Solving Shallow Water Waves With The Displacement Method And Time Domain FEM Integration In Three-gorge Ship-lift

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.