以權重最小二乘局部近似法應用於不恆定淺水流水理數值模擬之研究

Abstract

本論文主要係以修正有限配點法(Modified finite point method, MFPM)內的權重最小二乘局部近似(Weighted moving-least-square local approximation, WMLA)法，搭配多項式做為基底函數，來模擬不恆定淺水流問題。而此法之計算架構屬於無網格數值方法中的配點法，具有理論簡單、程序直接、佈點自由、適用於複雜與變動邊界問題等優點，且在擬合空間中的局部區域離散資料時，除了能準確地推算區域內任一位置的函數值外，更可以容易地求得所對應的函數偏導數值。本方法甚至可以靈活應用於給定不同偏導數形式之邊界條件，極具實用性。在時間方面，考量顯性法計算快速與隱性法具高穩定性之優點，採用顯性的Leap-frog method搭配隱性的Crank-Nicolson method形成的Predictor-corrector method，如此便可免於一般單獨使用隱性法求解時之矩陣運算流程，特別是對於求解非線性偏微分方程問題時，矩陣系統需要依靠迭代才得以求解，而因為Predictor-corrector method仍屬於顯示法，不需要經過迭代即可求解，能大量地節省程式運算耗時，且亦能藉由其中包含的隱性法來改善運算之穩定性，大幅地提升整體計算效率。 本研究之控制方程為淺水波方程組(Shallow water equations, SWE)，屬於非線性又聯立的偏微分方程組，當使用WMLA法時，便能快速且準確地求解。另外，文中將應用此法所建立之不恆定淺水流水理數值模式，配合多種情況下適用的初始條件、邊界條件、乾溼地形與區域分割處理方法，進行包含理想與實際等多個案例全域自由液面及流場之模擬，並經由解析解、實驗量測數據或其他數值方法的模擬結果驗證後，皆獲得良好的成果，適合成為未來從事大型水利工程或相關研究之參考依據。

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.