Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures

Abstract

Despite decades of progress, structural topology optimization is still one of the most important areas of engineering optimal design. The intensive research within this area has been stimulated by the development of efficient methods and algorithms on one side and the needs and demands of contemporary engineering on the other. Over the years, the practical aspect of topology optimization has become one of the most significant issues within the design community. Simultaneously, the range of design applications has been broadening. Among many research areas where topology optimization is present, attention has been paid to the design of multi-material structures. The gradation of the material properties has a significant influence on the final layout of the structure, so this problem can be treated as an extension of the classical task of the topology optimization of structures made of a material with uniform distributions of properties. While working with multi-material structures, the important role plays an interface between parts made of materials with different properties. In this paper, the implementation of interfaces made of functionally graded materials (FGM) is proposed. A functionally graded interface means that continuous and smooth changes of properties are assigned to a particular direction from one material surface to another. This paper presents the idea of topology optimization of graded multi-material structures using a simple, fast convergent technique based on the Cellular Automata approach. The proposal is to take the advantage of the versatility of efficient professional finite element-based structural analysis software and the simplicity of the original heuristic topology generator, to build a tool for the optimization of FGM structures as well as multi-material structures including the FGM interface.

1. Introduction

The concept of topology optimization can be dated back even 150 years and within this period topology optimization has built its fascinating history, from theoretical discussions to real-life advanced engineering, as described in [1]. However, the idea of structural topology optimization was raised only in the late 1980s of the 20th Century with the paper by Bendsoe and Kikuchi [2], treated as the pioneering one within this field. Over the years, research on topology optimization has been conducted having a significant influence on modern engineering design and also indicating directions for research development. Nevertheless, despite decades of progress, structural topology optimization is still one of the most important and intensively developing areas of engineering optimal design.

Topology optimization is understood as searching for the distribution of material within the design domain so as to meet the implied optimality criteria. The new shape and material layout are generated, while some parts of the material are relocated and others are selectively removed. In the classical approach, the structure compliance is minimized subject to imposed constraints within the specified volume fraction of the resulting topology. This formulation originally has been applied to single-material structures but can easily be extended towards topology optimization of multi-material structures, including those of functionally graded material properties.

The concept of gradation in material composition started in 1972 for polymeric materials. Since then, Japanese researchers began to develop that idea and established the concept of FGM in 1985. Since the late 1990s, the FGMs have evolved dynamically, induced by fundamental investigations on the physical and chemical properties of the newly invented materials. Their unique properties, but also the development of design methods and implementation of manufacturing techniques have led to the great success of the FGM concept. The detailed, impressive history of FGM, including manufacturing, modeling, and designing, together with a literature review, one can find, for example, in recently published papers [3].

The multi-material topology optimization can involve two approaches. The first one regards searching for the optimal distribution of multiple materials in the specified design domain in order to meet design criteria, e.g., [4,5,6,7,8,9,10,11], designing interfaces made of functionally graded materials (FGM) between them [12,13] or generation of optimal layout and gradation of the FGM in the specified design domain, e.g., [14,15,16,17,18]. The extended literature overview on these subjects one can find also in [6] or [19]. As to the related studies, the wide topic of topology optimization of functionally graded cellular materials can also be mentioned, e.g., [20,21].

The second approach assumes that the optimal layout of the structure is sought for a predefined material properties distribution, e.g., a predefined FGM layout. This approach was investigated first in [22,23,24,25] and has been continued for topology optimization of functionally graded piezoelectric structures, which can be illustrated by papers [24,26]. While mentioning the above approach, the paper [27] can also be pointed out, where optimization of multi-material and FGM structures including cracks was investigated. Although the optimal distribution of two and three materials is the main part of the article, examples of predefined FGM are also investigated [27].

Based on the above-mentioned information, one can observe that the number of papers discussing topology optimization of graded multi-material structures is rather low. Additionally, according to the authors’ knowledge, papers on topology optimization of graded multi-material structures using efficient heuristic methods are practically absent in the literature. Meanwhile, heuristic techniques are easy for numerical implementation, do not require gradient calculations, and can easily be combined with any finite element structural analysis code. These features caused these techniques to have gained popularity within the research community. It is therefore proposed to implement the efficient heuristic algorithm based on the concept of Cellular Automata into the optimization of FGM structures.

