# 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
Depending on the metal and ceramic materials being used those two objective functions can be contradictory, or not. The solutions not fulfilling the Tsai-Hill failure criteria are rejected. In addition, the natural frequency of the fundamental mode is required to fall in the range: \(2000\) rad/s \(\leq\omega_{1}\leq 4500\) rad/s and the solutions not fulfilling this condition are equally rejected. In a first optimization run, the design variables are: the metal material; the ceramic material; the power law index, \(p\); the thickness of the FGM plate, \(h_{FGM}\). The first two design variables are discrete. The material can be chosen from a list of available metal and ceramic materials. The \(p\)-index is defined as: \(0\leq p\leq 10\), with possible increments of 0.1. Constraints can be imposed on the materials which can be combined to form the FGM, considering manufacturing limitations for example. The FGM thickness variable has a lower bound of 5 mm and an upper bound of 20 mm, with possible increments of 0.1 mm. In a second optimization run, two design variables are added: the thickness of the metal lower face, \(h_{m}\); and the thickness of the ceramic upper face \(h_{c}\). The thicknesses are defined as in Figure 1. The metal and ceramic faces have a lower bound of zero and an upper bound of 20 mm, also with increments of 0.1 mm. The total thickness cannot exceed the value of 20 mm. The optimization results obtained with SA algorithm are presented in Table 6, considering four design variables, i.e., the metal and ceramic materials, the \(p\)-index and the FGM thickness, \(h_{FGM}\). In Table 7, are shown the optimization results considering six design variables. The results obtained are very similar to the previous ones, since the optimal results have converged to full ceramic or full metal solutions, the optimal thicknesses \(h_{M}\) and \(h_{C}\) are both zero in all cases. The optimal solutions for minimum mass are either the SUS304/Si\({}_{3}\)N\({}_{4}\) FGM with \(p=0\), with 7.3 mm thickness (solution S1) or the Ti6 Al4V/Si\({}_{3}\)N\({}_{4}\) FGM with \(p=0\), and 7.3 mm thickness (solutions S5 and S5\({}^{\prime}\)). These full ceramic silicon nitride optimal solutions are consequence of the highest Young modulus and lowest density of this ceramic among the considered set, which lead to an optimal solution with lower mass fulfilling the lower limit imposed on the natural frequency constraint. The optimal solution for minimum cost is the SUS304/Al\({}_{2}\)O\({}_{3}\) FGM with \(p=10\) and 15.8 mm thickness (solutions S4 and S4\({}^{\prime}\)). This is mostly a metal solution, which has the highest mass from all obtained solutions, since the steel SUS304 is the material having the lower cost and the highest density from the considered set. The optimal solutions listed in Tables 6 and 7 are represented in Figure 2. In some metal and ceramic combinations, the minimization of mass and minimization of cost are conflicting objectives, and therefore the solution to be adopted should be based on a compromise between both objectives. In this case the Ti6 Al4V/Si\({}_{3}\)N\({}_{4}\) with 7.3 mm thickness and \(p=0\), which means a Si\({}_{3}\)N\({}_{4}\) full ceramic (solutions S5 and S6), would be the best choice. Whenever metal and ceramic materials have similar prices and similar densities, the mass minimization and cost minimization are not conflicting objectives, and therefore similar solutions are obtained in both optimization problems.