CARMA—Cellular Automata with Refined Mesh Adaptation—The Easy Way of Generation of Structural Topologies

Abstract

This paper is focused on the development of a Cellular Automata algorithm with the refined mesh adaptation technique and the implementation of this algorithm in topology optimization problems. Traditionally, a Cellular Automaton is created based on regular discretization of the design domain into a lattice of cells, the states of which are updated by applying simple local rules. It is expected that during the topology optimization process the local rules responsible for the evaluation of cell states can drive the solution to solid/void resulting structures. In the proposed approach, the finite elements are equivalent to cells of an automaton and the states of cells are represented by design variables. While optimizing engineering structural elements, the important issue is to obtain well-defined solutions: in particular, topologies with smooth boundaries. The quality of the structural topology boundaries depends on the resolution level of mesh discretization: the greater the number of elements in the mesh, the better the representation of the optimized structure. However, the use of fine meshes implies a high computational cost. We propose, therefore, an adaptive way to refine the mesh. This allowed us to reduce the number of design variables without losing the accuracy of results and without an excessive increase in the number of elements caused by use of a fine mesh for a whole structure. In particular, it is not necessary to cover void regions with a very fine mesh. The implementation of a fine grid is expected mainly in the so-called grey regions where it has to be decided whether a cell becomes solid or void. The benefit of the proposed approach, besides the possibility of obtaining high-resolution, sharply resolved fine optimal topologies with a relatively low computational cost, is also that the checkerboard effect, mesh dependency, and the so-called grey areas can be eliminated without using any additional filtering. Moreover, the algorithm presented is versatile, which allows its easy combination with any structural analysis solver built on the finite element method.