Abstract
A frequency response function (FRF) based surrogate model for finite element model updating (FEMU) is presented in this paper. Extreme learning machine (ELM) is introduced as the surrogate model of the finite element model (FEM) to construct the relationship between updating parameters and structural responses. To further improve the generalization ability, the input weights and biases of ELM are optimized by Lévy flight trajectorybased whale optimization algorithm (LWOA). Then, LWOA is also applied to obtain the best updating results, where the objective function is defined by the difference between analytical FRF data and experimental data. Finally, a plane truss is used to demonstrate the performance of the proposed method. The results show that, compared with secondorder response surface (RS), radial basis function (RBF), traditional ELM, and other optimized ELM, a LWOAELM model has higher prediction accuracy. After updating, the FRF data and frequencies have a significant match to the experimental model. The proposed FEMU method is feasible.
1. Introduction
Due to the capacity for structural identification and health monitoring, finite element method has attracted much more attentions in the past few years. However, an initial finite element model (FEM) cannot reflect the actual structure precisely, because of simplifications and idealizations (ideal boundary conditions, material properties, etc.) while constructing the FEM [1–3]. In order to be used for further analysis, the initial FEM should be updated to minimize the error between the analytical responses and the experimental ones, which is called finite element model updating (FEMU) [4]. Several methods for FEMU have been developed. Among these methods, vibrationbased techniques are widely used. This kind of technique can be classified into two main categories: the methods based on modal data and the ones based on frequency response function (FRF) data [5–9]. For the former, natural frequencies and mode shapes are introduced to define the objective function [10]. For the latter, FRF data provide enough response information directly without modal analysis, which can increase the accuracy of model updating. The FRFbased methods have been gradually applied in FEMU [11, 12].
In addition, the model updating requires dozens of iterations to compute the analytical responses, which is timeconsuming. The surrogate models, such as response surface (RS), Kriging model, radial basis function (RBF), neural network (NN), and support vector machine (SVM), are widely used to construct the relationship between the input parameters and output responses to replace the original FEM. Marwala [13] introduced multilayer perceptron (a neural network) as the surrogate model to update an Htype structure. Ren and Chen [14] presented a quadratic polynomial response surfacebased method. A simply supported beam and a concrete bridge demonstrated the applicability and efficiency of the proposed method. Wang et al. [15] applied the Kriging model in a honeycomb Sandwich beam to reduce the solving time. Zhang and Hou [16] presented a SVM based response surface model for vehicle chassis components. With good generalization performance and fast learning speed, a learning method called extreme learning machine (ELM) for singlehidden layer feedforward neural networks (SLFNs) has received many applications in data analysis, protein sequence classification, feature selection and clustering learning, etc. [17–19]. Although ELM does not need to adjust so many parameters during training process, generalization performance of ELM could be affected by the input weights and the biases. To improve the performance, several methods, such as genetic algorithm (GA) and particle swarm optimization (PSO), have been proposed to optimize the input weights and biases of ELM [20, 21].
Besides computational cost issue, the efficiency of the algorithms is another crucial step. Model updating is an optimization problem to minimize the difference between the analytical responses and experimental ones. Gradientbased techniques have a wide range of applications. However, the methods may be computationally expensive and are not feasible to achieve complicated engineering problems [22]. Intelligent algorithms (GA, PSO, etc.) can be used to reduce the difference between the FEM and the measured behavior [23]. Guo et al. [24] conducted the updating procedure by the hybrid genetic/patternsearch optimization algorithm. Boulkaibet et al. [25] adopted fish school search algorithm (FSS) and PSO to update two real structures. Whale optimization algorithm (WOA), which mimics the hunting behavior of humpback whales in nature, is a swarmbased optimization algorithm introduced by Mirjalili [26]. An improved whale optimization algorithm (WOA) based on Lévy flight (LWOA) is applied to infinite systems identification and classic engineering design problems [27, 28].
To the authors’ best knowledge, ELM and LWOA have not been explored to solve the FEMU problem. To expand their applications and improve the efficiency of FEMU, in this paper, LWOA is firstly used to optimize input weights and the biases of ELM. Then, a LWOAELM based surrogate model is established by the updating parameters and corresponding FRF data to replace the initial FEM. The objective function is established using the residual between the analytical values and the experimental ones. Finally, the LWOA is employed to search the best updating parameters of FEMU.
The rest of the paper is organized as follows. Section 2 presents the FRF and the objective function. The concepts of ELM, LWOA, and optimized ELM are provided in Section 3. In Section 4, an example is given to illustrate the FEMU based on the new method. The paper is concluded in Section 5.
2. Frequency Response Function and Objective Function
The motion equation of structure with structural damping can be written aswhere , , and are matrices of mass, damping, and stiffness, respectively; , , and are vector of the structural displacement, velocity, and acceleration; and is a vector of applied force.
Assuming a harmonic input, the corresponding analytical frequency response matrix is given bywhere is the frequency of excitation.
