哈密顿动力系统的高性能保结构数值方法研究

项目名称： 哈密顿动力系统的高性能保结构数值方法研究

项目编号： No.11272076

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 高强

作者单位： 大连理工大学

项目金额： 78万元

中文摘要： 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。

中文关键词： 哈密顿系统；非线性；保结构；周期结构；数值方法

英文摘要： 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 correspon

英文关键词： Hamiltionian systems；nonlinear；structure-preserving；periodic structure；numerical methods