A New Scheme of Capturing Diagonal Terms in the Mfs for Potential Problems

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Project title

Code/計畫編號

MOST106-2221-E019-019

Translated Name/計畫中文名

一套新的基本解法對角線項捕捉技術於勢能問題之應用

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=12226938

Year

2017

Start date/計畫起

01-08-2017

Expected Completion/計畫迄

31-07-2018

Bugetid/研究經費

620千元

ResearchField/研究領域

土木水利工程

"本計劃擬發展一套新的基本解法對角線項捕捉技術來求解勢能問題。在傳統的基本解法中，為了避免當場源點碰在一起時所產生的奇異行為，源點總是被放置於領域之外。若將場源點佈置於同一個位置時，由於數學上基本解( ln r )的緣故，將使得影響係數矩陣的對角線項變成無限大，然而就真實計算或是物理的觀點而言，影響係數矩陣的對角線項應該是有限值而非無窮大。在此，我們擬提出一套藉由滿足力平衡條件與剛體運動的新方法來計算當場源點碰在一起時影響係數矩陣中對角線項的值。這是一套自然、直接且容易的方法。首先，擬將力平衡方程式離散化，用來計算曳引力場中影響係數矩陣的對角線項。接著通過滿足剛體運動的測試，求出勢能場中的影響係數矩陣的對角線項。本計劃所提的方法不只適用於計算基本解法中的單層勢能法，也能用於雙層勢能法的求解。此外，針對圓形的例子，我們將使用解析的方法來導得其對角線項的解析表示式。最後，我們將舉幾個數值算例來論證本計劃所提方法的可行性、正確性與精確性。" "In this proposal, a new scheme of capturing diagonal terms in the method of fundamental solution (MFS) will be proposed to solve the potential problems. In the MFS, the source points are always located outside of the domain for avoiding the singularity of source point and collocation point coinciding together. When the source point and collocation points are distributed at the same position, the diagonal terms in the influence matrices have infinite values due to the fundamental solution ( ln r ), but they must be finite values in the real implementation or in the physical point of view. Here, we will propose a new scheme of satisfying the force equilibrium condition and rigid body motion to determine the diagonal term values of the influence matrices when the source points are close to the collocation points. This is a natural, straightforward and easy way. First, the force equilibrium equation will be discretized and used to determine the diagonal-term values of the influence matrix for traction. Then, through satisfying the rigid body test, the diagonal-term values of the influence matrix for potential will be captured. The proposed scheme will be used not only for single-layer approach but also for the double-layer approach. In addition, for circular case the diagonal-term values will be derived in an analytical manner. Finally, several numerical examples will be taken to illustrate the feasibility, validity and accuracy of the proposed scheme in this project."

Keyword(s)

基本解法
力平衡
剛體運動
拉普拉斯方程
勢能問題
method of fundamental solutions
equilibrium of force
rigid body motion
Laplace equation
potential problem

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A New Scheme of Capturing Diagonal Terms in the Mfs for Potential Problems

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