Lagrange multipliers, penalty factors, and basis transformation

Abstract

Abstract In the preceding chapter energy-based finite elements were discussed, paying most attention to the displacement method as this appears to have become dominant in solids mechanics applications. Moreover stress-type elements can be obtained more directly by applying the Galerkin method directly to the governing differential equations (as is done in Section 17.6) whilst the main incentive for the use of hybrid and mixed elements is to increase interelement stress and/or displacement continuity and this can be accomplished more generally, in any physical application, with the use of the techniques to be discussed in this chapter. Indeed the penalty factor and Lagrange multiplier techniques form a linear minimization problem which is a special case of the use of such techniques in non-linear optimization. The latter ar<: further discussed in Chapter 22 where many other useful relationships and applications to the finite element method are to be found.