最 小 平 方 有 限 元 素 法 在二維淺水波方程式之數值模擬

Simulation of Two-Dimension Shallow Water Equation By The Least-Squares Finite Element Method

賴慶賢(Qing-Xian Lai)

指導教授 ： 鄧志浩

中原大學/工學院/土木工程研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200393

摘要

摘要
本文主要為數值模擬不連續自由水面，數值模擬的方法，利用最小平方有限元素法(LSFEM) 解二維非線性淺波方程式。模擬不連續自由水面變化，主要分為兩種不同的形式：一為超臨界流在不同斷面的變化，所會造成的結果；一為潰壩所造成的上下游水深變化。
在超臨界流部分，模擬單邊歪斜收縮、單邊歪斜擴大、一邊歪斜收縮一邊歪斜擴大、曲線收縮擴大和曲線收縮斷面等五種不同的斷面。前三種和理論值比較非常穩合。收縮擴大斷面則和Yang和 Hsu(1993)相差不多。至於曲線收縮斷面和實驗值比較，較Causon等人(1999)以有限體積方法和實驗值比較結果更接近。在潰壩數值模擬也很有效首先模擬一維潰壩和理論值比較；在部分潰壩比較上下游水深比為10/9、2、和1的水面及流場變化圖。
本研究發現計算結果和理論值及利用其他研究方法相吻合，利用LSFEM方法求解淺水波方程式，可以有效地呈現出在不連續水面、在不同斷面的變化情形。
關鍵字:最小平方有限元素法、淺水波方程式、超臨界流、部分潰壩

關鍵字

淺水波方程式 ； 部分潰壩 ； 最小平方有限元素法 ； 超臨界流

並列摘要

ABSTRACT
This study is focused on the numerical simulation of discontinuous free surface problems such as the supercritical shock waves flowing on various geometries and the flow field of a broken dam. The Least-square finite element method (LSFEM) is adopted for the simulation of two-dimensional nonlinear shallow water equation. The simulation of supercritical shock waves in the following five different geometries: one-side oblique contraction channel, one-side oblique expansion channel, a oblique contraction-expansion channel, curved contraction expansion and curved contraction channel are evaluated by the theoretical solution, and the simulated results of previous published data. The dam break problems are also simulated with the reservoir water depth ratio of 10/5 in one dimension dam break and 10/9, 2, and 1 in partial dam break.
The simulated results are shown to be in good agreement with analytical solution and numerical results of other methods. This paper demonstrates that the LSFEM can effectively simulate the supercritical shock waves and dam break flow with discontinuous free surface.

Key words：least-square finite element method、shallow water equation
、supercritical flow、partial dam break

並列關鍵字

least-square finite element method ； shallow wate

參考文獻

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