以同幾何分析解鏡片上的橢圓方程

Abstract

等幾何分析(IGA)是計算輔助設計和有限元分析的整合。 IGA 可以縮短 設計和分析之間的差距，它還有望成為未來工業製成的重要工具。 文章分成三個主要部分，第一部分使用不同次方的B樣條對函數進行逼近誤 差分析。 第二部分要對一個鏡片上進行逼近分析，首先我討論二維空間的圓，我將圓 分成五個貼片以避免奇異點的發生。其次，利用相同的方法將圓柱分成五個貼 片的網格在三維空間上。另外，我希望使用 NURB 來表現鏡片幾何模型，所以 我將鏡片在一般座標系切成我所需的五個貼片。依據每一個貼片的形狀找到對 應得弧度，再利用 B-spline 曲線公式解線性方程組求得相對應的控制點。 第三部分採用 Galerkin 方法和 Galerkin 配置法求解高階 B-spline 透鏡幾何 中的橢圓問題，包括泊松方程和線彈性方程。 我對高階方案進行準確性研 究。 最後，我以圓，圓柱和透鏡的例子來進行橢圓方程數值解的誤差分析， 並且獲得所需的精度。

Isogeometric analysis (IGA) is an integration of computer-aided design(CAD) for representing geometric objects and finite element analysis(FEA) for analyzing mechanical properties of the underlying objects. IGA is aiming at bridging the gap between design and analysis. It is expected to be an important tool for future industrial processing. This article is composed of three parts. The first part reviews the approximation error analysis for functions on real line by B-splines and NURBS with different degrees. In the second part, we construct a hexahedral mesh on a lens by tensor product of standard 1-D B-spline. The representation of surface approximate by B-spline. The lens is partitioned into 5 patch with each homeomorphic to a cubic. The B-spline are materially connected across the patch boundary. In the third part, the Galerkin method and Galerkin-collocation method are implemented for solving Poisson equation and linear elasticity equation in disk, cylinder and lens geometry with high order basis. We perform accuracy study for high order schemes and achieve desired accuracy for high order scheme with even coarse grid.