# An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}~10

^{4}evaluations of the Green function are needed for each incident wave period, compared to those $\mathcal{O}$(10

^{6}) evaluations in the three-dimensional cases (see [12]). Furthermore, by taking advantage of the contemporary computational technologies, some special technique may be applied to parallelize the algorithm on multi-processor machines.

## 2. Mathematical Theory and Algorithms

#### 2.1. Governing Equation and Boundary Conditions

_{F}, an up-side open boundary S

_{U}, a lee-side open boundary S

_{L}, and a wetted surface boundary S

_{B}on the structure.

_{j}represents displacement of the body motion in each mode (sway, heave, or roll), and φ

_{j}stands for the corresponding radiation potential to each motion mode.

_{j}(j = 1~4) must satisfy the Laplace equation:

_{F}:

_{B}of the structure:

_{U}and S

_{L}:

_{1}= n

_{x}, n

_{2}= n

_{z}, n

_{3}= (z − z

_{c})n

_{x}− (x − x

_{c})n

_{z}, where n

_{x}and n

_{z}are the x and z components of the unit inward normal, respectively, and (x

_{c}, z

_{c}) 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.

#### 2.2. Numerical Techniques

**x**

_{0}= (ξ, ζ); r is the distance between field point and source point, and r

_{1}is the distance between field point and the image of source point with respect to the free surface.

_{B}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:

_{ij}(j ≠ 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.

#### 2.3. Direct Calculation of Free-Surface Green’s Function

_{1}(x, z) for all relevant values of input parameters (x, z) of possible physical interest [17]. Using the identity:

#### 2.4. Fast Evaluation by the Analytical Method

_{1}(X, Y), F

_{2}(X, Y), and F

_{3}(X, Y). Figure 2, Figure 3 and Figure 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 F

_{1}(X, Y) and F

_{3}(X, Y) are even functions in symmetric with respect to the Y axis, while F

_{2}(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 variation becomes slow-varying with the increase of Y in the region of Y ∈ [3, 9+].

## 3. Numerical Results and Discussion

_{mn}which is troublesome for calculation due to its high singularity in the integrand. In this case, the radius of the cylinder is a, and the submergence (vertical distance from its centroid to the mean free surface) is f/a = 1.5. The 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 function highly agrees with both of the other two methods. In addition, there is no “irregular frequencies” phenomenon, which proves the knowledge that for submerged bodies, the solution is always unique.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

RKG_BEM | Rankine Green function based Boundary Element Method |

FSG_BEM | Free-surface Green function based Boundary Element Method |

DrG_BEM | Boundary Element Method based on direct integration of the free-surface Green function |

AlG_BEM | Boundary Element Method based on analytical solution of the free-surface Green function |

## Appendix A

_{i}are the Gauss nodes and ω

_{i}the corresponding weights, ${x}_{k}^{*}$ and ${\omega}_{k}^{*}$ denote the Kronrod nodes and corresponding weights, respectively.

_{n}represents the approximation of the initial Gaussian rule, and K

_{2n + 1}the approximation of its Kronrod extension.

_{n}and K

_{2n + 1}, will be obtained, as well as Equation (A3). If the error estimation is smaller than a prescribed tolerance Eps, the more accurate approximation K

_{2n + 1}is accepted as the final integral value; otherwise, go to Step 2.

^{i}will be attained:

^{i}is smaller than Eps, accept ${K}_{2n+1}^{i}$ 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

_{mn}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:

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**Figure 2.**Comparison of the singular function F

_{1}(X, Y) calculated by the two methods: (

**a**) contour plot by direct integration; (

**b**) oblique view by direct integration; (

**c**) contour plot by analytical solution; and (

**d**) oblique view by analytical solution.

**Figure 3.**Comparison of the singular function F

_{2}(X, Y) calculated by the two methods. For captions of the subplots please refer to Figure 2.

**Figure 4.**Comparison of the singular function F

_{3}(X, Y) calculated by the two methods. For captions of the subplots please refer to Figure 2.

**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 value of G

_{x}; (

**d**) imaginary part value of G

_{x}; (

**e**) real part value of G

_{z}; and (

**f**) imaginary part value of G

_{z}.

**Figure 6.**Comparison of two methods for calculating Wehausen’s Green function and its derivatives (K = 0.001 m

^{−1}, ζ = −0.1 m, z = −0.2 m). For captions of the subplots please refer to Figure 5.

**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 mass; (

**e**) sway added damping; and (

**f**) heave added damping.

**Figure 8.**Comparison of hydrodynamic characteristics of a submerged cylinder. For captions of the subplots please refer to Figure 7.

**Figure 9.**CPU time of the two boundary element methods based on different calculation schemes of the free-surface Green function in sequential or parallel mode: (

**a**) comparison for the direct integration method; and (

**b**) comparison for the analytical solution method.

**Figure 10.**Computation time (μs) for each implementation of the Green’s kernel, by the two methods, respectively, as a function of point distance |x − ξ|. The figure is obtained based on averaged CPU time of 1 million evaluations of the two codes, respectively, for every input of the wave period ω.

**Figure 11.**Convergence test of the present method to the exact solution [21] with respect to the number of elements: (

**a**) sway exciting force; (

**b**) heave exciting force.

**Figure 12.**Variation of hydrodynamic characteristics of a floating rectangular cylinder (f/a = 0.0) 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 one). For captions of the subplots please refer to Figure 7.

Method | L_{F} | L_{U} | L_{L} | L_{B} | N_{F} | N_{U} | N_{L} | N_{B} |
---|---|---|---|---|---|---|---|---|

FSG_BEM | / | / | / | πa | / | / | / | 10 |

RKG_BEM | 60a | 20a | 20a | πa | 240 | 90 | 90 | 30 |

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Liu, Y.; Gou, Y.; Teng, B.; Yoshida, S.
An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function. *Computation* **2016**, *4*, 36.
https://doi.org/10.3390/computation4030036

**AMA Style**

Liu Y, Gou Y, Teng B, Yoshida S.
An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function. *Computation*. 2016; 4(3):36.
https://doi.org/10.3390/computation4030036

**Chicago/Turabian Style**

Liu, Yingyi, Ying Gou, Bin Teng, and Shigeo Yoshida.
2016. "An Extremely Efficient Boundary Element Method for Wave Interaction with Long Cylindrical Structures Based on Free-Surface Green’s Function" *Computation* 4, no. 3: 36.
https://doi.org/10.3390/computation4030036