The Extended Free Formulation of Finite Elements in Linear Elasticity

The free formulation of Bergan and Nyga˚rd (1984) has been successfully used in the construction of high-performance finite elements for linear and nonlinear structural analysis. In its original form the formulation combines nonconforming internal displacement assumptions with a specialized version of the patch test. The original formulation is limited, however, by strict invertibility conditions linking the assumed displacement field to the nodal displacements. The present paper lifts those restrictions by recasting the free formulation within the framework of a mixed-hybrid functional that allows internal stresses, internal displacements, and boundary displacements to vary independently. This functional contains a free parameter and includes the potential energy and the Hellinger-Reissner principles as special cases. The parameter appears in the higher-order stiffness of the element equations.