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.
翻译:在本文中, 我们展示了两种基于Hermite 的多元复合方法, 用于为机械系统生成一步数融合器。 这些方法基于使用 Hermite 多元合成器来将配置分解, 从而在配置和速度方面导致数字轨迹的连续性。 首先, 我们将 Hermite 多元合成器纳入时间分解, 并得出一步变异方法, 方法是在一个时间步骤中将 Lagrange- d' Alembert 原则分解。 其次, 我们展示了 Galerkin 方法, 通过设定一个时间步骤到零的动作方程式剩余数的加权平均值来生成一步数融合器。 我们考虑三个数字例子来理解一步变异和加勒金方法的数值性能。 我们首先研究一个双倍的粒子, 并将变异方法与加勒金方法的相应结果进行比较。 我们然后研究 解动振动振荡振荡振荡器, 以了解存在分裂力的数值行为。 最后, 我们用一个步骤方法将一个步骤和一个步骤的不精确的轨迹变动方法应用一个方向, 一种更精确的轨变动方法, 以不高的 度的 度 度 度 度 度 度 度 度 度 度 度 度 度