E. Haugeneder, H.A. Mang:
"Admissible and Inadmissible Simplifications of Variational Methods in Finite Element Analysis";
in: "Proceedings of the IUTAM Symposium on Variational Methods in the Mechanics of Solids", Pergamon, 1980, 187 - 198.
In the analysis of thin plates and shells by the finite element method, principles of stationary value of modified potential energy - variational principles with subsidiary conditions from a mathematical viewpoint - permit derivation of algebraic constraint conditions for the components of the displacement vector. In this way, geometric boundary and interelement continuity conditions which otherwise would be violated, can be satisfied. The main advantage of such finite element techniques is the possibility to choose element displacement functions of a lower order than would be required within the framework of the principle of minimum potential energy.More recently, a number of researchers has suggested simplifications of these techniques, characterized by "a priori elimination" of Lagrangian multipliers in the functionals subject to variation, representing additional independent unknowns.
This paper contains a theoretical and numerical investigation of the consequence of such inadmissible simplifications of principles of stationary value of modified potential energy. Moreover, an admissible simplification of such variational principles is presented. It is based on approximating interelement slope-continuity with the help of sufficiently stiff rotational springs at the interelement boundary lines. This effectiveness approximation (classified as a penalty function approach) does not require formulation of algebraic constraint conditions.