# Optimization of Metal-Ceramic Functionally Graded Plates Using the Simulated Annealing Algorithm
## 3 Finite Element Model  
The displacement components \(u,\ v\) and \(w\) at any point in the laminate domain along the \(x,\ y\) and \(z\) directions, respectively, are expanded in a Taylor's series powers of the thickness coordinate \(z\). Each component is a function of \(x,\ y,z\) coordinates and time, t. The following higher order displacement field is used [44]:  
\[u=u_{0}+z\theta_{x}+z^{2}u_{0}^{*}+z^{3}\varphi_{x}^{*}v=v_{0}+z\theta_{y}+z^{ 2}v_{0}^{*}+z^{3}\varphi_{y}^{*}w=w_{0}+z\varphi_{z}+z^{2}w_{0}^{*} \tag{4}\]  
where \(u_{0},v_{0},w_{0}\), are the displacements of a generic point in the reference surface, \(\theta_{x},\theta_{y}\) are the rotations of normal to the reference surface about the y and x axes respectively (Figure 1) and \(\theta_{z},u_{0}^{*},v_{0}^{*},w_{0}^{*},\theta_{x}^{*},\varphi_{y}^{*}\) are the higher order terms defined at the reference surface, in a total of 11 nodal parameters (HSDT11). Based on this displacement field, the elastic strains can be written as: \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Materials** & **Properties** & \(\boldsymbol{P_{0}}\) & \(\boldsymbol{P_{-1}}\) & \(\boldsymbol{P_{1}}\) & \(\boldsymbol{P_{2}}\) & \(\boldsymbol{P_{3}}\) \\ \hline Aluminum & \(E\) (Pa) & \(349.55\times 10^{9}\) & 0 & \(-3.853\times 10^{-4}\) & \(4.027\times 10^{-7}\) & \(-1.673\times 10^{-10}\) \\ oxide & \(\
u\) & 0.26 & 0 & 0 & 0 & 0 \\ (Al\({}_{2}\)O\({}_{3}\)) & \(\rho\) (kg/m\({}^{3}\)) & 3800 & 0 & 0 & 0 & 0 \\  & \(\alpha\) (1/K) & \(6.8269\times 10^{-6}\) & 0 & \(1.838\times 10^{-4}\) & 0 & 0 \\  & \(\kappa\) (W/m K) & \(-14.087\) & \(-1123.6\) & \(-6.227\times 10^{-3}\) & 0 & 0 \\ \hline Silicon Nitride & \(E\) (Pa) & \(348.43\times 10^{9}\) & 0 & \(-3.070\times 10^{-4}\) & \(2.160\times 10^{-7}\) & \(-8.946\times 10^{-11}\) \\ (Si\({}_{3}\)N\({}_{4}\)) & \(\
u\) & 0.24 & 0 & 0 & 0 & 0 \\  & \(\rho\) (kg/m\({}^{3}\)) & 2370 & 0 & 0 & 0 & 0 \\  & \(\alpha\) (1/K) & \(5.8723\times 10^{-6}\) & 0 & \(9.095\times 10^{-4}\) & 0 & 0 \\  & \(\kappa\) (W/m K) & \(113.723\) & 0 & \(-1.032\times 10^{-3}\) & \(5.466\times 10^{-7}\) & \(-7.876\times 10^{-11}\) \\ \hline Titanium alloy & \(E\) (Pa) & \(122.56\times 10^{9}\) & 0 & \(-4.586\times 10^{-4}\) & 0 & 0 \\ (Ti-66 Al-4V) & \(\
u\) & 0.2884 & 0 & \(1.121\times 10^{-4}\) & 0 & 0 \\  & \(\rho\) (kg/m\({}^{3}\)) & 4429 & 0 & 0 & 0 & 0 \\  & \(\alpha\) (1/K) & \(7.5788\times 10^{-6}\) & 0 & \(6.638\times 10^{-4}\) & \(-3.147\times 10^{-7}\) & 0 \\  & \(\kappa\) (W/m K) & 1.0 & 0 & \(1.704\times 10^{-2}\) & 0 & 0 \\ \hline Stainless steel & \(E\) (Pa) & \(201.04\times 10^{9}\) & 0 & \(3.079\times 10^{-4}\) & \(-6.534\times 10^{-7}\) & 0 \\ (SS304) & \(\
u\) & 0.3262 & 0 & \(-2.002\times 10^{-4}\) & \(3.797\times 10^{-7}\) & 0 \\  & \(\rho\) (kg/m\({}^{3}\)) & 8166 & 0 & 0 & 0 & 0 \\  & \(\alpha\) (1/K) & \(12.330\times 10^{-6}\) & 0 & \(8.086\times 10^{-4}\) & 0 & 0 \\  & \(\kappa\) (W/m K) & 15.379 & 0 & \(-1.264\times 10^{-3}\) & \(2.092\times 10^{-6}\) & \(-7.223\times 10^{-10}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Temperature-dependent properties for metal and ceramic materials [42; 43]. \[\epsilon=\left\{\begin{array}{c} \begin{array}{c}\epsilon_{xx}\\ \epsilon_{yy}\\ \epsilon_{zz}\\ \gamma_{xy}\\ \gamma_{yz}\\ \gamma_{xz}\end{array} \right\}=\left\{ \begin{array}{c}\begin{array}{c} \frac{\frac{\partial u_{0}}{\partial x}}{\partial y}\\ \frac{\partial v_{0}}{\partial y}\\ \phi_{z}\\ \frac{\partial u_{0}}{\partial y}+\frac{\partial v_{0}}{\partial x}\\ \theta_{y}+\frac{\partial v_{0}}{\partial y}\\ \theta_{x}+\frac{\partial v_{0}}{\partial x}\end{array}\end{array} \right\}+z \left\{\begin{array}{c} \begin{array}{c}\frac{\frac{\partial u_{0}^{*}}{ \partial x}}{\partial y}\\ \frac{\partial v_{0}}{\partial y}\\ \frac{\partial v_{x}}{\partial y}\\ 0\\ 2v_{0}^{*}+\frac{\partial v_{0}^{*}}{\partial x}\\ \partial\phi\\ 2u_{0}^{*}+\frac{\partial v_{0}}{\partial x}\end{array} \right\}+z^{2}\left\{