An Extension of the Partition of Unity Finite Element Method

Here, we propose an extension of the Partition of Unit Finite Element Method (PUFEM) and a numerical procedure for the solution of J2 plasticity problems. The proposed method is based in the Moving Least Square Approximation (MLSA) and is capable of overcoming singularity problems, in the global shape functions, resulting from the consideration of linear or higher order base functions, in the classical PUFEM. The classical PUFEM employs a single constant base function and results in the so-called Sheppard functions. In order to avoid the presence of singular points, the method considers an extension of the support of the classical PUFEM weight function. Moreover, by using a single constant base function, the proposed method reduces in the limit, to the classical PUFEM. Since the support of the global shape functions do overlap, the method becomes closely related to the Element Free Galerkin (EFG) method. The most important characteristic of the proposed method is that it can be naturally combined with the EFG method allowing us to impose, in some limiting sense, the essential boundary conditions, avoiding the usage of the penalty and/or multiplier methods. In order to obtain higher order global shape functions a hierarchical enhancement procedure was implemented.