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論文中文名稱:發展新一族逐步積分法及其應用 [以論文名稱查詢館藏系統]
論文英文名稱:Development of a Novel Family of Direct Integration Methods in Nonlinear Time History Analysis and Its Application [以論文名稱查詢館藏系統]
英文姓名:Tran Ngoc Cuong
指導教授英文名:Shuenn-Yih Chang
口試委員中文名:張順益; 尹世洵; 楊元森; 鍾立來; 吳俊霖;
中文關鍵詞:無條件穩定; 數值耗散; 非線性動力分析; 二階精度; 結構相關的積分方法
英文關鍵詞:Unconditional stability; numerical dissipation; nonlinear dynamic analysis; second-order accuracy; structure-dependent integration method;

在本研究中將提出新一族具有結構相依特性的逐步積分法。這族積分法具有一般所需要的數值特性,如無條件穩定、外顯式、二階精確度,可控制的數值阻尼和自啟動機制等,並且是由單一的一個自由參數p來控制其數值特性。此自由參數可以視為是高頻振態之數值阻尼指標參數。一般而言,當p在0和1之間時,此族積分法可以具有理想的數值阻尼,其中p = 1時,可獲得零數值阻尼,而p = 0時,可得到最大的數值阻尼。特別值得注意的是當振動頻率趨近於無窮大時,p是此積分法的頻譜半徑。
由於本研究所提出的這族具有結構相依特性的逐步積分法可以同時具有無條件穩定和外顯式的特性,因此其計算效率將明顯優於與其相對應而由學者Wood, Bossak和Zienkiewicz所發展出來的WBZ內隱式積分法。無條件穩定性意味著積分時間步長的擇定並不需要考慮穩定條件的要求,而外顯式意味著每一步的計算並不需要使用到非線性迭代。為了進一步驗證此族積分法的計算效率,特別將此族積分法增加到結構模擬分析軟體OpenSees中。由於OpenSees本身即可模擬各種不同的材料行為以及各式各樣不同的結構物,因此可以用來模擬複雜的結構系統及其非線性行為。利用數值例子來驗證此族積分法的可行性及其數值特性。最後,經由使用CPU時間的比較來確認其極為優異的計算效率。
論文英文摘要:Although many families of integration methods with desired numerical properties, such as unconditional stability and numerical dissipation, have been successfully developed for structural dynamics, they generally are implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. Consequently, some structure-dependent integration methods have been developed to improve the computational efficiency since they can integrate unconditional stability and explicit formulation together. In general, the structure-dependent integration methods are very computationally efficient for solving a general structural dynamic problem, where the total response is dominated by low-frequency modes while the high-frequency responses are of no interest. However, each of the currently available structure-dependent methods may still have its disadvantages, such as no numerical dissipation, conditional stability for instantaneous stiffness hardening systems and no self-starting. On the other hand, since the structure-dependent integration methods were successfully developed in the near recent, they are not widely known and commonly adopted although they can simultaneously possess some favorable numerical properties.

A novel family of structure-dependent integration methods is proposed in this work. This family method can have desired numerical properties, such as unconditional stability, explicit formulation, second-order accuracy, controllable numerical dissipation, and self-starting. A free parameter p is used to control the numerical properties. In addition, it can be considered as an indicator variable of numerical dissipation for high-frequency modes. In general, the proposed family method can have zero damping if p = 1 is adopted while it can have the largest numerical damping for the case of p = 0. Notice that p is the asymptotic value of the spectral radius as the natural frequency tends to infinity.

Since the proposed family method can simultaneously integrate unconditional stability and explicit formulation together, many computational efforts can be saved when compared to its corresponding implicit family of integration methods, i.e., the family method developed by Wood, Bossak, and Zienkiewicz. Unconditional stability implies that there is no limitation on the choice of step size based on stability consideration. An explicit formulation implies that it involves no nonlinear iteration for each time step. To numerically confirm the computational efficiency of the proposed family method, it is implemented into OpenSees software, which is an open-source software framework. As a result, the responses of various types of structural systems subject to almost any dynamic loading can be simulated by using OpenSees software. These structural systems can be mimicked by a large number of degrees of freedom and their structural properties can be simulated by a variety of mathematical models to account the very complicated nonlinear behaviors. Numerical examples are used to examine the feasibility and confirm the numerical properties of this proposed family method. Finally, the computational efficiency of the proposed family method is also numerically investigated by comparing the consumed CPU time with that involved by the other integration methods.
論文目次:Chapter 1 Introduction 1
1.1 Problem Statement 1
1.1.1 Explicit and Implicit 2
1.1.2 Structure-Dependent Methods and Its Applications 5
1.2 Objective of the Research 6
1.3 Overview of Dissertation 7
Chapter 2 Literature Review 9
2.1 Algorithmic Measures of Integration Schemes 9
2.2 Recursive Matrix Form 11
2.3 Convergence 13
2.3.1 Consistency and Local Truncation Error 13
2.3.2 Stability 15
2.4 Numerical Properties 16
2.4.1 Spectral Radius 16
2.4.2 Relative Period Error and Numerical Damping Ratio 18
2.4.3 Overshooting 19
2.4.4 Instantaneous Degree of Nonlinearity 20
2.5 Procedure to Explore an Integration Method 21
Chapter 3 Some Integration Methods in Structural Dynamic 27
3.1 History of Integrator Transient Algorithms 27
3.2 Newmark Family Method (NFM) 34
3.3 HHT Method 36
3.4 WBZ Method 38
3.5 Chung & Hulbert Method 40
3.6 Chang Newmark Family Method 43
3.7 Chang HHT Two-Step Method (CHHT2) 46
3.8 Chang WBZ Two-Step Method (CWBZ2) 49
3.9 KR Method 52
3.10 Summary 54
Chapter 4 Propose a Novel Family of Methods 71
4.1 Proposed Family Methods 71
4.2 Study of Parameters 75
4.2.1 Recursive Matrix Form 76
4.2.2 Consistency 77
4.2.3 Stability 77
4.2.4 Spectral Radius 78
4.2.5 Relative Period Error 79
4.2.6 Numerical Damping Ratio 80
4.3 Improving Stability Property 81
4.3.1 Numerical Properties 82
4.3.2 Overshooting 84
4.4 Implementation Details 85
4.5 Summary 88
Chapter 5 Implementation of CWBZ1 into OpenSees 97
5.1 OpenSees Introduction 97
5.2 OpenSees Class Hierarchy and Workflow 100
5.2.1 Modeling Classes 101
5.2.2 Finite-Element Model Classes 101
5.2.3 Analysis Classes 103
5.2.4 Numerical Classes 105
5.3 The Incremental Solution of Nonlinear Finite-Element Equation 106
5.4 CWBZ1 in OpenSees 109
5.4.1 Implementation Details 109
5.4.2 Command Manual 110
5.4.3 Modified files 111
Chapter 6 Numerical Examples 131
6.1 Basic Examples 132
6.1.1 Mathematical Example 132
6.1.2 An Elastoplastic System 133
6.1.3 Free Vibration Responses of 6-story Building 135
6.1.4 Seismic Responses of the 6-story Softening System 138
6.1.5 Harmonic Responses of 6-story Hardening System 139
6.1.6 Computational Efficiency 140
6.2 Practical Examples 141
6.2.1 A Reinforced Concrete (RC) Frame 141
6.2.2 A Ten-Story RC Building 143
Chapter 7 Conclusions and Future Directions 171
7.1 Summary 171
7.2 Future Direction 172
Appendix A OpenSees Codes 175
A.1 CWBZ1.h 175
A.2 CWBZ1.cpp 178
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