed in all the plots for the region of Y ∈ [0, 3]. It is also important to see that the becomes slow-varying with the increase of Y in the region of Y ∈ [3, 9+]. Computation 2016 , 4,variation 36 9 of 21 9
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(c)
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(d)
Figure 2. Comparison ofofthe functionF1F(X, 1(X, calculated bytwo themethods: two methods: (a) plot contour Figure 2. Comparison thesingular singular function Y)Y) calculated by the (a) contour plot by bydirect direct integration; (b) oblique direct integration; (c) by contour plot by analytical integration; (b) oblique view byview directby integration; (c) contour plot analytical solution; and (d) oblique view by analytical solution; and (d) oblique view bysolution. analytical solution. 9
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1
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Y
(c)
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(d)
Figure 2. Comparison of the singular function F1(X, Y) calculated by the two methods: (a) contour plot by 4,direct integration; (b) oblique view by direct integration; (c) contour plot by analytical Computation 2016, 36
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solution; and (d) oblique view by analytical solution. 9
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4 0
Y
(c)
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-10
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0
X
(d)
Figure 3. Comparison of the singular function F2(X, Y) calculated by the two methods. For captions
Figure 3. Comparison of the singular function F2 (X, Y) calculated by the two methods. For captions of of the subplots please refer to Figure 2. the subplots Computation 2016,please 4, 36 refer to Figure 2. 10 of 21 9
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(d)
Figure 4. Comparison of the singular function F3(X, Y) calculated by the two methods. For captions
Figure 4. Comparison of the singular function F3 (X, Y) calculated by the two methods. For captions of of the subplots please refer to Figure 2. the subplots please refer to Figure 2.
3. Numerical Results and Discussion
Based on the direct calculation method and the analytical solution method described above, values of the free-surface Green function and its derivatives are compared, as shown in Figures 5 and 6, against variation of the physical horizontal distance |x − ξ| between the source and the field points. Both the real part and the imaginary part are compared, showing that they coincide fairly well with
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3. Numerical Results and Discussion Based on the direct calculation method and the analytical solution method described above, values of the free-surface Green function and its derivatives are compared, as shown in Figures 5 and 6, against variation of the physical horizontal distance |X − ξ| between the source and the field points. Both the real part and the imaginary part are compared, showing that they coincide fairly well with each other. It should be noted that calculation of the imaginary parts is relatively straightforward since Computation 2016 , 4, 36 troublesome principal values of the singular integrals. 11 of 21 the real parts contain 0.15
0.15 Direct Integration Analytical Solution
0.1 0.1
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( f)
Figure 5. Comparison of two methods for calculating Wehausen’s Green function and its derivatives Figure 5. Comparison of two methods for calculating Wehausen’s Green function and its derivatives (K = 1.2 m−1, ζ = −1.0 m, z = −1.0 m): (a) real part value of G; (b) imaginary part value of G; (c) real part (K = 1.2 m−1 , ζ = −1.0 m, z = −1.0 m): (a) real part value of G; (b) imaginary part value of G; (c) real value of Gx; (d) imaginary part value of Gx; (e) real part value of Gz; and (f) imaginary part value of Gz. part value of Gx ; (d) imaginary part value of Gx ; (e) real part value of Gz ; and (f) imaginary part value of Gz .
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-1 -0.9995 -0.9995 Direct Integration Analytical Solution
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x 10
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Figure Comparisonofoftwo twomethods methodsfor forcalculating calculating Wehausen’s Wehausen’s Green Figure 6. 6. Comparison Greenfunction functionand andits itsderivatives derivatives −1 (K = 0.001 m , ζ = −0.1 m, z = −0.2 m). For captions of the subplots please refer to Figure 5. − 1 (K = 0.001 m , ζ = −0.1 m, z = −0.2 m). For captions of the subplots please refer to Figure 5.