The concept of Cellular Automata (CA) was proposed by von Neumann and Ulam in the late 1940s. The idea of CA is to mimic the behavior of complex systems by applying simple rules built based on local information. This method, thanks to its efficiency and simplicity, has been found useful by researchers representing various branches of modern engineering, including topology optimization. The Cellular Automata approach was discussed within the topology optimization area for the first time by Inou et al. [28], but the majority of papers appeared only during the last two decades, see [29,30,31,32,33,34,35,36]. This paper presents the idea of Cellular Automata utilized in the topology optimization of graded multi-material structures, that is optimization of FGM structures as well as multi-material structures including the FGM interface. The optimal topologies of structures are sought under predefined material properties distribution.

The outline of the paper is as follows. In Section 2, the topology optimization basics together with the classical formulation of the problem are presented. Next, in Section 3, the characteristic features of functionally graded materials are briefly outlined. The concept of Cellular Automata as the optimization tool is introduced and the flexible algorithm built based on this idea is described in Section 4. Its implementation into the topology generation process is illustrated by selected numerical examples in Section 5 and Section 6. While performing computations, Cellular Automaton has been combined with finite element-based structural analysis software ANSYS 14. Based on the results of the performed tests, the paper ends with concluding remarks in Section 7.

2. Topology Optimization Basics

While generating structural topologies, the material is redistributed within the design domain. The idea is to utilize the material in the best possible way, according to the assumed design criteria. Finally, it leads to removing material unnecessary from these criteria point of view. The classical and widely recognized formulation of the structural topology optimization problem is the minimization of the structure compliance c, Equation (1) while satisfying the constraint imposed on the assumed volume fraction κ [37], Equation (2). Using the finite element approach, one calculates the following:

$$\mathrm{minimize}c\left(\mathit{d}\right)=u^{T}ku={\sum }_{n=1}^N{d}_n^p{u}_n^T{k}_n{u}_n,$$

(1)

$$\mathrm{subject}\mathrm{to}V\left(d\right)=\kappa V_0,$$

(2)

$$k u=f$$

(3)

$$0<d_{min}\le d_n\le 1$$

(4)

The quantity $u_n$ denotes displacement vector,$k_n$ stands for the stiffness matrix and both are defined for N elements. The design variable $d_n$ represents the relative material density assigned to each element. In Equation (3) $k$ represents the global stiffness matrix, $u$ stands for the global displacement vector and $f$ is the vector of forces. Due to the simple bounds imposed on the design variables in Equation (4), with $d_{min}$ as a non-zero minimal value (e.g., 10⁻⁹) of relative density, singularity is avoided.

The quoted formulation requires relating the design variables to the mechanical properties of the material so that the modulus of elasticity $E_i$ of each finite element has to be a function of the design variable $d_i$. The most common approach the material representation is SIMP, i.e., Solid Isotropic Material with Penalization [38,39] in the form of a power law, see Equation (5):

$$E_n=d_n^p{E}_0$$

(5)

In Equation (5) the quantity $E_0$ stands for modulus of elasticity, defined for solid material, whereas $p$ (typically $p$ = 3) is responsible for the penalization of intermediate densities. This allows controlling the design process and leads to obtaining black-and-white resulting structures. An alternative interpolation scheme for minimum compliance topology optimization problem was proposed in [40] as RAMP interpolation (Rational Approximation of Material Properties), see Equation (6):

$$E_n=d_n/\left(1+q\left(1-d_n\right)\right)E_0.$$

(6)

The effect of penalization induced by the parameter $q$ in RAMP is analogous to that of $p$ in the SIMP scheme. The extended studies on material interpolation schemes in topology optimization are delivered by papers [41,42,43].

In the present paper, both SIMP and RAMP schemes defined above are utilized and the results of topology optimization of graded multi-material structures for both of them are presented.

3. Topology Optimization of Graded Multi-Material Structures: The Basic Introduction

The main concept of present investigations, which needs to be emphasized, is the idea of topology optimization of multi-material structures with a predefined gradation of material properties. This can be treated as the extension of the discussion included in [44]. The two tasks are under consideration, namely topology optimization of the FGM structures and design of sub-domain oriented multi-material structures, including FGM interfaces. Functionally graded materials are composites consisting of two or more constituent phases with continuous variation of properties along one direction, which is directly related to changes in the material properties throughout the volume.

