Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L2-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.