A Cellular Automata Modeling Approach for Continuum Elastodynamics

Abstract

This article details a physics-based Cellular Automata (CA) modeling approach for studying the dynamic response of a solid elastic continuum. The domain of interest is discretized into rectangular cellular automata and a rule set is developed for evolving each cell’s state based on its present state and its neighbors’ states. A cell’s state is comprised of its displacement and velocity components, and its external force. It is shown that the choice of rectangular cells yields discrete equations equivalent to the centered-difference finite difference (FD) approach. However, the discrete equations are arrived at from the ‘bottom up’ using local rules vice ‘top-down’ discretization of global partial differential equations. A further distinction between the two methods concerns the location of stresses and its impact on boundary conditions: the CA approach assigns stresses to the cell faces while the FD approach assigns stress collocated with displacement components at a single node. These differences may provide important perspective on modeling arbitrary geometry with a finite difference-like approach based on cell assembly, similar to finite element analysis. Implementation of the CA paradigm using autonomous, local cells fits naturally with object-oriented programming practices and lends itself readily to distributed computing. Results are provided for an example ground-shock simulation in which a differentiated Gaussian pulse on the surface of a half-space generates the expected pressure, shear, and surface waves. Comparisons to waves computed using a staggered-grid finite difference approach demonstrate very good agreement. In addition, the simulation results suggest that the CA approach may exhibit less ‘ringing’ as waves pass, and more symmetry in left-ward and right-ward moving waves.