臺灣博碩士論文加值系統

對於分離體的動態接觸問題，由於分離體間的大位移和滑動使得分離體的接觸位置會隨著計算步程而改變，因而造成傳統有限元素法模擬的困難。為了處理分離體的運動行為而有分離元素法的產生，其中不連續變形分析法可歸類於分離元素法的一種，但是早期的不連續變形分析法(DDA)在模擬問題時每一個分離體只是單純的常應變假設，為了希望能夠更適切的模擬複雜的接觸行為，本研究將結合不連續變形分析法(DDA)和有限元素法(FEM)的優點來處理分離體的接觸問題，並進一步的改善。
在處理分離體時首先改變不連續變形分析法常應變的假設，每個分離體改用有限元來處理，目前所發展的程式分離體內可用常應變三角形、四邊形等參元或其混合形式來模擬。同時為了希望能夠處理更複雜的問題，在推導運動方程式時改用虛功原理和更新式的拉格蘭治描述來推導分離系統的運動方程式。
為了適合不連續變形分析法和有限元結合的接觸分析，分離體間的接觸偵測採用多層接觸偵測和不連續變形分析法的接觸判斷精神，並配合有限元的特性做改善。對於接觸區域不可相互貫入的接觸行為目前採用懲罰函數法來處理，並直接利用結點力的推導概念導出和元素形狀函數無關的正向接觸和剪向接觸矩陣，再配合摩擦力兩階段的處理方式可導出和勁度矩陣無關的接觸力向量，再結合分離體的接觸開閉迭代技巧，可有效的處理分離體的接觸問題。
對於分離體的慣性改用配點時間積分法來處理，並且也探討了配點時間積分法的穩定性、精確性及高頻消散能力，同時導出了配點時間積分法的位移增量式公式並做了部份參數研究，同時提出了建議參數以做為分離系統動、靜態模擬的參考。
為了能夠節省計算量和記憶體容量，採用動態記憶體的儲存方式，並利用有限元變寬帶的處理概念和分離體的接觸分析，提出分離系統的動態接觸變寬帶解法，利用此方法可處理分離系統的動、靜態接觸問題，再配合分離體結點的重新編號可以更有效節省計算量和記憶體容量，並可以為往後的接觸及開裂問題做準備。
對於分離體不可入侵的限制目前採用懲罰函數法，其中懲罰函數值可想像成一個接觸彈簧，為了獲得適當的接觸彈簧值，現階段利用每一組接觸組合彈簧能量不變的原則和容許的貫入量來估算每一個接觸組合的建議使用接觸彈簧值，利用此方式可避免接觸彈簧值直接選取的困擾，並且對許多的接觸問題都可以獲得不錯的結果。
分離系統利用上述的方式來處理應該可以更適切、有效的模擬其動、靜態行為，並且可為接觸力學研究者提供一個有效、可靠的分析模擬方法。

In the analysis of the dynamic contact problem of a multibody system, contact locations among discrete bodies may change due to the sliding and the translation of blocks. This variation of contact locations during motion leads to some computational difficulties in the analysis by applying the conventional finite element method (FEM). To handle the kinematics of multibody system, a method so-called the Discontinuous Deformation Analysis (DDA) method was introduced by Shi in the past decade. However, the constant strain field inside each block modeled by the DDA method is not capable to simulate complicated deformation behaviors of blocks of irregular shapes. Therefore, a numerical method by combining the FEM and the DDA method is developed in this thesis to well simulate the dynamic contact behaviors of two-dimensional deformable discrete system.
Finite element mesh of mixed constant strain triangle elements and four-node isoparametric elements can be patched on each block to approximate the displacement field. The system of governing equations is derived by the principle of virtual work and the updated Lagrangian description. This kind of formulation is prepared for the analysis of the multibody dynamic with material and geometrical nonlinearities.
A multi-layer contact searching technique is developed based on the contact detection techniques of DDA method and characters of the finite elements patched on blocks. Penalty method is adopted to maintain the in-penetrability constraint between contacting blocks. A normal contact matrix and a shear contact matrix, which are unrelated to the element shape function, are derived based on the nodal force concept. These two contact matrices together with a two-stage friction force treatment technique can effectively model the contact and separation conditions during motion.
To handle the inertial effect, a collocation time integration scheme that is suitable for the discrete system, is also derived in the thesis. Sets of collocation time integration parameters for static and dynamic analyses are suggested after a through study of the stability, accuracy and high frequency dissipation characters of the proposed time integration scheme.
Based on the character of the node-to-edge contact detection basic unit used in the contact analysis, a special type contact element is formed once the contact condition occurs. This type of element provides the stiffness expressed by the elements of the normal and shear contact matrices to the stiffness matrix of the global system. Since the contact locations are changing during the motion, the bandwidth of the coefficient matrix of the governing equations is also changed passively. Hence, an adaptive skyline solver combined with a nodal renumbering technique is proposed to effectively reduce the memory size and computation time for the simulation. The nodal renumbering technique can be used for the analysis of problems with fracture and fragmentation features.
The selection of a suitable penalty number for the contact analysis is an art during the past decades. In this thesis, an energy consistency technique is proposed to adaptively adjust the penalty value of the contact spring during the iteration process within a time step. This technique is valid for both the normal and the shear contact conditions.
A computational code, which is capable of simulating the dynamic and static contact behaviors of two-dimensional deformable discrete system, is developed based on all the forementioned techniques. The efficiency and accuracy of this code are verified by the results of numerical simulations of some benchmark problems.

