臺灣博碩士論文加值系統

本文應用無網格法（Meshless method）中的具對稱系統矩陣之局部最小二乘無網格法來分析探討彈性動力問題。本方法是以離散點的方式來建構數值模型，再用局部最小二乘法(Local least-square method, LLS)建立局部聯立方程組並使其對稱化，再加總為全域聯立方程組，即為對稱化局部最小二乘法(Symmetrize local least-square method, SLLS)，之後再配合Newmark- 法和Wilson- 法以對一維之彈性振動問題進行分析。

本文的數值範例所得結果和解析解進行比較，並與微分再生核近似法(DRKM)之數據進行比較，其結果令人滿意，驗證了具對稱微分矩陣之局部最小二乘無網格法在彈性振動分析上的可行性。

In this paper we introduce the meshless method of local least-square with symmetric system matrix to solve the elastic-dynamical problems. The numerical model is established on discretization points. We use the local least-square method(LLS) to establish a system of equations and improve it to be symmetrical,then combine the local system of equations to a global system of equation. Finally,we use the Newmark- method and the Wilson- method to analyze the 1-D elastic-dynamical problems.

In the numerical example we comparing the data analyzed in this article with the exact solution, also compared the result with results of differential reproducing kernel approximation method (DRKM). It shows that SLLS can be used on the 1-D elastic-dynamical analysis.

摘要 …………………………………………………………………………………Ⅰ
ABSTRACT …………………………………………………………………………Ⅱ
致謝 …………………………………………………………………………………Ⅲ
目錄 …………………………………………………………………………………Ⅳ
表目錄 ………………………………………………………………………………Ⅵ
圖目錄 ………………………………………………………………………………Ⅹ
第一章 緒論 …………………………………………………………………………1
1.1 前言 ………………………………………………………………………………1
1.2 無元素法的發展 …………………………………………………………………2
1.3 本文架構 …………………………………………………………………………4
第二章 局部最小二乘法理論推導 …………………………………………………5
2.1 局部最小二乘法 …………………………………………………………………5
2.2 微分運算矩陣之對稱化 …………………………………………………………9
2.3 加權函數和鄰近點的選取………………………………………………………12
第三章 動力問題與歷時分析………………………………………………………15
3.1 一維彈性力學公式………………………………………………………………15
3.2 邊界條件處理……………………………………………………………………18
3.3 時域直接積分法之探討─Newmark-method……………………………………19
3.4 時域直接積分法之探討─Wilson-method………………………………………22
第四章 數值分析結果………………………………………………………………25
4.1 兩端固定的彈性線受初始位移之行為…………………………………………25
4.1.1 例一……………………………………………………………………………25
4.1.2 例二……………………………………………………………………………30
4.1.3 例三……………………………………………………………………………33
4.2 兩端固定的彈性線受初始速度之行為…………………………………………35
4.2.1 例四……………………………………………………………………………36
4.2.2 例五……………………………………………………………………………38
4.3 一維懸臂樑自由端受集中壓力作用之行為……………………………………41
4.3.1 例六之位移解析解……………………………………………………………41
4.3.2 例六之數値分析………………………………………………………………43
第五章 結論…………………………………………………………………………46
參考文獻 ……………………………………………………………………………49
自述…………………………………………………………………………………105

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