臺灣博碩士論文加值系統

Although many families of integration methods have been successfully developed with desired numerical properties, such as unconditional stability, numerical dissipation and second order accuracy, they are generally 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. In general, the structure-dependent integration methods are very computationally efficient for solving a general structural dynamic problem. However, each of the currently available structure-dependent methods may still have its disadvantages, such as order of accuracy, conditional stability for instantaneous stiffness hardening systems, overshoot, weak instability 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 new 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, no overshoot in both displacement and velocity. A free parameter ρ is used to control the numerical properties. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as WBZ-α method, HHT-α method and generalized-α method.
Since the proposed family method can simultaneously integrate unconditional stability and explicit formulation together, many computational efforts can be saved when compared to average acceleration method (AAM). 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.

Although many families of integration methods have been successfully developed with desired numerical properties, such as unconditional stability, numerical dissipation and second order accuracy, they are generally 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. In general, the structure-dependent integration methods are very computationally efficient for solving a general structural dynamic problem. However, each of the currently available structure-dependent methods may still have its disadvantages, such as order of accuracy, conditional stability for instantaneous stiffness hardening systems, overshoot, weak instability 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 new 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, no overshoot in both displacement and velocity. A free parameter ρ is used to control the numerical properties. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as WBZ-α method, HHT-α method and generalized-α method.
Since the proposed family method can simultaneously integrate unconditional stability and explicit formulation together, many computational efforts can be saved when compared to average acceleration method (AAM). 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.

Contents
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 7
1.3 Overview of Dissertation 8
Chapter 2 Literature Review 9
2.1 Algorithmic Measures of Integration Schemes 9
2.2 Recursive Matrix Form 12
2.3 Convergence 14
2.3.1 Consistency and Local Truncation Error 14
2.3.2 Stability 17
2.4 Numerical Properties 18
2.4.1 Spectral Radius 18
2.4.2 Relative Period Error and Numerical Damping Ratio 20
2.4.3 Overshooting 22
2.4.4 Instantaneous Degree of Nonlinearity 23
2.5 Procedure to Explore an Integration Method 25
2.6 Some Integration Methods 26
2.6.1 Overview of Integration methods 26
2.6.1.1 Newmark Family Method (NFM) 29
2.6.1.2 HHT-α Method 31
2.6.1.3 WBZ-α Method 33
2.6.1.4 Generalized -α Method 34
Chapter 3 Propose a Novel Family of Methods 42
3.1 Proposed Family Methods 42
3.2 Study of Parameters 45
3.2.1 Recursive Matrix Form 45
3.2.2 Consistency 46
3.2.3 Stability 46
3.3 Improving Stability Property 48
3.3.1 Numerical Properties 49
3.3.2 Overshooting 50
3.3.3 Overshooting in traansient response 51
3.3.4 Overshooting in steady state response 52
3.4 Implementation Details 54
Chapter 4 Implementation of CVM into OpenSees 62
4.1 OpenSees Introduction 62
4.2 OpenSees Class Hierarchy and Workflow 65
4.2.1 Modeling Classes 66
4.2.2 Finite-Element Model Classes 66
4.2.3 Analysis Classes 67
4.2.4 Numerical Classes 69
4.3 The Incremental Solution of Nonlinear Finite-Element Equation 71
4.4 CVM in OpenSees 73
4.4.1 Implementation Details 74
4.4.2 Command Manual 75
4.4.3 Modified files 75
Chapter 5 Numerical Examples 95
5.1 Basic Examples 96
5.1.1 Free Vibration Response to a Highly Nonlinear System 96
5.1.2 An Elastoplastic Structure 97
5.1.3 Forced Vibration Nonlinear SDOF System with Damping 98
5.1.4 Free Vibration Responses of 8-story Building 98
5.1.5 Forced Vibration Responses 8-story Building 99
5.1.6 Seismic Responses of 8-story Hardening System 100
5.1.7 MDOF Nonlinear Spring Mass System 100
5.2 Practical Examples 102
5.2.1 A Ten-Story RC Building 102
5.2.2 5-MW Offshore Wind Turbine 104
Chapter 6 Conclusions and Future Directions 125
6.1 Summary 125
6.2 Future Direction 127
References 128
Appendix A OpenSees Codes 136
A.1 CVM.h 136
A.2 CVM.cpp 139

 
List of Tables

Table 2.1 A wish list of characteristics for an algorithm 38
Table 5.1 Ground acceleration 101
Table 5.2 Time integration data 101
Table 5.3 Initial and highest natural frequencies 102
Table 5.4 CPU time 102
Table 5.5 Eigen analysis result 105

