光學設計中Monge-Ampere 方程的數值計算

Abstract

幾何光學設計的問題是一個古老的問題卻也是一個歷久彌新的問題。 在當代許多應用領域包括汽車車燈，微型投影機，液晶螢幕背光板還有太陽能集光板等的設計都會面臨到同樣的非成相幾何光學的設計問題。 對非成像幾何光學中自由曲面的設計問題，基於能量守恆的原理可以推導出著名的非線性偏微分Monge-Ampere 方程。 Monge-Ampere 方程也同時出現在 differential geometry, mass transportation and geostrophic fluid 等研究領域 在本計畫中我們將運用 Feng and Neilan 所提出的 vanished moment method 與 BCIZ 的有限元素法來離散線性化的Monge-Ampere 方程。並利用條件優化的 conjugate gradient 方法來計算線性系統的解。其中為了加速conjugate gradient的收斂性與計算效能，我們將以雙調合算子來作為優化的preconditioner。 並用多重網格法來計算迭代過程中雙調合算子的解。 我們將在許多的標準測試問題中比較Feng and Neilan (2009), Froese and Oberman (2009), and Dean and Glowinski (2006). 等所得到的數值解。也將運用我們在這裡所提出的方法來重建一些應用方面有興趣的簡單的光學自由曲面。

The geometric optics has been studied for 100 years. The optical design problems are concerned in many applications from automobile head-light design, solar energy reflector, LED illumination, micro projector and LCD backlight, etc. The optical design problems in this area are generally characterized into image optic design and non-image optic design. The image optical design mainly concerns the image quality generated from the designed optical system and the non-image optic design mainly considers transportation of energy flow in the designed optical system. The design principle of an optical system in practice can further be categorized into (i) spherical geometry (ii) aspherical geometry and (iii) free-form surface. In this proposal, we are particularly interested in the free-from surface (FFS) design problems in non-image optics. From the energy conservation principle, the model of freeform surface problems leads to a well known nonlinear partial differential equation: the Monge-Ampere equation. The Monge-Ampere equation also arises in various research fields including differential geometry, mass transportation, geostrophic fluid. In this paper, we introduce some of known theoretical results and numerical methods of the Monge-Ampere equation. Particularly, the vanished-moment method proposed by Feng and Neilan in 2007 has been successfully applied in solving the Monge-Ampere equation arising from geostrophic fluids. In this paper, we employee the vanished-moment method to solve the Monge-Ampere equation arising from the FFS design problem. To ensure the FFS from numerical computation is continuous and to be able to compute the curvature of the FFS, we discretize the linearized Monge-Ampere equation by BCIZ finite element method. The discrete linear system is then solved by the conjugate gradient method (CG) in which multigrid approximation of the discrete biharmonic operator is used as the preconditioner of the CG method. Accuracy and robustness of our approach are demonstrated on several benchmark examples. We compare our numerical results with the results obtained by Feng and Neilan (2009), Froese and Oberman (2009), and Dean and Glowinski (2006).