Nonlinear Elastoplastic Analysis Using Nonconforming Quadtree Mesh Based on FEM with Wachspress Shape Functions

In this paper, a nonconforming quadtree finite element method (FEM) with Wachspress shape functions (WSFs) is proposed for the nonlinear analysis of elastoplastic behavior. The adaptive quadtree mesh technique used for bilinear quadrilateral (Q4) elements can result in hanging nodes along refined edges, which necessitates the construction of multiple corresponding shape functions. This paper employs the WSF within the polygonal framework to construct a unified shape function for nonconforming quadtree meshes, thereby eliminating issues related to hanging nodes. Within the framework of continuum mechanics, an implicit implementation of the von Mises elastoplastic model is proposed, and the stiffness matrix, formulated using the FEM with WSFs, is divided into elastic and plastic components, with only the plastic matrix updated during Newton–Raphson iteration, improving computational efficiency. Both homogeneous and multi-material slope stability examples are presented, validating the high accuracy and convergence of the FEM with WSFs in comparison to traditional FEM techniques. Moreover, the method exhibits excellent adaptability and robustness in handling adaptive quadtree mesh refinement with hanging nodes.