# Optimization of Metal-Ceramic Functionally Graded Plates Using the Simulated Annealing Algorithm
## 6 Conclusions  
Design optimization is an important engineering tool for the proper design of FGM structures. The use of the simulated annealing optimization algorithm associated with a finite element model for the structural analysis of through-the-thickness FGM metal-ceramic plate structures, subjected to thermomechanical loadings, has been shown in this paper, as an extension of previous works developed by the authors. The accuracy of the finite element model was validated in previous works from the authors, namely in references [37, 38], against alternative solutions, being exact, numerical or experimental, published by other authors, and therefore this demonstration was omitted here. The optimization results obtained with the SA algorithm have been compared, for validation purposes, in a few numerical cases, with previously published works from the authors, namely with optimization results obtained using a direct search derivative-free algorithm--global and local optimization using direct search (GLODS) and also with optimization results published in [33] obtained with the feasible arc interior point (FAIPA) gradient-based algorithm. The validation of the SA optimization results was excellent, but the computation efforts involved were not comparable since the results obtained with GLODS were included in a multiobjective optimization scheme and, regarding the FAIPA results, the computation time was not available. In this work, besides the \(p\)-index and the thickness of the FGM plate, the choices of metal and ceramic materials have been added as design variables of the optimization problem, with the aim of finding the best metal-ceramic combination from a discrete set of available materials to fulfill an optimization objective, namely the mass minimization, cost minimization and other structural behavior objectives. These design variables have been pointed out in recent reviews as the most important for the design of FGM structures, alongside the need to address the manufacturing limitations which can be easily implemented with the SA. The convergence of the SA to the global optima cannot be guaranteed, but for the validation cases which have been carried out, a good agreement has been verified with respect to the alternative solutions mentioned before. It has been verified as a good performance of the SA algorithm for problems with lower number of design variables. For the optimization problems with higher number of design variables, which involved the \(p\)-index, thicknesses, and the materials all together, the computation time required by the SA algorithm was quite high due to the high number of function evaluations which were necessary to obtain convergence. Whenever a lower number of design variables are used in the optimization problem, the SA algorithm showed a good performance. The numerical examples presented in this paper, involving the optimal design of FGM plate structures subject to thermomechanical loadings, can be used as benchmark solutions, knowing that these studies are, so far, scarce in the literature. Manufacturing constraints, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design Variables** & **Tsai-Hill Criteria** &
\begin{tabular}{c} **Elastic Strain Energy** \\ **(J/m2)** \\ \end{tabular} & **Mass (kg)** & **Cost (USD)** \\ \hline \(p=0.6\) & 0.10368 & 47.66 & 7.17 & 298.7 \\ \(\text{Ti6 Al4V}/\text{Si}_{3}\text{N}_{4}\) & & & & \\ \hline \(p=0.6\) & 0.10425 & 47.88 & 9.01 & 261.5 \\ \(\text{Ti6 Al4V}/\text{Al}_{2}\text{O}_{5}\) & & & & \\ \hline \(p=0.6\) & 0.27673 & 39.10 & 12.41 & 211.0 \\ \(\text{SUS304}/\text{Al}_{2}\text{O}_{3}\) & & & & \\ \hline \(p=0.6\) & 0.27738 & 399 & 10.37 & 296.1 \\ \(\text{SUS304}/\text{Si}_{3}\text{N}_{4}\) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Optimal design for the simply supported square plate with circular hole subject to transversal pressure load, for minimization of the Tsai–Hill failure criteria, considering different metal–ceramic materials.