# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 1 Introduction  
As is well known, the determination of the true elastic constants is an essential step for design, fabrication, and the structural health monitoring of structures. Therefore, the development of efficient and effective methods for determining the actual elastic constants of materials and structures has become an important topic of research. However, for engineering applications, having simple testing procedures that can produce good estimations of the elastic constants of composite materials in an efficient way is always desirable. In recent years, measured mechanical characteristics, together with optimization techniques such as the genetic algorithm, particle swarm optimization method, stochasticglobal optimization method, simulated annealing algorithm, hybrid method, etc., have been used to determine the material constants of different types of structural parts/specimens [1]. For example, a number of researchers have used optimization techniques to extract elastic constants of laminated composite plates from vibration test data [2; 3; 4; 5; 6; 7; 8]. Most of the previously proposed vibration-based elastic constants identification techniques used a number of measured, structural natural frequencies to formulate the minimization problem, which is solved by using a minimization algorithm to determine the material constants. It is noted that the vibration testing techniques that involve the selection of plate restraints, installation of sensors/accelerometers, exertion of excitation force, extraction of modal characteristics (natural frequencies and mode shapes), etc., may be time consuming and expensive. Furthermore, the uncertainties of, for instance, the measured modal characteristics, the mathematical model for analyzing composite plate vibration, modeling the actual boundary conditions, etc., may significantly affect the accuracy of the identified elastic constants. Therefore, at this stage, the techniques of using vibration data for elastic constants identification of composite materials/structures may not be completely viable for practical applications. However, a number of papers are devoted to the use of measured strains/displacements to identify the elastic constants of isotropic/composite materials [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. In particular, the full-field measurements of strains/displacements have been used to identify the elastic constants of solids. For instance, [PERSON] et al. [10] proposed an experimental procedure for the direct identification of the in-plane elastic properties of T-shaped orthotropic composite plates from measured heterogeneous strain fields using an optical method. In their study, the accuracy of the elastic constants extracted from the experimental data was not verified. A review of different full-field methods has been conducted by [PERSON] et al. [11]. The full-field strain measurement is a relatively complicated process in which the identification of parameters requires the use of suitable computational strategies to analyze the experimental data. Furthermore, the capability of the full-field methods to identify accurate elastic constants of composite materials/structures has yet to be demonstrated. However, without the need to perform any full-field measurements, [PERSON] and his associates [12; 17; 18; 19] used optimization methods and measured strains/displacements at some particular points on composite plates to identify the elastic constants of the constituent composite materials. For instance, [PERSON] and [PERSON] [12] constructed a constrained minimization problem to identify the material constants of laminated composite plates subjected to point or uniformly distributed loads using measured strains and/or displacements at some particular points on the plates.