哈密顿动力系统的高性能保结构数值方法研究

In this project, the high performance and structure-preserving methods are studied for the nonlinear Hamiltonian dynamic systems with no constrains, holonomic constrains and non-holonomic constrains. For the above three kinds of dynamic systems, based on the dual variational principle and generating functions, a unified theory for constructing structure-preserving numerical algorithms will be established and the numerical algorithms with arbitrary order of precision and with symplectic-preserving and symmetric-preserving properties are constructed. For Hamiltonian dynamic systems with holonomic constrains, the structure-preserving algorithms which can satisfy precisely the displacement and velocity constrains are studied, which can overcome the numerical difficulty result from violating constrains. For Hamiltonian dynamic systems with non-holonomic constrains, how to satisfy the non-holonomic constrains is a key problem for numerical integration. In this project, the structure-preserving algorithms which can satisfy precisely non-holonomic constrains are studied, which can resolve the difficult problem in constructing numerical method for dynamic sysmtems with non-holonomic constrains. Moreover, in this project, the symmetric property is used to analyze the algebraic structure of the matrix exponential corresponding to the periodic linear Hamiltonian dynamic systems and an accurate, efficient and structure-preserving algorithm for computing the dynamic responses of periodic linear Hamiltonian dynamic systems is established. This method can overcome the difficulties of huge computational costs and memory requirements, and gives an effective way for analyzing the periodic linear Hamiltonian dynamic systems.

研究无约束、完整约束和非完整约束非线性哈密顿动力系统高精度保结构数值算法。针对上述三类系统，基于对偶变量变分原理和生成函数方法，建立一套统一的保结构数值算法构造理论，构造具有任意阶精度的保持哈密顿系统辛结构、可逆系统对称结构的数值算法。对完整约束哈密顿动力系统，研究同时保持系统结构特性和在积分点严格满足位移和速度约束的高精度算法，解决约束违约带来的数值积分困难。对非完整约束哈密顿动力系统，精确满足非完整约束是构造数值方法的难点，本项目研究同时保持系统结构特性和严格满足非完整约束的高精度算法，解决非完整约束对数值积分方法构造提出的严重问题。利用线性周期哈密顿系统的对称性，分析周期结构对应矩阵指数的特殊代数结构，研究线性周期哈密顿系统的高精度、高效率并同时保持辛结构的数值算法，解决由于周期结构自由度数巨大导致动力分析计算量大和存贮空间要求高的关键问题，实现线性周期哈密顿系统的有效分析手段。

本项目开展了非线性哈密顿动力系统和线性周期哈密顿动力系统高性能保结构数值方法研究工作。主要工作包括： (1) 无约束非线性哈密顿系统保持辛结构和可逆结构的高精度数值方法研究。以对偶变量变分原理为基础，采用不同类型的生成函数构造不同类型的保结构数值方法，并实现任意阶精度算法的构造。(2) 完整约束非线性哈密顿系统保结构数值方法研究。采用对偶变量变分原理，选择两端状态向量的不同组合作为独立变量，以生成函数为基础构造保辛数值方法，可同时精确满足位移和速度约束，对广义位移和广义动量的高阶近似可直接得到高精度数值方法，解决约束违约带来的数值积分困难。(3) 非完整约束非线性哈密顿系统保持可逆结构、精确满足积分节点处非完整约束的高精度数值方法研究。采用两类变量的拉格朗日－达朗贝尔变分原理，广义位移和动量可独立变分，避免了用位移近似速度的问题，可以实现非完整约束的精确满足。(4) 线性周期哈密顿系统的高精度、高效率并同时保持辛结构的数值算法。利用线性周期哈密顿系统的对称性，分析周期结构对应矩阵指数的特殊代数结构，解决由于周期结构自由度数大导致动力分析计算量大和存贮空间要求高的关键问题，实现线性周期哈密顿动力系统的有效分析手段。本项目已发表刊论文16篇，其中SCI检索 12 篇，参加国际学术会议3次，培养博士生5人、硕士生1人。