In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.

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Summary 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.

KEYWORDS finite strain plasticity, mixed least-squares finite element method, small strain plasticity HIGHLIGHTS • Mixed LSFEMs for small and finite strain plasticity • Construction of a continuous modified weak form for applying the Newton-Raphson method • Comparison to established methods and validation Int J Numer Methods Eng. 2019;117:141-160. wileyonlinelibrary.com/journal/nme

The main goal of this contribution is the solution of geometrically nonlinear problems using the mixed least-squares finite element method (LSFEM). An investigation of a hyperelastic material law based on logarithmic deformation measures is performed. The basis for the proposed LSFEM is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation, see e.g. Cai and Starke [1]. For the interpolation of the solution variables vector-valued Raviart-Thomas functions for the approximation of the stresses and standard Lagrange polynomials for the displacements are used. In order to show the performance of the presented formulations a numerical example is investigated, where we compare the different interpolation combinations used.