封面
摘要
Abstract
目錄
表目錄
圖目錄
第一章 前言
1-1 研究動機
1-2 研究方法
第二章 文獻回顧
2-1 有限元素法(FEM)
2-2 分離元素法( DEM)
2-3 不連續變形分析法(DDA)
2-4 有限元素法和不連續變形分析的比較
第三章 不連續變形分析法(DDA)及塊體附貼有限元的理論
3-1 不連續變形分析的理論
3-2 增量式的描述及虛功方程
3-3 有限元三角形常應變、四邊形等參元的形狀函數及對應的線性和非線性應變-位移矩陣
3-4 有限元的彈性矩陣及幾何勁度矩陣
3-5 限制條件的推導
3-6 外力向量的推導
3-7 結論
第四章 塊體附貼有限元的接觸模擬
4-1 塊體的接觸預估
4-2 塊體的接觸、資料建立和迭代的處理
4-3 正向接觸矩陣
4-4 剪向接觸矩陣
4-5 摩擦力向量
4-6 結論
第五章 塊體附貼有限元的慣性處理
5-1 DDA的慣性處理
5-2 配點時間積分法的慣性處理
5-3 時間積分的穩定性
5-4 分離系統的靜態模擬
5-5 時間積分的正確性
5-6 配點時間積分法的位移增量公式
5-7 數值算例
5-8 結論
第六章 塊體附貼有限元動態接觸變寬帶的處理
6-1 變寬帶的定義及分解
6-2 對稱變寬帶矩陣的克雷斯基(Cholesky)分解法及SOR、共軛梯度法的上三角形方式儲存
6-3 矩陣變寬帶的儲存、容量計算及元素引用
6-4 有限元變寬帶的處理法
6-5 塊體貼附有限元動態接觸變寬帶的處理
6-6 結論
第七章 塊體附貼有限元動態接觸變寬帶的結點重新編號
7-1 結點的編號與對應矩陣元素的排列關係
7-2 圖形資料的表示法
7-3 矩陣的帶寬及包圍量
7-4 RAM( Reverse Cuthill-McKee)法的結點重新編號
7-5 反向排列的原因和包圍量的求法
7-6 有限元網格的結點排序法
7-7 塊體附貼有限元動態接觸變寬帶的結點重新編號
7-8 結論
第八章 程式驗證及數值算例
8-1 懸臂梁自由端受集中載重的變位模擬
8-2 含圓洞平板受軸向均勻拉力作用的孔邊應力集中問題
8-3 基礎承載問題的模擬
8-4 懸臂梁受到均佈載重和剛性面接觸的變位模擬
8-5 長方形桿件受均勻拉力的動、靜態模擬
8-6 一維直桿受持續外力作用和兩直桿追撞的模擬
8-7 下滑塊體位移和速度的驗證
8-8 動態接觸接觸變寬帶和上三角滿矩陣處理的比較
8-9 塊體附貼有限元動態接觸變寬帶結點重新編號的有效性
8-10 七個塊體所組成拱的動態模擬
8-11 斜面上多個塊體受重力作用的動態模擬
第九章 結論與建議
9-1 結論
9-2 建議和未來發展方向
參考文獻
附錄 A 接觸彈簧的處理
附錄 B 接觸判斷流程圖
附錄 C 塊體附貼有限元處理流程圖

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