Coupling Method Of FEM And CA In 2D Elastostatics

Cellular automaton(CA)is a dynamic system of discrete space and time and has been applied in various fields as a promising simulation tool.In this study,a meshless algorithm based on the concept of CA is proposed for 2D elastostatics.This algorithm is then incorporated with the Finite Element Method(FEM)to develop a hybrid scheme to calculate the displacement and stress in 2D solids.According to the hybrid scheme,a 2D elastic domain is divided into two sub-domains,including the zone near the boundary meshed by FEM and the rest discretized into a grid of random CA nodes.In the CA sub-domain,each arbitrary node is assumed to be the centre of a virtual hexagonal cover that may enclose several neighbouring nodes.Among them,six nodes are selected and their displacements can be expressed as the displacements of the hexagon's centre and vertexes by FEM.Then,the displacement of the arbitrary node is related to its selected neighbours through an inverse process.While in the near-boundary sub-domain where FEM elements exist,the displacement relationship between a typical node and its neighbouring nodes are derived from the FEM stiffness matrix directly.The two types of displacement relationship are combined under the CA frame and they are defined as the local rule of the CA.In this way,FEM and CA are coupled seamlessly.To solve a 2D elastostatics,the nodal displacement is calculated by CA's dynamical evolution,while the nodal stress at the specific can be evaluated by conventional FEM scheme.The CA's evolution for solution can start and proceed from any node in any updating sequence,so the current numerical method has great advantage in parallel computing.Therefore,a parallel scheme is proposed in this study.The correctness of the hybrid numerical approach is verified by several examples.