H.A. Mang, G. Hofstetter, R.H. Gallagher:

"Simplified Variational Principles - An Inadequate Theoretical Concept Resulting in Inadequate Finite Element Methods";

in: "Proceedings of the International Conference on Numerical Methods in Engineering - Theory and Applications (NUMETA '85)", J. Middleton, G.N. Pande (ed.); Balkema, Rotterdam, 1985, 39 - 50.

The attempt to formulate variational principles with subsidiary conditions without introducing Lagrange multipliers representing additional independent variables has led to the development of so-called Simplified Variational Principles (SVP). They are characterized by expressing Lagrange multipliers in terms of original field variables by means of Euler equations providing the physical interpretation of the multipliers. For systems with infinitely many degrees of freedom it is shown that the Euler equations of a Lagrange multiplier method (LMM) based on a modification of the principle of minimum of potential energy (PMIPE) do not hold for arbitrary subsidiary conditions for the corresponding SVP. By analogy, for systems with a finite number of degrees of freedom, representing finite-element approximations of systems with infinitely many degrees of freedom, the characteristics of the SVP, unlike to ones of the LMM, are found to be problem-dependent. Shortcomings of the SVP are listed and verified numerically. They include the possibility of an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement.

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