弹性力学弱形式广义基本方程的建立和应用

建立了弹性力学中的弱形式广义基本方程，并以此为基础，检验和简单综述了第一作者以前的有关离散算子、广义差分、拟协调元和弹性力学的哈密顿正则方程的工作。广义方程包括经典微分方程和边界条件在一起，如此不仅有限元法，而且差分法都具有自然边界条件，若干不同变分原理可以从弱形式方程导出，而且是它的特殊情况，给出了它们的限制范围，并给出在弱连续条件下的势能原理，而它是协调元和非协调元的共同基础。从弱形式方程运用局部函数可以导出离散算子方程，它包括有限元方程和差分方程同在一体，拟协调元法是广义协调方程的解，自然满足平衡对弱连续条件的要求，叙述了弱形式的弹性力学哈密顿正则方程，边界条件作为非齐次项，以便于采用数值、半解析和解析计算方法。

Establishment and applications of weak form generalized basic equations in elasticity

A system of weak form generalized basic equations in elasticity is established i n this paper. Some of the past work, mainly selected from the first author, on d iscrete operator, generalized finite difference, finite element methods and Hami lton canonical equations in elasticity, is reviewed and examined by the newly es tablished equations. The generalized equations contain the classical differentia l equation and boundary condition in one form so that not only the finite elemen t methods but also the finite difference methods can posses the natural boundary conditions. Various variational principles are special cases of the weak form e quations and can be deduced directly from them. The limitations of those princip les are stated and modified potential energy theorem with weak continuity is giv en which can be applied to non-conforming finite elements. The so-called discr ete operator equations can be deduced from weak form equation by using local fun ctions in them. It contains both the finite element equations and finite differe nce equations in one equation. The quasi-conforming element methods are the exa ct solutions of generalized compatibility equation and satisfy the weak continui ty requirement naturally. A new generalized finite difference method with natura l boundary conditions is also stated. At last, the weak form Hamilton canonical equations in elasticity are given with differential equation and boundary condit ion in one form which are available for adopting numerical or semi-analytical c omp utational methods and even for the analytical methods.