基于哈密顿体系辛几何算法求解空间地基问题

地基应力和位移场的求解是岩土工程中的基本问题之一，以往的求解方法是在一类变量范围内求解，属于拉格朗日体系。利用弹性力学的哈密顿理论，通过适当的变量代换，由力学的控制方程引入对偶变量，直接将方程导入到哈密顿体系，应用分离变量法求解。在哈密顿体系下，利用辛几何的性质，在完备的解空间内将方程的解用本征向量函数展开，讨论零本征值和非零本征值对应的不同本征解及其物理意义。数值算例表明，所得结果同以往结果一致。该方法不同于传统方法，为地基的研究提供了一条新途径和思路。

Hamilton system and symplectic algorithm for space foundation

It is a basic problem to solve the space foundation in geotechnical engineering. The problem is always solved with one kind of variables in traditional methodology, which is under Lagrange system. Dual variables were employed into the governing equations of mechanics via variable substitutions. Through this way, the governing equations were transmitted into Hamiltonian system. Therefore, the methods of separation of variables can be applied to solving the problem. In the completed solution space, the eigenfunction expansion of the solutions of the equations was obtained by the use of the properties of symplectic geometry. Discussed the zero and nonzero eigenvalues of the equations and their physical meanings. Different from the traditional methods, the paper gives a direct method to solve the problem of the half space foundation