对称格林函数简化计算的数学推导
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摘要
[目的]为了减少利用边界元法计算大型浮体运动的水动力系数时的面元数量,并提高计算速度,[方法]针对具有对称面的浮体,利用其在求解奇点源强时系数矩阵可以看成循环矩阵的特性,提出对称格林函数的简化计算方法。[结果]利用该方法可以显著地加快对称浮体在波浪中运动的水动力系数求解过程:对于具有1个对称面的浮体,源强系数矩阵只需求解1/2的元素,同时线性方程组的求解时间降为原来的1/4;对于具有2个对称面的浮体,源强系数矩阵只需求解1/4的元素,同时线性方程组的求解时间缩短为原来的1/16。[结论]数值计算结果表明给出的对称性简化方法是有效的,能够极大地提高计算效率。
[Objectives] This paper studies the way to reduce the elements used for calculating the hydrodynamic coefficient for movement of a large floating body with boundary element method and to quicken the calculation.[Methods]Based on the characteristic of the symmetrical floating body that the coefficient matrix can be considered as a circulant matrix during solvement of source intensity of singular point,a simplified calculation method of symmetrical Green function is proposed. Besides,detailed mathematical derivation is carried out and the physical meanings are explained.[Results]It is found that,through this method, the process of solving the hydrodynamic coefficient for wave movement of symmetrical floating body can be quickened significantly. For the floating body with one symmetrical plane,it only needs to solve half of the elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one quarter. And for the floating body with two symmetrical planes,it only needs to solve a quarter of elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one sixteenth.[Conclusions]The numerical results show that the proposed symmetrical simplification method is effective,and can greatly improve the calculation efficiency.
[Objectives] This paper studies the way to reduce the elements used for calculating the hydrodynamic coefficient for movement of a large floating body with boundary element method and to quicken the calculation.[Methods]Based on the characteristic of the symmetrical floating body that the coefficient matrix can be considered as a circulant matrix during solvement of source intensity of singular point,a simplified calculation method of symmetrical Green function is proposed. Besides,detailed mathematical derivation is carried out and the physical meanings are explained.[Results]It is found that,through this method, the process of solving the hydrodynamic coefficient for wave movement of symmetrical floating body can be quickened significantly. For the floating body with one symmetrical plane,it only needs to solve half of the elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one quarter. And for the floating body with two symmetrical planes,it only needs to solve a quarter of elements of the coefficient matrix of source intensity;meanwhile the time of solving linear equations is reduced by one sixteenth.[Conclusions]The numerical results show that the proposed symmetrical simplification method is effective,and can greatly improve the calculation efficiency.
引文
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[2]HAVELOCK T.Waves due to a floating sphere making periodic heaving oscillations[J].Proceedings of the Royal Society A:Mathematical,Physical and Engi.neering Sciences,1955,231(1184):1-7.
[3]JOHN F.On the motion of floating bodies II:simple harmonic motions[J].Communications on Pure and Applied Mathematics,1950,3(1):45-101.
[4]NEWMAN J N.Algorithms for the free-surface Green function[J].Journal of Engineering Mathematics,1985,19(1):57-67.
[5]王如森.三维自由面Green函数及其导数(频域-无限水深)的数值逼近[J].水动力学研究与进展,1992,7A(3):277-286.WANG R S.The numerical approach of three dimen.sional free-surface green function and its derivatives(frequency domain-infinite depth)[J].Journal of Hy.drodynamics,1992,7A(3):277-286(in Chinese).
[6]WU Y S.Hydroelasticity of floating bodies[D].Lon.don:Brunel University,1984:164-190.
[7]卢诚波.几类循环矩阵的快速算法研究[D].上海:上海大学,2011.