# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 4 Elastic Constants Identification of Composite Laminate
### Numerical Examples
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{**Level 2.**} & \multicolumn{5}{c}{**Material Constant**} \\ \cline{3-6}  & & \(E_{1}\) (GPa) & \(E_{2}\) (GPa) & \(\
u_{12}\) & \(G_{12}\) (GPa) \\ \hline Actual & 138.50 & 8.11 & 0.31 & 5.22 \\ \hline \multicolumn{6}{c}{Particle group No.} \\ \hline  & 1 & 138.28 & 8.302 & 0.331 & 5.271 \\  & 2 & 138.54 & 7.797 & 0.304 & 5.219 \\ Identified & 3 & 138.59 & 8.044 & 0.301 & 5.198 \\  & 4 & 138.52 & 8.259 & 0.309 & 5.210 \\  & 5 & 138.38 & 8.760 & 0.323 & 5.228 \\ \hline Mean (Quasi-optimal design variable \(\mu\)) & 138.46 (\(-\)0.03\(\%\)) * & 8.233 (1.51\%) & 0.314 (1.17\%) & 5.225 (0.10\%) \\ Standard deviation \(\sigma\) & 0.11 & 0.32 & 0.01 & 0.02 \\ Coefficient of variation & 0.08 & 3.88 & 3.61 & 0.48 \\ \hline \multicolumn{6}{c}{* Percentage difference = \(\left(\frac{L^{\text{initial}-\text{actual}}}{\text{actual}}\right)\times 100\%\).} \\ \end{tabular}
\end{table} 
Table 4: Solution of the second-level optimization for graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) laminate using actual strains. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{**Level 1**} & \multicolumn{5}{c}{**Material Constant**} \\ \cline{3-6}  & & \(E_{1}\) (GPa) & \(E_{2}\) (GPa) & \(\
u_{12}\) & \(G_{12}\) (GPa) \\ \hline No. & Actual & Actual & 8.11 & 0.31 & 5.22 \\ \hline \multirow{2}{*}{1} & Starting point & 323.83 & 5.470 & 0.240 & 8.163 \\  & Lowest local minimum & 138.75 & 10.126 & 0.298 & 5.080 \\ \hline \multirow{2}{*}{2} & Starting point & 440.60 & 6.235 & 0.428 & 5.819 \\  & Lowest local minimum & 138.62 & 8.057 & 0.298 & 5.190 \\ \hline \multirow{2}{*}{3} & Starting point & 423.37 & 6.992 & 0.164 & 9.102 \\  & Lowest local minimum & 138.65 & 8.262 & 0.296 & 5.173 \\ \hline \multirow{2}{*}{4} & Starting point & 38.41 & 6.472 & 0.432 & 1.908 \\  & Lowest local minimum & 138.61 & 8.283 & 0.300 & 5.183 \\ \hline \multirow{2}{*}{5} & Starting point & 174.31 & 8.895 & 0.451 & 4.871 \\  & Lowest local minimum & 137.84 & 9.625 & 0.366 & 5.335 \\ \hline \multirow{2}{*}{6} & Starting point & 28.28 & 11.457 & 0.158 & 6.655 \\  & Lowest local minimum & 139.08 & 9.948 & 0.270 & 4.994 \\ \hline \multirow{2}{*}{7} & Starting point & 466.51 & 4.205 & 0.460 & 2.226 \\  & Lowest local minimum & 138.78 & 7.244 & 0.275 & 5.174 \\ \hline \multirow{2}{*}{8} & Starting point & 368.32 & 6.533 & 0.163 & 3.714 \\  & Lowest local minimum & 138.45 & 6.739 & 0.309 & 5.284 \\ \hline \multirow{2}{*}{9} & Starting point & 23.15 & 2.677 & 0.209 & 7.166 \\  & Lowest local minimum & 137.72 & 7.448 & 0.384 & 5.442 \\ \hline \multirow{2}{*}{10} & Starting point & 300.78 & 9.360 & 0.367 & 7.618 \\  & Lowest local minimum & 138.55 & 7.955 & 0.304 & 5.212 \\ \hline \multirow{2}{*}{Mean (Quasi-optimal design variable \(\mu\))} & 138.50 & 8.369 & 0.310 & 5.207 \\  & (0.00\%) * & (3.19\%) & (0.03\%) & (\(-\)0.25\%) \\ \multicolumn{2}{c}{Standard deviation \(\sigma\)} & 0.40 & 1.11 & 0.03 & 0.12 \\ \multicolumn{2}{c}{Coefficient of variation} & 0.29 & 13.21 & 11.19 & 2.29 \\ \hline \multicolumn{6}{c}{* Percentage difference = \(\left(\frac{L^{\text{initial}-\text{actual}}}{\text{actual}}\right)\times 100\%\).} \\ \end{tabular}
\end{table} 
Table 3: Solution of the first-level optimization for graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) laminate using actual strains. The iteration between the two levels continued until the standard deviations of the quasi-true design variables were less than a prescribed value (\(\sigma_{\zeta}=10^{-3}\)). Finally, the solution of the optimization problem converges at the third level. The identified elastic constants associated with the global minimum are listed in Table 5 in comparison to the actual elastic constants. It is noted that the differences between the actual and identified elastic constants were less than or equal to 0.97%.