局部化無網格徑向基底函數數值法於淺水方程式之應用

The Meshless Localized Radial Basis Function Collocation Method for Shallow Water Equations

陳姿帆(Tzu-Fan Chen)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/碩士(2011年)

https://doi.org/10.6342/NTU.2011.10562

摘要

本研究將介紹如何利用局部化徑向基底函數數值方法(LRBF-DQ method)求解淺水方程式之應用，並探討其數值特性。此方法屬於一種具備高精確度優勢之無網格數值方法。本數值方法採用微分積分數值方法之架構，結合以徑向基底函數做為其計算權重測試方程式之技術，以達成無網格前提下逼近函數微分值之目的，是一個不需要建立網格而且在計算上非常有效率的數值方法，另外尚有改善計算矩陣非常條件不良以及不規則域的計算。本研究亦將介紹以此數值方法對淺水方程式進行離散，並透過求解數個案例進行分析探討。案例在此分成兩個部分，第一部分為一維淺水的驗證，包含不規則底床、潰壩及工程應用，透過解析解及傳統數值方法證實本數值方法的準確性，第二部分則為二維線性及非線性淺水的應用，本數值模式成功的應用於二維不規則計算域中的模擬。此篇論文所提出的數值方法都有能力去得到精確且穩定的數值結果。

關鍵字

淺水方程式 ； 區域化徑向基底函數數值方法 ； 微分積分數值方法 ； 徑向基底函數 ； 無網格法

並列摘要

In this thesis, the local radial basis function differential quadrature (LRBF-DQ) method is applied to solve the shallow water equations. Meshless methods have been suggested to solve hydraulic problems. The LRBF-DQ method is one of the new developed meshless methods. This localized approach is developed from the differential quadrature (DQ) method by employing the radial basis function (RBF) for the test function. Several hydraulic engineering cases are demonstrated for numerical analyses. First part is one dimensional shallow water problems. A one-dimensional dam break problem is adopted to verify the accuracy of this procedure. There is consistency between the numerical results and the analytical solutions. And the other case is the simulation of the inflow of the Yuanshantze Flood Diversion. The solutions by LRBF-DQ method are well verified with the HEC-RAS model. The second part is two-dimensional linear and non-linear shallow water problems. According to the numerical results it indicates that the LRBF-DQ method is stable, accurate and robust for solving the one-dimensional and two-dimensional shallow water equation problems.

並列關鍵字

shallow water equations (SWEs) ； multiquadric (MQ) ； radial basis functions (RBFs) ； local radial basis function differential quadrature (LRBF-DQ) ； meshless method

參考文獻

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