# Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures
## 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 \(E(x)\) varies in one direction as the function of variable \(x\). For this example, \(E_{0}=10\) stands for normalized Young's modulus and \(\alpha=-0.06\) 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  
Figure 1: (**a**) Evolution of function \(F(n)\) based on increasing \(N_{1}\), \(N_{2}\) fixed. (**b**) Function \(F(n)\) for selected values of \(\beta\). 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 \(\times\) 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 \(\kappa\) 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 Figures 3 and 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. Figure 3: Final topologies obtained for square structure for FGM-SIMP material interpolation scheme, \(x\)-direction of material gradation: (**a**) uniform material, minimal compliance: 2.588 Nm; (**b**) graded material according to Equation (7), minimal compliance: 15.427 Nm. Figure 2: The square structure (50 \(\times\) 50 elements) with applied loads and supports. To illustrate the influence of material gradation on the final shapes of structures, the results of calculations performed for \(\alpha=-0.02\), \(\alpha=-0.04\), \(\alpha=-0.06\), \(\alpha=-0.08\) are presented in Figure 5 for the FGM-SIMP material interpolation scheme and in Figure 6 for the FGM-RAMP one.