Eikonal方程式的快速數值解

A Fast Numerical Solution for Eikonal Equation

本文將提出一個描述相界面演遞的Eikonal方程式快速數值解法。在此方法中，配合模擬範圍內建置合宜的計算格點，並以界面到達計算點的時間作為計算變數。首先以界面位置預測界面到達計算點的時間，再根據此計算變數建構計算時間改變後所對應的界面位置，並以界面位置再修正界面到達計算點的時間，如此循環求解，直到界面抵達所有計算點。本文同時依據與界面的相對位置，將計算空間適當分類，使得在實際計算過程中，僅須針對界面附近的計算點，估算界面到達時間，從而得到相當快速、便捷的計算程序。經由若干案例計算，並與現有的計算方法比較驗證，說明此方法不僅可以非常快速地得到Eikonal方程式的數值解法，其模擬結果亦較現有的依點前進法或快速前進法精確。 In this paper, an efficient numerical method has been proposed to simulate internal ballistic of a solid motor with multiple propellants. The essential feature in this method can be depicted by transforming the original initial value problem for the solid propellant burning procedure into an associated boundary value problem, which can be described by the Eikonal equation. A fixed mesh covering simulation domain is established. The arrival time of the initial burning surface has been selected as the solution variable. At first, the arrival time is estimated by burn distance from the burning surface which is then reconstructed based on the arrival time as the simulation evolves. In this simple manner, all arrival time in the computational domain can be determined to serve as the essential element in the ballistic simulation. Meanwhile, special regions can be identified by the burning surface to enhance computational efficiency. By examining theoretical results from the physical model and computational results of test problems, the proposed numerical methodology in this research can be an effective tool for solid propellant motor design.