Isogeometric Analysis Dual Bem for the Degenerate Boundary with Laplace Equation

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Project title

Code/計畫編號

MOST106-2221-E019-010

Translated Name/計畫中文名

同幾何分析對偶邊界元素法用於拉普拉斯方程的退化邊界問題

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=12238274

Year

2017

Start date/計畫起

01-08-2017

Expected Completion/計畫迄

31-07-2018

Bugetid/研究經費

836千元

ResearchField/研究領域

土木水利工程

"同幾何分析(IGA)為近年來非常熱門的議題，是由Hughes 等人於2005 年首度提出的。同幾何分析主要概念為使用樣條曲線函數作為基底函數，並在偏微分方程的離散中來表達我們感到興趣區域形狀的幾何特性。而非均勻有理B 樣條(NURBS)是在電腦輔助設計(CAD)與電腦輔助工程(CAE)上常使用的定義幾何形狀的數學方法，非常合適來做為IGA 的基底函數。這個嶄新的方法已被用於描述計算區域的幾何特性及相關場變數，並在許多科學領域與應用領域中取得傑出的改良成果。IGA 有幾個明顯的優點:1.網格的自適性、2. 電腦輔助設計與電腦輔助工程的自動銜接所節省的成本、3.對工程師的友善介面、4. 因完美表現幾何所得到解的高精確度與收斂性。另一方面，對偶邊界元素法/對偶邊界積分方程法對於數值計算方面對於處理退化邊界問題扮演一個很重要的角色，包含達西滲流流經截水牆的分析問題、聲場的隔音牆分析問題以及彈性體的裂縫問題。為了來解由於退化邊界所造成的矩陣秩降問題，對偶邊界元素法引入超奇異積分方程來使影響係數矩陣滿秩，洪與陳[Journal of Engineering Mechanics-ASCE, 114, 6, 1988; Applied Mechanics Review-ASME, 52, 1, 1999]在此方面的進展以得到邊界元同行高度引用(單篇超過300 次)。本計畫中，我們結合同幾何分析與對偶邊界元素法來處理含退化邊界的拉普拉斯方程(Laplace equation)問題，而此法的概念在本計畫書中的圖1 顯示。以過去我們團隊成功處理退化邊界問題的經驗，我們相信我們在二維與三維問題中可以取得提高數值解精確度與工程應用的友善性。我們也將嘗試來提出一個方案，透過結合退化核與同幾何分析來提高數值方法的解之精確度。並將推廣此方法應用於非等向性問題，來嘗試取得相對於傳統邊界元素法而言更精準的成果。而所發展之程式在工程實務上來說為提高工程師使用介面的友善性。" "IsoGeometric Analysis (IGA) as a popular approach was initially proposed by Hughes et al. in 2005. The main concept of the IGA is to use the spline-based functions for the discretization of the partial differential equation to represent the geometry of the interested domain, and non-uniform rational B-Spline (NURBS) popular for the Computer-Aided Design (CAD) and the Computer Aided Engineering (CAE) is suitable as a basis function. This novel approach has been carried out for an outstanding improvement by matching the computational geometry well and solving the isogeometric form for the field variable in the numerical implementation from the scientific community and many engineering applications. There are several significant advantages for the IGA: 1. Mesh adaptivity, 2. The cost saving for the translation of data between the CAD and the CAE, 3. User-friendly interface to engineer, 4. High accuracy and convergence for the perfect geometry description. On the other hand, dual boundary element method/boundary integral equation method (BEM/BIEM) plays an important role in the numerical simulation to deal with the degenerate boundary problem, such as Darcy flow around the cutoff wall, a screen in an acoustic cavity and a crack in an elastic body. To solve the rank-deficiency problem due to the degenerate boundary, the hypersingular integral equation is introduced to promote the full rank for the influence matrix in the dual BEM. Two papers proposed by Hong and Chen [Journal of Engineering Mechanics-ASCE, 114, 6, 1988; Applied Mechanics Review-ASME, 52, 1, 1999] are highly cited by many BEM researchers (In Google citation, the citation of the paper in AMR is more than 300 times). In this project, we combine the IGA and the dual BEM to deal with degenerate boundary problems of the Laplace equation. The frame of this project is shown in Fig. 1 of the proposal. Based on the successful experience for solving the degenerate boundary problem by our group and many IGA applications in many areas, we hope that we can take a wonderful result to promote the accuracy of the numerical implementation in 2-D and 3-D degenerate boundary problems and make a user-friendly interface for engineers. We may try to determine a scheme from the ideas of the degenerate kernel and the IGA to level up the accuracy of the numerical solution. We may also extend to the anisotropic problem. By using the IGABEM, we may obtain the better result compared with the one of the conventional BEM. The developed program will be more user-friendly to engineers in engineering practice."

Keyword(s)

"對偶邊界元素法/邊界積分方程法
同幾何分析
非均勻有理B 樣條
退化邊界問題
拉普拉斯方程
Dual BEM/BIEM
IGA
NURBS
Degenerate boundary problem
Laplace equation

DSpace CRIS

Isogeometric Analysis Dual Bem for the Degenerate Boundary with Laplace Equation

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