The model of properties gradation has a significant influence on the behavior of the FGM structure. Following the considerations of [21,22,23,24,25,27], in the present paper, the exponential model has been chosen for one-dimensional material gradation, see Equations (7)–(9):

$$E\left(x\right)=E_0{e}^{\alpha x},$$

(7)

$$E_0=E_1{\left(\frac{E_1}{E_2}\right)}^{\frac{x_1}{x_2-x_1}}$$

(8)

$$\alpha =-ln\left(\frac{E_1}{E_2}\right)\frac{1}{x_2-x_1},$$

(9)

where x is a coordinate measured along the material variation direction, E₁ is the Young modulus of the material at position x₁, and E₂ is the Young modulus of the material at position x₂. Function E(x) describes the FGM material. It is assumed that the Poisson ratio varies only slightly between two materials, so its gradation can be omitted.

In the topology optimization of FGM structure material interpolation schemes, Equations (5) and (6) can be rewritten as FGM-SIMP, see Equation (10), [25] and FGM-RAMP, Equation (11):

$$E\left(x\right)=d_n^p{E}_0{e}^{\alpha x},$$

(10)

$$E\left(x\right)=d_n/\left(1+q\left(1-d_n\right)\right)E_0{e}^{\alpha x},$$

(11)

respectively.

The possibility of application of FGMs in engineering practice has induced the research on the numerical representation of FGM materials, particularly their implementation into the finite element method, for details see [23]. In this paper, the numerical model of the structure has been built as the set of layers, treated as the isotropic ones. The material properties have been evaluated in the middle of each layer by using power law grading, see Equation (7). For each layer, the individual material properties were defined, so as to approximate the functional variability of material properties for the whole structure [45,46]. The structure assembled from single-material layers resembles the traditional piecewise material distribution applied in standard topology optimization formulation. It is worth mentioning that in [21,22,23,24,25] the continuous approximation of material distribution CAMD [47,48] was adopted. The layer-based formulation is the suitable approach for the numerical modeling of FGM structures from the manufacturing process point of view because step-wise gradation is easier to produce than the continuous one.

4. Cellular Automata Approach

During the last forty years of research on topology optimization, the trend of seeking new, efficient, easy-to-implement algorithms has been noticeable. Hence, the computationally demanding gradient-based methods have been more often replaced by heuristics, which provided simple and efficient algorithms for optimal design, e.g., [49,50]. Heuristics are strategies derived based on experience or inspired by the observation of nature. Among many techniques based on heuristic concepts, there are Cellular Automata. Cellular Automata mimic the behavior of complex systems in a simple way by analogy to the biological tissues. Like in biology, while performing topology optimization, the design domain is decomposed into a lattice of discrete cells, which in engineering implementations, is usually equivalent to finite element mesh used in structural analysis. The local information gathered within each cell neighborhood is the basis for establishing local rules which govern the behavior of the system. The identical rules are applied to all cells simultaneously. Cells by interactions with their neighbors replace a complex problem with a sequence of simple decision making.

The concept of a CA-based algorithm as the topology generator is under constant development and many extensions and modifications have been already implemented. Based on the results obtained and the discussion conducted in the former authors’ papers, e.g., [35,36,44,51,52], one can point out some advantages of using such an algorithm. The Cellular Automata-based topology generator is versatile, easy to implement, and ready to be integrated with professional FEM-based structural analysis software. The algorithm does not require additional density filtering, the so-called gray elements are eliminated. It is because the update rules that govern algorithm performance are based on the information exchanged within cells forming this neighborhood, which can be compared to the filtering technique based on averaging sensitivities calculated for elements surrounding the central one. Moreover, in favor of the CA approach the neighborhood remains the same independent of the mesh size, whereas, in the case of the filtering technique, the neighborhood size has to be selected depending on the mesh size. It is also worth pointing out that the CA approach can be easily adapted to problems with irregular meshes. which has been also illustrated in [51]. In this paper, the extension of the algorithm by adding the self-adaptive update rule is proposed.

