# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 4 Numerical Results  
The proposed differential quadrature methods exhibit enhanced convergence and efficiency in analyzing the vibration of magneto-electro-thermo-elastic nanobeams. The boundary conditions were incorporated into the governing Equations (47)-(50) and solved iteratively using Equations (55)-(62). The computational parameters for each method were optimized to ensure accurate results with an error of at least \(10^{-10}\). The natural frequencies \(\omega\) can be determined using the following equation:  
\[\omega=\Omega L\sqrt{\frac{I_{0}}{\bar{A}_{11}}} \tag{63}\]  
where the natural frequency of the nanobeam is denoted by \(\Omega\). For the present results, material parameters for the composite are listed in Table 1.  
The PDQM utilized a non-uniform grid based on Gauss-Chebyshev-Lobatto points for discretization [42]. \[x_{i}=\frac{1}{2}\bigg{[}1-\cos\bigg{(}\frac{i-1}{N-1}\pi\bigg{)}\bigg{]},\quad(i =1:N) \tag{64}\]  
There were three to fifteen grid points, \(N\). As seen in Table 2, the outcomes were consistent with earlier analytical solutions [46; 47] for 11 grid points. The Sinc-Discrete Singular-Convolution Differential Quadrature (Sinc-DQ) method was employed in a regular grid with grid sizes ranging from 3 to 15. The numerical results obtained using the Sinc-DQ method converged to the exact solutions [25] for grid sizes greater than or equal to 9, as shown in Table 3. Additionally, the Sinc-DQ method exhibited superior computational efficiency compared to the PDQM method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Parameters** & & **BiTiO\({}_{\mathbf{3}}\)-COFe\({}_{\mathbf{2}}\)O\({}_{\mathbf{4}}\)** & **PTZ-5H-COFe\({}_{\mathbf{2}}\)O\({}_{\mathbf{4}}\)** \\ \hline Elastic parameters (GPa) & C\({}_{11}\) & 226 & 206 \\  & C\({}_{12}\) & 125 & 126.25 \\  & C\({}_{13}\) & 124 & 127.3 \\  & C\({}_{44}\) & 44.2 & 34.3 \\ \hline Piezoelectric parameters (C/m\({}^{2}\)) & \(e_{31}\) & \(-\)2.2 & \(-\)3.25 \\  & \(e_{33}\) & 9.3 & 11.65 \\  & \(e_{15}\) & 5.8 & 8.5 \\ \hline Dielectric parameters (C/Vm) \(\times\)\(10^{-9}\) & \(e_{33}\) & 5.64 & 7.555 \\  & \(\epsilon_{33}\) & 6.35 & 6.5465 \\ \hline Piezomagnetic parameters (N/Am) & \(q_{31}\) & 290.1 & 290.15 \\  & \(q_{33}\) & 349.9 & 349.9 \\  & \(q_{15}\) & 275 & 275 \\ \hline Magnetoelectric parameters (Ns/VC) \(\times\)\(10^{-12}\) & \(d_{11}\) & 5.367 & 16.5 \\  & \(d_{33}\) & 2737.5 & 20.7 \\ \hline Magnetic parameters (Ns\({}^{2}\)/C\({}^{2}\)) \(\times\)\(10^{-6}\) & \(\mu_{11}\) & \(-\)297 & \(-\)297 \\  & \(\mu_{33}\) & 83.5 & 83.5 \\ \hline Thermal module (N/m\({}^{2}\)K) \(\times\)\(10^{5}\) & \(\lambda_{1}\) & 4.74 & 5 \\  & \(\lambda_{3}\) & 4.53 & 4.85 \\ \hline Density (kg/m\({}^{3}\)) & \(\rho\) & 5550 & 6550 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key material properties of the investigated composites [45; 46]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Grid Size, N** & \(\mathbf{\omega_{1}}\) & \(\mathbf{\omega_{2}}\) & \(\mathbf{\omega_{3}}\) & \(\mathbf{\omega_{4}}\) & \(\mathbf{\omega_{5}}\) \\ \hline
3 & 14.5445 & 145.484 & 165.349 & — & — \\
5 & 7.6348 & 30.1832 & 44.1502 & 147.015 & 161.182 \\
7 & 7.6440 & 19.0072 & 32.8765 & 74.905 & 88.464 \\
9 & 7.6460 & 18.6790 & 32.4585 & 50.130 & 64.8814 \\
11 & 7.6460 & 18.6790 & 32.4772 & 48.095 & 64.0633 \\
13 & 7.6460 & 18.6790 & 32.4765 & 48.006 & 64.1910 \\ \hline Exact results [46] & 7.6473 & 18.6692 & 32.4618 & — & — \\ PDQM [47] & 7.6267 & 18.6229 & 32.3964 & — & — \\ \hline Computational time & 0.15 s & A non-uniform grid with more than 11 nodes \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normalized frequencies: PDQM vs. exact and numerical solutions (C-C METE nanobeam). (\(L=80\) nm; \(h=10\) nm; \(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0\); \(\mu=0\); \(k_{1}=k_{2}=k_{3}=0\)). The Discrete Singular Convolution Differential Quadrature Method utilizing the Delta Lagrange Kernel (DSCQM-DLK) was implemented in a uniform grid with sizes ranging from 3 to 11.