论文标题
机械系统的基于HERMITE的一步，变异和Galerkin时间集成器
Hermite-based, One-step, Variational and Galerkin Time Integrators for Mechanical Systems
论文作者
论文摘要
在本文中，我们提出了两种基于HERMITE多项式的方法，用于为机械系统提供一步数值积分器。这些方法基于使用Hermite多项式对配置的离散配置，这导致数值轨迹在配置和速度中均连续。首先，我们通过在一个时间步骤中离散lagrange-d'Alembert原理来融合使用时间限制的Hermite多项式，并得出一步变化方法。其次，我们通过将运动方程式的加权平均值在一个时间步骤中设置为零来提出盖尔金方法来得出一步数值积分器。 我们考虑三个数值示例，以了解一步变化和盖尔金方法的数值性能。我们首先研究了双孔电势的粒子，并将变异方法结果与Galerkin方法的相应结果进行了比较。然后，我们研究悬停振荡器，以了解存在耗散力的数值行为。最后，我们将提出的方法应用于具有两个自由度的非线性航空弹性系统。变分和盖尔金的一步方法都可以准确地捕获保守和非保守动力学，并具有出色的能量行为。一步盖尔金方法比一步变化方法和变分积分表现出更好的轨迹和能量性能。
In this paper, we present two Hermite polynomial based approaches to derive one-step numerical integrators for mechanical systems. These methods are based on discretizing the configuration using Hermite polynomials which leads to numerical trajectories continuous in both configuration and velocity. First, we incorporate Hermite polynomials for time-discretization and derive one-step variational methods by discretizing the Lagrange-d'Alembert principle over a single time step. Second, we present the Galerkin approach to derive one-step numerical integrators by setting the weighted average of the residual of the equations of motion over a time step to zero. We consider three numerical examples to understand the numerical performance of the one-step variational and Galerkin methods. We first study a particle in a double-well potential and compare the variational approach results with the corresponding results for the Galerkin approach. We then study the Duffing oscillator to understand the numerical behavior in presence of dissipative forces. Finally, we apply the proposed methods to a nonlinear aeroelastic system with two degrees of freedom. Both variational and Galerkin one-step methods capture conservative and nonconservative dynamics accurately with excellent energy behavior. The one-step Galerkin methods exhibit better trajectory and energy performance than the one-step variational methods and the variational integrators.
