Optimized Picard iteration methods for nonlinear dynamical systems with non-smooth nonlinearities, and orbital mechanics



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This work first compares the variational iteration method (VIM), the Adomian decomposition method (ADM) and the Picard iteration method (PIM) for solving a system of first order nonlinear ordinary differential equations (ODEs). A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM. It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM. The global variational iteration method is briefly reviewed, and further recast into a Local VIM, which is much more convenient and capable of predicting long term complex dynamic responses of nonlinear systems even if they are chaotic. A very simple and efficient local variational iteration method for solving problems of nonlinear science is further proposed in this work. The analytical iteration formula of LVIM is followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use. Another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate. Then a new class of time-integration-algorithms based on LVIM is presented for strongly nonlinear dynamical systems. These algorithms are far more superior to the currently common time integrators in computational efficiency and accuracy. Through examples of both smooth and non-smooth nonlinear dynamical systems, it is shown that the presently developed algorithms are far more superior to the fourth-order Runge-Kutta (RK4), HHT-α and ODE45 of MATLAB, in predicting the chaotic responses of very strongly nonlinear dynamical systems over long-periods of time. These algorithms can be gainfully employed in studying the long-term chaotic responses of beams, plates, and shells. In the area of non-smooth dynamical systems, a systematic approach is described to study the nonlinear dynamical behaviors of a two degree-of-freedom system induced by dry friction. Several dimensionless parameters related to the steady state responses of the system are derived first by using Buckingham’s π theorem. The effects of various physical parameters on the dynamical responses are investigated. For the first time in literature, we use the locus of sticking phase plotted in the sliding region to elucidate the bifurcations that exist in this system, which can be broadly divided into three main categories: the Border-Collision Bifurcation, the Sliding Bifurcations (Grazing-Sliding Bifurcation and Multi-Sliding Bifurcation), and the Fixed-Point Bifurcation. New nonlinear phenomena are found herein, including chaos caused by the Border-Collision Bifurcation and Sliding Bifurcations, the local and global change caused by the Fixed-Point bifurcation, as well as the effect of external harmonic force on the system, etc. In simulation, we adopted the OFAPI method proposed by the authors, which can be applied to the integration of non-smooth systems directly without using Henon’s method to counter the non-smoothness in the nonlinearities. Further, a new Feedback-Accelerated Picard Iteration method is presented for solving long-term orbit propagation problems and perturbed Lambert’s problems. The resulting iterative formulae are explicitly derived so that they can be directly adopted to solve problems in orbital mechanics. Several typical orbit regimes incorporating high-order gravity and air drag force are used to demonstrate the application of the proposed method in orbit propagation. In solving perturbed orbit transfer problems. The combination of it with a Fish-Scales-Growing Method successfully extends its convergence domain and provides a potential approach for solving long duration two-point boundary value problems in conservative systems. The numerical results show that the proposed method is highly precise and efficient.



Nonlinear dynamics, Computational mechanics