Monge-Amp□re方程的數值方法與其在非成像光學上的應用

Abstract

本論文介紹光學設計中自由型曲面的設計方法，我們探討了自由型曲面設計藉由偏微分方程來求得，其中的偏微分方程由Schruben推導而來，其偏微分方程的形式為Monge-Amp□re方程式，我們考慮簡化型Monge-Amp□re方程式，藉由馮教授所使用的方法，加上一個四次微分的消散項，可以使得原來的方程式是一個完全非線性的方程式轉變成類線性方程式改變了方程式的特性，使得在偏微分方程上有較好的特性，我們用有限元素法來做為我們的計算方法，挑選BCIZ有限元可以有效的處理四次微分項且並且可以簡單的求得曲面的曲率的計算，也可以滿足光學系統的所需要的一些特性，只需要解一個線性方程式和使用牛頓疊代法求做求解用的工具，以獲得較高的計算效率。

We consider the freeform surface design problem. Fully nonlinear partial differential equations as derived by the Schruben for model. The partial differential equation is the form of well knowMonge-Amp□re equation. We following Prof. Feng’s idea to solve Monge-Amp□re equation by adding a vanish moment biharmonic term. As a result the original fully nonlinear equation is change into quasi-linear equation. We using finite element method to solve this equation. Its well knows that the traditional BCIZ element can effectively deal with biharmonic item and compute the curvature of the solution. Which is usually required in a optical systems. We descritize the nonlinear equation by the Newton’s method. The numerical studies in this thesis show that our approach is efficient and accurate.