Due to different nature of the free-surface Green function when the source point locates at the Due to different of to theverify free-surface Green whenbodies the source point locatesbodies. at the In free free surface or not, nature we need the results forfunction both floating and submerged surface or not, we need to verify the results for both floating bodies and submerged bodies. In order order to compare with some existing analytical results, we select horizontal floating/submerged to circular compare with some existing analytical results, we the select horizontal floating/submerged circular cylinders for the benchmark examples. When cylinder is submerged, its wet body surface cylinders for the benchmark examples. When the cylinder is submerged, its wet body surface should should be considered as a completely immersed circle; when it is floating with its centroid located on beexactly considered as a completely immersed circle; when it is floating with its centroid located on exactly the mean water surface, whereas its wet body surface should be treated as a half circle. Their theanalytical mean water surface, its wet body shouldmultipole be treated as a half circle. Their analytical solutions arewhereas all obtained based onsurface the so-called expansion method, which were solutions arein all[2,20] obtained on thecircle so-called multipole expansion method, which published for a based submerged in water of infinite depth, and in [1,21] for were a halfpublished circle in in water [2,20] of forinfinite a submerged in water of depth, andfor in [1,21] for a half circle in depth. circle The RKG_BEM is infinite also implemented comparison, in which allwater of theof
boundaries should be taken into consideration; whereas for the FSG_BEM, only the body surface needs to be meshed. The computations are implemented on a SONY laptop (Sony Corporation, Tokyo, Japan), with an Intel(R) (Intel, Inc., Santa Clara, CA, USA) Core(TM) i7-2670QM CPU of 2.2 GHz, on 64-bit Windows operating system. The OpenMP parallelization technique has been applied in the parallel mode. Computation 2016, 4, 36 12 of 20 Figure 7 shows modulus of complex exciting force, added mass and added damping of a semi-immersed cylinder of radius a in comparison with those computed by the RKG_BEM and the analytical multipole expansionismethod [20]. The corresponding meshes used byallthe boundary infinite depth. The RKG_BEM also implemented for comparison, in which of two the boundaries element areconsideration; specified in Table 1. The takes 39.08 for the RKG_BEM should be methods taken into whereas forcomputation the FSG_BEM, only thesbody surface needs and to be 2.67 s for the FSG_BEM, both in parallel mode. In Figure 7, the present method based on meshed. The computations are implemented on a SONY laptop (Sony Corporation, Tokyo, Japan),the with evaluated free-surface Green achieves good CPU agreement with both of the other ananalytically Intel(R) (Intel, Inc., Santa Clara, CA, USA)function Core(TM) i7-2670QM of 2.2 GHz, on 64-bit Windows two methods. Noted thereparallelization are some odd points on the the FSG_BEM. This operating system. The that, OpenMP technique hascurve beencalculated applied inby the parallel mode. phenomenon should be attributed to the so-called “irregular frequencies” [10,22]. The irregular Figure 7 shows modulus of complex exciting force, added mass and added damping of a frequencies arecylinder given byof theradius eigenvalues associated with interior Dirichlet within the semi-immersed a in comparison withthe those computed byproblem, the RKG_BEM and fluid domain on which the integral equation is applied. The discrete boundary integral equation is the analytical multipole expansion method [20]. The corresponding meshes used by the two boundary ill-conditioned and not uniquely solvable at these frequencies. element methods are specified in Table 1. The computation takes 39.08 s for the RKG_BEM and 2.67 s for the FSG_BEM, both in parallel mode. In Figure 7, the present method based on the analytically Table 1. Mesh specifications for the case shown in Figure 4. LF, LU, LL and LB denote the length of the evaluated free-surface Green function achieves good agreement with both of the other two methods. boundaries as shown in Figure 1, respectively, and NF, NU, NL, and NB denote the number of elements Noted meshed that, there are some oddrespectively. points on the curve calculated by the FSG_BEM. This phenomenon on the boundaries, should be attributed to the so-called “irregular frequencies” [10,22]. The irregular frequencies are Method LFassociated LU with the LL interiorLDirichlet B Nproblem, F NUwithin the NL fluidNdomain B given by the eigenvalues on FSG_BEM / / / πa / / / 10 which the integral equation is applied. The discrete boundary integral equation is ill-conditioned and RKG_BEM 20a 20a πa 240 90 90 30 not uniquely solvable at60a these frequencies. 1.8
1.3 RKG-BEM
1.2 1.1
Analytical
FSG-BEM Analytical
1.4
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1.2 fz /(ρg)
0.9 fx /(ρg)
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(e)
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Figure 7. Comparison of hydrodynamic characteristics of a floating cylinder, in semi-immersed circle Figure 7. Comparison of hydrodynamic characteristics of a floating cylinder, in semi-immersed circle of radius a: (a) sway exciting force; (b) heave exciting force; (c) sway added mass; (d) heave added of radius a: (a) sway exciting force; (b) heave exciting force; (c) sway added mass; (d) heave added mass; (e) sway added damping; and (f) heave added damping. mass; (e) sway added damping; and (f) heave added damping.
