臺灣博碩士論文加值系統

為了求解雙調和方程式的正算和反算問題，本文先推導出雙調和方程式級數的展開式，並引入特徵長度的?念來提高數值解的穩定性和精度，然後使用配點法(collocation method)來求解待定係數，所求解後的級數解即為所求。此方法在單連通、多連通區域均適用。邊界可以是任意形狀，並且對於內域、外域問題均可做計算。由數值計算的結果可以看出我們的方法有著很高的精度，對於邊界資料的擾動也有一定的穩定性。

To solve the direct and inverse Biharmonic Equation, we obtain the Biharmonic Equation series expansions by introducing the characteristic length to improve the stability and the accuracy, and then apply the collocation method to calculate the coefficient of the series. At last we can solve the Biharmonic equation. This method can be use in both simply connected domain and multiply connected domain. The shape of boundary is arbitrary. This method can also be used in both interior problem and exterior problem. This method has high accuracy and stability even under the boundary data disturbance.

摘要…………………………………………………………………… i

Abstract……………………………………………………………….. ii

目錄…………………………………………………………………… iii

圖目錄………………………………………………………………… v

第一章 緒論………………………………………………………….. 1
1.1前言……………………………………………………………….. 1
1.2 文獻回顧…………………………………………………………. 1
1.3 研究目的…………………………………………………………. 2
1.4 本文架構…………………………………………………………. 3
第二章 理論基礎…………………………………………………….. 4
2.1 反問題定義………………………………………………………. 4
2.2 邊界條件的類型…………………………………………………. 5
第三章 理論架構…………………………………………………….. 7
3.1 雙調和方程的數學模型…………………………………………. 7
3.2 修正型的Trefftz方法……………………………………………. 9
第四章 配點法求解雙調和方程…………………………………….. 11
4.1 配點法……………………………………………………………. 11
4.2.2 LU分解法(LU Decomposition)………………………………… 14
4.3 數值算例…………………………………………………………. 16
4.3.1 範例一………………………………………………………….. 16
4.3.2 範例二………………………………………………………….. 17
4.3.3 範例三………………………………………………………….. 18
第五章 結論與未來展望…………………………………………….. 19
參考文獻……………………………………………………………… 20
誌謝…………………………………………………………………… 23

圖4.1：正算的邊界示意圖…………………………………………… 24
圖4.2：反算(一個動不給條件)邊界的示意圖……………………… 24
圖4.3：反算(二個動不給條件)邊界的示意圖……………………… 25
圖4.4：範例一 與 的關係圖………………………………………
26
圖4.5：範例一CTM與MCTM條件數比較………………………… 26
圖4.6：範例一正算問題解析解……………………………………… 27
圖4.7：範例一正算問題數值解……………………………………… 27
圖4.8：範例一正算問題，數值解計算誤差 ………………………… 28
圖4.9：範例一反算問題解析解(一個洞不給條件)………………… 29
圖4.10：範例一反算問題數值解(一個洞不給條件)……………… 29
圖4.11：範例一反算問題，數值解計算誤差(一個洞不給條件)… 30
圖4.12：範例一反算問題解析解(二個洞不給條件)……………… 31
圖4.13：範例一反算問題數值解(二個洞不給條件)……………… 31
圖4.14：範例一反算問題，數值解計算誤差(二個洞不給條件)… 32
圖4.15：範例二 與 的關係圖…………………………………… 33
圖4.16：範例二域外加佈源點示意圖……………………………… 33
圖4.17：範例二正算問題解析解…………………………………… 34
圖4.18：範例二正算問題數值解…………………………………… 34
圖4.19：範例二正算問題數值解誤差……………………………… 35
圖4.20：範例二反算問題解析解(一個洞不給條件)……………… 36
圖4.21：範例二反算問題數值解(一個洞不給條件)……………… 36
圖4.22：範例二反算問題，數值解計算誤差(一個洞不給條件)… 37
圖4.23：範例二反算問題解析解(二個洞不給條件)……………… 38
圖4.24：範例二反算問題數值解(二個洞不給條件)……………… 38
圖4.25：範例二反算問題，數值解計算誤差(二個洞不給條件)… 39
圖4.26：範例三 與 的關係圖……………………………………
40
圖4.27：範例三域外加佈源點示意圖……………………………… 40
圖4.28：範例三正算問題解析解…………………………………… 41
圖4.29：範例三正算問題數值解…………………………………… 41
圖4.30：範例三正算問題，數值解計算誤差……………………… 42
圖4.31：範例三反算問題解析解(一個洞不給條件)……………… 43
圖4.32：範例三反算問題數值解(一個洞不給條件)……………… 43
圖4.33：範例三反算問題，數值解計算誤差(一個洞不給條件)… 44
圖4.34：範例三反算問題解析解(二個洞不給條件)……………… 45
圖4.35：範例三反算問題數值解(二個洞不給條件)……………… 45
圖4.36：範例三反算問題，數值解計算誤差(二個洞不給條件)… 46

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