FINITE ELEMENT FORMULATION BY VARIATIONAL PRINCIPLES WITH RELAXED CONTINUITY REQUIREMENTS

Abstract

This chapter focuses on finite element formulation by variational principles with relaxed continuity requirements. The finite element method has long been recognized as an extension of the well-known Ritz procedure for constructing approximate solutions to the governing variational principle associated by a given boundary value problem. The method was originally used in the analysis of solid mechanics problems and several alternative variational principles in elasticity have been employed in the finite element formulation. As in the finite element method a continuum is subdivided by finite element mesh, it is possible to modify the variational principles by allowing discontinuous fields at the interelement boundaries and hence, to create the so-called hybrid models in finite element analyses. The chapter reviews various finite element formulations for two typical partial differential equations. The first one involves the harmonic equation which has its application in many continuum mechanics problems typical of which are torsion of prismatic bars, deflection of stretched membranes, heat conduction, and potential flow. The second one is associated with the bi-harmonic equation which is associated, for example, with the bending of thin plates.