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We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.

In the method for hydrodynamic analysis of floating bodies with forward speed, due to the complex boundary condition on the free surface, the integral equation involves not only the unknown velocity potential but also its first-order and second-order derivatives on the free surface. Unlike classical methods in which a waterline integral is present by using Stokes’ theorem, we apply directly the free-surface condition so that the troublesome waterline integral is avoided. The first- and second-order derivatives of the velocity potential in this new method are then approximated using B-Spline method.

In this paper, the incident wave on free surface is approximated by cubic B-spline and the relationship between incident wave and its first-order and second-order partial derivatives are derived and compared with the analytical value.

This method is approved to have sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.

Let the knot vector _{0} < = u_{2} < = u_{3} < = ... < = u_{m}. We define the kth degree B-spline curve as Equation (1) [

where

The ith B-spline basis function of degree k, written as

In the same way, we define the B-spline surface by B-spline tensor product expansion as Equation (3) [

where

Given a clamped B-spline curve of degree k defined by n + 1 control points _{0} = u_{1} = ... = u_{k} = 0, u_{k}_{+1}, u_{k}_{+2}, ..., u_{m−k−}_{1}, u_{m−k} = u_{m−k}_{+1} = ... = u_{m} = 1, we can compute the point on the B-spline curve by the de Boor algorithm as Equation (4) [

where

(5)

Given a clamped B-spline curve of degree k, we can compute r order derivatives

where

B-spline curve approximation can be stated as the problem of constructing a B-spline curve passing through a set of fixed points

where

We choose a set of points

where A is the amplitude of the incident wave, k is wave number,

Let

When the incident wave is given as

We choose cubic B-spline to approximate the free surface. The free surface is divided into two patches, then each patch is described by cubic B-spline tensor product expansion as Equation (10).

The points on the surface can be described as Equation (11):

The first and second order partial derivatives of incident wave

We can calculate the value of

tion 2.2 and 2.3. Then the value of and _{r} is defined as Equation (14):

We choose the same points

Then we use the cubic B-spline to approximate the free surface as we do in Section 3.1.

Then we can calculate the value of

In the same way, the value of

u | v = 0.5 | |||||
---|---|---|---|---|---|---|

0 | 3.1385E−01 | 3.1400E−01 | 4.7431E−04 | 7.6716E−06 | −9.8143E−05 | 4.2804E−03 |

0.1 | −4.8933E−02 | −4.8813E−02 | 3.8206E−04 | −2.4175E−02 | −2.4349E−02 | 7.0685E−03 |

0.2 | −3.1167E−01 | −3.1164E−01 | 7.3223E−05 | 3.0150E−03 | 3.0139E−03 | 4.3419E−05 |

0.3 | 2.5385E−02 | 2.5265E−02 | 3.8278E−04 | 2.4606E−02 | 2.4569E−02 | 1.5064E−03 |

0.4 | 3.1375E−01 | 3.1376E−01 | 3.9942E−06 | −9.7945E−04 | −9.7195E−04 | 3.0321E−04 |

0.5 | −3.3066E−03 | −3.2561E−03 | 1.6085E−04 | −2.5005E−02 | −2.4648E−02 | 1.4440E−02 |

0.6 | −3.1394E−01 | −3.1394E−01 | 2.2539E−06 | −4.8887E−04 | −4.7954E−04 | 3.7742E−04 |

0.7 | −1.8954E−02 | −1.8852E−02 | 3.2335E−04 | 2.4721E−02 | 2.4605E−02 | 4.6962E−03 |

0.8 | 3.1236E−01 | 3.1235E−01 | 2.4578E−05 | 2.5142E−03 | 2.5244E−03 | 4.1486E−04 |

0.9 | 4.2845E−02 | 4.2365E−02 | 1.5289E−03 | −2.5056E−02 | −2.4424E−02 | 2.5599E−02 |

1 | −3.1388E−01 | −3.1400E−01 | 3.7208E−04 | −4.5533E−06 | −9.8143E−05 | 3.7858E−03 |

v | u = 0.1 | |||||
---|---|---|---|---|---|---|

0 | −2.0700E−01 | −2.0704E−01 | 1.8920E−01 | 4.5000E−03 | 4.4454E−03 | 1.2133E−02 |

0.1 | −1.8170E−01 | −1.8552E−01 | 1.8520E−01 | 6.7000E−03 | 6.7666E−03 | 9.9432E−03 |

0.2 | −1.5930E−01 | −1.6145E−01 | 1.7000E−01 | 8.4000E−03 | 8.4564E−03 | 6.7086E−03 |

0.3 | −1.3700E−01 | −1.3712E−01 | 1.4140E−01 | 9.7000E−03 | 9.6894E−03 | 1.0950E−03 |

0.4 | −1.1370E−01 | −1.1319E−01 | 1.1110E−01 | 1.0600E−02 | 1.0599E−02 | 9.3605E−05 |

0.5 | −8.9700E−02 | −8.9578E−02 | 8.6100E−02 | 1.1200E−02 | 1.1273E−02 | 6.5371E−03 |

0.6 | −6.5400E−02 | −6.5674E−02 | 6.8200E−02 | 1.1700E−02 | 1.1769E−02 | 5.9234E−03 |

0.7 | −4.0800E−02 | −4.1040E−02 | 4.3700E−02 | 1.2100E−02 | 1.2108E−02 | 6.9538E−04 |

0.8 | −1.5600E−02 | −1.5557E−02 | 1.2100E−02 | 1.2300E−02 | 1.2290E−02 | 7.7338E−04 |

0.9 | 1.1000E−02 | 1.1159E−02 | −1.5700E−02 | 1.2200E−02 | 1.2305E−02 | 8.6232E−03 |

1 | 4.1000E−02 | 4.1014E−02 | −2.4200E−02 | 1.2100E−02 | 1.2109E−02 | 7.1767E−04 |

value of

In this paper, we adopt cubic B-spline approximating the free surface in different incident waves, calculate the first and second order derivatives of incident wave based on B-spline theory and compare the numerical value to the analytical value. This method is approved to have sufficient accuracy but it depends on the selection of points. In addition, it also can be applied to approximate the velocity potential on the free surface.

This paper is funded by the International Exchange Program of Harbin Engineering University for Innovation- oriented Talents Cultivation and MOST 2011CB013703 plan.

Fang Li,Hui Li,Huilong Ren, (2015) B-Spline Approximation of Ship Waves on the Free Surface. Journal of Applied Mathematics and Physics,03,81-85. doi: 10.4236/jamp.2015.31011