约束Hamilton系统的辛算法及其在多体系统动力学中的应用
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摘要
约束Hamilton系统的辛算法是近几年出现的新的数值方法，约束Hamilton系统的一个重要领域就是非耗散的约束机械系统，所以约束Hamilton系统的辛算法的应用自然与某些类的多体系统动力学相关。为此，本文联系多体系统进行研究。非树形（或带约束）多体系统动力学模型一般都可具有微分/代数方程组形式，这样的方程称为指标-3问题，它的求解是一难题。在求解这样的微分/代数方程组的过程中，经常出现所谓的‘违约问题’,以及刚性方程问题，这都是数值方法求解的难点。目前流行的数值积分方法都没有很好地解决这些问题以满足现代工程技术中计算机辅助工程发展的需要。
在本文的研究中，主要基于:微分/代数方程组，可以看成代数约束流形上的常微分方程组这一现代数学的思想方法，以及常微分方程的数值解，可以看成相应的摄动常微分方程的精确解,这一简单思想。我们以此两方面为本文的主线，首先系统地阐述了辛算法赖以存在的基础?现代微分流形理论特别是辛几何理论；然后较为详细地论述流形上动力学方程的辛算法理论；接着讨论了约束Hamilton系统动力学的约束正则形式方程模型、多体统动力学与辛算法、算法优劣的标准问题；最后，利用流形上的辛算法理论进行仿真研究。本文包括以下八章：
1.回顾了流形上的动力学方程的辛算法的发展情况，最后，提出了本文的研究课题；
2.介绍了约束Hamilton系统动力学方程数值算法的基础的数学理论。主要包括拓扑学理论，现代微分流形理论两个方面；
3.较为系统地阐述了约束Hamilton系统动力学的系统的几何理论，但就哈密顿系统理论而言，关于它的数学基础理论，必需是辛几何理论。所以在这一章的引言部分介绍辛算法的历史；然后阐述一般的辛流形理论;接着建立辛流形M上的算法理论；利用此理论，就流形上动力学方程用辛Runge-Kutta法给出辛算法的具体的算式；为了保持辛算法理论的完整性，也较为系统地阐述了R2n空间中的辛算法的构造理论；最后总结本章内容；
4. 辛算法的基础是Hamilton正则方程，在本章中推导出了约束Hamilton系统动力学的约束的正则方程形式，同时也详细地阐述了将一般约束形式多体系统动力学方程化为约束正则形式的动力学方程的方法。在最后，介绍了一般的微分/代数方程组的一些重要概念，如指标等；
5. 多体系统动力学方程数值方法的发展历程及探讨辛算法的应用；
6.在本章系统地阐述了目前流行的关于判断约束Hamilton系统动力学方程算法优劣的两个标准：几何约束误差、能量约束误差。
7.在本章第一部分，通过一个经典的算例曲柄连杆机构，利用辛算法与WU S. D.，Chiou J. C.and Lin Y. C.等在Mechanics Structures & Machines上发表的算法的仿真结果进行比较；在第二部分又通过对三极摆这一算例，利用我们构造的合成辛算法与现在较为流行的违约校正法、R2n空间辛算法进行比较分析，其中重点采用了计算系统的位置约束误差、速度约束误差、Hamiltonian 量等进行仿真比较;第三部分用一个三连杆机构做算例，计算了较多的量，说明用约束正则方程计算约束力的的优点，并简单说明建立约束Hamilton正则方程的动力学的一个困难等。
8.总结了全文的工作，提出了几个有价值的研究课题。
Symplectic methods of the constrained Hamiltonian systems are new numerical methods occurring in recent years. The constrained Hamiltonian systems are relative to the multibody systems, for example, the mechanical systems where dissipative forces can be neglected. The mathematical model of the multibody systems of the constrained or closed loops is of differential- algebraic equations forms. It is a set of differential algebraic equations of index-3, and we call for high index questions. Presently, it is a problem for us to solve the equations. Stiffness question and drift-the errors in the original constraint often occur during the process of solving them. And popular numerical methods have well not solved the questions.
