使用WENO方法來研究在尤拉座標下彈性流體之數值模擬

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.