The solid variational principles of the discrete from—The variational principles of the discrete analysis by the finite element method

Abstract

This paper suggests a new solid variational principle of discrete form. Basing on the true case of the discrete analysis by the finite element method and considering the variable boundaries of the elements and the unknown functions of piecewise approximation, the unknown functions have various discontinuities at the interfaces between successive element.

Thus, we have used mathematical technique of variable boundary with discontinuity of the unknown functions, based on the conditions that the first variation vanishes immediately, to establish the solid variation principles of discrete form. It generalizes the classical and non-classical variational principles. Successive equations that have to be satisfied by the unknown functions are the convergency necessary conditions for the finite elements method (including conforming and non-conforming). They expand that convergency necessary conditions of the compatibility conditions in the internal interfaces.