A New Computational Approach for Solving the Nonlinear Boundary Value Problem in an Arbitrary Plane Domain

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Project title

Code/計畫編號

NSC100-2221-E019-008

Translated Name/計畫中文名

求解任意領域非線性邊界值問題的新計算方法

Project Coordinator/計畫主持人

Jiang-Ren Chang

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Systems Engineering and Naval Architecture

Website

https://www.grb.gov.tw/search/planDetail?id=2339584

Year

2011

Start date/計畫起

01-08-2011

Expected Completion/計畫迄

31-07-2012

Bugetid/研究經費

372千元

ResearchField/研究領域

機械工程

本研究計畫擬採用一多流式、指數收斂型算則的新有限差分法來求解非線性、橢圓型、任意平面領域的邊界值問題。一如所熟知的，以傳統的有限插分法來求解非線性且幾何形狀複雜的問題是很困難的。為克服這些問題，本研究將提出「虛擬長方領域內部與邊界殘留」的觀念與藉引入一虛擬時間座標而引出的形狀函數來建立此一理論架構。因引入有虛擬時間座標，吾人將得以避開處理複雜的邊界條件，在僅需使用型狀涵數且不須計算代數方程之反矩陣問題下，多流式、指數收斂型算則之有限差分法即可有效建立。此外，為增加該方法數值之穩定性，本研究將應用保群算法來處理虛擬時間積分之問題。由於保群算法與此「多樣且指數收斂型算則的新有限差分法」都具有光錐結構與多樣性之特性，吾人得以引入權重因子使得光錐結構保有多樣路徑，更使得此「多流式指數收斂型算則的新有限差分法」在其每一虛時間步驟都呈現出李代數、光錐與群之性質。最後，本研究將設計一些數值算例，來驗證本計畫所提出方法的準確性與收斂性。In this research, a novel finite difference method (FDM) in conjunction with the manifold-based exponentially convergent algorithm (MBECA) will be adopted to solve a nonlinear elliptic boundary value problem defined in an arbitrary plane domain. As well known, it is quite difficult to solve nonlinear and geometric complexity problems by conventional FDM. To overcome these problems, the concepts of internal residual and boundary residual in a fictitious rectangular domain and shape function that we can easily detour by adding a fictitious time coordinate will be introduced in this research work. Here, by introducing a fictitious time coordinate, we do not need to directly treat complex boundary conditions only by using the shape function, and without solving an inverse matrix of algebraic equation by the MBECA. Besides, in order to increase the numerical stability of the MBECA, a group-preserving scheme (GPS) will be introduced to address fictitious time integration. Given the cone structure of the GPS and MBECA and their manifold properties, we can preserve the manifold path on the cone structure by a weighting factor such that the MBECA must also exhibit a cone construction, Lie algebra, and group properties at each fictitious time step. Finally, the accuracy and convergence behavior of this present method will be demonstrated in several numerical examples.

Keyword(s)

虛擬時間積分法
邊界值問題
有限差分法
保群算則

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A New Computational Approach for Solving the Nonlinear Boundary Value Problem in an Arbitrary Plane Domain

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