【作者】 陈晓辉；

【导师】 钟万勰；

【作者基本信息】 大连理工大学 ， 固体力学， 2007， 硕士

【摘要】 本文以三峡升船机的浅水波为工程背景，提出了浅水波的位移法理论，研究了保守体系的时程积分和非线性保辛算法，并将保辛算法应用于浅水波位移法理论问题的数值求解中。首先本文探讨了保守体系的时程积分，对区段内的Hamilton矩阵进行二次插值，基于混合能的变分原理，采用时间有限元应用辛矩阵正则变换的乘法摄动。针对非线性微分方程，运用坐标变换，提出了求解非线性微分方程的保辛积分算法。这为浅水波位移法理论的数值计算提供了强有力的工具。其次系统地提出了浅水波的位移法理论。浅水波的速度分布与水深无关，位移也与水深无关，相当于杆件的刚性横截面假定。本文采用物质坐标，用水平位移作为基本未知量，将浅水波分析按固体力学中杆件非线性大位移问题处理。于是，分析力学的变分原理都可运用了，这样可采用保守体系的时程积分和非线性保辛算法等有效手段进行数值求解。对于实际工程课题，当升船机船厢箱体晃动较大时，导致波幅较高，本文进一步考虑垂直方向速度对动能的贡献。当箱底泥沙沉积时，本文提出了缓变水域，水底凸起或者是凹陷的情况下的浅水波位移法理论模型。以上的浅水波位移法理论中的有限元离散是将时间和空间分别离散。本文进一步将有限元分析方面，不用对时间、空间分别离散而是组成混和的时空混和有限元网格应用于浅水波位移法理论。上述的理论是最基本的一维模型理论。本文进一步考虑平面二维浅水波问题，二维问题需要考虑速度势函数与雅可比行列式，势能部分的处理是关键，我们对势能部分进行泰勒展开，有限元离散，从而得到平面问题的刚度阵，同样应用保辛算法进行数值求解。数值结果说明了本文模型与保辛算法的正确性。本文的研究工作受到国家自然科学基金(#10421002，#10372019，#10272026，#10632030)支持和国家重点基础研究专项经费资助项目2005CB321704的支持。更多还原

【Abstract】 We solve shallow water waves with the displacement method and time domain FEM integration in conservative systems based on the model of Three Gorge Ship-lift in this dissertation.First, we do some research on time integration in conservative systems. Time integration of an initial value problem uses the finite difference scheme quite often. The characteristic of conservative system is symplectic conservation. However, the usual FDM schemes disregard the symplectic conservation for conservative systems. The FEM method is naturally symplectic conservative, even for perturbation method. First the method of coordinate transformation and the polynomial approximation of mixed energy density are applied in this paper, which need only fewer steps of integration. Numerical results demonstrate the effective of present method.Second, we provide the theory of Displace method. The horizontal velocity distribution of shallow water wave theory is independent on the vertical coordinate z, it implies that the horizontal displacement is independent on z too, which corresponds to the rigid transverse cross section assumption of the rod theory in structural mechanics. Using the material coordinate as in analytical mechanics, the horizontal displacement is selected as the fundamental function to solve, therefore the shallow water wave analysis becomes the large displacement non-linear problem in structural mechanics. Analytical mechanics and the corresponding variational principles can be applied now. The powerful methodology, such as canonical transformation and symplectic conservative approximate FEM discretization and so on, is quite beneficial to numerical solution. The water waves in the pool of three-gorge ship-lift are used as numerical examples, and the symplectic conservative time integration gives satisfactory result.We should consider the contribution of vertical velocity to the kinetic energy in the condition that the pool of three gorge ship lift shakes too much. We should also consider the model which has a slow change base when some sand and soil deposited on the base of the pool.Furthermore, the mathematical theory is the same for analytical dynamic mechanics and analytical structural mechanics. This paper presents that it is not necessary to discrete time and space domain separately for the finite element analysis and that the time and space domain can be discrete mixed to form harmony finite element mesh. We also use Time-Space Domain in Shallow water waves.In addition, we also consider two dimension shallow water waves with displacement method.The numerical results provide strong support to the theory in this dissertation.Acknowledgements: The financial supports by Natural Science Foundation (Contract 10421002, 10372019, 10272026) and Chinese National Research Fund (Contract 10632030, 2005CB321704) are gratefully acknowledged.更多还原