# Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures
## 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  
Figure 16: The final topologies for the FGM structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, FGM—yellow color; SIMP interpolation scheme: (**a**) E\({}_{1}\) = 10\({}^{11}\) Pa E\({}_{2}\) = 2 \(\times\) 10\({}^{11}\) Pa, minimal compliance: 11.514 \(\times\) 10\({}^{-4}\) Nm; (**b**) E\({}_{1}\) = 10\({}^{11}\) Pa E\({}_{2}\) = 10\({}^{12}\) Pa, minimal compliance: 6.532 \(\times\) 10\({}^{-4}\) Nm. Figure 15: The final topologies for the FGM structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, FGM—yellow color; RAMP interpolation scheme: (**a**) E\({}_{1}\) = 10\({}^{11}\) Pa E\({}_{2}\) = 2 \(\times\) 10\({}^{11}\) Pa, minimal compliance: 9.330 \(\times\) 10\({}^{-4}\) Nm; (**b**) E\({}_{1}\) = 10\({}^{11}\) Pa E\({}_{2}\) = 10\({}^{12}\) Pa, minimal compliance: 3.706 \(\times\) 10\({}^{-4}\) Nm. 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.