As a non-gradient method, PTO is simple to The implementation of non-uniform cells of Cellular Automaton requires a reformulation of standard local rules, for which the influence of the neighborhood on the current cell is independent of sizes of the neighboring cells.Ī new topology optimization method called the Proportional Topology Introducing an irregular lattice of cells allows to reduce the number of design variables without loosing the accuracy of results and without an excessive increase of the number of elements caused by using a fine mesh for a whole structure. The aim of the present paper is to extend the concept of CA towards the implementation of unstructured grid of cells related to non-regular mesh of finite elements. However, a practical engineering analysis and design in many cases require using highly irregular meshes for complicated geometries and/or stress concentration regions. It is assumed that the interaction between cells takes place only within the neighborhood and the states of cells are updated synchronously in subsequent time steps according to some local rules.The majority of results that have been obtained so far were based on regular lattices of cells. In the engineering implementation of CA, the design domain is decomposed into a lattice of cells, and a particular cell together with the cells to which it is connected forms a neighborhood. In recent years the Cellular Automata (CA) concept has been successfully applied to structural topology optimization problems. We see, that both algorithms converge to almost same topology. Convergence of the topologies are illustrated at Fig. Thus, if the size of the element is less or equal that this minimal value, no refinement will be made regardless to (24). To compare the results, we take this mesh size as the minimal size of the element in Algorithm 2. For the Algorithm 1, which is non-adaptive, we choose the mesh size empirically. We solve the problem with both Algorithm 1 and Algorithm 2. Empirically we take the following values of the parameters: M = 0.5, α = η = 0.1, E = 0.05.

The mortar FEM allows using any method of mesh refinement, so we choose the uniform dividing of the rectangular element into four new elements, as shown at Fig. We use rectangular finite elements with bilinear approximations. Material parameters are: Young modulus equals 21000, Poisson coefficient equals 0.3. Numerical investigation of the algorithms was made using the test problem, illustrated in Fig.

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