Abstract Energy conserving and dissipative algorithm designs in Hamilton’s canonical equations via Petrov–Galerkin time finite element methodology are proposed in this paper that provide new avenues with high-order convergence rate, improved solution accuracy, and controllable numerical dissipation. Lagrange quadratic shape function in time with flexible interpolation points are considered to approximate the solution over a time interval. Instead of specifying the weight functions, two algorithmic parameters, namely, a principal root ( ρ q ∞ ) and a spurious root ( ρ p ∞ ), are introduced to formulate a generalized weight function, which enables us to introduce controllable numerical dissipation with respect to displacement and momenta while preserving high-order convergence, and features with improved solution accuracy. A family of third-order accurate time finite element algorithms with controllable dissipation and improved solution accuracy is presented in both the homogeneous and non-homogeneous dynamic problems; and via setting ( ρ q ∞ , ρ p ∞ ) = ( 1 , 1 ) , this third-order family of algorithms directly leads to a new family of fourth-order accurate non-dissipative algorithms in general homogeneous problems; and the fourth-order accuracy is also preserved in non-homogeneous problems when the third-order time derivative of the external excitation has the order of O ( Δ t n ) ( n ≤ 1 ) . Numerical examples are performed to demonstrate the pros/cons for the conserving properties of various schemes in the proposed Petrov–Galerkin time finite element of algorithms.

中文翻译：

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哈密​​顿力学的广义能量守恒/耗散时间有限元方法

摘要 本文通过Petrov-Galerkin 时间有限元方法提出了Hamilton 经典方程中的能量守恒和耗散算法设计，提供了具有高阶收敛速度、提高求解精度和可控数值耗散的新途径。具有灵活插值点的拉格朗日二次形状函数被认为是在一个时间间隔内逼近解。不是指定权重函数，而是引入两个算法参数，即主根 ( ρ q ∞ ) 和伪根 ( ρ p ∞ ) 来制定广义权重函数，这使我们能够引入可控的数值耗散到位移和动量，同时保持高阶收敛性，以及具有更高求解精度的特征。在齐次和非齐次动力学问题中，提出了一系列具有可控耗散和提高求解精度的三阶精确时间有限元算法；并且通过设置 ( ρ q ∞ , ρ p ∞ ) = ( 1 , 1 ) ，这个三阶算法族在一般齐次问题中直接导致了一个新的四阶精确非耗散算法族；当外部激励的三阶时间导数为 O ( Δ tn ) ( n ≤ 1 ) 阶时，四阶精度也保持在非齐次问题中。执行数值示例以证明在所提出的 Petrov-Galerkin 时间有限元算法中各种方案的守恒性质的优缺点。