Locking and boundary layer in hierarchical models for thin elastic structures

In the analysis of plate-and shell-like structures using the finite element method, we may encounter difficulties owing to locking and boundary layer effects, which lead to numerical simulations with a large error. There have been considerable advances in finding out the causes and remedies of locking. However a posteriori detection of this phenomenon in the numerical results without comparing with other reference data is another issue with a great importance. Here, one simple but reliable detection method is introduced. On the other hand, boundary layer is a non-smooth singular part in the solution of the singularly perturbed boundary value problems and restricted within very thin region in the neighborhood of boundaries. If this rapidly varying behavior could not have been captured enough, approximated solution greatly deviates from the exact solution. In this study, a guideline for the optimal mesh design is provided together with mathematical analysis of this effect.