## 4 Numerical Results  
Figure 3a presents a metal-ceramic FG square plate under the action of uniform vertical distributed load \(q_{0}=1.0\) N/m\({}^{2}\), where the side length \(a\) is 0.1 m and the thickness \(h\) are variables for the parametric investigation. The elastic modulus \(E\), Poisson's ratio \(\
u\) and the density \(\rho\) are 70 GPa, 0.3 and 2707 kg/m\({}^{3}\) for metal and 380 GPa, 0.3 and 3800 kg/m\({}^{3}\) for ceramic. The simply supported condition is enforced as follows: \(\overline{U}^{0}_{z}=0\) for all the four sides, \(\overline{U}^{0}_{x}=\cdots=\overline{U}^{3}_{x}=0\) for two sides 1 and 3, and \(\overline{U}^{0}_{y}=\cdots=\overline{U}^{3}_{y}=0\) for the other sides 2 and 4. The full in-plane integration on the mid-surface for the stiffness and mass matrices and the load vector was performed using 7 Gauss points, while the thickness-wise integration was made using the trapezoidal rule by dividing the thickness into 50 uniform segments. The solutions of linear matrix Equation (16) were obtained by the frontal solver. The evaluation points and the calibrated deflection and stresses used in this study are defined by  
\[\overline{w}=\frac{10E_{c}h^{3}}{q_{0}a^{4}}w(0,0,0),\ \overline{\sigma}_{x}=\frac{h}{q_{0}a} \sigma_{x}(0,0,z^{*}),\ \overline{\tau}_{xz}=\frac{h}{q_{0}a}\tau_{xz}\Bigl{(}-\frac{a}{2},0,z^{*} \Bigr{)} \tag{22}\]  
where, \(z^{*}\) is the vertical position where the stresses show their maximum absolute values. Figure 3: (**a**) A square FG porous plate under the uniform vertical distributed load \(q_{0}\), (**b**) an \(11\times 11\) uniform NEM grid. First, the variation in \(\overline{w}\) and \(\overline{\sigma}_{x}\) was examined with respect to the NEM grid density, which is illustrated in Figure 3b. The width-thickness ratio \(a/h\) was 10 and the ceramic index \(n\) and the porosity parameter \(e\) were set by zero. The numerical results are presented in Table 1, where the values within the parentheses are the relative differences to the values obtained using the \(19\times 19\) NEM grid. It is seen that two calibrated quantities converge to the reference values proportional to the grid density and a \(15\times 15\) uniform grid shows the relative differences less than 1.0%, so according to this convergence test, a \(15\times 15\) uniform grid was chosen for the remaining numerical experiments, unless stated otherwise. The calibrated central deflection \(\overline{w}\) was computed for various combinations of \(a/h\), \(n\) and \(e\) and compared with those of [PERSON] and [PERSON] [36] as given in Table 2. The reference solutions were analytically solved using a four-variable refined plate theory. It is seen that the present results coincide well with the reference solutions such that the peak relative difference is 5.336% at the combination of \(a/h=20,n=0.5\) and \(e=0.2\) for the uneven porosity distribution. In addition, the free vibration of the simply supported FG porous plate was performed using Lanczos transformation and Jacobi iteration methods. The lowest 10 natural frequencies for the previous combinations of \(a/h\), \(n\) and \(e\) were computed using the \(15\times 15\) uniform NEM grid. In Table 3, the calibrated fundamental frequencies \(\overline{\omega}_{(1,1)}=\omega_{(1,1)}h\sqrt{\rho_{m}/E_{m}}\) are compared with those in reference [36]. It is seen that the calibrated fundamental frequencies predicted by the present method coincide well with those of reference [36]. The detailed numerical values informs that the maximum relative difference is 3.954% at the combination of \(a/h=20,n=1.0\) and \(e=0.4\) for the even porosity distribution. Hence, it has been justified that the present (3,3,2) hierarchical model is reliable for the free vibration analysis of FG porous plates. Thus, the present numerical method using the (3,3,2) hierarchical model has been verified such that thepeak relative difference between the present method and the reference is 5.336%.