| Summary: | 碩士 === 國立臺灣大學 === 土木工程學研究所 === 104 === In this study, the weighted moving-least-square local approximation (WMLA) which is included in modified finite point method (MFPM) is used to simulate unsteady shallow-water flow problems with polynomial basis function. The structure of computational algorithms employed in this method is based on collocation of meshless numerical method that is simple in theory, direct in programing, flexible in setting computational nodes and easy to use in complex and variable boundary problems. When using the WMLA fits discrete data in local domain of space, it can not only calculate the function values of any position in the region accurately, but also obtains the corresponding partial derivative values easily. This method can even specify boundary conditions of the partial derivative form flexibly, so it is very practical. In time marching, adopting Predictor-corrector method which composes of explicit Leap-frog method and implicit Crank-Nicolson method simulates. Both the computational efficiency and numerical stability are maintained. Especially, when we solve nonlinear partial differential equations, matrix systems need to be solved by iterations. The Predictor-corrector method is still explicit method and therefore, does not require any time iterations. Not only does it save much computational time, but it also improves the stability.
The governing equations of this study are shallow water equations (SWE), which belong to nonlinear and combined partial differential equations. Using the WMLA can solve these equation efficiently. In addition, this thesis will use the numerical models of unsteady shallow-water hydrodynamics which is developed by WMLA to simulate many idealized and realistic cases. It can obtain free surface and flow field in global domain under different initial conditions, boundary conditions, dry or wet topography and regional segmentation approaches. The simulated results are verified with analytical solutions, experimental measurement data and other numerical method simulations. Very good agreement is observed. Solutions obtained in this study can be applied to large-scale hydraulic engineering problems or other related researches.
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