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標題: | 四元數三維拉普拉斯方程之邊界元素法 Boundary element method for quaternion valued Laplace equation in three dimensions |
作者: | Yi-Chuan Kao 高怡絹 |
指導教授: | 洪宏基 |
關鍵字: | 邊界元素法,邊界積分方程,奇異性,柯西主值,三維拉普拉斯方程,四元數, boundary element method,boundary integral equations,singularity,Cauchy principal value,three-dimensional Laplace equation,quaternion, |
出版年 : | 2016 |
學位: | 碩士 |
摘要: | 本論文旨在發展四元數邊界元素法,以求解三維空間的實數場、三維向量場、四元數場的拉普拉斯方程式問題。不論在域內點、邊界點以及域外點,我們都推導出它們的四元數邊界積分方程。不僅止在光滑邊界,在角點或稜邊也得到四元數奇異邊界積分方程。對此奇異邊界積分方程,奇異積分存在於柯西主值。它可以透過一個簡單的調和函數解析地算出,其餘則已無奇異性,可交由數值法處理,並適用於任意幾何形狀。四元數邊界元素法的特色是,可以整合單位法向量及普通的表面元素,成為具有方向性的四元數表面元素。而當域內點非常靠近邊界時,也會發生近奇異性。我們也同樣透過調和函數去減緩這個近奇異的邊界層現象。最後,我們以靜磁、功能梯度材料熱傳以及格林函數的問題驗證四元數邊界元數法的適用性。 In this thesis, a quaternion boundary element method (BEM) for solving three-dimensional problems governed by scalar, vector and quaternion Laplace equations is developed. To derive quaternion valued boundary integral equations (BIEs) for both the domain point and the out-of-domain point, the quaternion valued Stokes’ theorem is utilized. For smooth boundary points and points at corners and edges of nonsmooth boundary, the singular quaternion valued BIEs are all obtained; the integrals are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). Here, we develop an analytical scheme to evaluate the CPV by introducing a simple quaternion valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “boundary layer” phenomena appear and are remedied by a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. Finally, several numerical examples including the problems of magnetostatics, heat conduction in functionally graded materials and Green’s function, are considered to demonstrate the validity of the present approach. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/3710 |
DOI: | 10.6342/NTU201602179 |
全文授權: | 同意授權(全球公開) |
顯示於系所單位: | 土木工程學系 |
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