Cellular automata as an approximate method in structural analysis

Abstract

This thesis deals with the mathematical idealization denoted cellular automata (CA) and the applicability of this method to structural mechanics. When using CA, all aspects such as space and time are discrete. This discrete nature of CA allows for ease of interaction with digital computers, while physical phenomena which are essentially discrete in nature can be simulated in a realistic way. The application of such a novel numerical method opens up new possibilities in structural analysis. In this study, the fundamentals of CA are studied to determine how the parameters of the method are to be evaluated and applied to the established field of structural analysis. Attention is given to the underlying mathematics of structural mechanics, as well as approximate methods currently used in structural analysis, e.g. the finite element method (FEM) and the boundary element method (BEM). For structural simulations performed with the CA implemented in this study, machine learning based on a genetic algorithm (GA) is used to determine optimum rules for the CA, using finite element, boundary element and analytical approximations as the basis for machine learning. Rather unconventionally, symmetric problems in structural analysis are analyzed using asymmetric rules in the machine learning process, where the symmetry of the solution found is used as a quantitative indication of the quality of the solution. It is demonstrated that the quality of the asymmetric rules is superior to the quality of symmetric rules, even for those problems that are symmetric in nature. Finally, exploiting the inherent parallelism of CA, it is shown that distributed computing can greatly improve the efficiency of the CA simulation, even though the speed-up factor is not necessarily proportional to the number of sub lattices used. The distributed computing device itself is constructed by combining 18 obsolete Pentium computers in a single cluster. In terms of CPU performance the constructed distributed computer is not state-of-art, but it is constructed with no hardware costs whatsoever. In addition, the software used in assembling the cluster is in the public domain, and is also available free of charge. Such a parallel configuration is also known as the poor man’s computer. However, faster and more modern machines can simply be added to the existing cluster as and when they become available. While CA are recent additions to the “tools” used in structural analysis, increased use of CA as distributed computing becomes more widely available is envisaged, even though the CA rules are at this stage not transferable between different problems or even between meshes of varying refinement for a given problem.