Evolving rules for a self-organizing finite element mesh generation algorithm

Using only simple stimuli from the local environment to coordinate the activity of the group, social insects such as termites build highly complex structures. This was discovered by Grasse (1959) and is known as stigmergy. We employ such a distributed approach for meshing finite element domains. In two dimensions, a triangular mesh is a discretization of a domain into a set of approximately equilateral triangles. The finite element method can then be used to solve fluid flow problems at the vertices in discrete time. The size of the triangles is related to rate of change of the solution values and hence meshing is a complex task. We use a genetic algorithm to evolve a set of rules which can be used by a colony of termite-like agents to mesh an arbitrary domain. Work on building activity of swarms has shown that stigmergic algorithms can produce more complex structures than sequential algorithms which are dependent on past building activity. We compare our stigmergic algorithm with a similar sequential algorithm. Whilst the sequential algorithm generalizes poorly to new domains, our stigmergic approach generalizes well and produces better quality meshes on the examples considered. In this approach, meshing is in response to conditions in the local environment, hence changing conditions can easily be accommodated, making it particularly useful for applications such as injection moulding where the mesh must 'grow' with the domain.