Unlocking the secrets of locking: Finite element analysis in planar linear elasticity - Ainsworth, Mark; Parker, Charles (Computer methods in applied mechanics and engineering, 2022)
Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking” or “non-robustness”, is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p-version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p≥4. Numerical examples are provided throughout to illustrate the theoretical results.