| Summary: | 碩士 === 國立海洋大學 === 機械與輪機工程學系 === 91 === The first step in investigating the dynamics of a continuous time system subjected to one or multiple constraints described by a set of ordinary differential equations is to integrate to obtain trajectories that can retain the constraints. In this thesis we first convert the nonlinear dynamical system into an augmented dynamical system of the Lie type locally. In doing so, the inherent symmetry group and the (null) cone structure of nonlinear dynamical system are brought out; then the Cayley transformation and the exponential transformation are utilized to develop group preserving schemes in the augmented space. The schemes are capable of updating the augmented state point to locate automatically on the cone at the end of each time increment. By projection we thus obtain the numerical schemes on state space , which have the form similar to the Euler scheme but with stepsize being adaptive.
Then we use the updating state technique to produce the orientation vector . We also use the forth-stage Runge-Kutta method to calculate the and produce the orientation vector . In order to match the constraint we assume a new . Substituting into and solving it by the Newton-Raphson method, we may obtain . For some cases exact solution of can be obtained. With the new we update to a new , which preserves the symmetry and also retains the constraints.
Furtheremore, the schemes are shown to have the same asymptotic behavior as the original continuous system and do not induce spurious solutions or ghost fixed points. Some examples are used to test the performance of the schemes. Because the numerical implementations are easy and parsimonious and also have high computational efficiency and accuracy, theses schemes are recommended to use in the physical calculations.
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