# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 6 Conclusions  
An efficient and effective multi-level optimization method, consisting of several levels of optimization, was presented to solve different types of minimization problems. The proposed optimization method  
Figure 4: Dimensions of laminated specimens for the tensile test.  
included two major parts. Regarding the first part, the initial minimization problem was tackled via the solutions of a series of level-wise minimization problems, in which the level-wise feasible region of any level was derived from that of the previous level via a statistical approach. As for the second part, the multi-start global search technique (MGST) and the particle swarm optimization technique (PSOT) are used, respectively, to solve two consecutive level-wise minimization problems. The locations of the global minima of several functions were identified using the proposed method, as well as several commonly used global minimization algorithms. The total number of starting points generated in the proposed method was much less than those generated in each of the selected minimization methods. It has been shown that the proposed method could produce more accurate results than the selected methods. To demonstrate its applications, the proposed optimization method has been applied to the elastic constants identification of composite laminates subjected to a uniaxial load. Three measured strains in 0\({}^{\circ}\), and 45\({}^{\circ}\), and 90\({}^{\circ}\) directions have been used to establish the objective function. It has been shown that the measured normal strain in the 45\({}^{\circ}\) direction can work as a constraint to regulate the variations of the theoretical axial and lateral strains, and in turn make the design variables move efficiently toward their optimal values. Therefore, more accurate results can be obtained if the measured 45\({}^{\circ}\) strain is taken into account in establishing the objective function. The elastic constants of [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]s laminates made of different composite materials have been identified using the 0\({}^{\circ}\), 45\({}^{\circ}\), and 90\({}^{\circ}\) normal strains on the top layer of each laminate. For comparison, the selected global minimization techniques have also been used to perform the elastic constants identification. Again, the total number of starting points generated in the proposed method was much less than those generated in each of the selected global minimization methods. It has been shown that the proposed method with the use of three levels and a few starting points randomly generated at each level can produce more accurate results than the selected global minimization techniques. The absolute errors of the elastic constants identified are less than or equal to 0.97%, 0.06%, and 0.99% for the graphite/epoxy, carbon/epoxy, and glass/epoxy, respectively. Finally, the tensile test of a graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]s laminate was performed. The proposed method was applied to identify the elastic constants of the laminate using three measured strains. In the face of existing uncertainties, excellent results with a percentage of differences equal to or less than 1.15% have been obtained to reveal the potential of the proposed method for practical applications.  
[PERSON] did the mathematical programming, analyses, experimental work, and data collection. [PERSON] set the goal, constructed the method, and wrote the paper.  
This research was funded by the Ministry of Science and Technology of the Republic of China grant number MOST 108-2221-E-009-060.  
This research work was supported by the Ministry of Science and Technology of the Republic of China under Grant No. MOST 108-2221-E-009-060.  
The authors declare no conflict of interest.