Development of the partition of unity based solid finite elements free from the linear dependence problem

Abstract

The partition of unity based 2D 4-node quadrilateral and 3D 8-node hexahedral, 6-node prismatic, 5-node pyramidal elements are presented. To resolve the linear dependence problem, sets of piecewise linear shape functions are proposed and adopted for the geometry and displacement interpolations. The rank deficiency was not observed with various mesh patterns and excellent convergence behaviors were observed, even when distorted meshes are used. The feasibility of the automatic procedure that improves the solution accuracy using the adaptive use of cover functions is also illustrated through several problems. Based on the error indicator proposed, the scheme automatically selects the order of cover functions in the procedure. The procedure provides good predictions for strain energy and stress with small increment of DOFs instead of any traditional local mesh refinement.