Figure 8 shows modulus of complex exciting forces, added mass and added damping of a Table 1. Mesh specifications for the case shown in Figure 4. L , L , L and LB denote the length of the submerged cylinder in comparison with those computedF byU theL RKG_BEM and the analytical boundaries as shown in Figure 1, respectively, and NF , NU , NL , and NB denote the number of elements multipole expansion method [21]. The computation takes 49.21 s for the RKG_BEM and 2.65 s for the meshed on the boundaries, respectively. FSG_BEM, both in parallel mode. In the solution of multipole expansion method, we derive a new exact formulation (see Appendix B) for the multipole expansion coefficient Amn which is troublesome Method LF LU LL LB NF NU NL NB for calculation due to its high singularity in the integrand. In this case, the radius of the cylinder is a, FSG_BEM / distance / / its πa / surface) 10 is f/a = 1.5. The and the submergence (vertical from centroid/to the /mean free RKG_BEM 60a 20a 20a πa 240 90 90 30 meshes used by the two boundary element methods are specified in Table 2. In Figure 8, similar to the case of semi-circle, the present method based on the analytically evaluated free-surface Green Figurehighly 8 shows modulus of complex exciting added mass and added damping of a function agrees with both of the other two forces, methods. In addition, there is no ‘irregular frequencies’ phenomenon, which with proves the computed knowledgebythat submerged the solution is submerged cylinder in comparison those thefor RKG_BEM andbodies, the analytical multipole always unique. expansion method [21]. The computation takes 49.21 s for the RKG_BEM and 2.65 s for the FSG_BEM, both in parallel mode. In the solution of multipole expansion method, we derive a new exact Table 2.(see MeshAppendix specifications thethe casemultipole shown in Figure 5. Refer to Table 1 for the notes of theis symbols. formulation B)forfor expansion coefficient Amn which troublesome for calculation Method due to its high singularity in the integrand. In this case, the radius of the LF LU LL LB NF NU NL NBcylinder is a, and the submergence (vertical distance from its centroid to the mean free surface) is f/a = 1.5. The FSG_BEM / / / 2πa / / / 10 meshes used by the two boundary in 2. In Figure 8, RKG_BEM 60a element 20a methods 20a are specified 2πa 240Table 90 90 60similar to the case of semi-circle, the present method based on the analytically evaluated free-surface Green function highlyFigure agrees9 with of the other two methods. addition, there is nowave “irregular frequencies” showsboth a comparison of computation timeIn(unit: s) for 60 incident periods between phenomenon, which proves the knowledge that bodies,and thethe solution is always unique. the BEMs (boundary element methods) based onfor thesubmerged direct integration analytical solution of the free-surface Green function, in either sequential mode or parallel mode. In Figure 9, a remarkable Table 2. Mesh specifications case shown in Figure 5. Refer to Table 1 fortends the notes the symbols. trend of reduction in CPU timefor isthe shown by using the parallel mode, which to beofmore apparent with increasing number of the total elements, in both Figure 9a,b. On the other hand, the analyticalMethod LU LL LBof computation NF N U time N L forNthe B based Green function has savedLFa significant amount BEM analysis. FSG_BEM / the computation / / 2πa for / around / 27~36 / times 10 in the sequential Roughly speaking, it has improved speed 60a 20a in 20a 2πa to 240the direct-integration 90 90 60 Green function mode, and 12~60 RKG_BEM times in the parallel mode, comparison method, depending on the number of total elements in the input mesh of geometry. The computation timeFigure is further compared in Figure for each calling of thes)subroutine, involving for 9 shows a comparison of 10, computation time (unit: for 60 incident wave calculation periods between both the value of Green’s function and its derivatives. It is evidently shown that the direct integration the BEMs (boundary element methods) based on the direct integration and the analytical solution of the method consumes much more time than the analytical solution In Figure addition, computation free-surface Green function, in either sequential mode or parallelmethod. mode. In 9, athe remarkable trend time tends to increase proportionally as the increase of the point distance |x − ξ|, which is different of reduction in CPU time is shown by using the parallel mode, which tends to be more apparent with from that of the series expansion method as reported in the three-dimensional problem in [17,23–25]. increasing number of the total elements, in both Figure 9a,b. On the other hand, the analytical-based Green function has saved a significant amount of computation time for the BEM analysis. Roughly speaking, it has improved the computation speed for around 27~36 times in the sequential mode,
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and 12~60 times in the parallel mode, in comparison to the direct-integration Green function method, depending on the number of total elements in the input mesh of geometry. The computation time is further compared in Figure 10, for each calling of the subroutine, involving calculation for both the value of Green’s function and its derivatives. It is evidently shown that the direct integration method consumes much more time than the analytical solution method. In addition, the computation time tends to increase proportionally as the increase of the point distance |x − ξ|, which is different from Computation 2016, 4expansion , 36 15 of 21 that of the series method as reported in the three-dimensional problem in [17,23–25]. 1.6
1.6
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1 fz /(ρg)
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Figure 8. Comparison of hydrodynamic characteristics of a submerged cylinder. For captions of the Figure 8. Comparison of hydrodynamic characteristics of a submerged cylinder. For captions of the subplots please refer to Figure 7. subplots please refer to Figure 7.
