# 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  
The optimization of an FGM square plate (\(a\times a\)) with a circular hole with diameter \(d\) is now considered as a design optimization case, with \(a=0.5\) m and \(d=a/3\). The plate is \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{2}{c}{**Tasi-Hill Criteria**} & \multicolumn{2}{c}{**Mass (kg)**} & \multicolumn{1}{c}{**Cost (USD)**} \\ \cline{2-6}  & \(p\) & \(h_{\text{FGM}}\)**(mm)** & & & \\ \hline Objective: _min Mass_ & & & & & \\ SA & 0.2 & 34.2 & 0.999 & 38.69 & 886 \\ \hline [PERSON] et al. [38] & 0.2 & 34.2 & 0.9991 & 38.69 & 885.75 \\ \hline Objective: _min Cost_ & & & & & \\ SA & 2.9 & 53.2 & 0.999 & 93.72 & 752 \\ \hline [PERSON] et al. [38] & 2.9 & 53.2 & 1.0 & 93.72 & 752.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal designs for the FGM SUS304/Al2O3 simply supported square plate for the mentioned objective functions subject to Tsai-Hill failure constraints with \(0.2\leq p\leq 10\). Two design variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{2}{c}{**Tasi-Hill Criteria**} & \multicolumn{2}{c}{**Mass (kg)**} & \multicolumn{1}{c}{**Cost (USD)**} \\ \cline{2-6}  & \(p\) & \(h_{\text{FGM}}\)**(mm)** & & & \\ \hline Objective: _min Mass_ & & & & & \\ Simulated Annealing (SA) & 0.2 & 37.8 & 0.999 & 31.5 & 1279 \\ \hline [PERSON] et al. [38] & 0.2 & 37.8 & 0.999 & 31.5 & 1279.24 \\ \hline Objective: _min Cost_ & & & & & \\ SA & 10 & 20.2 & 0.993 & 38.59 & 227 \\ \hline [PERSON] et al. [38] & 10 & 20.2 & 0.993 & 38.59 & 226.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal designs for the functionally graded material (FGM) SUS304/Si3N4 simply supported square plate for the aforementioned objective functions subject to Tsai-Hill failure constraints with \(0.2\leq p\leq 10\). Two design variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Objective: _max._\(\omega_{1}\)} & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{1}{c}{**\(\omega_{1}\)(rad/s)**} & \multirow{2}{*}{**Mass (kg)**} & \multirow{2}{*}{**Cost (USD)**} \\ \cline{2-2} \cline{6-6}  & \(p\) & \(h_{\text{FGM}}\)**(mm)** & & & \\ \hline SA & 3.4 & 59.1 & 8000 & 101.20 & 1076.50 \\ \hline [PERSON] et al. [37] & 3.4 & 59.1 & 8000 & 101.20 & 1076.50 \\ \hline [PERSON] et al. [33] & 3.8 & 59.5 & 8000 & 103.74 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Maximum fundamental natural frequency \(\omega_{1}\) for the FGM SUS304/Si3N4 simply supported square plate. Two design variables. Bounds imposed on the design variables: 5.0 mm \(\leq h_{\text{FGM}}\leq\) 60 mm; 0.2 \(\leq p\leq 10\). simply supported all round and is subjected to a thermal load through the thickness of the plate, such that the metal side is kept at a stress-free temperature and at the ceramic side the temperature is raised up to a \(\Delta T=120\) K, while the temperature distribution in the \(x-y\) plane is uniform. In this problem, we are interested in finding the best material choice, from a list of available ceramic and metal materials, the power law index \(p\), and the thicknesses of the FGM material, in order to minimize the material cost of the plate or minimize the mass of the plate. The optimization process is constrained with Tsai-Hill failure criteria and natural frequency constraints. The material properties are defined in Table 1, being temperature dependent. The following, indicative material costs are considered: stainless steel--3 USD/kg; titanium alloy (Ti6 Al4V)--30 USD/kg; silicon nitride (Si\({}_{3}\)N\({}_{4}\))--50 USD/kg; alumina (Al\({}_{2}\)O\({}_{3}\))--27.4 USD/kg, based in approximated average market costs. Two objective functions are considered: the minimization of the overall mass of the plate; and the minimization of the material cost of the plate.