Using Local Radial Basis Function Collocation Method to Approximate Radiation Boundary Conditions

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Project title

Code/計畫編號

MOST104-2221-E019-032

Translated Name/計畫中文名

利用局部輻射基礎方程組合法近似輻射邊界

Project Coordinator/計畫主持人

Tai-Wen Hsu

Co-Investigator(s)/共同執行人

蔡加正

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=11573123

Year

2015

Start date/計畫起

01-08-2015

Expected Completion/計畫迄

01-07-2016

Bugetid/研究經費

1118千元

隨著資訊科技的快速發展，數值方法也變得越來越強大，許多過去被認為難以解答的問題，已經被現代的技術所克服，特別是無網格數值方法，這幾年已經成為一個熱門的領域，基本解方法作為一種無網格數值方法，主要是用來解偏微分方程式的齊次解，它源自於間接法邊界積分方程，有別於邊界積分方程，基本解方法可以免於數值積分和奇異性。基本解方法已經成功地應用到許多的偏微分方程式，包括波以松方程式、擴散方程式、史托克斯方程組、柯西－內維拉方程組、雙調和方程式、赫姆霍茲方程式等。理論上，只要欲求解的偏微分方程式或方程組存在解析的基本解，就可以用基本解法來求得數值解。再來關於緩坡方程式的邊界處理上是使用一般離散的技巧，但邊界處裡上仍會面臨許多限制與困難，例如，計算域不規則或是高階展開等。近年來此等問題已有適合處理之法，如全域徑向基底函數配點法(Kansa,1990a,b)已普遍被使用。此法之好處在於可有效處理任意或是複雜的場域，但容易產生系統矩陣之病態系統，尤其是計算波浪運動的相關問題上。本而在本計畫當中，吾人將利用區域徑向基底函數配點法(LRCM, Lee et al., 2003)來處理邊界值問題。經由區域化過程使密矩陣系統轉化為稀疏矩陣，提升計算效率。本計畫的主要目的是利用無網格技巧解析橢圓形態緩坡方程式，搭配LRCM 處理高階邊界條件，達到通用解析大角度入射的平面波場的問題。本計畫之內容，同時也涵蓋此方法的程式撰寫，同時，最後的數值方法也將與解析解或文獻上之其他數值方法的解答比較。 It was well known that environments of the ocean would be usually concerned for the Sea-Island country in the world. In the ocean, there were so many physical phenomenons interesting us. However, the most important phenomenon was wave transmition and dissipation. For analyzing wave motions, developing theory and establishing numerical model would be a reliable way for engineering applications. The method of fundamental solutions (MFS) is one of the popular methods in the boundary-type category for applications of partial differential equations, such as the Poison equation, the diffusion equation, the Navier Stoles equation, the Cauchy remen equation, the Helmholtz equation, etc. This meshless method, the MFS, is going to be applied to analyze the mild-slope equation. The other relevant issue is the boundary domain. The global radial basis functions collocation method(GRCM, Kansa, 1990a, b) is widely used by dealing with high-order expansion of boundary. The advantage of GRCM is its effectiveness in dealing with arbitrary and complex domains. However, GRCM usually results in ill-conditioned system matrices for the case where high resolutions are required, such as the wave problems in which detailed description of wave configuration is required. More recently, a localization procedure has been proposed to transform the prescribed dense system matrices into sparse matrices. Lee et al. (2003) are the pioneers to propose the local RBF collocation method (LRCM) to solve the boundary value problems. The main purpose of this project is that we use the meshless method to analyze mild-slope equation. We also consider treatments of higher-order boundary conditions by LRCM. As expected, a wide-range incident problem of a wave field can be solved generally. A reliable validation compared with other methods will be provided in this project. All the effort of this project would be published on the international journals.

Keyword(s)

基本解方法
對偶法
瑞斯納板
柯西荷夫板
溫克爾基礎
雙參數基礎
無網格
Radiation boundary condition
Mild-slope equation
Mesh-less numerical method
Local radial basis function collocation method

DSpace CRIS

Using Local Radial Basis Function Collocation Method to Approximate Radiation Boundary Conditions

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