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(a) (b) 10 400 (a) (b) 8 based on different calculation schemes of Figure 9. CPU time of the two boundary element methods Figure 9.300CPU time of the two boundary element methods based on different calculation schemes of the free-surface Green function in sequential parallel mode: comparison for schemes the direct Figure 9. CPU time of the two boundary elementormethods based on (a) different calculation of 6 the free-surface Green function in sequential or parallel mode: (a) comparison for the direct integration integration method; and (b) comparison for the analytical solution method. the 200 free-surface Green function in sequential or parallel mode: (a) comparison for the direct 4 method; and (b) comparison for the analytical solution method. integration method; and (b) comparison for the analytical solution method. 3
10
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CPU( time ( s) CPU time μs ) μ
(a)
(b)
DrG AlG DrG AlG
1
10 Figure 9. CPU time of the two boundary element methods based on different calculation schemes of 1 10 the free-surface Green function in sequential or parallel mode: (a) comparison for the direct integration method; and (b)10comparison for the analytical solution method. 0 0
103 10 -1
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CPU time ( μs )
Figure 10. Computation time (μs) for each implementation of the Green’s kernel, by the two methods, respectively, as a function of distance |x − ξ|. The figure isGreen’s obtained basedbyon CPU Figure 10. Computation time (μs) eachimplementation implementation of kernel, the two methods, 10point Figure 10. Computation time (µs) forfor each ofthe the Green’s kernel, byaveraged the two methods, time of 1 million evaluations of the two codes, respectively, for every input of the wave period ω. respectively, a function of point distance|x |x−− ξ|. ξ|. The based on averaged CPUCPU respectively, as a as function of point distance Thefigure figureisisobtained obtained based on averaged time of 1 million evaluations twocodes, codes,respectively, respectively, for thethe wave period ω. ω. time of 1 million evaluations of10of thethe two forevery everyinput inputofof wave period x 10 x 10 1
0
-3
-3
14
14
10 elements 30 elements 10elements elements 60 30 elements 2 3 4 60 elements
-3
14 12
x 10
10 elements 30 elements
-3
x 10
14 12
10elements elements 60 Figure 11 demonstrates the 10convergence rate of the present FSG_BEM using analytical solution 30 elements 12 0 1 5 612 7 8 9 10 10 10 60 elements Green function. It is shown that with the increase of |x-ζ| number of elements, the numerical solution gets 10 10 8 close quickly8to the exact solution [21], especially from 30 to 60 elements in the implemented case the Figure 10. Computation time (μs) for each implementation 8 8 of the Green’s kernel, by the two methods, 6 differencerespectively, is 6almost negligible. This may suggest that, for such cross-sections in simple geometries, as a function of point distance |x − ξ|. The figure is obtained based on averaged CPU 6 6 4 4 only a fewtime elements areevaluations sufficientoftothe obtain a high accuracy. of 1 million two codes, respectively, for every input of the wave period ω.
