浮体非线性波浪力求解的泰勒展开边界元法

Nonlinear wave hydrodynamics are becoming a hot research topics in ship and offshore engineering research field. For hydrodynamics analysis of free surface wave interaction with arbitrary form floating bodies, potential flow and the corresponding Laplace equation is most solved numerically through Boundary Element Method(BEM), or named panel method. Mixed distribution of sources and dipoles integral equation For lower order BEM, direct BEM which is based on mixed sources and dipoles distribution BIE is superior to the indirect BEM which is based on sources only distribution BIE from the view points of accuracy and efficiency for solving the velocity potentials on the boundary. Nevertheless, numerical differentiation have to be used to obtain tangential velocity followed from the direct BEM , where indirect BEM is more robust for obtain the induced velocity. So there are two common routines adopted for solving first order and second order wave hydrodynamics. The first way is using direct BEM for solving potentials and using indirect BEM for obtaining velocity. The second way is using indirect BEM for solving both potential and velocity. For smoothed boundary without corners like a sphere boundary both of the two ways may give satisfied numerical solutions, although the first way converge better than the second way for obtaining the potential. For no-smoothed boundary like a vertica finite draft circular cylinder or box like floating bodies, the velocity is not easy to converge near the corners by the two ways. Especially for solving the tangential velocity around the corner region, direct BEM with numerical differentiation has lower accuracy than that on other smoothed part region panels, while much bad tangential velocity in the corner region relatively to the smoothed region results from indirect BEM no matter how fine panel discretization is used. These characteristics of lower order BEM or constant panel method leads to lower convergences of prediction for second order wave forces and second order potential obtained from the results of the corresponding tangential velocity calculations..To improve the accuracy of BEM for the solution of no-smoothed boundary problem, a new numerical scheme named TEBEM is presented by Duan. TEBEM is the abbreviation of Taylor Expansion Boundary Element Method. TEBEM is shown most accurate than the general IDBEM and DBEM for simple infinite fluid flow. More detailed investigation of this method on solving nonlinear wave hydrodynamics problem for practical marine structures will be done in this project.

在船舶与海洋工程水动力性能分析中，海浪的非线性波浪力效应及其利用正在成为科学界和工程界的研究热点，而提高浮体湿表面流体速度求解的精度成为非线性波浪力分析的关键技术。目前水动力学势流模拟常用的边界元方法，在求解带有拐角的非光滑边界处流体速度时，或者存在计算精度低的问题、或者存在稳定性和鲁棒性差的问题。这严重阻碍了非线性波浪力求解精度的提高。本项目抓住了这一关键问题，在理论创新方面，发现了边界函数及其切向导数可以联合构成第二类弗雷德霍姆边界积分方程组，独创了泰勒展开边界元（TEBEM）这一高精度新方法。本项目拟开展浮体在波浪中水动力问题求解的各类边值问题TEBEM求解方法及其精度验证研究。从而推进解决复杂外形浮体的非线性波浪力准确模拟问题。

本项目研究背景：由于海洋浮式结构物的工作环境及自身特点，波浪对结构物的二阶波浪力是不可忽视的。传统常值边界元方法计算非光滑边界处的流体质点切向诱导速度误差极大，导致浮体水动力分析存在较大误差，甚至导致时域模拟结果发散。此外，传统高阶元方法在处理角点奇异性和计算角点高阶导数存在较大困难。本项目创造性提出了泰勒展开边界元法，并利用该方法对浮体二阶波浪载荷问题进行研究。..针对本项目研究背景及目标，本项目主要研究内容包括：（1）泰勒展开边界元方法理论研究；（2）利用本项目提出的泰勒展开边界元方法预报零航速海洋浮体平均漂移载荷，计算二阶辐射及绕射速度势；（3）利用该方法预报有航速船舶运动响应；（4）进行三大主力船型（油船，散货船，集装箱船）波浪增阻预报研究。..由上述研究内容，本项目取得了一些重要研究成果，可分为理论成果和应用成果。理论成果主要是原创性提出了一阶、二阶泰勒展开边界元方法的基础理论和公式，并验证其精度。（参见报告正文进展1）该方法核心思想是基于格林第三公式构建边界积分方程。在面元中点对偶极强度作泰勒展开并保留一阶或二阶导数项，对源强作泰勒展开只保留一阶导数项。并引入场点的切向一阶、二阶及混合偏导数来封闭方程组，从而构成了关于偶极强度、偶强的一阶、二阶和混合偏导数，源强的一阶偏导数为未知数，源强为已知变量的线性代数方程组，形成泰勒展开边界元方法。..应用成果包括（1）利用一阶泰勒展开边界元法计算任意海工浮体的平均漂移载荷，数值结果表明近场和远场公式预报结果吻合度较好，而现有商业软件只有中场和远场公式吻合较好，近场公式发散。（参见报告正文进展2）（2）利用二阶泰勒展开边界元方法计算海工浮体二阶绕射和辐射速度势以及二阶倍频载荷。（参见报告正文进展3）（3）利用脉冲响应函数方法预报有航速船舶六自由度运动。（参见报告正文进展7）（4）利用泰勒展开边界元法结合阻尼区法和匹配边界元法分别预报船舶波浪增阻（参见报告正文进展4，5和6）。