Material Property-Estimate Technique Based on Natural Frequency for Updating Finite Element Model of Orthotropic Beams

2.1 Optimization Parameter Selection

To accurately identify the mechanical properties of composite structures, material properties need to be derived via standardized material tests, such as tensile, compression, and bending tests. Mechanical properties can vary significantly depending on the stacking environment and expertise of workers, even if the same base material is used and stacked with the exact same specifications. As an alternative to address this challenge, an inverse method can be applied to calculate the material’s properties using the dynamic properties of target structure, such as modal test results. This study updates the finite element model based on an optimization process that involves minimizing the difference between the natural frequencies obtained via the finite element model and modal test results, respectively. In the optimization process, density, thickness, number of layers, elastic and shear moduli, as well as Poisson’s ratio can be employed as the targets of optimization parameters. In this study, the elastic moduli, E₁₁ and E₂₂, and shear moduli, G₁₂, G₁₃, and G₂₃, which are considered to significantly influence the mechanical properties of orthotropic plate materials, were selected as optimization parameters.

2.2 Finite Element Model Development

The finite element model is saved in the bulk data file format (*.bdf), which is an MSC/NASTRAN input file. Fig. 2 illustrates an example of a bdf-file of the finite element model for an orthotropic beam used in this study. The modal analysis adopts the SEMODES (SOL 103) module of MSC/NASTRAN. The finite element is formulated using CQUAD4, a quadrilateral shell element. The material properties of the composite structure were defined using PCOMP. Although the orthotropic material properties were defined by MAT8, they were saved as separate MAT.INC files for defining the elastic and shear moduli to enable the easy control of optimization parameters. The file was designed to be retrieved by the “include” command from the bulk data file of MSC/NASTRAN during the optimization process.

2.3 Finite Element Model Update

The finite element model update for the composite structure is conducted such that the natural frequency of the target structure is induced to approach the actual value. The optimization process for the finite element model update adopted the discrete genetic algorithm, which is a typical global optimization algorithm.

The discrete genetic algorithm is accompanied by the following procedure:

More details on the finite element model update using the discrete algorithm are presented in Chapter 3.

3.1 Optimization Parameter Discretization and Coding

The optimization parameters are discretized and coded in the discrete genetic algorithm. The elastic and shear moduli, which are optimization parameters, are discretized in equal intervals for the desired range and stored as databases with unique numbers.

In the discrete genetic algorithm, the unique number of the database is coded as a chromosome in the form of an array composed of bits.

Given the number of databases discretized for each optimization parameter as N_ij, the selected database is expressed as an array L_ij composed of N_ij,b(=log₂ N_ij) number of bits (N_ij,b-bit), as expressed in Eq. (1).

where lijk denotes the k-th bit value in the coded array of the optimization parameter. For example, when the optimization parameters, l11k, l22k, l12k, l13k, and l23k, are discretized and coded, each of E₁₁, E₂₂, G₁₂, G₁₃, and G₂₃ becomes the k-th bit value of the N_ij,b-bit array, respectively.

The chromosome array G, consisting of all database information of the optimization parameters, can be expressed in Eq. (2), which links the chromosome array to each of the optimization parameters. The total number of bits, N, of the chromosome array is expressed in Eq. (3).

3.2 Fitness Assessment

Fitness is a quantitative index that indicates the proximity of the calculated natural frequency to the actual natural frequency when the material properties corresponding to individuals of each generation are reflected in the finite element model. A cost function was employed in this study to assess fitness, with lower costs representing higher fitness. The cost function J is expressed by Eq. (4) and was defined by considering the weight of the relative squared error for each natural frequency of the modal test and finite element analysis results (Zahari et al. 2016).

where fifem and fiexp, W_i, and n represent the i-th natural frequency derived from finite element analysis and modal test, weight for the -th natural frequency (“1” was applied as the weight in this study), and number of natural frequencies used in the optimization process, respectively. The natural frequency fiexp of the target beam model utilized the modal test results.

