A Method of Fundamental Solutions with Source on the Real Boundary for Solving Scattering Problems of Water Wave

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Project title

Code/計畫編號

MOST109-2221-E019-004

Translated Name/計畫中文名

使用源點佈於真實邊界的基本解法求解水波散射問題

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=13537361

Year

2020

Start date/計畫起

01-08-2020

Expected Completion/計畫迄

31-07-2021

Bugetid/研究經費

757千元

ResearchField/研究領域

土木水利工程

本計劃擬使用源點佈於真實邊界的基本解法來求解水波散射問題。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與正在執行的科技部計畫至求解外域亥姆霍茲方程問題(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與計畫編號: MOST 108-2221-E-019-003，名稱: 新基本解法於薄膜自由振動問題之研究)。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。。然而就真實計算或物理觀點而言，對角線項的值應為有限值而非無窮大。在這計畫中，我們將引入漢克爾函數在微小變量下的極限公式來計算影響係數矩陣的對角線項。在極限情況下的對角線項的核函數將被分解為常規項跟奇異項。常規部分可被直接計算求得。我們將透過給ㄧ點源產生的一個基本解來計算出外域問題的對角線的奇異項。這是一套簡單的方法並且保留了一些基本解法重要的特徵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本計畫書所提方法的可行性、正確性與精確性。 In this proposal, the method of fundamental solutions (MFS) with source on the real boundary will be used to solve scattering problems of water wave. 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. The proposal is an extended work of the previous and ongoing MOST projects to solve the Helmholtz problem with exterior domain. 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 approach to infinity due to the fundamental solution ( ). However, in the real implementation physical viewpoint, these values must be finite values. In this proposal, the limiting formula of Hankel function for small argument will be introduced to determine the diagonal terms of influence matrices. The kernel functions of diagonal terms in the limiting case will be decomposed into regular part and singular part. The regular part can be directly determined. Here, we will propose a diagonal-term capturing scheme for exterior problem by putting a source to generate a fundamental solution to calculate singular part. This is simple to be implemented and the prominent features of the MFS such as meshless and free of numerical integration are still maintained. Besides, the proposed scheme will be used not only for single-layer approach but also for the double-layer approach. Finally, several numerical examples will be taken to illustrate the feasibility, validity and accuracy of the proposed method in this project.

Keyword(s)

基本解法
無網格法
亥姆霍茲方程式
散射問題
水波問題
method of fundamental solutions
meshless method
Helmholtz equation
scattering problem
water wave problem

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A Method of Fundamental Solutions with Source on the Real Boundary for Solving Scattering Problems of Water Wave

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