While performing the optimization process, the local rules are responsible for updating cell states represented by design variables. The Jacobi update scheme has been utilized, meaning that updating is based on the states of the neighboring cells determined in the previous iteration, see Equation (12):

$$d_n^{new}=d_n+\mathrm{∆}{d}_n$$

(12)

The design variable $d_n$ is the relative density of the material, while $\mathrm{∆}{d}_n$ stands for its correction introduced in each iteration. While building the updating scheme its flexibility and versatility are very important since the appropriate selection of $\mathrm{∆}{d}_n$ value is crucial for the effectiveness of the topology generation process. In this paper, the extended version of the rule proposed and discussed in detail in the paper [52] is utilized. First, the structural analysis is performed, and based on the results, the values of local compliances for all cells/elements are calculated. The compliances of N cells are sorted in ascending order and those having the lowest and the highest values are identified which is represented by the number N₁ of cells with the lowest and the number N-N₂ of cells with the highest values. To those cells, the constant values of F(n) = −C if n < N₁ and F(n) = C if n > N₂ are assigned, where n is a cell index and C is a constant coefficient usually defined as 1. This means, that for the cells, for which the algorithm is sure, that material should be removed, i.e., cells with low compliances, n < N₁, or added, i.e., cells with high compliances, n < N₂, the change of design variable value can be assigned as the admissible limit value. For intermediate interval N₁ ≤ n ≤ N₂ a monotonically increasing function f(n) representing elements compliances has to be specified, see Equation (13):

$$F\left(n\right)=\left\{\begin{array}{c}-C \mathrm{if} n<N_1\\ f\left(n\right) \mathrm{if} N_1\le n\le N_2\\ C \mathrm{if} n>N_2\end{array}\right..$$

(13)

Under the assumption that the update is based on the states of the neighboring cells, the rule takes the following final form:

$$d_n^{new}=d_n+\left[F\left(n\right)+{\sum }_{k=1}^{M}F\left(k\right)\right]\frac{m}{M+1}.$$

(14)

In Equation (14) the quantity m plays the role of admissible change (move limit) of the design variable values and F(k) is the analogously constructed function for the M neighboring cells. Depending on the result of selection controlled by Equation (13) material is added to or removed from the cell n.

The threshold values N₁ and N₂ can be adjusted so that the width of the interval [N₁, N₂] can be modified during the iteration process. The numerical experiments confirmed that it starts with a relatively wide initial interval [N₁, N₂] and then its successive reduction leads to the fast elimination of cells of intermediate densities. The large design domain at the beginning allows for a preliminary outline of the structure layout by eliminating void cells. Then, along with the iteration process, the solution is tuned by subsequent reduction of the [N₁, N₂] interval, which manifests by the elimination of so-called gray cells of intermediate densities. Finally, it leads to a distinct solid/void structure. This resembles the well-known simulated annealing strategy. The described approach results in modifying the shape of the F(n) function, which is illustrated in Figure 1a. One can find a detailed discussion regarding the implemented strategy in the paper [52]. Based on those investigations, the following strategy for the proposed update rule implementation can be proposed. The calculations start with $N_1=N·0.1,$ and then from iteration 25 $N_1=N·0.5$, while $N_2=N·0.9$ is kept for all iterations.

As to the above-introduced function f(n), the one defined by Equation (15) is proposed:

$$f\left(n\right)=C\frac{\tan \mathrm{h}\left[\beta \left(\frac{n-N_1}{N_2-N_1}-\frac{1}{2}\right)\right]}{\tan \mathrm{h}\left(\frac{1}{2}\beta \right)}.$$

(15)

The parameter β determines the form of the f(n) function. In what follows, β tending to zero f(n) tends to have linear functions, whereas large values of β f(n) tend to have a step function, see Figure 1b.

The flexibility of update rules, induced by adjusting the value of β, influences algorithm performance; therefore, in order to control the effectiveness of this process, implementation of the self-adaptive technique is proposed. The design variables are updated at each iteration step using three values of β, i.e., a small, middle, and large one, and then the algorithm proceeds to the next iteration with the solution for which the compliance has been found as the smallest one. Based on the test performed, it can be recommended to choose: β = 0.1, β = 4 and β = 8. This scheme of updating guarantees tracking the lowest value of an objective function in each iteration, without the necessity of adjusting the control parameters of the algorithm.

