A Study on the Degenerate Scale by Using the Dimensionless 2d Fundamental Solution

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基本資料

Project title

Code/計畫編號

MOST107-2221-E019-003

Translated Name/計畫中文名

無因次二維基本解之退化尺度研究

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=12618088

Year

2018

Start date/計畫起

01-08-2018

Expected Completion/計畫迄

31-07-2019

Bugetid/研究經費

756千元

ResearchField/研究領域

土木水利工程

邊界元素法/邊界積分方程法處理Dirichlet邊界條件的二維拉普拉斯問題時，若問題邊界為特殊的尺度，會產生解不唯一的情況，該特殊的尺度稱作退化尺度。二維拉普拉斯方程之退化尺度會導致值域缺損常數項，其發生的機制在於此類方法引入基本解所致，故基本解加入剛體運動項是最初處理退化尺度的最簡單方法。然而，此法未能有效解決退化尺度的問題，而是將退化尺度移往別處。以因次分析的觀點考量，本計劃擬引入一個無因次具客觀性基本解與自適性特徵長度取代剛體運動項，特徵長度將隨問題的邊界大小而改變，應能有效解決退化尺度問題，而非僅將其移往別處。本計畫擬以解析證明與數值兩方面來驗證此想法之可行性，並結合正在進行的科技部計劃：『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』(106-2221-E-019 -009 -MY3)將此法由二維內域問題延伸至二維外域問題。解析方面，針對特殊的幾何外型如圓、橢圓與無限域平面含兩顆圓洞，閉合型基本解ln(r)可以分別用極座標、橢圓座標與雙極座標展開成分離核的型式，我們將會以自適性特徵長度修正基本解，並用相應的分離核證明積分操作元(BIEM)與影響係數矩陣(BEM)因退化尺度而導致的值域缺損可以藉此補足。針對任意幾何外形，我們結合過去對數容量以及複變數保角映射的研究(102-2221-E-019 -034 )，嘗試證明採用自適性的特徵長度後，任意幾何外型通過黎曼保角映射所對應的對數容量不會等於1，因此不會有退化尺度的產生。同理，我們將利用伸縮尺度的概念，來證明任意幾何外型的問題皆可用本方法解決。數值方面，我們將使用邊界元素法程式來數值驗證所有因退化尺度導致問題的解不唯一現象，可以被本計畫之方法所解決。最後，將此方法延伸至二彈力問題及平板的問題。 The special size of domain results in the nonunique solution when the boundary element method (BEM) and the boundary integral equation method (BIEM) are used to solve 2D Laplace problems subjected to the Dirichlet boundary condition. This special size of boundary geometry is termed a degenerate scale. The degenerate scale in the 2D Laplace equation causes the range deficiency in the BEM/BIEM by a constant term. Since the degenerate scale results from the fundamental solution in BEM/BIEM, the simplest method by adding a rigid body mode was used to solve the degenerate scale problem in the literature. However, the degenerate scale is moved to another one, and this problem still exists. In the view of the dimensional analysis, we will introduce the non-dimensional objective fundamental solution with an adaptive characteristic length instead of a constant term in the proposal. The adaptive characteristic length depends on the domain. It may effectively avoid the degenerate scale problem rather than moving it to another one. We will verify the validity by the analytical proof and the numerical experiment in the BEM/BIEM. In this proposal, we combine the ongoing MOST project, ” Study of degenerate scale for 2D exterior problems: analytical derivation using bipolar coordinates and numerical experiment using boundary element method ” (NSC 106-2221-E-019 -009 -MY3), to extend the interior problem to the exterior problem. For the special geometric domain such as a circle, an ellipse and an infinite plane with two circular holes, the closed-form fundamental solution of the 2D Laplace equation, ln(r), can be analytically expanded into degenerate kernel by the corresponding polar, elliptical and bipolar coordinates, respectively. We will enrich the fundamental solution by using the adaptive characteristic length and analytically complete the range of the integral operator (BIEM) or the influence matrices (BEM). For an arbitrary geometric domain, we will link the study (NSC 102-2221-E-019 -034) for the logarithmic capacity and the theory of complex variable. We will prove that the corresponding logarithmic capacity is not equal to 1 while the adaptive characteristic length is introduced in the fundamental solution. By using the scaling technique (expansion ratio), we will prove for the problem with a general shape. In the numerical implementation, we will also employ the BEM program to examine whether the problem of nonunique solution due to the degenerate scale is solved for all cases by employing the proposed method or not. Finally, we may extend this method to the 2D elasticity problem and the 2D plate problem.

Keyword(s)

邊界元素法/邊界積分方程
退化尺度
特徵長度
對數容量
無因次基本解
BEM/BIEM
degenerate scale
characteristic length
logarithmic capacity
non-dimensional fundamental solution

DSpace CRIS

A Study on the Degenerate Scale by Using the Dimensionless 2d Fundamental Solution

基本資料

Description