# Optimization of Metal-Ceramic Functionally Graded Plates Using the Simulated Annealing Algorithm
## 5 Numerical Applications
### Design Optimization Studies with a Square FGM Plate with Circular Hole Subjected to Thermal Loading
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{2}{c}{\(\delta_{max}\) (10\({}^{-3}\) m)} & \multicolumn{2}{c}{\(\omega_{1}\) (rad/s)} & \multicolumn{2}{c}{**Mass (kg)**} & \multicolumn{2}{c}{**Cost (USD)**} \\ \hline SUS304/Si3N4 & & & & & & & \\ \hline
**objective**: _min. Mass_ & 7.3 & 0 & 2.1331 & 2015 & 3.95 & 197 & S1 \\ \hline
**objective**: _min. Cost_ & 15.7 & 10 & 1.0457 & 2010 & 27.37 & 161 & S2 \\ \hline SUS304/Al2O3 & & & & & & & \\ \hline
**objective**: _min. Mass_ & 9.3 & 0 & 1.5913 & 2003 & 8.06 & 221 & S3 \\ \hline
**objective**: _min. Cost_ & 15.8 & 10 & 1.7380 & 2003 & 28.01 & 131 & S4 \\ \hline Ti6 Al4V/Si3N4 & & & & & & & \\ \hline
**objective**: _min. Mass_ & 7.3 & 0 & 2.1331 & 2014 & 3.95 & 197 & S5 \\ \hline
**objective**: _min. Cost_ & 7.3 & 0 & 2.1331 & 2014 & 3.95 & 197 & S6 \\ \hline
**TiO6 Al4V/Al2O3 & & & & & & & \\ \hline
**objective**: _min. Mass_ & 9.3 & 0 & 1.5573 & 2014 & 8.06 & 221 & S7 \\ \hline
**objective**: _min. Cost_ & 9.3 & 0 & 1.5573 & 2014 & 8.06 & 221 & S8 \\ \hline \hline \end{tabular} In all problems constraints are: natural frequency: \(2000\leq\omega_{1}\leq 4500\) rad/s.  
\end{table} 
Table 6: Optimal designs for the simply supported square plate with circular hole, considering different metal–ceramic materials. Bounds imposed on the design variables: \(5.0\) mm \(\leq h_{FGM}\leq 20\) mm; \(0\leq p\leq 10\). Fundamental natural frequency: \(\omega_{1}\). Maximum displacement: \(\delta_{max}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{2}{c}{\(p\)} & \multicolumn{2}{c}{\(\delta_{max}\) (10\({}^{-3}\) m)} & \multicolumn{2}{c}{\(\omega_{1}\)(rad/s)} & \multicolumn{2}{c}{**Mass (kg)**} & \multicolumn{2}{c}{**Cost (USD)**} \\ \hline \multicolumn{2}{c}{\(\mathbf{h_{M}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{FGM}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{C}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{C}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{FGM}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{C}}\) (mm)} & \multicolumn{2}{c}{\(\mathbf{h_{C}}\) (mm)} \\ \hline SUS304/Si3N4 & & & & & & & \\ \hline
**objective**: _min. Mass_ & 0 & 6.3 & 1.0 & 0 & 2.2277 & 2014 & 3.95 & 197 & S1\({}^{\prime}\) \\ \hline
**objective**: _min. Cost_ & 5.7 & 5.9 & 0.5 & 0 & 1.706 & 2004 & 14.08 & 205 & S2\({}^{\  
Excellent improvements in the objective functions were obtained, with respect to the initial designs, in all optimization cases. The SA algorithm denoted lower performance when dealing with many design variables (namely for the present problem having 6 design variables), since a huge number of function evaluations are required (more than 50 k). The performance was much more acceptable for problems with lower number of design variables. The ability to deal with both discrete and continuous design variables in the same optimization problem was a remarkable advantage. Design Optimization Studies with a Square FGM Plate with Circular Hole Subjected to Mechanical Loading  
The optimization of an FGM simply supported square plate (\(a\times a\)) with a circular hole, as in the previous section, is now considered for the minimization of the transversal displacement (\(\delta_{\text{max}}\)) subject to Tsai-Hill stress criteria constraints and also for the minimization of the Tsai-Hill criteria, thus maximizing the strength safety factor. The plate is subjected to a transversal uniform pressure of \(2\times 10^{5}\) Pa, causing compression in the metal side and tension in the ceramic side. In these simple optimization problems, the design variables are: the FGM \(p\)-index comprised in the range \(0\leq p\leq 10\); the metal material; the ceramic material. The available metal and ceramic materials are the ones listed in Tables 1 and 2 and the plate thickness is fixed as \(h_{FGM}=8\) mm. Table 8 shows the optimal design for minimum transversal displacement.