分析结构力学与有限元

分析力学历来是在动力学范围内论述的，结构力学与最优控制模拟关系的共同基础就是分析力学．这表明在结构力学与最优控制理论的架构内也应有分析力学的整套理论．本文就结构力学讲述分析力学，称分析结构力学．保守体系可用Hamilton体系的方法描述，其特点是保辛．保辛给出保守体系结构最重要的特性．有限元法是从结构力学发展的，有限元的单元刚度阵应保持对称性，其实这就是保辛．根据区段单元变形能只与其两端位移有关，就可通过数学分析得到Lagrange括号与Poisson括号，展示了其辛对偶体系、正则方程、正则变换等的内容．

Analytical structural mechanics and finite element

Traditionally, analytical mechanics is described by means of dynamics, and the common foundation for structural mechanics and optimal control is analytical mechanics. So, under the framework of structural mechanics or optimal control theory, there should also be a whole set of analytical mechanics theory, which we define as analytical structural mechanics. A conservative system can be described with the Hamilton system methodology, and its characteristic is the symplectic conservation, which is the most important feature of conservative system. The finite element method was initiated from structural mechanics, and its element stiffness matrices should be symmetric, which is, in fact, the symplectic conservation. Based on the fact that the interval deformation energy depends only on the two end displacements vector, we derive the Lagrange and Poisson brackets analytically, the symplectic duality system, the canonical equations, and the canonical transformations, etc.