Mathematics

Some of the very well-known locking issues that were studied for many years by the finite element community arise again for recently introduced isogeometric discretizations. These discretizations present some characteristics that prevent old recipes for locking problems to be applied: they do not solve the problem, are far from optimal or present stability issues. We present here new developments for dealing with some locking difficulties present in continuum mechanics problems. For nearly-incompressible elasticity problems, we propose a new method based on the projection of volumetric stresses onto new discontinuous spaces built upon coarser grids. With the proper choice of the projection spaces, this approach is able to tackle the locking problem, guarantees the optimal convergence of the method and its stability. Moreover, the locality of the projector allows to preserve the sparsity of the stiffness matrix, that is, efficiency. In a similar way, solid shell locking problems are alleviated also by the proper projection of the involved strain components onto local spaces.