Error of fz Error of fz
Error of fx Error of fx
-1
4 2
4 2 -3
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x 10
2 014 0 12 -2 0
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10 elements 30 elements 60 5elements
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(b) (b)
8 Error of fz
Error of fx
120 -2 0
6
(a) (a)
8
x 10 1402
10 elements 30 elements 60 5 elements 6
Figure6 11. Convergence test of the present method to 6the exact solution [21] with respect to the 4 number of elements: (a) sway force; (b) heave exciting force.solution [21] with respect to the Figure 11. Convergence test exciting of the present method to4 the exact number of elements: (a) sway exciting force; (b) heave exciting force. 2 2 0 -2
0
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3 Ka
(a)
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11. Convergence test of the present method to the exact solution [21] with respect to the FigureFigure 11. Convergence test of the present method to the exact solution [21] with respect to the number number of elements: (a) sway exciting force; (b) heave exciting force. of elements: (a) sway exciting force; (b) heave exciting force.
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To further demonstrate the proposed FSG_BEM, wave interactions with a horizontal rectangular cylinder have been examined, for either floating or submerged cases. For simplicity, both of width and length of the rectangular cross-section shape are taken as 2a, respectively. f /a is the submergence of its centroid, as defined in the circular cylinder case. We examine four different submergences (particularly, f /a = 0.0 stands for the floating cylinder), and the results are shown in Figure 12. From the results, it is apparent that the floating cylinder is very different from those submerged in the water, especially in the low frequency region. In general, the peak value of the forces and its corresponding peak frequency tend Computation to decrease as4,the 2016, 36 submergence increases. 17 of 21 2
1.8 f/a = f/a = f/a = f/a =
1.6 1.4
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f/a = f/a = f/a = f/a =
1.8 1.6
0.0 2.0 4.0 8.0
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1.2 fz /(ρg)
fx /(ρg)
1 0.8
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0.5
1
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(a) f/a = f/a = f/a = f/a =
0.0 2.0 4.0 8.0
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f/a = f/a = f/a = f/a =
0.8
a33 /(ρS)
a22 /(ρS)
2.5
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3
0
0.5
1
(c)
1.5 Ka
2
2.5
3
(d) 0.7
0.45 f/a = f/a = f/a = f/a =
0.4 0.35
0.0 2.0 4.0 8.0
f/a = f/a = f/a = f/a =
0.6
0.0 2.0 4.0 8.0
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b 33 /(ρSω)
b 22 /(ρSω)
2
(b)
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0
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0.2 0.15 0.1
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0
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(e)
2
2.5
3
0
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1
1.5 Ka
2
2.5
3
(f)
Figure 12. Variation of hydrodynamic characteristics of a floating rectangular cylinder (f/a = 0.0) and
Figure 12. Variation of hydrodynamic characteristics of a floating rectangular cylinder (f /a = 0.0) submerged rectangular cylinder for different submergences (f/a = 2.0, f/a = 4.0 and f/a = 8.0). and submerged rectangular cylinder for different submergences (f /a = 2.0, f /a = 4.0 and f /a = 8.0). S represents the cross-section area (note that S of a floating rectangular cylinder is half of a submerged S represents cross-section area (note that S of floating one). Forthe captions of the subplots please refer to a Figure 7. rectangular cylinder is half of a submerged one). For captions of the subplots please refer to Figure 7.
Figure 11 demonstrates the convergence rate of the present FSG_BEM using analytical solution Green function. It is shown that with the increase of number of elements, the numerical solution gets close quickly to the exact solution [21], especially from 30 to 60 elements in the implemented case the difference is almost negligible. This may suggest that, for such cross-sections in simple geometries, only a few elements are sufficient to obtain a high accuracy. To further demonstrate the proposed FSG_BEM, wave interactions with a horizontal rectangular
