Least-Squares Mixed Finite Element Formulations for Isotropic and Anisotropic Elasticity at Small and Large Strains

The performance of least-squares finite element formulations for geometrically linear and nonlinear problems is investigated in this work. We consider different elastic material behaviors as, e.g., quasi-incompressibility and transverse isotropy. Basis for the provided element formulations is a first-order system of differential equations consisting of the residual forms of the balance of momentum, a constitutive relation, and a (redundant) residual enforcing a stronger control of the balance of moment of momentum. The sum of the squared \(L^2(\mathcal{B})\)-norms of the residuals leads to a functional, which is the basis for the related minimization problem. As unknown fields the displacements (approximated in \(W^{1,p}(\mathcal{B})\)) and the stresses (approximated in \(W^{q}({\text {div}},\mathcal{B})\)) are chosen. Here, the choice of the polynomial orders of the interpolation functions for the displacements and stresses is not restricted by the so-called LBB condition; they can be chosen independently. Numerical examples for the proposed formulations are presented and compared to standard and mixed Galerkin formulations.