3.3 Discrete Genetic Algorithm Development

4.1 Verification of the Finite Element Model Update Method

The feasibility of the inverse method on the material properties of the composite structure was evaluated by performing inverse operation on the material properties of an orthotropic beam. In general, cantilevered beam type specimens are widely used to determine the dynamic properties of structures. Therefore, a cantilevered orthotropic beam was selected as the target structure for verification. Natural frequencies corresponding to the exact solution were calculated after conducting modal analysis on the target structure. Table 1 summarizes the material properties of the target cantilevered orthotropic beam.The material direction was assumed to be longitudinal. The finite element model for the target structure was designed with 240 rectangular shell elements and 305 nodes (Fig. 8). The length, width, and thickness of the finite element model were 600, 40, and 10 mm, respectively while the clamp fixture boundary condition was applied to the 0–40 mm area in the longitudinal direction. The natural frequencies of the target structure were extracted for a total of six generations, from first to sixth, and the values obtained are summarized in Table 2 together with the mode shape information.

The inverse operation was performed to calculate the material properties of the target structure in reverse solely via the natural frequency information presented in Table 2. In this process, elastic moduli, E₁₁ and E₂₂, and shear modulus, G₁₂, were selected as optimization parameters. Table 3 summarizes the ranges of material properties, as well as those of the discrete genetic algorithm parameters. The ranges were configured to include the material properties from Table 1. As a reference, 16 (4-bit) optimization parameter databases were configured for each parameter, and the moduli, G₂₃ and G₁₃ were fixed at 6.900E+09 N/m². However, the number of individuals for each generation was set as 50, and the probabilities of crossover and mutation were set as 0.9 and 0.1, respectively. The convergence criteria were established by setting 1E-10 and 100 as the required cost and maximum number of iterations, respectively.

Fig. 9 presents a convergence plot graph illustrating the minimum cost function value for each generation. The sixth generation’s minimum cost function J converged to zero, and the material properties derived via the inverse method matched the material properties of Table 1 without displaying any error. This indicates that the material property inversion method implemented in this study was successfully executed.

4.2 Application to Cantilevered CFRP Beam

To evaluate the practical applicability of the proposed finite element model update method, the inverse operation was performed to calculate the material properties of a random composite beam. The target structure was a unidirectional CFRP specimen together with the cantilevered CFRP beams used in the study conducted by Hwang et al. (2018). The specimen’s specifications and fixing methods are similar to those used in numerical analysis model presented in Section 4.1.

As shown in Fig. 10, the natural frequencies of the target structure corresponding to the exact solution were calculated via a modal test using an impact hammer. The natural frequencies obtained from the first to fourth generations were 50.87 Hz (1st vertical bending), 255.30 Hz (1st twisting), 273.72 Hz (1st lateral bending), and 326.65 Hz (2nd vertical bending), respectively (Hwang et al., 2018).

Elastic moduli, E₁₁, and E₂₂, and shear moduli, G₁₂, G₁₃, and G₂₃, were selected as optimization parameters. Table 4 summarizes the ranges of the material properties, as well as those of the discrete genetic algorithm parameters. As a reference, the number of discrete data for the elastic moduli, E₁₁ and E₂₂, was set to 64 (6-bit) and 32 (5-bit), respectively, and the number of discrete data for the shear moduli was set to 8 (3-bit). However, the number of individuals for each generation was set to 40, and the probabilities of crossover and mutation were set to 0.9 and 0.1, respectively, same as that of the previous section. The convergence criteria were established by setting 1E-6 as the required cost and 50 as the maximum number of iterations.

Fig. 11 presents a convergence plot graph showing the function values of the minimum cost for each generation in the optimization process. As illustrated, the required cost was not fulfilled until the vertical bending (3rd) maximum number of iterations. The final cost function value obtained is 6.904E-4 and Table 5 summarizes the final material properties of the orthotropic beam, which are the target optimization parameters.

Table 6 summarizes the comparison of natural frequencies measured via the modal test on the target orthotropic beam model, as well as the natural frequencies obtained from the updated finite element model. Additionally, the table presents the fifth and sixth natural frequencies obtained using the finite element model, as well as the first to sixth mode shapes. As presented in Table 7, the results obtained from the experiment and numerical estimation agreed optimally. Accordingly, via the updated finite element model, auxiliary information, such as high-order natural frequencies and mode shapes, can be estimated, as well as the material properties of the target structure.