## 3 Topology Optimization Using the Cellular Automata Method  
The concept of topology optimization can be dated back to 1904 and is attributed to the paper by [PERSON] [15], but the papers by [PERSON] and [PERSON] [16], and [PERSON] [17] published in the late 1990s sparked the development of this research area. The idea of topology optimization is to find a material layout which meets the assumed criteria, and in this sense gives the optimal solution. During the optimization process, the material is redistributed within the desired design domain, which leads to the creation of a new shape of optimized structure. In topology optimization, the computational design domain is discretized and represented by finite elements, for which design variables called relative densities \(d_{i}\) are selected. The material representation can be defined using the well-known and broadly accepted solid isotropic material with penalization (SIMP) approach, e.g., [18; 19], where the elastic modulus \(E_{i}\) of each element is a function of relative density \(d_{i}\):  
\[E_{i}\ =\ d_{i}^{p}E_{0},\quad d_{min}\leq d_{i}\leq 1. \tag{1}\]  
In Equation (1), \(E_{0}\) is the Young modulus of a solid material, whereas the power \(p\) (usually equal to 3) is introduced to penalize intermediate densities. The constraints are imposed on the design variables, and \(d_{min}\) in Equation (1) is a non-zero minimum relative density introduced in order to avoid singularity. Figure 3: The appearance of thin and slender parts of structural topology in the case of a fine mesh has to be controlled by filtering techniques: (**a**) example of a fine mesh without filtering; (**b**) example of a fine mesh with filtering. For the design domain discretized and represented by finite elements, the topology optimization problem aimed at minimizing the function of compliance \(U(\mathbf{d})\) of a structure subject to a total volume constraint can be expressed as (see Equation (2)):  
\[ \begin{array}{ll}\text{minimize}&U(\mathbf{d})\ =\ \mathbf{u}^{T}\mathbf{k} \mathbf{u}=\sum\limits_{i\,=\,1}^{N}d_{i}^{p}\mathbf{u}_{i}^{T}\mathbf{k}_{i} \mathbf{u}_{i}\\ \text{subject to}&V(\mathbf{d})=\kappa V_{0}\\ &\mathbf{k}\mathbf{u}=\mathbf{f}\end{array}  \tag{2}\]  
In (2), \(\mathbf{u}\) and \(\mathbf{f}\) are the global displacement and force vectors, \(\mathbf{k}\) is the global stiffness matrix, \(\mathbf{u}_{i}\) and \(\mathbf{k}_{i}\) are the element displacement vector and stiffness matrix, respectively, and \(N\) is the number of elements. Simultaneously, a constraint imposed on global volume of a structure, represented by the function \(V(\mathbf{d})\) of the design variables \(d_{i}\), has been applied for a specified volume fraction \(\kappa\), and \(V_{0}\) standing for the initial design domain volume. A broad discussion of the principles of topology optimization can be found, for example, in [16; 17; 18; 20]. Simultaneously with the development of the topology optimization area, especially within engineering applications, numerous numerical optimization techniques aimed at solving newly formulated problems efficiently and quickly have been proposed. Some of those optimization techniques are based on gradient information, e.g., the optimality criteria method [21; 22] or level set methods [21; 22; 23], some others implement non-gradient evolutionary techniques [23]. Among many proposals one can also find Cellular Automata (CA). This method has become more and more popular since the introduction of the concept by [PERSON] [24] and [PERSON] [25]. However, the application of Cellular Automata to structural optimization has occurred only in the last two decades. In particular, this concept has been successfully applied to topology optimization, which was illustrated in a series of published papers, e.g., [26; 27; 28; 29; 30; 31]. In particular, the efficiency of the Cellular Automata method is discussed in [32].