Chapter 7: Variational Methods

Abstract

Publisher Summary This chapter focuses on the use of variational methods in the study of the energy methods of solid mechanics. The Galerkin method can be shown to produce element matrix integral definitions that would be identical to those obtained from a variational form, if one exists. Most nonlinear problems do not have a variational form, yet the Galerkin method and other weighted residual methods can still be used. There are several reasons for using variational methods. One is that if the variational integral form is known, one does not have to derive the corresponding differential equation. Also, most of the important variational statements for problems in engineering and physics have been known for over 200 years. Another important feature of variational methods is that often dual principles exist that allow one to establish both an upper bound estimate and a lower bound estimate for an approximate solution. These can be very helpful in establishing accurate error estimates for adaptive solutions.