Using Boundary Element Methods to Study and Simulate Two-Dimensional Sloshing Problems

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簡歷

基本資料

Project title

Code/計畫編號

NSC96-2628-E019-031-MY3

Translated Name/計畫中文名

邊界元素法對沖激問題之研究與模擬

Project Coordinator/計畫主持人

Jiang-Ren Chang

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Systems Engineering and Naval Architecture

Website

https://www.grb.gov.tw/search/planDetail?id=1753101

Year

2009

Start date/計畫起

01-08-2009

Expected Completion/計畫迄

31-07-2010

Bugetid/研究經費

573千元

ResearchField/研究領域

機械工程

在本研究計畫中，吾人擬提出一三年期研究計畫，分別推導出Trefftz直接法、Trefftz間接法、改良式Trefftz法與複變數邊界元素法，來求解與分析液貨艙中液體受到外力作用時所產生的沖激問題。此外，並針對容槽液體受到外力作用，內部液體的振盪行為，發展出一套數值程式來加以模擬。本研究乃假設液體須滿足勢流理論，且有足夠的邊界條件可以求解自由液面之勢位，但是自由液面的速度與邊界條件必須隨著時間不斷更新。由於Trefftz邊界元素法係採用正規函數為基底函數，得以避開傳統邊界元素法之奇異行為，俟求出未知係數之後，即可求出其它相關之物理量。在第一年與第二年中，由於分別使用Trefftz直接法與間接法，將一如過往之研究結果，會衍生有數值不穩定的劣化問題。吾人擬採用Tikhonov正規化方法與SVD法來加以克服。而在第二年中，也擬研發出改良式Trefftz法，利用特徵長度的技巧，來改善基底函數中因幾何參數改變所造成之數值發散問題，並藉此大幅降低條件數，這將是傳統 Trefftz法截至目前為止所無法解決之重大問題。此外， Trefftz方法雖有不具奇異性的積分核，但是卻因此使得欲解的線性方程式系統隸屬於第一類Fredholm方程式，這種第一類方程式為大家所熟知的一種基本特性，就是它的病態行為，需要使用一些特別的技巧來克服病態的問題。在第三年中亦將針對二維的沖激問題，使用一種新的邊界元素法，也就是複變數邊界元素法來模擬。在該法當中，方程式表達中所使用的物理量為勢位的法向導微以及勢位的切向導微。因此，使用該法可以克服傳統邊界元素法中的困境，而該法的積分核函數為奇異的，所以欲解的積分方程式乃是屬於第二類的Fredholm積分方程，在數值解上已經被證明是自動穩定的，並不會有Trefftz法的盲點。然而，該法的一個麻煩處是我們必須要處理複變數中的多值問題，也就是分歧路徑的問題，這也將是本研究將克服之重點。In this three-year research proposal, the direct Trefftz method, indirect Trefftz method, and the modified Trefftz method with the complex boundary element method will be developed to deal with sloshing problems existing in the liquid cargo tank. Aimed at the sloshing phenomenon, a numerical package will also be implemented to simulate the oscillation of the liquid in the tank. Assuming the potential theory is satisfied and the boundary conditions are well enough, the potential of the free surface can be determined; however, the velocity of the free surface and boundary conditions should be updated as time changes. Since the Trefftz boundary element methods (BEMs) adopt the regular functions as basis functions such that it can avoid the singular behaviors as appear in the conventional BEM and once the unknown coefficients are decided, the corresponding physical quantities can be obtained. In the first and second years, the direct Trefftz method and the indirect one will be respectively developed. As found in previous researches, the numerical instability will occur. The Tikhonov’s regularization method and the SVD method, as usual, will be adopted to overcome the difficulties. In the third year, a modified Trefftz method, which makes use of the characteristic length technique, will be proposed to improve the numerically divergent problem owing to changing geometric parameters of the basis functions and further, reduce the condition number of the resulting matrix. This is also the key problem to be resolved in the development of the Trefftz methods. Besides, a newly developed complex BEM will also be used to solve and simulate sloshing problems because its kernel function is singular and the integral equation is categorized into the Fredholm integral equation of the second kind. Such a characteristic of the complex BEM is proved to be automatically numerical stable and without the deficit of the Trefftz methods. However, the branch-cut problem of the complex variables should be overcome and this is also one of the key problems to be resolved in the research.

Keyword(s)

沖激問題
Trefftz邊界元素法
病態問題
分岐路徑
Sloshing problems
Trefftz method
ill-posed problem
branch-cut

DSpace CRIS

Using Boundary Element Methods to Study and Simulate Two-Dimensional Sloshing Problems

基本資料

Description