Adaptive mesh-free lower bound limit analysis using non-linear programming

An adaptive mesh-free approach is developed to compute the lower bounds of limit loads in plane strain soil mechanics problems. There is no pre-defined connectivity between nodes in the mesh-free techniques, and this property facilitates the implementation of h-adaptivity. Nodes may be added, moved, or discarded without complex changes in the data structures involved. In this regard, the Shepard mesh-free method is used in conjunction with the nodal stress rate smoothing technique and the lower bound limit analysis theory to establish a non-linear optimization problem. This problem is solved by the second-order cone programming technique, and the result is a stress field that satisfies the lower bound requirements in a non-rigorous manner. The lack of rigorousness arises from relaxation during nodal stress rate smoothing process. An error estimator is introduced by the application of Taylor series expansion, and by controlling the local error via a user-defined tolerance, the adaptive refinement strategy is established. To demonstrate the effectiveness of the proposed method, the procedure is applied to the examples of purely cohesive and cohesive-frictional soils.