# Optimization of Metal-Ceramic Functionally Graded Plates Using the Simulated Annealing Algorithm
## 5 Numerical Applications  
### Design Optimization Studies with a Simply Supported FGM Plate Subjected to Mechanical Loading  
The design optimization of a FGM simply supported plate (\(a\times b\)) is considered for validation purposes. Two metal-ceramic combinations are considered: first, stainless steel as the metal and silicon nitride as the ceramic (SUS304/Si\({}_{3}\)N\({}_{4}\)); and secondly, stainless steel and aluminum oxide as ceramic (SUS304/Al\({}_{2}\)O\({}_{3}\)). The optimization results are compared with alternative single objective optimization results published by [PERSON] et al. [38], obtained with the same finite element model, employing global and local optimization using direct search (GLODS), as described in [PERSON] and Madeira [49]. The FGM plate (\(a=b=0.5\) m) is subjected to a uniformly distributed transverse loading of \(10^{5}\) N/m\({}^{2}\). The temperature on the metal side is 300 K, the stress-free temperature, and the temperature on the ceramic side 400 K. The material properties are those from Table 1. The material cost for the metal SUS304 is assumed to be 3 USD/kg. The assumed material costs for the ceramics are 50 USD/kg for the Si\({}_{3}\)N\({}_{4}\) and 27.4 USD/kg for the Al\({}_{2}\)O\({}_{3}\), as in the reference [38]. For comparison purposes with the alternative results [38], two design optimization cases are considered: minimization of the mass; and minimization of the material cost. The optimization is constrained with the Tsai-Hill stress failure criteria. The isotropic tensile and compressive strengths, \(X_{T}\) and \(X_{C}\), respectively, and the shear strength, \(X_{S}\), are given in Table 2. The constraint equation is expressed as:  
\[\left(\frac{\sigma_{xx}}{X_{T/C}}\right)^{2}+\left(\frac{\sigma_{yy}}{X_{T/C}} \right)^{2}+\left(\frac{\sigma_{xy}}{X_{S}}\right)^{2}-\frac{\sigma_{xx} \sigma_{yy}}{X_{T/C}{}^{2}}\leq 1 \tag{11}\]  
being \(X_{T/C}\) either the material strengths in tension or in compression, ie. \(X_{T}\) or \(X_{C}\) respectively, depending on the sign of the corresponding stress. During the optimization process with the SA algorithm, all intermediate solutions corresponding to non-feasible designs are rejected. In both optimization cases, results of which are presented in Tables 3 and 4, two design variables are considered: the power law index, \(p\), assuming that: \(0.2\leq p\leq 10\) with possible increments of 0.1; and the total thickness of the FGM graded material, \(h_{FGM}\), considering \(5\,\mathrm{mm}\leq h_{FGM}\leq 60\,\mathrm{mm}\), with possible increments of 0.5 mm. From Tables 3 and 4, a complete validation of the optimal solutions obtained using the simulated annealing algorithm can be verified with the alternative single objective optimal designs obtained by [PERSON] et al. [38] using the same finite element model but different optimization algorithms, as mentioned above. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(X_{T}\) & \(X_{C}\) & \(S\) \\ \hline Aluminum Oxide (Al\({}_{2}\)O\({}_{3}\)) & 210 & 2200 & 330 \\ Silicon Nitride (Si\({}_{3}\)N\({}_{4}\)) & 360 & 689 & 207.8 \\ Stainless Steel (SUS304) & 215 & 215 & 124 \\ Titanium alloy (TioAl4V) & 880 & 880 & 550 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Metal and ceramic material strengths in tension, compression and shear (MPa). In a second validation case, whose results are presented in Table 5, the objective is the maximization of the fundamental natural frequency, with the desired frequencies falling in the range: \(3000\frac{\text{trad}}{\text{s}}\leq\omega_{1}\leq 8000\frac{\text{trad}}{ \text{s}}\). Two design variables are considered: the thickness of the FGM plate graded material, with a lower bound of 5 mm and an upper bound of 60 mm, with possible increments of 0.1 mm; and the power law index, \(p\), assuming again that: \(0.2\leq p\leq 10\), with possible increments of 0.1 As before, the solutions not falling in the feasible region are rejected.