A MIXED VARIATIONAL PRINCIPLE FOR FINITE ELEMENT ANALYSIS

MICHAEL LEE DAY, Purdue University

Abstract

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of the potential and complementary energy principles. The principle evolves from alleviating two drawbacks associated with more traditional approaches. These include the problems with interelement continuity and the favoring of displacements over stresses, or vice versa. Although various formulations, such as the hybrid and previous mixed models, have touched upon parts of this idea, they are still lacking in some respect. In order to achieve the desired effect of relaxing the troublesome continuity requirements and to put the displacements and stresses on a more equal footing, it was necessary to modify the potential and complementary principles with Lagrange multipliers. Then both modified principles are simultaneously employed so as not to bias either the compatibility or equilibrium conditions. This gives birth to a finite element formulation which abandons the nodal concept and instead has generalized parameters as its degrees of freedom. As a direct consequence of the manner in which it was derived, this mixed principle has several advantages. The relaxed continuity requirements allow a more convenient choice of trial functions. Also, equally good displacements and stresses result. Among the favorable by-products of the derivation are that proper sophistication of the displacements and stresses allows the solution to lie between the upper and lower bound properties of the pure models. Furthermore, the degrees of freedom are not restricted to nodal quantities, allowing easier construction of higher order elements. Finally, all compatibility and equilibrium conditions are satisfied in the element interior, which is not so in previous mixed models.

Degree

Ph.D.

Subject Area

Aerospace materials

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