Preface 1. Introduction 1.1 The weighted residual methods 1.1.1 Problem description 1.1.2 Primal methods 1.1.3 Mixed methods 1.2 Application of weighted residual methods 1.2.1 Transient motions 1.2.2 Periodic motions 1.3 Finite difference methods 1.3.1 Explicit methods 1.3.2 Implicit methods 1.4 Asymptotic methods 1.4.1 Perturbation method 1.4.2 Adomian decomposition method 1.4.3 Picard iteration method References 2. Harmonic Balance Method and Time Domain Collocation Method 2.1 Time collocation in a period of oscillation 2.2 Relationship between collocation and harmonic balance 2.2.1 Harmonic balance method 2.2.2 High dimensional harmonic balance method 2.2.3 Equivalence between HDHB and collocation 2.3 Initialization of Newton-Raphson method 2.3.1 Initial values for undamped system 2.3.2 Initial values for damped system 2.4 Numerical examples 2.4.1 Undamped Dulling equation 2.4.2 Damped Duffing equation Appendix A Appendix B References 3. Dealiasing for Harmonic Balance and Time Domain Collocation Methods 3.1 Governing equations of the airfoil model 3.2 Formulation of the HB method 3.2.1 Numerical approximation of Jacobian matrix 3.2.2 Explicit Jacobian matrix of HB 3.2.3 Mathematical aliasing of HB method 3.3 Formulation of the TDC method 3.3.1 Explicit Jacobian matrix of TDC 3.3.2 Mathematical aliasing of the TDC method 3.4 Reconstruction harmonic balance method 3.5 Numerical examples 3.5.1 RK4 results and spectral analysis 3.5.2 HBEJ vs. HBNJ 3.5.3 Aliasing analysis of the HB and TDC methods 3.5.4 Dealiasing via a marching procedure Appendix References 4. Application of Time Domain Collocation in Formation Flying of Satellites
4.1 TDC searching scheme for periodic relative orbits 4.2 Initial values for TDC method 4.2.1 The C-W equations 4.2.2 The T-H equations 4.3 Evaluation of TDC search scheme 4.3.1 Projected closed orbit 4.3.2 Closed loop control 4.4 Numerical results Appendix References 5. Local Variational Iteration Method 5.1 VIM and its relationship with PIM and ADM 5.1.1 VIM 5.1.2 Comparison of VIM with PIM and ADM 5.2 Local variational iteration method 5.2.1 Limitations of global VIM 5.2.2 Variational homotopy method 5.2.3 Methodology of LVIM 5.3 Conclusion References 6. Collocation in Conjunction with the Local Variational Iteration Method 6.1 Modifications of LVIM 6.1.1 Algorithm-1 6,1.2 Algorithm-2 6.1.3 Algorithm-3 6.2 Implementation of LVIM 6.2.1 Discretization using collocation 6.2.2 Collocation of algorithm-1 6.2.3 Collocation of algorithm-2 6.2.4 Collocation of algorithm-3 6.3 Numerical examples 6.3.1 The forced Duffing equation 6.3.2 The Lorenz system 6.3.3 The multiple coupled Duffing equations 6.4 Conclusion References 7. Application of the Local Variational Iteration Method in Orbital Mechanics 7.1 Local variational iteration method and quasi-linearization method 7.1.1 Local variational iteration method 7.1.2 Quasi-linearization method 7.2 Perturbed orbit propagation 7.2.1 Comparison of local variational iteration method with the modified Chebyshev picard iteration method 7.2.2 Comparison of FAPI with Runge-Kutta 12(10) 7.3 Perturbed Lambert's problem 7.3.1 Using FAPI 7.3.2 Using the fish-scale-growing method 7.3.3 Using quasilinearization and local variational iteration method 7.4 Conclusion References 8. Applications of the Local Variational Iteration Method in Structural Dynamics
8.1 Elucidation of LVIM in structural dynamics 8.1.1 Formulas of the local variational iteration method 8.1.2 Large time interval collocation 8.1.3 LVlM algorithms for structural dynamical system 8.2 Mathematical model of a buckled beam 8.3 Nonlinear vibrations of a buckled beam 8.3.1 Bifurcations and chaos 8.3.2 Comparison between HHT and LVIM algorithms 8.4 Conclusion Appendix A Appendix B Appendix C References Index
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