 
List of Figures

Figure 2.1 Algorithmic measures for integration method 40
Figure 2.2 Period distortion and amplitude decay 40
Figure 2.3 Procedure to explore an algorithm 41
Figure 3.1 Variations of eigenvalues of A with as tends to infinity 57
Figure 3.2 Variation of upper stability limit with δi for ρ = 1 57
Figure 3.3 Variation of upper stability limit with δi for ρ = 0.5 58
Figure 3.4 Variation of spectral radii with Δt/T for different ρ and δi values 58
Figure 3.5 Variation of relative period error with Δt/T for different ρ and δi values 59
Figure 3.6 Comparison of overshooting response of CVM for first initial conditions 59
Figure 3.7 Comparison of overshooting response of CVM for second initial conditions 60
Figure 3.8 Steady-state response to sine load for CVM 61
Figure 4.1 Main interface of OpenSees 77
Figure 4.2 Model of the closed-source program 77
Figure 4.3 Software design approach for finite-element analysis (OpenSees, 2015) 77
Figure 4.4 Key of symbols (McKenna F. T., 1997) 78
Figure 4.5 OpenSees framework (OpenSees, 2015) 78
Figure 4.6 Hierarchy diagram of the finite-element Model class (Jiang, 2012) 79
Figure 4.7 Class diagram of the Analysis framework (McKenna F. T., 1997) 79
Figure 4.8 Work-flow of the Analysis class for an explicit method 80
Figure 4.9 The SolutionAlgorithm classes 80
Figure 4.10 The Integrator class and its subclasses 81
Figure 4.11 Procedure of the solution of an explicit integration method 82
Figure 4.12 The incremental solution procedure for the Newmark method 83
Figure 4.13 The incremental solution procedure for the Newmark explicit method 84
Figure 4.14 Hierarchy for CVM class in OpenSees 85
Figure 4.15 The incremental solution procedure of CVM 86
Figure 4.16 Declare an integrator tags of CVM class to the classTags.h file 87
Figure 4.17 Declare an included file to the commands.cpp file 87
Figure 4.18 Add some commands to invoke CVM class from the commands.cpp file 87
Figure 4.19 CVM.h file 88
Figure 4.20 Interface for constructor function of the CVM class 89
Figure 4.21 Interface for destructor function of the CVM class 89
Figure 4.22 Interface for the newStep() function 90
Figure 4.23 Interface for the firstStep() function 91
Figure 4.24 Interface for the predictU() function 92
Figure 4.25 Interface for the formR() function 93
Figure 4.26 Interface for the update() function 94
Figure 5.1 Free vibration responses to mathematical nonlinearity example 106
Figure 5.2 An SDOF with a bilinear inelastic behavior 106
Figure 5.3 The ground acceleration record of CHY028 106
Figure 5.4 Displacement response to CHY028 and corresponding hysteretic loops 107
Figure 5.5 Displacement response to CHY028 and corresponding hysteretic loops 107
Figure 5.6 Displacement response to CHY028 and corresponding hysteretic loops 108
Figure 5.7 Displacement responses for nonlinear damped SDOF system 108
Figure 5.8 A 8 story shear-beam type building 109
Figure 5.9 Bottom story responses of 8 story building under free vibration 109
Figure 5.10 A 8 story shear beam building with external force 110
Figure 5.11 Forced vibration bottom story responses of 8 story building for LS 110
Figure 5.12 Forced vibration bottom story responses of 8 story building for SS 111
Figure 5.13 Seismic responses of top story of 8 story building subjected to earthquake record CHY028 111
Figure 5.14 A new spring mass system 112
Figure 5.15 Displacement responses to the spring mass system for 500-DOF 112
Figure 5.16 Displacement responses to the spring mass system for 1000-DOF 113
Figure 5.17 Displacement responses to the spring mass system for 2000-DOF 113
Figure 5.18 Displacement responses to the spring mass system for 3000-DOF 114
Figure 5.19 Comparison of CPU time of the spring-mass system for 500 DOF 114
Figure 5.20 Comparison of CPU time of the spring-mass system for 1000 DOF 115
Figure 5.21 Comparison of CPU time of the spring-mass system for 2000 DOF 115
Figure 5.22 Comparison of CPU time of the spring-mass system for 3000 DOF 116
Figure 5.23 The 10-story building 117
Figure 5.24 Hysteretic behavior of materials used for the 10-story building 117
Figure 5.25 Cross-sections of the elements of the 10-story RC building 118
Figure 5.26 Displacement response at of the 10-story building 119
Figure 5.27 Comparison of CPU time of AAM and CVM for solving the 10-story building 119
Figure 5.28 Flow chart of analysis of 5MW offshore wind turbine 120
Figure 5.29 Input data and its locations in FAST 120
Figure 5.30 Loading on the jacket support structure 121
Figure 5.31 Opensees model for offshore wind turbine jacket support structure 121
Figure 5.32 wind velocity data from FAST 122
Figure 5.33 Applying platform nodal load in OpenSees 122
Figure 5.34 Applying towertop nodal load in OpenSees 123
Figure 5.35 Tower top displacement responses 123
Figure 5.36 Platform displacement responses 124

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