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毕业论文-非线性水波的边界元计算方法，共52页，26572字
数值计算方法已成为研究波浪在近岸的传播变形及其与障碍物之间的相互作用的主要方法之一。与实验方法和现场观测相比，它有着成本低、可以避免比尺效应、工况选择灵活等极为突出的优点。本文在势波理论的基础上，建立了一个能够精确计算立面二维非线性水波的数值模型。所采用的数值方法为混合欧拉-拉格朗日方法。欧拉步采用边界积分方法，并利用以累积弦长为参数的B样条插值方法结合线性最小二乘方法处理边界积分；拉格朗日步采用4级4阶龙格库塔方法。
本文采用流函数理论确定入射波，通过计算强非线性波浪在平底地形上的传播过程，对模型进行了有效的验证。同时对计算过程中出现的锯齿不稳定现象进行分析，指出了拉格朗日插值方法在欧拉步插值中的不足，加深了对强非线性波浪计算过程中可能出现的局部不稳定现象的认识，改进计算方法后避免了计算误差引起的假破碎现象。此外，本文还对波浪在斜坡地形上的传播变形过程以及波浪和水面障碍物相互作用情况进行了计算，并对波浪经过堤坝时出现的变形、分裂、破碎和波浪遇到障碍物产生反射和透射现象进行了初步的分析。
Numerical computation is now the most popular method in the study of nonlinear wave propagation and transformation. How to deal with the free surface has been a difficulty if the wave is highly nonlinear. The present mathematical model is based on the potential wave theory and the mixed Euler-Lagrangian method (MEL) is employed for discretization. The Eulerian step is solved with the boundary integral equation method while the free surface is approximated by B-spline with the accumulated chord as the parameter and the linear least square approach is applied to the variables. The Runge-Kutta Method is used in the Lagrangian step. The incident wave is given by the stream function wave theory. The numerical model is proved to be accurate, there is no Saw-tooth instability or artificial breaking. It is shown that the effects of the bottom slope and an obstacle on the wave motion can be very significant.
目录
中文摘要................................................................................................................I
ABSTRACT ...................................................................................................... III
目录.............................................................................................................. V
第一章 引言 ........................................................................................................ 1
1.1 选题依据 ........................................................................................................................ 1
1.2 相关研究现状 ................................................................................................................ 1
1.3 本文主要工作 ................................................................................................................ 2
第二章 数学物理模型 ........................................................................................ 3
2.1 问题描述 ........................................................................................................................ 3
2.2 控制方程及边界条件 .................................................................................................... 3
2.3 造波理论 ........................................................................................................................ 4
第三章 数值方法 ................................................................................................ 7
3.1 流函数波理论 ................................................................................................................ 7
3.2 传统边界积分 ................................................................................................................ 9
3.2.1 边界积分方程 .......................................................................................................... 9
3.2.2 单元离散 ................................................................................................................ 10
3.2.3 奇异积分 ................................................................................................................ 11
3.3 B 样条边界积分 ........................................................................................................... 13
3.3.1 B 样条基函数.......................................................................................................... 14
3.3.2 插值参数和算法 .................................................................................................... 16
3.3.3 B 样条的合理计算单位 .......................................................................................... 18
3.3.4 奇异积分 ................................................................................................................ 18
3.4 自由表面绝对速度 ...................................................................................................... 21
3.5 线性最小二乘.............................................................................................................. 22
3.6 龙格－库塔方法.......................................................................................................... 23
第四章 模型验证与优化 ...................................................................................25
4.1 二次单元插值的边界元计算结果和锯齿不稳定现象 .............................................. 25
4.2 锯齿不稳定现象的成因分析...................................................................................... 28
4.3 累积弦长的 B 样条边界元计算 ................................................................................. 31
4.4 强非线性波的局部破碎现象的数值处理方法 .......................................................... 35
4.5 小结 ............................................................................................................................. 39
第五章 地形和障碍物对波浪作用初步研究 ...................................................41
5.1 波浪爬坡...................................................................................................................... 41
5.2 障碍物对波浪的作用.................................................................................................. 44
第六章 结论 .......................................................................................................47
插图索引 ............................................................................................................... I
参考文献 ............................................................................................................III
致谢 ..............................................................................................................V
声明 .............................................................................错误！未定义书签。
附录 A 外文资料原文 ...................................................................................... IX
附录 B 外文资料的调研阅读报告或书面翻译 ..........................................XVII
在学期间参加课题的研究成果 .................................................................... XXV
论文主要内容及进度安排：

论文主要内容包括：
1．流函数造波；
2．自由水面采用 B 样条插值的边界元理论；
3．利用混合欧拉-拉格朗日方法建立高精度数值模型；
4．初步分析波浪爬坡引起的变形、分裂和破碎；
5．初步研究水面障碍物对波浪的作用。