5. Introductory Example

Following the papers [22,23,24,25] the square structure presented in Figure 2 has been chosen as the introductory example illustrating the implementation of the proposed CA approach to the topology optimization of graded multi-material structures. The FGM material defined by Equation (7) was utilized, where the Young modulus $E\left(x\right)$ varies in one direction as the function of variable x. For this example, $E_0=10$ stands for normalized Young’s modulus and α = −0.06 is the coefficient which defines a change in the material properties within the domain. While dealing with the FGM topology optimization, the material interpolation schemes, i.e., SIMP in Equation (5) and RAMP in Equation (6) can be rewritten as FGM-SIMP and FGM-RAMP in the form of Equations (10) and (11), respectively. The structure is loaded with the force P = 1 N distributed in the nodes along the edge of 2 elements (see Figure 2), and the Poisson ratio value for all materials is set to 0.35 for all experiments.

We meshed the structure with 50 × 50 regular elements (Plane 182, 4 nodes). As the optimal topology generator, the Cellular Automata algorithm introduced in Section 5 was utilized. The Moore-type neighborhood, i.e., the one formed by the cells having common vertices with the central one, has been adapted. As the structural analysis tool, the finite element analysis package Ansys 14 was selected. The sequential approach has been implemented which means that structural analysis performed for each iteration is followed by the application of the local update rules defined by Equations (12)–(14). Simultaneously, in each iteration, the volume constraint, Equation (2) has been applied to the updated design elements. The volume fraction κ equal to 0.3 was selected. As the stopping criterion, the assumed change of the objective function value for subsequent iterations has been adapted, but in general, it can also be defined as performing a selected number of iterations.

Final topologies obtained for FGM-SIMP and FGM-RAMP material interpolation schemes are presented in Figure 3 and Figure 4, respectively. These are compared with the results of topology optimization performed for structures with uniform material distributions.

For final topologies, the values of compliances were slightly different for SIMP and RAMP interpolation schemes. The SIMP interpolation gave 2.588 Nm and the RAMP one 2.560 Nm for one uniform material, whereas for graded material SIMP gave 15.427 Nm, and 15.173 Nm was obtained using the RAMP interpolation scheme.

The obtained results are closely related to the ones reported in [22]. As can be seen in the figures, the graded material highly influences the final structure layout which is different from the one obtained for the structure with homogeneous material distributions. The implementation of material gradation leads to the manifestation of the characteristic reinforcement and thickening of the resulting structure, observed within areas where the structure has lower stiffness.

To illustrate the influence of material gradation on the final shapes of structures, the results of calculations performed for α = −0.02, α = −0.04, α = −0.06, α = −0.08 are presented in Figure 5 for the FGM-SIMP material interpolation scheme and in Figure 6 for the FGM-RAMP one.

A similar analysis has been performed for the same test structure but this time with a material gradation in the y-direction, according to Equation (16):

$$E\left(y\right)=E_0{e}^{\beta y}$$

(16)

To illustrate the influence of the orientation of material gradation on final topologies, the calculations have been performed for β = −0.02, β = −0.04, β = −0.06, β = −0.08. The obtained results are presented in Figure 7 for the FGM-SIMP material interpolation scheme and in Figure 8 for the FGM-RAMP one.

With a view to complete the discussion, the structure with material gradation in both x and y directions, according to Equation (17), has been considered:

$$E\left(x,y\right)=E_0{e}^{\alpha x+\beta y}$$

(17)

The final topologies generated for the four sets were α = −0.02 and β = −0.02; α = −0.04 and β = −0.04; α = −0.06 and β = −0.06; α = −0.08 and β = −0.08, which are presented in Figure 9 for the FGM-SIMP material interpolation scheme and in Figure 10 for the FGM-RAMP one.

As one can notice, final compliances for α = −0.08 and β = −0.08 are extremely high. That means that this configuration of material gradation needs to be improved. The first approach, mentioned in Section 1, i.e., searching for the optimal distribution of multiple materials in the specified design domain, can be utilized for this kind of problem.

The results of the performed tests confirmed that the CA algorithm has the ability to cope with the generation of topologies for graded multi-material structures. The discussion continues in the next section.

6. Implementation of Sub-Domain Oriented Multi-Material Topology Optimization including FGM Interface

The sub-domain-oriented multi-material topology optimization is under consideration. While working with multi-material structures, the interface between parts made of materials having different properties plays an important role. The macroscopic boundary between different materials introduces material discontinuity which can be eliminated by the implementation of the FGM interface. In this way, the mechanical, physical, and chemical properties of materials change continuously from one material to the other. In this section, the implementation of interfaces made of FGM is discussed using a simple supported beam loaded by a concentrated force acting according to two load schemes, as presented in Figure 11.

