准坐标描述的拉格朗日方程及其在非线性动力学系统中的能量约束型数值积分方法

Quasi-Lagrangian equations and its energy-conservative numerical integration for nonlinear dynamic systems

在动力学问题的数值仿真中,保持离散后的系统与原始系统相同的特性和结构是非常重要的.对于使用哈密顿方程建模并使用辛积分器求解的保守系统,其系统的能量可以在求解过程中守恒.然而,有许多系统是在跟随坐标系下建模的,且模型中包含非线性耗散项,除非将耗散造成的能量损失一并考虑在内,否则很难保持系统的能量守恒特性。因此,在数值模拟期间保持总能量守恒是至关重要的。为了解决这个问题,本研究使用准坐标描述的拉格朗日方程来描述建立在跟随坐标系下的动力学系统,并提出了一种新的能量约束型欧拉积分器来求解建立的系统.该积分器将能量守恒定律作为约束方程,并从最初的常微分方程中构建一个新的微分代数方程通过隐式地求解该微分代数方程,获得保持系统总能量守恒的数值结果.为了将所提出的方法与常用的辛积分器和普通数值积分器进行比较,分别将各个积分器用于求解线性和非线性系统的初值问题。常用的辛积分器由于准坐标描述的拉格朗日方程的非对称结构,无法保持此类模型的总能量守恒特性.与之相对的,本文提出的能量约束型欧拉积分器能较严格地保持系统的能量守恒属性,且对保守和非保守系统都有较好的效果.

In dynamic simulations,it is important to maintain the same properties and structure as the original system.For conservative systems modeled using Hamiltonian equations and solved with symplectic numerical solvers,the energy-conservative property of the original system can be maintained.However,there are also many dynamic systems modeled under body-fixed frames and contain nonlinear dissipation,making it difficult to maintain energy conservation unless the energy loss from dissipation is also considered.Maintaining total energy conservation during numerical simulations is therefore crucial.To address this problem,this research uses quasi-Lagrangian equations to model dynamic systems described in body-fixed frames.A novel energy-constraint Euler integrator is proposed to solve the quasi-Lagrangian equations of the dynamic model.This integrator takes the energy conservation law as an algebraic constraint and forms a differential algebra equation(DAE)system from the original ordinary differential equation(ODE)system.By solving the DAE system implicitly,the numerical results that maintain the total energy conservation of the system are obtained.To compare the proposed method to common symplectic and non-symplectic numerical integrators,initial value problems for the proposed models are solved.The common symplectic solvers cannot maintain the total energy conservation property of the dynamic models described in quasi-coordinates due to the asymmetric structure of the quasi-Lagrangian equations.On the other hand,the proposed energy-constraint Euler integrator strictly maintains the energy conservation property of the system,working well for both conservative and non-conservative dynamic systems.