弹性力学Hamilton方法广义解的适定性

WELL POSEDNESS FOR GENERALIZED SOLUTION OF ELASTICITY IN HAMILTON SYSTEM

首先引入了Ｈａｍｉｌｔｏｎ体系中平面应力弹性力学问题正则方程的Ｇａｌｅｒｋｉｎ变分方程，证 明了Ｇａｌｅｒｋｉｎ变分方程和目前文献中所用的Ｒｉｔｚ。变分方程的等价性，以及相应广义解的适定性，从而为目前的数值方法提供了理论基础．从证明过程中还可以看到广义解实际上是Ｒｉｔｚ 变分泛函的一个鞍点．

A solving method of elasticity based on Hamilton system has been widely applied, specially, its semi-analytical solution of mixed state Hamiltonian element is a great success. But the existence, uniqueness and stability of generalized solution has not been proved. In this paper, plane stress elastic problem is taken for example, Galerkin variational equation of canonical equation of its is firstly introduced. Then it is proved that Galerkin variational equation and Rity variational equation used in the present references are equivalent in a similar way to equivalent proof of energy principle and virtual work principle in finite element method. Finally, well posedness for generalized solution of elasticity in Hamilton system is proved by combining Galerkin variational principle with Rity variational principle, which differ from well posedness proof generalized solution in finite element method in that it is proved by Lax-milgram theorem. It is also seen in the process of proof of that generalized solution is actually a saddle point of Ritz function. These provide theoretical basis of numerical method used in Hamilton system now.