# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 4 Elastic Constants Identification of Composite Laminate
### Numerical Examples  
As an example, the strains of a Graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) composite laminate were used in the proposed multi-level optimization method to identify the elastic constants of the constituent composite material. The actual elastic constants of the composite material were set as \(E_{1}\) = 138.5 GPa, \(E_{2}\) = 8.1 GPa, \(G_{12}\) = 5.2 GPa, and \(\
u_{12}\) = 0.31. When subjected to axial force \(F_{x}\) = 0.5 kN, the actual strains of the laminate computed from Equations (16)-(18) were \(\varepsilon_{x}\) (\(\mu\epsilon\)) = 210.49, \(\varepsilon_{y}\) (\(\mu\epsilon\)) = -151.81, and \(\varepsilon_{45}\) (\(\mu\epsilon\)) = 29.34. Herein, the actual strains were treated as the \"given\" strains, which were used to identify the true elastic constants. To cover different kinds of fiber-reinforced composite materials, the upper and lower bounds of the elastic constants were initially chosen as  
\[ \begin{array}{c}0\leq E_{1}\leq 1000\ GPa\\ 0\leq E_{2}\leq 50\ GPa\\ 0\leq G_{12}\leq 20\ GPa\\ -1.0\leq\
u_{12}\leq 0.5\end{array}  \tag{20}\]  
At the first level of optimization, the values of the weighting factors \(\xi_{i}\) (i = 1-3) in Equation (19) could be chosen arbitrarily so long as these values were positive and the weighting factors of the strain difference terms in Equation (19), containing the axial strain, had larger values than that of the strain difference term that only contained the lateral strain. Herein, based on engineering judgment, the values of the weighting factors \(\xi_{i}\) (i = 1-3) were set to be 10\({}^{6}\), 10\({}^{4}\), and 10\({}^{6}\), respectively. This choice could have led to more accurate prediction of \(E_{1}\). On the contrary, at the second level of optimization, the values of the weighting factors \(\xi_{i}\) (i = 1-3) in Equation (19) were set to be 10\({}^{4}\), 10\({}^{6}\), and 10\({}^{4}\), respectively. At this level, the larger weight for the lateral strain may have led to more accurate predictions of \(E_{2}\), while only a small adjustment of \(E_{1}\) might occur. The procedure of the elastic constants identification of the graphite/epoxy laminate was first described in detail to illustrate the application of the MLOM. At the first level, the constrained minimization problem of Equation (19) was converted to an unconstrained minimization one, as stated in Equation (2). The unconstrained minimization problem was then solved using the MGST, in which ten starting points were randomly generated in the initial feasible range to determine the quasi-global minima and their associated quasi-optimal design variables. The starting points and the statistics of the quasi-optimal design variables are listed in Table 3. It is noted that the means of the quasi-optimal design variables were very close to the actual values, with errors less than or equal to 3.19%. The contracted feasible region adopted for the next level was determined as [\(\mu_{i}\pm\sigma_{i}\)]. At the second level where PSOT was applied to solve the optimization problem, five sets, of which each set contained ten randomly generated starting points, were used to search for the quasi-global minima. The statistics of the quasi-optimal design variables are listed in Table 4. The comparison between the previous and current levels shows that the means of the quasi-optimal elastic constants of the current level were much closer to the actual values, with errors less than or equal to 1.51%. The contracted feasible region adopted for the next level was also determined as [\(\mu_{i}\pm\sigma_{i}\)].