The regular mesh of 6400 (160 × 40) elements (Plane 42, 4 nodes) has been generated to perform structural analysis and topology optimization for the data: P = 1000 N, κ = 0.5. As to the generation of topologies using the CA algorithm, the Moore-type neighborhood has been applied. The five cases are investigated, namely, one material topology optimization, two material structures, and two material structures with two different values of thickness of FGM interface and pure FGM structure. All cases are illustrated in Figure 12.

Two sets of different E₁ to E₂ ratios were examined: E₁ = 10¹¹ Pa, E₂ = 2·10¹¹ Pa and E₁ = 10¹¹ Pa, E₂ = 10¹² Pa. The Poisson ratio value for all materials is set to 0.35 for all experiments. The final topologies obtained for all the cases indicated in Figure 6 are presented in Figure 13 and Figure 14.

Those calculations were repeated for the FGM-RAMP material interpolation scheme. The results for FGM structure are presented in Figure 15 for two sets of data: E₁ = 10¹¹ Pa, E₂ = 2 × 10¹¹ Pa and E₁ = 10¹¹ Pa, E₂ = 10¹² Pa.

To complete the considerations and illustrate the influence of material gradation on the final topologies, the experiment involving the reordering of materials in the structure was performed. Now, the weaker material is on the edges of the structure, while the stiffer one is around the central axis of the structure. As it was expected, the compliances calculated for the optimal topologies obtained for this material distribution are greater than the ones found for the primary distribution. The resulting topologies are presented in Figure 16 for both sets of data: E₁ = 10¹¹ Pa, E₂ = 2·10¹¹ Pa and E₁ = 10¹¹ Pa, E₂ = 10¹² Pa.

Based on the performed calculations, one can conclude that the final topologies depend on the stiffer and the weaker material distribution and this dependence is stronger if the difference in stiffness of the involved materials is large. Predefined distribution of the material and FGM interface has a significant influence when comparing Figure 13e and Figure 16a or Figure 14e and Figure 16b.

7. Concluding Remarks

Attention has been paid to the topology optimization of multi-material structures. As to the numerical implementation of topology generators within this area, one can point out the gradient-based optimality criteria method, e.g., [22,25,27], level set method, e.g., [18,21], or very rare, but efficient, heuristic approaches, e.g., [19]. Hence, the concept of topology optimization of graded multi-material structures using a simple, fast convergent technique based on the Cellular Automata approach has been introduced and implemented. Taking advantage of the efficiency of the original heuristic topology generator, not suffering from the checkerboard effect, and exploiting the versatility of professional finite element-based structural analysis software made it possible to build a useful tool for the optimization of FGM structures as well as multi-material structures including the FGM interface. While working with multi-material structures, the idea of sub-domain-based multi-material topology optimization has been implemented as the alternative to the “free” multi-material design, e.g., [4,5,6,7,8,9,10,11,12,13,16]. The optimal topologies have, therefore, been generated under the restriction that the redistribution of material can be performed only within sub-domains selected for employed materials. The two material representations, namely, SIMP and RAMP schemes, were utilized and the results of topology optimization of graded multi-material structures for both of them are presented. The results of the paper confirmed that the gradation of the material properties has a significant influence on the final layouts of the discussed structures. The wide spectrum of design cases was investigated, including one material topology optimization, two material structures, two material structures with two different values of thickness of FGM interface, and pure FGM structures. In order to illustrate the influence of material gradation on the final topologies more deeply, the experiment involving the reordering of materials in the structure was also performed.

The main point of the paper is to introduce the efficient heuristic topology generator into topology optimization of graded multi-material structures. According to the authors’ knowledge, papers on the topology optimization of graded multi-material structures using efficient heuristic methods are practically absent in the literature. The proposed paper attempts to fill this gap. The aspects which can be raised in favor of the heuristic techniques are as follows: gradient calculations are not needed, numerical implementation is easy, and FEM-based structural analysis codes can be linked with such an algorithm. This refers also to the Cellular Automata-based algorithm introduced in this paper. Another issue worth pointing out is the introduction and implementation of an extended version of the CA algorithm. The extension regards the implementation of the self-adaptive technique, which is described in Section 4.

Based on the results obtained in the paper, it can be concluded that the introduced and discussed approach can serve as a useful tool while working with topology optimizations of multi-material structures and may be an alternative to other approaches proposed within this research area.