臺灣博碩士論文加值系統

本研究建構一個三度空間的顆粒模擬系統，使得該系統可以透過電腦模擬的程序模擬自然界中顆粒間排列方式，進而作為各領域中模擬求解顆粒堆積相關應用問題的基礎。本顆粒堆積模擬系統，在不考慮力學因素下，以作業研究的技術分析求解，探討一組顆粒間可能存在的排列型式。對於顆粒的排列組合的探討分成兩部份，第一部份為求解給定顆粒間可能存在的最緊密堆積方式，這個技術在應用上可以對應到給定級配的砂質土壤顆粒的最大乾密度問題。第二個部份為從給定堆積顆粒試體的剖面資訊推算求解大小顆粒的組成級配，這個技術可以應用於混擬土試體的骨材級配分析。
本論文主要可分為四個部份(1)三維顆粒數學模型的建立，與操控技術：如顆粒平移、旋轉、接觸判斷等。顆粒模型分為橢球及多面體兩類。(2)模擬顆粒緊密堆積的過程亦即利用最佳化技術，調整各顆粒的相對位置，使顆粒間趨向緊密的狀態。(3)推導及建立三維顆粒與二維剖面的相對關係。(4)利用線性規劃的技術從顆粒堆積成的試體之剖面，推算各種類顆粒間的個數比例關係。我們並利用發展出來的模擬及求解技術，作許多相關的模擬測試，發現此技術可為顆粒堆積模擬應用的基礎，且模擬測試結果與真實物理實驗結果比對，也得到合理的結果。

In this thesis we present a 3-D model that is able to simulate the packing process of granular particles. We use 3-D ellipsoids as well as polyhedra as the surrogate to simulate natural particle. Assuming mechanical forces non-existant, we model and solve the packing of granular particles as an optimum problem. The research is presented in two aspects. First we use the model to estimate of the maximum packing density for a set of given particles, which has application in determinating the maximum density of granular soil. Secondly we attempt to estimate the particle composition based on cross section information. This technique has application in estimating aggregate composition in cemented materials such as concrete.
This thesis can be divided into four parts: (1) Formulation of 3-D system of particles, including mathematical techniques in particle manipulating such as shifting, rotation, and contact testing of particles. Models for ellipsoid and polyhedra are both developed. (2) A model to simulate the packing process. Procedures involved are particle filling, followed by position and attitude adjusting in order to reach a higher density. (3) Theories to link ellipsoids and the elliptical slices obtained by slicing an ellipsoid with planes. (4) A linear system to solve for a set of composition that can be best represented by the cross sections observed.
We verify the effectiveness of the model by testing the model against analytical results for some special cases, and also with particle distribution derived from a real life standard sand. All results are reasonable, and we verify that the system can serve as the foundation for further partical research.

摘要1
ABSTRACT2
誌謝3
目錄4
表目錄6
圖目錄7
0.概說9
1.文獻回顧10
2.顆粒數學理論及推導11
2.0橢球相關理論12
2.0.0橢球通式12
2.0.1橢球平移與旋轉12
2.0.2橢球接觸判斷13
演算法A：任意兩橢球接觸判斷14
演算法B：原點到橢球的最短距離15
演算法C：原點到橢圓的最短距離16
2.0.3橢球與平面16
演算法D：橢球W與平面P接觸判斷16
2.1多面體相關理論17
2.1.0多面體通式17
2.1.1多面體平移與旋轉19
2.1.2多面體接觸判斷19
2.1.3多面體與平面19
3. 模式建立與求解20
3.0 顆粒緊密堆積20
3.0.0電腦模擬程序20
3.0.1模擬退火法21
3.1顆粒剖面與級配23
3.1.0橢球剖面資訊23
3.1.1數值方法26
3.1.2線性規劃求解27
4.電腦模擬驗證29
4.0橢球緊密堆積測試29
4.0.0 模擬系統密度計算29
4.0.1 標準砂模擬測試30
4.0.2 正球緊密堆積測試33
4.1 多面體堆積測試34
4.2橢球剖面推算級配36
4.2.0 剖面資訊之建立37
4.2.1 由剖面資訊求解試體級配37
5.結論39
6.參考文獻41
附錄A 座標系統轉換與橢球方程式43
A.0座標系統轉換43
A.1座標軸旋轉、與收縮膨脹43
A.2橢球與橢圓方程式的處理44
A.3求解橢球剖面橢圓的雙軸長46
A.4橢球與多面體接觸判斷47
附錄B 多面體相關數學47
B.0線性規畫簡介48
B.1求解起始可行基底解與多面體接觸判斷50
B.2多面體與平面的交集51
附錄C 程式設計與繪圖51
C.0顆粒模擬系統51
C.0.0資料結構52
C.0.1顆粒繪圖53
C.1演算法特例處理54
C.1.0黃金分割法54
C.1.1多面體的參考中心55
C.1.2 Simplex演算法的退化解55
附錄D 應用軟體56
D.0 AUTOCAD DXF圖檔格式56
D.0.0 DXF檔案結構56
D.0.1 DXF 基本圖元的組成57
D.1數學規劃軟體CPLEX58
D.1.0 MPS FORMAT58
(A) ROWS section58
(B) COLUMNS section59
(D) RANGES section59
(E) BOUNDS section60
(F) OBJSEN and OBJNAME 選項60
(G)整數變數60
D.1.1 CPLEX的函數庫61
自述64

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