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Title: | 使用WENO方法來研究在尤拉座標下彈性流體之數值模擬 Numerical Simulations of Elastic Flows in Eulerian Coordinate Using WENO Scheme |
Authors: | Yu-Ting Hong 洪毓廷 |
Advisor: | 陳宜良(I-Liang Chern) |
Keyword: | 權重無震盪法,彈性,相容性條件, WENO,elasticity,compatibility condition, |
Publication Year : | 2021 |
Degree: | 碩士 |
Abstract: | 在本論文中,我們使用五階WENO 方法搭配二階Runge-Kutta 方法,來數值求解尤拉座標中的等向性超彈性(isotropic hyper-elastic)模型。我們在尤拉座標系中將超彈性材料方程式公式化為雙曲守恆系統;然而除了方程式以外,仍有額外的兩個條件需要被滿足,這和氣體動力學的情況並不相同,一個條件是反變形梯度(inverse deformation gradient)的相容性條件,另一個則是密度的一致性條件。當使用WENO 方法求解該系統時,相容性條件會作為擴散項加入反變形梯度的演化方程式,而一致性條件亦作為鬆弛項加入該條方程式,我們透過數值模擬來驗證方法的可行性。 本研究包含了四項數值模擬實驗,第一項實驗顯示:我們的方法對於平滑解達到了二階收斂性;後三項實驗則是對於黎曼問題的測試,這些結果展現了WENO方法對於求解不連續性解的準確性。 In this thesis, we apply a fifth-order weighted essentially non-oscillatory(WENO-5) scheme together with a second-order Runge-Kutta method to solve isotropic hyper-elastic models in Eulerian coordinate numerically. We formulate equations of hyper-elastic materials in Eulerian frame of reference as a system of hyperbolic conservation laws. However, additional constraints should be satisfied, which are different from the gas dynamics. One is a compatibility condition for inverse deformation gradient. The other is a consistency condition for density. In applying WENO method to solve this system, the compatibility condition is added as a pseudo-diffusion term in the evolution equation of the inverse deformation gradient, whereas the consistency condition is also added in this evolution equation as a relaxation term. The feasibility is shown by numerical tests. Four numerical simulations are included in this thesis. The first case demonstrates that our method attains second-order convergence for smooth solutions. And the last three cases, which are the tests for Riemann problems, show the accuracy of our method for solving solutions with discontinuities. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/64738 |
DOI: | 10.6342/NTU202100401 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 應用數學科學研究所 |
Files in This Item:
File | Size | Format | |
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U0001-0202202116422000.pdf Restricted Access | 4.45 MB | Adobe PDF |
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