新基本解法於薄膜自由振動問題之研究

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基本資料

Project title

新基本解法於薄膜自由振動問題之研究

Code/計畫編號

MOST108-2221-E019-003

Translated Name/計畫中文名

新基本解法於薄膜自由振動問題之研究

Project Coordinator/計畫主持人

Ying-Te Lee

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=13116851

Year

2019

Start date/計畫起

01-08-2019

Expected Completion/計畫迄

31-07-2020

Bugetid/研究經費

761千元

ResearchField/研究領域

土木水利工程

在此結案報告中，我們發展一套新對角線項捕捉技術的基本解法來求解薄膜自由振動問題。不同於傳統基本解法之處在於本法可將源點分布於真實邊界上。本計畫書內容乃延伸先前科技部計畫至求解亥姆霍茲問題(計畫編號: MOST 106-2221-E-019-019，名稱: 一套新的基本解法對角線項捕捉技術於勢能問題之應用)。在傳統基本解法中，為了避免當場點與源點碰在一起時所產生的奇異性，源點總是被放置於待解問題的領域之外。針對亥姆霍茲問題，若將場點與源點佈置於同一個位置時，由於基本解( )的緣故，影響係數矩陣的對角線項將變成無窮大。然而就數值計算與物理觀點而言，這些對角線項的值應為有限值而非無窮大。在此，我們將引入漢克爾函數在微小變量下的極限公式結合先前計畫成功擷取拉普拉斯問題對角線項的方法來計算亥姆霍茲問題的影響係數矩陣對角線項。這是一套自然且容易的方法並且一些基本解法重要的特徵仍被保留，如:無網格與免數值積分等。此外，本計劃書所提的技巧，不僅可用於單層勢能法，亦可使用於雙層勢能法中。最後，我們將利用幾個數值算例來示範所發展方法的可行性、正確性與精確性。 In this report, the method of fundamental solutions (MFS) of new diagonal terms capturing scheme is proposed to solve the free vibration of membrane. It is different from the conventional method of fundamental solution that the source points can be put on the real boundary in the proposed method. This work is extended the previous MOST project to solve the Helmholtz problem. In the conventional MFS, the source points are always placed outside of the physical domain to avoid the singularity when source point and collocation point coincide together. For the Helmholtz problem, when the source point and collocation points are located at the same position, the values of diagonal terms in the influence matrices are infinity due to the fundamental solution ( ). However, in the numerical and physical point of view, these values have to be finite values. Here, we introduce the limiting formula of Hankel function for small argument connected with the successful approach of capturing diagonal terms for Laplace problems in the previous project to determine the diagonal term values of the influence matrices for the Helmholtz problem. This is a natural and easy way, and the prominent features of the MFS such as meshless and free of numerical integration are still maintained. Besides, the proposed scheme is used not only for single-layer approach but also for the double-layer approach. Finally, several numerical examples are taken to illustrate the feasibility, validity and accuracy of the proposed method.

Keyword(s)

基本解法
微小變量極限公式
亥姆霍茲方程式
無網格法
薄膜振動
method of fundamental solutions
limiting formula for small argument
Helmholtz equation
meshless method
membrane vibration

DSpace CRIS

新基本解法於薄膜自由振動問題之研究

基本資料

Description