The acceleration FRF (AFRF) can be calculated by
FEMU utilizes information from the actual structural responses to update the parameters of the FEM. Model updating problem can be formulated in the following form:where is the objective function; is the updating parameter; and and are the lower and upper bound of updating parameters, respectively.
Based on the AFRF data, the research sets the residual between the FEM data and experimental ones as the objective function. It can be described aswhere is the number of selected frequency; is the selected frequency points; is the experimental response at ; indicates the corresponding response of the FEM. Considering the peak values at specified frequencies, which may overwhelm the influence of the smaller values, the logarithmic () is adopted.
3. Basic Theory
3.1. Extreme Learning Machine
Extreme learning machine (ELM) for singlehidden layer feedforward neural networks (SLFNs) proposed by Huang has been used extensively and successfully in different fields with fast learning speed and good generalization capability. Just like the traditional feedforward network algorithm, the ELM consists of three layers of neurons: an input layer, a hidden layer, and an output layer (as shown in Figure 1). is the input weight vector between the input nodes and the hidden nodes; is the output weight vector between the hidden nodes and the output ones.
Suppose the training dataset , , where is the input vector and is the output vector. The output is given by
Equation (7) can be written as
The hidden layer output matrix is given bywhere denotes the number of hidden nodes; is the bias; is the activation function. The minimum norm least squares of equation (8) can be calculated bywhere is the Moore–Penrose generalized inverse of .
Given the training samples, active function , and hidden nodes number , the specific process of ELM can be divided into three steps: Step 1: randomly assign input weight vector and biases Step 2: calculate the hidden layer output matrix Step 3: calculate the output weight
3.2. Lévy FlightBased Whale Optimization Algorithm
Whale optimization algorithm (WOA) is a new metaheuristic algorithm, which simulates the social behavior of humpback whales. The basic WOA is divided into three operations: encircling, Bubblenet attacking behavior (exploitation phase), and search for prey (exploration period), where Bubblenet attacking consists of two approaches: shrinking encircling mechanism and spiral. Due to a lack of population diversity, WOA is easy to fall into local optimum. Lévy flight trajectory, which is the random walk step drawn from a Lévy distribution, is adapted to prevent WOA from local optimum and enhance the solution accuracy. The procedure of the LWOA is shown in Algorithm 1.

Firstly, humpback whales try to update their position towards the best search agent. The behavior is given as follows:where ; ; ; is a random vector in ; is the current iteration; is the maximum iteration number; is the position vector; is the position vector of the best solution obtained so far.
Then, the humpback whales choose shrinking encircling mechanism or the spiral model with a probability of 50% to update the position, according to the following equation:where indicates the distance between the best solution obtained so far and the current position; is a constant for defining the shape of the logarithmic spiral; is a number in ; is a random number in .
And then, humpback whales search randomly to update the position in the exploration phase. The mathematical model is as follows:where is a random position vector chosen from the current population.
is the parameter to choose mechanism to update position. If , a random search agent is performed by equations (15) and (16). And if , humpback whales update the position of the current search agent according to equations (12) and (13).
Finally, the Lévy flight trajectory is employed to balance the exploitation and exploration of WOA. The mathematical model can be formulated bywhere ; and obey normal distribution, and ; and ; means entrywise multiplications; is an index.
3.3. Optimized ELM
The prediction performance of ELM model can be assessed by root mean square error (RMSE) criterion and criterion:where is the number of observations; is the analytical value calculated by FEM; is the response value predicted by ELM; indicates the average value of . A value of close to 1 and RMSE close to 0 are always preferred, which reflect high accuracy of ELM.
In order to further improve accuracy of updating results, more reliable ELM models need to be created. The input weights and the biases are randomly generated in the ELM, which affect the prediction accuracy of the model. LWOA is utilized to optimize the input weights and biases of ELM. The minimized RMSE obtained by training samples is taken as the fitness function. It can be expressed as
In the LWOA, the input weights and the biases are considered to be the humpback whales. The dimension of search space is , where denotes the number of hidden nodes. The optimal parameters and corresponding to the minimum fitness value are obtained.
4. Model Updating Based on ELM and LWOA
4.1. Model Updating Procedure
FEMU is an inverse problem to update the parameters by minimizing the discrepancy between the measured responses and the analytical ones from FEM. In this study, LWOAELM model is constructed to replace the FEM, and LWOA is also applied to minimize the objective function and obtain the results. The flowchart of model updating using LWOAELM and LWOA is shown in Figure 2. The main steps can be illustrated as follows: Step 1: create the FEM of the structure by the ANSYS software. Step 2: select updating parameters (modulus of elasticity and material density are selected as the updating parameters in this paper). Step 3: construct the LWOAELM model by training samples, and check whether the accuracy criterion is satisfied (, RMSE < 1.5 × 10^{−4}) by the model. Step 4: search the global minimum solution of equation (5) by LWOA, and check whether the convergence criterion (maximum iterations) is satisfied. Step 5: obtain the model updating results.
