Algorithm Of Cellular Automaton Based On Polygonal Neighbourhood

Numerical methods provide a powerful tool for the modelling of the problems which are difficult to be solved analytically.Up to today,various numerical methods such as finite element method,boundary element method and meshless method have been widely used in engineering.With the aid of numerical methods,a large number of practical problems have been solved.In this study,a polygonal neighbourhood cellular automaton(PNCA)is proposed for the simulation of solid mechanics.In the PNCA,the interior of an elastic domain is discretized into a grid of nodes distributed randomly in the domain when solving a concerned problem.Each of the nodes is considered as a cell.The nodal displacements are designated as the state quantities of the cells.According to the algorithm,each cell is defined to have a polygonal influence zone which contains several neighbouring nodes.The local rule between a cells and its neighbours is established in terms of displacement relationship by employing the interpolation scheme used in finite element method.Therefore,the proposed PNCA can be coupled with finite element method seamlessly.In an elastic problem modelled by the current method,the region near the boundary of the domain is meshed into regular finite elements directly so that boundary conditions can be applied conveniently.During solving,the finite element nodes and the random nodes in the interior of the domain are proceeded under the same PNCA frame.Following the local rule,the PNCA evolves in time series till the moment that the values of the state quantities of all the cells converge.The steady state of the PNCA is regarded as the solution of the elastic problem.Typical two-dimensional mechanics examples are used to validate the current PNCA.By comparing the analytical and numerical results,the reasonableness of the propose method is verified.The PNCA has natural parallel computability,so a parallel scheme was developed to improve the computational efficiency.In order to reduce the number of evaluation generation,a factor which is related to the heredity of the nodal displacement is introduced into the local rule.Numerical experiment prove that the computational time can be shortened significantly.The proposed PNCA is a meshless method,therefore it has great potential in the simulation of large deformation and fracture mechanics,etc.As an attempt,a preliminary algorithm which may be used to deal with material nonlinearity is also presented.