时间有限元与保辛

TIME DOMAIN FEM AND SYMPLECTIC CONSERVATION

初值问题的时间积分经常采用差分近似。保守体系的特点是保辛。但通常的差分格式并不考虑保辛的性质,即使对保守体系。但有限元是自动保辛的,即使采用小参数摄动时,仍能保辛。文中先验证时间积分有限元的数值效果。再考虑齐次有阻尼Duffing方程的积分,它也可变换到时变的Hamilton系统,采用正则变换法摄动然后保辛积分可得满意结果。进一步又数值积分了二自由度齐次有阻尼Duffing方程的积分。

Time integration of an initial value problem uses the finite difference scheme quite often. The characteristic of conservative system is symplectic conservation. However, the usual FDM(finite difference method) schemes disregard the symplectic conservation for conservative systems. The FEM (finite element method) method is naturally symplectic conservative, even for perturbation method. The time integration by means of FEM is verified numerically first. Then the numerical integration of an initial value problem of homogeneous damping Duffing equation, which can be transformed to a time variant Hamilton system, is considered. Satisfactory numerical result is shown via the symplectic conservative integration method based on using canonical transformation perturbation approach. A damping non-linear vibration system with two degrees of freedom is numerically integrated for demonstration.