Maximum-Entropy Meshfree Method for the Reissner-Mindlin Plate Problem based on a Stabilised Mixed Weak Form

Meshless methods, such as the Element Free Galerkin (EFG) method, hold various advantages over mesh-based techniques such as robustness in large-deformation problems and high continuity. The Reissner-Mindlin plate model is a particularly popular choice for simulating thin structures.
It is well known in the Finite Element and Meshless literature that the simplest numerical treatments of the Reissner-Mindlin model lead to shear-locking which in turn produces erroneous results. This is due to the inability of the approximation functions to satisfy the Kirchoff constraint in the thin-plate limit.
A recent advance in the area of meshless approximation schemes are Maximum-Entropy (MaxEnt) approximants. MaxEnt schemes provide a weak Kronecker-delta property on convex node sets which allows the direct imposition of Dirichlet (essential) boundary conditions.
In this work, we derive a shear-locking free meshless method using MaxEnt approximants by consider- ing a stabilised mixed weak form. We include a scalar parameter which splits the energy from the shear bilinear form into two parts; the first is formed from the displacement fields only and the second from the independently interpolated shear strain field and the displacement fields. This splitting greatly eases the satisfaction of the LBB stability condition. We then eliminate the independently interpolated shear strain field using a localised projection operator, related to the “volume-averaged pressure” technique, which produces a final system of equations in the original displacement unknowns only. We show the good performance of the method for a variety of test problems.