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 trajectory-based 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 second-order response surface (RS), radial basis function (RBF), traditional ELM, and other optimized ELM, a LWOA-ELM 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.

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 [

In addition, the model updating requires dozens of iterations to compute the analytical responses, which is time-consuming. 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 [

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. Gradient-based techniques have a wide range of applications. However, the methods may be computationally expensive and are not feasible to achieve complicated engineering problems [

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 LWOA-ELM 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

The motion equation of structure with structural damping can be written as

Assuming a harmonic input, the corresponding analytical frequency response matrix is given by

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:

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 as

Extreme learning machine (ELM) for single-hidden 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

Extreme learning machine.

Suppose the

Equation (

The hidden layer output matrix

Given the training samples, active function

Step 1: randomly assign input weight vector

Step 2: calculate the hidden layer output matrix

Step 3: calculate the output weight

Whale optimization algorithm (WOA) is a new meta-heuristic algorithm, which simulates the social behavior of humpback whales. The basic WOA is divided into three operations: encircling, Bubble-net attacking behavior (exploitation phase), and search for prey (exploration period), where Bubble-net 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

Initialize relevant parameters

Generate a population

Update

Update the position of the current search agent by equation (

Select a random search agent

Update the position of the current search agent by equation (

Update the position of the current search agent by equation (

Update the position of the current search agent using Lévy flight by equation (

Check if any search agent goes beyond the search and amend it

Calculate the fitness of each whale

Update

Return

Firstly, humpback whales try to update their position towards the best search agent. The behavior is given as follows:

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:

And then, humpback whales search randomly to update the position in the exploration phase. The mathematical model is as follows:

Finally, the Lévy flight trajectory is employed to balance the exploitation and exploration of WOA. The mathematical model can be formulated by

The prediction performance of ELM model can be assessed by root mean square error (RMSE) criterion and

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

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, LWOA-ELM 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 LWOA-ELM and LWOA is shown in Figure

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 LWOA-ELM model by training samples, and check whether the accuracy criterion is satisfied (^{−4}) by the model.

Step 4: search the global minimum solution of equation (

Step 5: obtain the model updating results.

Flowchart of model updating.

A plane structure with modulus of elasticity of the element

FEM of the plane truss.

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

Parameters of experimental model and FEM.

Updating parameters | Experimental value | Initial FEM value |
---|---|---|

Modulus of elasticity ( | 210 | 231 |

Material density ( | 7850 | 7065 |

Latin hypercube sampling (LHS) [

Comparison of ELM with RS and RBF model. (a) Initial FEM values and values predicted by ELM, RS, and RBF model. (b) RMSE of ELM, RS, and RBF model.

To further improve the prediction ability of ELM model, the proposed LWOA-ELM model is constructed. Moreover, LWOA-ELM is compared with GA-ELM, PSO-ELM, and WOA-ELM. GA is a widely used optimization algorithm inspired from evolutionary process. PSO is a bio-inspired 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

Parameters of algorithms.

Algorithm | Parameters | Values |
---|---|---|

GA | Crossover probability | 0.9 |

Mutation probability | 0.1 | |

PSO | Learning factors | 2 |

Learning factors | 2 |

The fitness convergence curves of GA-ELM, PSO-ELM, WOA-ELM, and LWOA-ELM are shown in Figure

Convergence curves for LWOA, WOA, PSO, and GA.

RMSE of the optimized ELM models.

As seen in Figure

Comparison results of the four algorithms.

Model | RMSE | |
---|---|---|

GA-ELM | 2.3252 × 10^{−4} | 1 |

PSO-ELM | 1.1490 × 10^{−4} | 1 |

WOA-ELM | 1.4250 × 10^{−4} | 1 |

LWOA-ELM | 1.0263 × 10^{−4} | 1 |

After building the LWOA-ELM model, LWOA is applied to find the best solution of the objective function of equation (

Values and errors of the updated parameters without noise.

Updating parameters | Experimental value | Initial value | Error (%) | ELM | Error (%) | LWOA-ELM | Error (%) |
---|---|---|---|---|---|---|---|

Modulus of elasticity ( | 210 | 231 | 10 | 210.0131 | 0.0063 | 210.0045 | 0.0021 |

Material density ( | 7850 | 7065 | −10 | 7851.2 | 0.0153 | 7850.4 | 0.0051 |

Actually, the experimental responses usually contain noise. 5% random noise is added to experimental FRF. The

Result of FEMU.

FRFs of initial FEM, experimental model, and updated model. (a) Values of the real part. (b) Values of the imaginary part.

Values and errors of the updated parameters with 5% noise.

Updating parameters | Experimental value | Initial value | Error (%) | LWOA-ELM | Error (%) |
---|---|---|---|---|---|

Modulus of elasticity ( | 210 | 231 | 10 | 211.8 | 0.86 |

Material density ( | 7850 | 7065 | −10 | 7911.2 | 0.78 |

From Tables

Natural frequencies of the FEM, experimental, and updated model.

Mode order | Initial FEM | Experimental model | Updated model | Error | Error | Error reduction (%) |
---|---|---|---|---|---|---|

1 | 19.3290 | 17.4837 | 17.4904 | 10.55 | 0.038 | 99.63 |

2 | 62.6328 | 56.6535 | 56.6753 | |||

3 | 83.8904 | 75.8817 | 75.9109 | |||

4 | 126.4108 | 114.3429 | 114.3869 | |||

5 | 195.6685 | 176.9864 | 177.0545 |

Note:

As presented in Table

A LWOA-ELM 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, LWOA-ELM 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 LWOA-ELM has better accuracy performance than that by the second-order RS, RBF, traditional ELM, GA-ELM, PSO-ELM, and WOA-ELM. 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.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was supported by the Natural Science Foundation of China (No. 51768035) and Collaborative Innovation Team Project of Universities in Gansu Province (No. 2018C-12).