Study of Degenerate Scale for 2d Exterior Problems: Analytical Derivation Using Bipolar Coordinates and Numerical Experiment Using Boundary Element Method

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Project title

Code/計畫編號

MOST106-2221-E019-009-MY3

Translated Name/計畫中文名

二維外域問題退化尺度之研究: 雙極座標解析推導與邊界元素法數值實驗

Project Coordinator/計畫主持人

Jeng-Tzong Chen

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=12390000

Year

2018

Start date/計畫起

01-08-2018

Expected Completion/計畫迄

01-07-2019

Bugetid/研究經費

1134千元

ResearchField/研究領域

土木水利工程

本三年案報告中，我們研究在邊界元素法或邊界積分方程法中二維外域問題的退化尺度，從二維內域問題的退化尺度延伸至二維外域問題的退化尺度是本報告的主要焦點之一。在第一年中，我們著重於二維拉普拉斯方程的外域問題，首先我們考慮含單圓或橢圓洞的無限域平面問題，為了能夠解析推導上述兩個問題的退化尺度解析解，二維拉普拉斯方程的閉合型基本解ln(r)使用極座標與橢圓座標展開成退化核的形式。關於含多圓孔洞的無限域平面問題，其兩顆圓洞的例題是本報告的主要焦點，這是因為閉合型的基本解可在雙極座標下被展開成退化核的形式，所以兩顆圓洞問題的退化尺度可被解析地導得。我們也探討邊界條件的型態是否會影響二維外域問題的退化尺度，不僅考慮含多孔洞的無限域平面問題，同時也考慮了含多個鋼線的無限域平面問題。此外，對於退化尺度發生時所對應的零場與非零場也在此報告中探討。接著，我們使用同樣的想法分別在第二年與第三年的計畫裡研究二維雙調和方程與Navier方程的外域問題，我們也將在極座標與橢圓座標中去推導二維雙調和方程閉合型基本解r2ln(r)與二維彈力基本解的退化核形式。此外，也討論二維外域問題其退化尺度數目的議題。最後，我們使用通用的邊界元素法程式去數值驗證所有問題的退化尺度。In this three-years report, we will study the problems of degenerate scale for 2D exterior problems in boundary element method/boundary integral equation method (BEM/BIEM). Extending 2D interior problem to 2D exterior problem is the main focus of this report. In the first year, we focused on 2D exterior Laplace problems. First, we consider an infinite plane with a single circular or elliptical hole. To analytically derive the solution of degenerate scale of above two shapes, the closed-form fundamental solution of the 2D Laplace equation, ln(r), can be expanded into degenerate kernel by using the polar and elliptical coordinates. Regarding an infinite plane with single or two holes, the case of an infinite plane containing two circular holes is the major example in this report. Since the closed-form fundamental solution can also be expanded by a degenerate kernel in terms of the bipolar coordinates, the degenerate scale for an infinite plane with two circular holes may be analytically derived. We have also discussed whether the degenerate scale of 2D exterior problems depends on the type of boundary condition or not. Exterior problems containing not only multiple holes but also multiple rigid lines are also considered. In addition, the null field and non-zero field for the field solution corresponding to the degenerate scale is also discussed in this report. Next, we employ the same idea to study 2D exterior problems governed by the biharmonic equation and Navier equation in the second year and third year projects, respectively. We have also derived the degenerate kernels for the closed-form fundamental solution of the 2D biharmonic equation, r2ln(r), and the Kelvin solution in the polar and elliptical coordinates. In addition, the issue of the number of degenerate scales for 2D exterior problems is discussed. Finally, we have also employed the general BEM program to numerically examine the degenerate scale for all cases.

Keyword(s)

邊界元素法
退化尺度
二維外域問題
退化核
雙極座標
BEM
degenerate scale
2D exterior problems
degenerate kernels
bipolar coordinates

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Study of Degenerate Scale for 2d Exterior Problems: Analytical Derivation Using Bipolar Coordinates and Numerical Experiment Using Boundary Element Method

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