## 3. Tabasco Algorithm--Two-Level Sorting and Update Rules Inspired by the Grey Wolf Optimizer  
Formulating the topology optimization problem, in the present stud, the attention is focused on the minimization of structural compliance \(c\) (see Equation (1)). This is the classical approach (a discussion of the problem can be found in the early paper [39]). In this paper, the approach based on finite elements has been applied and the structural topology optimization problem formulation takes the form known from the well-recognized paper by [PERSON] [40]. Hence, it is assumed that only a specified volume fraction \(\kappa\) of a material is distributed within the final structure. This results in the constrained volume \(V\), Equation (2):  
\[\text{minimize }c(\mathbf{d})=\mathbf{u}^{\mathsf{T}}\mathbf{ku}=\sum_{i=1}^{N}d_{ i}^{\mathsf{T}}\mathbf{u}_{i}^{\mathsf{T}}\mathbf{k}_{i}\mathbf{u}_{i}, \tag{1}\]  
\[\text{with respect to }V(\mathbf{d})=\kappa V_{0}, \tag{2}\]  
\[\mathbf{k}\,\mathbf{u}=\mathbf{f} \tag{3}\]\[0<d_{min}\leq d_{i}\leq 1. \tag{4}\]  
The following quantities have been introduced at the element level: displacement vector \(\mathbf{u}_{i}\), stiffness matrix \(k_{i}\), design variable \(d_{i}\), defined as the relative material density, and number of elements \(N\), as well as the global quantities: stiffness matrix \(\mathbf{k}\), displacement vector \(\mathbf{u}\), and force vector \(f\) from Equation (3). In order to avoid singularity, the simple bounds related to the design variables with minimal non-zero density \(d_{min}\) (e.g., \(10^{-9}\)) have been introduced in Equation (4). In the above formulation, the convenient finite element approach has been adopted. The broadly accepted material model SIMP--Solid Isotropic Material with Penalization [35]--has been adopted, meaning that the elasticity modulus \(E_{i}\) for each cell/element is the function of the design variable \(d_{i}\) as shown in Equation (5):  
\[E_{i}=d_{i}^{p}E_{0}. \tag{5}\]  
The quantity \(E_{0}\) is the elasticity modulus of a solid material, whereas \(p\), usually equal to 3, plays the role of penalization power. The SIMP approach is the most common material interpolation scheme, and it is widely discussed and exploited in literature. The majority of papers on topology optimization use SIMP (or SIMP modifications) while performing calculations, using at the same time new techniques for updating the design variables--i.e., those conventional or newly developed heuristics. It is worth mentioning here the various modifications of the SIMP method (e.g., [41; 42; 43] and many more) created to prevent some difficulties in the application of SIMP in selected complex problems such as multi-material topology optimization, dynamic problems, design-dependent loads, thermodynamic problems. While performing the optimization process, the values of the design variables associated with cells are updated based on the local rules. In order to solve the problem, a heuristic local CA update rule is proposed, utilizing the information gathered from a selected neighborhood. New values of design variables can be calculated based on the Gauss-Seidel iteration mode, i.e., a cell updates its state based on updated values found for cell neighbors in the current and previous iteration, or on the Jacobi iteration mode, i.e., a cell updates its state based on the states of the surrounding cells obtained in the previous iteration only. This scheme was chosen for this paper. The Jacobi scheme of updating is utilized, which means that the states of neighboring cells known from the previous iteration are the basis for the update process, as in Equation (6):  
\[d_{i}^{new}=d_{i}+\Delta d_{i}. \tag{6}\]  
The quantity \(\Delta d_{i}\) represents the update component implemented in each iteration. Its selection decides the effectiveness of the updating scheme; therefore, this is the important research issue. The original proposal is included in this paper. It can be treated as an extension and modification of the rule discussed in the paper [44].