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4. Conclusions In this paper, we presented a FSG_BEM, which applies a three-node higher-order scheme, and an analytical algorithm of the free-surface Green function. Various numerical results show that the present method deserves high accuracy, good convergence, and fast computation efficiency. Acknowledgments: The authors gratefully acknowledge the financial support provided by National Natural Science Foundation of China (Grant No. 11072052 and No. 51221961), and the financial support provided by the National Basic Research Program of China (973-Program) (Grant No. 2011CB013703). The authors gratefully appreciate Phil McIver of Loughborough University, Maureen McIver of Loughborough University, and R. Porter of Bristol University, for their most valuable suggestions on the issue of removing the singularity. Author Contributions: Yingyi Liu developed the algorithm of free-surface Green’s function, performed all the calculations in the paper and wrote the manuscript. Ying Gou was the assistant supervisor of Yingyi Liu’s Master study in DUT, who provided guidance, did proofreading and polished the English of this manuscript. Bin Teng was the supervisor of Yingyi Liu’s Master study in DUT, who provided guidance and developed the two-dimensional HOBEM code used in this manuscript. Shigeo Yoshida is Yingyi Liu’s current supervisor for postdoctoral research, who supplies salary and freedom for Yingyi Liu’s research, including summarization of this work. All authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations RKG_BEM FSG_BEM DrG_BEM AlG_BEM
Rankine Green function based Boundary Element Method Free-surface Green function based Boundary Element Method Boundary Element Method based on direct integration of the free-surface Green function Boundary Element Method based on analytical solution of the free-surface Green function
Appendix A Let [a, b] be the integration interval, f be a Riemann integrable function, the following target integral: I=
Z b a
f ( x )dx
(A1)
can be approximated by adding n + 1 Kronrod points to the n-point Gauss quadrature rule, so that the function values produced by the lower-order rule can be re-used. This formula is called as Kronrod extension of Gaussian rules [26], which has a maximum degree of exactness 3n + 1, i.e.,: I=
n
n +1
i =1
k =1
∑ wi f ( x i ) +
∑ wk∗ f (xk∗ ) ,
f ∈ P3n+1 ,
(A2)
where xi are the Gauss nodes and ω i the corresponding weights, xk∗ and ωk∗ denote the Kronrod nodes and corresponding weights, respectively. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error suggested by Piessens et al. [27]: ε = (200 | Gn [ a, b] − K2n+1 [ a, b]|)1.5 ,
(A3)
where Gn represents the approximation of the initial Gaussian rule, and K2n + 1 the approximation of its Kronrod extension. Using the Gauss–Kronrod rule, an adaptive integral algorithm can be developed [28]: Step 1. Firstly, use n-point Gauss rule and (2n + 1)-point Gauss–Kronrod rule to integrate f (x) on the interval [a, b], respectively. Two approximations of the integral, i.e., Gn and K2n + 1 , will be obtained, as well as Equation (A3). If the error estimation is smaller than a prescribed tolerance Eps, the more accurate approximation K2n + 1 is accepted as the final integral value; otherwise, go to Step 2.
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Step 2. Divide interval [a, b] into two equal parts, i.e., [a, m] and [m, b], where m = (a + b)/2, and compute the two sub-integrals independently: I=
Z m a
f ( x )dx +
Z b m
f ( x )dx.
(A4)
i i Again, the two respective approximations Gni and K2n +1 , as well as the local error estimate ε will be attained: 1.5 i εi = 200 Gni [ a, b] − K2n a, b , (A5) [ ] +1 i where the superscript denotes the ith sub-interval. If εi is smaller than Eps, accept K2n +1 as the final integral value on ith sub-interval, and stop the circulation; if not, continue to subdivide the sub-intervals and repeat Step 2.
Appendix B In the multipole expansion method, Linton and McIver [20] gives an expression for wave scattering potential of a submerged horizontal cylinder in series form: ∞
∑ an+1 αn φn ,
(B1)
∞ e−inθ + ∑ Amn rm e−imθ , rn m =0
(B2)
φ=
n =1
where: φn =
(−1)m+n m! (n − 1)!
Amn =
I ∞ µ + K m+n−1 2µζ µ e dµ, 0
(B3)
µ−K
where a is the radius and ζ the submergence. Calculation of the multipole expansion coefficients Amn is not a trivial task, since usually a direct integration method will be adopted which leads to some substantial numerical errors. By the Newton’s binomial theorem and through integration by parts, we derive the following series representations for its accurate calculation: Amn = Re ( Amn ) + iIm ( Amn ),
(B4)
where: Re ( Amn ) = " m + n −3
∑
i =0
(−1)m+n m!(n−1)!
1 − 2ζ
m+n
( m + n −1) ! K i e−2Kζ (m+n−1−i )i! (−2ζ )m+n−1−i
(m + n − 1)!+ 2Ke2Kζ · ! 1+
m + n −2− i
∑
j =1
(2Kζ ) j j!
and: Im ( Amn ) =
−
#) (m+n−1) m+n−2 −2Kζ K e 2ζ
(−1)m+n 2πK m+n e2Kζ m! (n − 1)!
− K m+n−1 Ei (−2Kζ )
,
(B5)
(B6)
where m ≥ 0, n ≥ 1, m, n ∈ Z ∗ . References 1. 2.
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