This paper base mainly on the idea that the differential algebraic equation is an ordinary differential equation on a manifold, and the simple idea a numerical solution of ordinary is analytic solution of a perturbed differential equation. First we discuss differential manifold. Second, to state the theory of symplectic integrator. Third, we construct model about the constrained Hamiltonian system dynamics, and discuss rulers of algorithm. At last, we are simulating. This paper is organized as follow:
1. To review the symplectic methods, and present some topics of research for us;
2. To state the mathematical theory of dynamics;
3. To construct the theory of symplectic method of equations of the constrained Hamiltonian systems dynamics;
4. To construct canonical equation for the constrained Hamiltonian systems dynamics;
5. To review the numerical methods of the multibody systems dynamics and applications of the symplectic methods;
6. To construct rulers of the alglorithm;
7. To illustrate;
8. Conclusion.
在本文的研究中，主要基于:微分/代数方程组，可以看成代数约束流形上的常微分方程组这一现代数学的思想方法，以及常微分方程的数值解，可以看成相应的摄动常微分方程的精确解,这一简单思想。我们以此两方面为本文的主线，首先系统地阐述了辛算法赖以存在的基础?现代微分流形理论特别是辛几何理论；然后较为详细地论述流形上动力学方程的辛算法理论；接着讨论了约束Hamilton系统动力学的约束正则形式方程模型、多体统动力学与辛算法、算法优劣的标准问题；最后，利用流形上的辛算法理论进行仿真研究。本文包括以下八章：
1.回顾了流形上的动力学方程的辛算法的发展情况，最后，提出了本文的研究课题；
2.介绍了约束Hamilton系统动力学方程数值算法的基础的数学理论。主要包括拓扑学理论，现代微分流形理论两个方面；
3.较为系统地阐述了约束Hamilton系统动力学的系统的几何理论，但就哈密顿系统理论而言，关于它的数学基础理论，必需是辛几何理论。所以在这一章的引言部分介绍辛算法的历史；然后阐述一般的辛流形理论;接着建立辛流形M上的算法理论；利用此理论，就流形上动力学方程用辛Runge-Kutta法给出辛算法的具体的算式；为了保持辛算法理论的完整性，也较为系统地阐述了R2n空间中的辛算法的构造理论；最后总结本章内容；
4. 辛算法的基础是Hamilton正则方程，在本章中推导出了约束Hamilton系统动力学的约束的正则方程形式，同时也详细地阐述了将一般约束形式多体系统动力学方程化为约束正则形式的动力学方程的方法。在最后，介绍了一般的微分/代数方程组的一些重要概念，如指标等；
5. 多体系统动力学方程数值方法的发展历程及探讨辛算法的应用；
6.在本章系统地阐述了目前流行的关于判断约束Hamilton系统动力学方程算法优劣的两个标准：几何约束误差、能量约束误差。
7.在本章第一部分，通过一个经典的算例曲柄连杆机构，利用辛算法与WU S. D.，Chiou J. C.and Lin Y. C.等在Mechanics Structures & Machines上发表的算法的仿真结果进行比较；在第二部分又通过对三极摆这一算例，利用我们构造的合成辛算法与现在较为流行的违约校正法、R2n空间辛算法进行比较分析，其中重点采用了计算系统的位置约束误差、速度约束误差、Hamiltonian 量等进行仿真比较;第三部分用一个三连杆机构做算例，计算了较多的量，说明用约束正则方程计算约束力的的优点，并简单说明建立约束Hamilton正则方程的动力学的一个困难等。
8.总结了全文的工作，提出了几个有价值的研究课题。
Symplectic methods of the constrained Hamiltonian systems are new numerical methods occurring in recent years. The constrained Hamiltonian systems are relative to the multibody systems, for example, the mechanical systems where dissipative forces can be neglected. The mathematical model of the multibody systems of the constrained or closed loops is of differential- algebraic equations forms. It is a set of differential algebraic equations of index-3, and we call for high index questions. Presently, it is a problem for us to solve the equations. Stiffness question and drift-the errors in the original constraint often occur during the process of solving them. And popular numerical methods have well not solved the questions.
This paper base mainly on the idea that the differential algebraic equation is an ordinary differential equation on a manifold, and the simple idea a numerical solution of ordinary is analytic solution of a perturbed differential equation. First we discuss differential manifold. Second, to state the theory of symplectic integrator. Third, we construct model about the constrained Hamiltonian system dynamics, and discuss rulers of algorithm. At last, we are simulating. This paper is organized as follow:
1. To review the symplectic methods, and present some topics of research for us;
2. To state the mathematical theory of dynamics;
3. To construct the theory of symplectic method of equations of the constrained Hamiltonian systems dynamics;
4. To construct canonical equation for the constrained Hamiltonian systems dynamics;
5. To review the numerical methods of the multibody systems dynamics and applications of the symplectic methods;
6. To construct rulers of the alglorithm;
7. To illustrate;
8. Conclusion.
引文
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