10 - Mixed formulation and constraints — complete field methods

Abstract

The aim of this chapter is to discuss mixed formulation and constraints. The set of differential equations from which the discretization process is started determines whether the formulation is referred to as mixed or as irreducible. This chapter demonstrates how elasticity problems can be dealt with in mixed form and shows how such formulations are essential in certain problems. If the operator specifying the mixed form is symmetric or self-adjoint, the formulation can proceed from the basis of a variational principle that can be directly obtained for linear problems. This chapter essentially outlines the mixed form of finite element formulation, which opens a new range of possibilities, many with potentially higher accuracy and robustness than those offered by irreducible forms. However, an additional advantage arises even in situations where, by the principle of limitation, the irreducible and mixed forms yield identical results. Therefore, the study of the behavior of the mixed form can frequently reveal weaknesses or lack of robustness in the irreducible form that otherwise would be difficult to determine. The mixed approximation expands the potential of the finite element method and presents almost limitless possibilities of detailed improvement.