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Different approaches for mixed LSFEMs in hyperelasticity: Application of logarithmic deformation measures

We present geometrically nonlinear formulations based on a mixed least‐squares finite element method. The L2‐norm minimization of the residuals of the given first‐order system of differential equations leads to a functional, which is a two‐field formulation dependent on displacements and stresses. Based thereon, we discuss and investigate two mixed formulations. Both approaches make use of the fact that the stress symmetry condition is not fulfilled a priori due to the row‐wise stress approximation with vector‐valued functions belonging to a Raviart‐Thomas space, which guarantees a conforming discretization of H(div). In general, the advantages of using the least‐squares finite element method lie, for example, in an a posteriori error estimator without additional costs or in the fact that the choice of the polynomial interpolation order is not restricted by the Ladyzhenskaya‐Babuška‐Brezzi condition (inf‐sup condition). We apply a hyperelastic material model with logarithmic deformation measures and investigate various benchmark problems, adaptive mesh refinement, computational costs, and accuracy.