運用HCT有限元素法重建三維光學反射面

Abstract

Monge-Ampere 方程式為一個完全非線性的偏微分方程式 (PDE) ，我們將利用這個 PDE 來塑造光學設計的模型。首先，我們會討論一些被 Caraffelli、Oliker 和 Wang 等人[1,2,3,4]所證明的理論與數值方法，而且也會介紹一些數值方法用於解 Monge-Ampere 方程式。我們根據 Feng 和 Neilan 提出的方法[5]，利用 vanish moment method 解出建構在 Monge-Ampere 方程式的自由型態曲面 (freeform surface) 的光學設計問題。在幾何光學的領域，能量守恆對於光學設計來說是一個重要的限制條件。在 PDE 模型中，它必須遵守這個定律而且與數值的計算相互關聯。為了確保能量守恆定律和平滑的照度分布，在重建 2D 自由型態曲面時，光角與目標面的局部能量分割和曲面法向量的連續性是必須的。最後，我們透過一些典型的例子來呈現這個數值方法之精確性與穩定性。

Monge-Ampere equation is a fully nonlinear partial differential equation (PDE) and we use this PDE to model the optical design problem. First, we discuss some of the well-known theorems proved by Caraffelli, Oliker, and Wang et al. [1,2,3,4] and we introduce several numerical methods to solve the Monge-Ampere equation. Particularly, we employee the vanish moment method proposed by Feng and Neilan [5] to solve the Monge-Ampere equation arising from the freeform surface design problem. In geometric optics, energy conservation is a crucial constrain for optical design, and should be maintained in PDE modeling and the associated numerical computation. In order to ensure the energy conservation and smooth illuminance design. Local energy partition on source and target plane, and the continuity of normal vector of reconstructed 2D freeform surface are essential. Finally, accuracy and stability of our approach are demonstrated by several benchmark examples.