# Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures
## 5 Introductory Example
The regular mesh of 6400 (160 \(\times\) 40) elements (Plane 42, 4 nodes) has been generated to perform structural analysis and topology optimization for the data: P = 1000 N, \(\kappa\) = 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 3:2  
Two sets of different E\({}_{1}\) to E\({}_{2}\) ratios were examined: E\({}_{1}\) = 10\({}^{11}\) Pa, E\({}_{2}\) = 2\(\cdot\)10\({}^{11}\) Pa and E\({}_{1}\) = 10\({}^{11}\) Pa, E\({}_{2}\) = 10\({}^{12}\) 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 Figures 13 and 14. Figure 3:3 The materials distribution for initial rectangular structure: (**a**) one material structure; (**b**) two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, E\({}_{1}\)! E\({}_{2}\); (**c**) two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, E\({}_{1}\)! E\({}_{2}\) including 4 layers of FGM interface—yellow color; (**d**) two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, E\({}_{1}\)! E\({}_{2}\) including 8 layers of FGM interface—yellow color; (**e**) FGM structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, FGM—yellow color. Figure 3:4 _Cont._  
Figure 3:5 The final topologies generated for the rectangular structure for the 6 distributions of materials: E\({}_{1}\) = 10\({}^{11}\) Pa and E\({}_{2}\) = 10\({}^{12}\) Pa, indicated in Figure 3:6 SIMP interpolation scheme: **(a)** one material structure, E\({}_{1}\)—red color, minimal compliance: 14.624 \(\times\) 10\({}^{-4}\) Nm; **(b)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, minimal compliance: 5.644 \(\times\) 10\({}^{-4}\) Nm; **(c)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, including 4 layers of FGM interface—yellow color, minimal compliance: 5.112 \(\times\) 10\({}^{-4}\) Nm; **(d)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, including 8 layers of FGM interface—yellow color, minimal compliance: 4.390 \(\times\) 10\({}^{-4}\) Nm; **(e)** FGM structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, FGM—yellow color, minimal compliance: 3.688 \(\times\) 10\({}^{-4}\) Nm. Figure 3:7 The final topologies generated for the rectangular structure for the 6 distributions of materials: E\({}_{1}\) = 10\({}^{11}\) Pa and E\({}_{2}\) = 2 \(\times\) 10\({}^{11}\) Pa, indicated in Figure 3:8 SIMP interpolation scheme: **(a)** one material structure, E\({}_{1}\)—red color, minimal compliance: 14.624 \(\times\) 10\({}^{-4}\) Nm; **(b)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, minimal compliance: 10.506 \(\times\) 10\({}^{-4}\) Nm; **(c)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, including 4 layers of FGM interface—yellow color, minimal compliance: 10.110\({}^{-4}\) Nm; **(d)** two materials structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, including 8 layers of FGM interface—yellow color minimal compliance: 9.748 \(\times\) 10\({}^{-4}\) Nm; **(e)** FGM structure: E\({}_{1}\)—red color, E\({}_{2}\)—gray color, FGM—yellow color, minimal compliance: 9.338 \(\times\) 10\({}^{-4}\) Nm. 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: E\({}_{1}\) = 10\({}^{11}\) Pa, E\({}_{2}\) = 2 \(\times\) 10\({}^{11}\) Pa and E\({}_{1}\) = 10\({}^{11}\) Pa, E\({}_{2}\) = 10\({}^{12}\) 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.