# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 4 Elastic Constants Identification of Composite Laminate
### Numerical Examples
To demonstrate the advantage of the proposed method, the commonly used global minimization methods were also used to identify the elastic constants of the graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) composite laminate. The values of the parameters used in the selected global minimization methods have been given in Section 2. The elastic constants identified using the selected global minimization methods are listed in Table 6 in comparison to those identified using the proposed method. It is clear that the accuracy of the proposed method was much higher than the other methods. Again, it is noted that the total number of starting points generated in the proposed method was much less than those of the adopted popular global minimization methods. This revealed the fact that the proposed method was efficient and capable of producing accurate results. To further demonstrate the capability of the proposed MLOM, the elastic constants of different composite materials were also extracted from their given strains. The actual elastic constants, together with the given strains used for elastic constants identification of two composite materials, namely, carbon/epoxy and glass/epoxy, respectively, are listed in Table 7. The solutions of the first and second levels of optimization problems are listed in Table 8. Again, it is noted that at the second level, the mean of the quasi-optimal elastic constants were very close to the actual elastic constants with errors less than or equal to 0.09% for the carbon/epoxy and 1.5% for the glass/epoxy. At the third level, the identified elastic constants for the composite materials are listed in Table 9 in comparison to the actual ones. It is noted that the errors of the identified elastic constants were less than or equal to 0.06% and 0.99% for the carbon/epoxy and glass/epoxy, respectively. The small errors of the identified elastic constants revealed the capability of the proposed MLOM for elastic constants identification of different composite materials. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Material Constant**} \\ \cline{2-5}  & \(\mathbf{E_{1}}\) (GPa) & \(\mathbf{E_{2}}\) (GPa) & \(\mathbf{\
u_{12}}\) & \(\mathbf{G_{12}}\) (GPa) \\ \hline Actual & 138.5 & 8.11 & 0.31 & 5.22 \\ MSGO & 138.29 (\(-\)0.15\%) * & 7.678 (\(-\)5.33\%) & 0.329 (6.03\%) & 5.291 (1.35\%) \\ PS & 140.09 (1.15\%) & 10.048 (23.90\%) & 0.180 (\(-\)41.98\%) & 4.677 (\(-\)10.39\%) \\ GA & 136.81 (\(-\)1.22\%) & 7.232 (\(-\)10.82\%) & 0.467 (50.72\%) & 5.657 (8.38\%) \\ Present method (3 Levels) & 138.47 (\(-\)0.02\%) & 8.185 (0.93\%) & 0.313 (0.97\%) & 5.225 (0.09\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Solutions of different optimization methods for graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) laminate using actual strains. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Level 3**} & \multicolumn{4}{c}{**Material Constant**} \\ \cline{2-5}  & \(\mathbf{E_{1}}\) (GPa) & \(\mathbf{E_{2}}\) (GPa) & \(\mathbf{\
u_{12}}\) & \(\mathbf{G_{12}}\) (GPa) \\ \hline Actual & 138.50 & 8.11 & 0.31 & 5.22 \\ Identified & 138.47 (\(-\)0.02\%) * & 8.185 (0.93\%) & 0.313 (0.97\%) & 5.225 (0.09\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Solution of the third-level optimization for graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) laminate using actual strains.