4.2. Numerical Study
A plane structure with modulus of elasticity of the element , material density is used to test the proposed method for model updating. The initial FEM is established using ANSYS software. Figure 3 shows the FEM of the plane truss. The structure has 26 nodes and 49 finite elements. Each node (except 1 and 25) includes two translations DOFs. In this paper, nodes 24 and 19 are selected as the exciting and measuring points, respectively.
Modulus of elasticity and material density of the FEM are selected as the updating parameters. To simulate initial FEM value, increase the modulus of elasticity by 10%, and decrease density by 10%. The parameters are shown in Table 1.
Latin hypercube sampling (LHS) [29] is used to get the design points of updating parameters. Modulus of elasticity and material density change within ±20% of the FEM values. 100 random samples are obtained for ELM. The corresponding response data are generated by equation (3). Thus, ELM model is established based on the input variables (updating parameters) and corresponding outputs (AFRF data). According to literature [17] and cross validation method, the activation function is sigmoidal function, and the number of hidden nodes is set to 80. For verification, ELM model is compared with frequently used surrogate models, RBF model and secondorder RS model [14]. RBF neural network is a threelayer feedforward network with the hidden layer containing RBF units. RS is a mathematical and statistical technique and often applied in regression analysis. The models are constructed based on the same sample points. RBF is conducted in MATLAB neural network toolbox. The parameter spread is set to 25 by trial and error to keep a stable and minimum value of RMSE. Each algorithm is simulated 20 independent runs in MATLAB software. The results are presented as the average of 20 runs. Comparison of ELM with RS and RBF model is shown in Figure 4. It can be seen that the prediction accuracy of ELM is better than RS and RBF in Figure 4. RMSE of ELM is 0.0012 and is 1, RMSE of RBF is 0.0161 and is 0.9989, and RMSE of RS is 0.0314 and is 0.9963, which indicate that ELM model builds an accurate relationship between updating parameters and responses.
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To further improve the prediction ability of ELM model, the proposed LWOAELM model is constructed. Moreover, LWOAELM is compared with GAELM, PSOELM, and WOAELM. GA is a widely used optimization algorithm inspired from evolutionary process. PSO is a bioinspired stochastic algorithm derived from biological entities. In order to make a fair comparison, the population size and the maximum iteration number are set to the same data. And other parameters are the best sets after trial and error. Finally, the population size is set to 20. The maximum iteration number is set to 100. Other parameters of these algorithms are listed in Table 2.
The fitness convergence curves of GAELM, PSOELM, WOAELM, and LWOAELM are shown in Figure 5. From Figure 5, it is apparent that LWOA has a fast convergence than others. The smallest fitness value is LWOA. Figure 6 reflects RMSE of the four optimized ELM models.
As seen in Figure 6, the prediction accuracy of the four optimized ELM models is improved. LWOAELM is significantly better than GAELM, PSOELM, and WOAELM. The comparison results of the four algorithms are given in Table 3.
After building the LWOAELM model, LWOA is applied to find the best solution of the objective function of equation (5). At each time of iteration, the analytical values of FRF are predicted by LWOAELM instead of FEM. For the algorithm, the dimension is set as 2 (two updating parameters), and the maximum iteration number is set to 500. The population size is set to 50. The updated parameters are listed in Table 4. The parameters are closer to experimental parameters, which prove the validity of this method.
Actually, the experimental responses usually contain noise. 5% random noise is added to experimental FRF. The before and after updating compared with the experimental data is given in Figure 7. And Figure 8 shows the FRF of initial FEM, experimental, and updated model. As shown in Figures 7 and 8, the model has a good agreement with the experimental one. The results are given in Table 5.
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From Tables 4 and 5, it can be observed that the noise affects the updating results. The maximum error is 0.86%. The error is still acceptable. To further illustrate the proposed method, the first five frequencies of the FEM, experimental, and updated model are shown in Table 6.
As presented in Table 6, the relative errors decrease from 10.55% to 0.038%. It is verified that the proposed method is capable of updating the model, and the natural frequencies match the experimental model well.
5. Conclusion
A LWOAELM model based method for FEMU is proposed in this paper. LWOA is applied to optimize the input weights and biases of ELM. During the optimization process, LWOAELM model is introduced as the surrogate model to replace the initial FEM. The parameters are updated using LWOA, which obtain minimum of the objective function based on the AFRF data. The example results show that LWOAELM has better accuracy performance than that by the secondorder RS, RBF, traditional ELM, GAELM, PSOELM, and WOAELM. With 5% noise, the errors of updated parameters are still less than 0.9%. The proposed method is suitable for FEMU. For future work, complex structure will be developed, and modal test will be performed.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Natural Science Foundation of China (No. 51768035) and Collaborative Innovation Team Project of Universities in Gansu Province (No. 2018C12).