Using Fictitious Time Integration Method and Meshless Numerical Method for Solving Nonlinear Problems

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Project title

Code/計畫編號

NSC99-2221-E022-007

Translated Name/計畫中文名

虛擬時間積分法結合無網格數值方法於非線性場之研究

Project Coordinator/計畫主持人

Chia-Cheng Tsai

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Marine Environmental Engineering,NKUST

Website

https://www.grb.gov.tw/search/planDetail?id=2112776

Year

2010

Start date/計畫起

01-08-2010

Expected Completion/計畫迄

31-07-2011

Bugetid/研究經費

594千元

ResearchField/研究領域

土木水利工程

本計畫擬結合虛擬時間積分法與無網格數值方法，用以解析某些非線性場的問題。台大土木系劉進賢教授與S.N. Atluri，於2008年共同發展出虛擬時間積分法，它是一種很強大的數值方法，特別適用於非線性問題與噪音問題。本研究將結合虛擬時間積分法與基本解法與柴比雪夫多項式，用以解析多種非線性場：這些問題包含非線性波以松方程式、非線性彈性力學問題、非線性輻射問題等。基本解法與柴比雪夫多項式的結合，產生一種很準的(指數收斂)數值方法，只要基本解和特解知道，這個方法能用來解析非齊次的偏微分方程式，在吾人過去數年的國科會計畫中，已經法展很完備了。本研究計畫，主要延伸基本解法與柴比雪夫多項式，用以解非線性問題。在使用虛擬時間積分法的時候，非線性偏微分方程式會先轉換為一系列的非齊次線性偏微分方程式，之後用過去發展的基本解法與柴比雪夫多項式，就能解析接下來的問題。除了理論的推導，我們也將發展模式，參加國際研討會，訓練學生發表國際期刊，相信有助於國家基礎研究的發展。 In this project, we are going to combine the fictitious time integration method with a meshless numerical method to solve certain nonlinear problems. The fictitious time integration method (FTIM) previously developed by C.S. Liu and S.N. Atluri in 2008 is combined with the method of fundamental solutions and the Chebyshev polynomials to solve several nonlinear PDEs. There nonlinear problems include the nonlinear Poisson's equation, nonlinear elasticity, nonlinear radiation problems, etc. The method of fundamental solutions with Chebyshev polynomials (MFS-CP) is an exponentially-convergent meshless numerical method which is able to solving nonhomogeneous partial differential equations if the fundamental solution and the analytical particular solutions of the considered operator are known. In the previous projects, the MFS-CP is completely developed by the author. In this project, we are going to extend the MFS-CP to solve nonlinear PDEs by using the FTIM. In the solution procedure, the FTIM is introduced to convert a nonlinear PDE into a sequence of linear nonhomogeneous modified PDEs which are then formally solved by the MFS-CP. Several numerical experiments were carried out to validate the proposed methods. We will participant international conferences and publish international papers.

Keyword(s)

虛擬時間積分法
非線性
篇微分方程式
網格數值方法
fictitious time integration method
nonlinearity
partial differential equation
meshless numerical method

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Using Fictitious Time Integration Method and Meshless Numerical Method for Solving Nonlinear Problems

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