# Cellular Automata Approach to Topology Optimization of Graded Multi-Material Structures
## 5 Introductory Example
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):  
\[E(y)=E_{0}e^{By} \tag{16}\]  
To illustrate the influence of the orientation of material gradation on final topologies, the calculations have been performed for \(\beta=-0.02\), \(\beta=-0.04\), \(\beta=-0.06\), \(\beta=-0.08\). The obtained results are presented in Figure 7 for the FGM-SIMP material interpolation scheme and in Figure 8 for the FGM-RAMP one. Figure 4: Final topologies obtained for square structure for FGM-RAMP material interpolation scheme, \(x\)-direction of material gradation: (**a**) uniform material, minimal compliance: 2.560 Nm; (**b**) graded material according to Equation (7), minimal compliance: 15.173 Nm. Figure 5: Final topologies obtained for square structure using the FGM-SIMP material interpolation scheme, \(x\)-direction of material gradation: (**a**) \(\alpha=-0.02\), minimal compliance: 4.386 Nm; (**b**) \(\alpha=-0.04\), minimal compliance: 7.919 Nm; (**c**) \(\alpha=-0.06\), minimal compliance: 15.427 Nm; (**d**) \(\alpha=-0.08\), minimal compliance: 31.656 Nm. 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:  
\[E(x,y)=E_{0}e^{xx+\beta y} \tag{17}\]  
Figure 6: Final topologies found for square structure using the FGM-RAMP material interpolation scheme, \(x\)-direction of material gradation: (**a**) \(\alpha=-0.02\), minimal compliance: 4.338 Nm; (**b**) \(\alpha=-0.04\), minimal compliance: 7.794 Nm; (**c**) \(\alpha=-0.06\), minimal compliance: 15.173 Nm; (**d**) \(\alpha=-0.08\), minimal compliance: 31.512 Nm. Figure 7: Final topologies obtained for square structure for the FGM-SIMP material interpolation scheme, \(y\)-direction of material gradation: (**a**) \(\beta=-0.02\), minimal compliance: 4.863 Nm; (**b**) \(\beta=-0.04\), minimal compliance: 9.328 Nm; (**c**) \(\beta=-0.06\), minimal compliance: 18.112 Nm; (**d**) \(\beta=-0.08\), minimal compliance: 35.210 Nm. As one can notice, final compliances for \(\alpha=-0.08\) and \(\beta=-0.08\) 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.  
Figure 3:0 The rectangular structure (16 m \(\times\) 4 m) applied to two load schemes and support. Figure 3:1 Final topologies obtained for square structure for the FGM-RAMP material interpolation scheme and \(x\) and \(y\)-directions of material gradation: (a) \(\alpha=-0.02\), \(\beta=-0.02\), minimal compliance: 8.333 Nm; (b) \(\alpha=-0.04\), \(\beta=-0.04\), minimal compliance: 32.581 Nm; (c) \(\alpha=-0.06\), \(\beta=-0.06\), minimal compliance: 152.936 Nm; (d) \(\alpha=-0.08\), \(\beta=-0.08\), minimal compliance: 819.221 Nm.