An edge-based finite element method (ES-FEM) with adaptive scaled-bubble functions for plane strain limit analysis

Abstract This paper deals with critical roles of cubic bubble functions for the edge-based finite element method (ES-FEM) formulation within the framework of the kinematic theorem for predicting the plastic collapse loads of structures. We show that the bubble function can be scaled by a non-zero coefficient alpha, while the volumetric locking is entirely eliminated. Based on this finding, the present method is designed, in a very small value, with a bubble function that is maximum at the center of the element. This can lead to the loss of the partition of unity. For easy reference, the present method is termed as α bES-FEM. The significant advantage of the present method lies in the rigorous treatment of the volumetric locking-free that can occur in the fully plastic range. The α bES-FEM works well with high efficiency by using triangular meshes. It achieves both simplicity and computational efficiency for implementation into packages of plastic limit analyses. In case of α bES-FEM, the locking issue can be solved conveniently by two schemes: (1) the usual piecewise linear displacements enriched with a cubic bubble function on a primal mesh of triangular elements and (2) a projection operator of strain rates through a dual mesh associated with the edges in the mesh. The optimization formulation of finite element limit analysis is written in the form of a second-order cone programming (SOCP). The well-established interior-point solvers can be exploited efficiently. The α bES-FEM using a small number of integration points enables to solve the large-scale optimization problems efficiently. In addition, an adaptive meshing procedure based on an alternative indicator of dissipation is also derived to further enhance the quality of the solution without increasing significantly the number of degrees of freedom of the model. Numerical results show the robustness of the proposed method.