# Improved Metaheuristic Algorithm Based Finite Element Model Updating of a Hybrid Girder Cable-Stayed Railway Bridge

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Updating Assisted with the Kriging Model

## 3. Improvement Strategy of Metaheuristic Algorithm

#### 3.1. Metaheuristic Algorithm

#### 3.2. Improvement Strategy

_{max}is the maximum generation; t is the current generation; and e is the natural index. It can be concluded from Equation (13) that with a generation’s increase, the probability of implementing random crossover and mutation operation also increases.

#### 3.3. Algorithm Procedure

#### 3.4. Benchmark Functions

_{1}and f

_{2}, respectively. The f

_{1}is a multimodal function that has several extreme points but only one global optimal point. The f

_{2}is a unimodal function. The objective functions f

_{1}and f

_{2}can be defined as:

_{i}denotes the i-th design variable; and e is the natural index.

_{1}obtained from standard GSA are 6.71 × 10

^{−1}and 1.83 × 10

^{−2}, respectively, while corresponding values obtained from IGSA are 5.50 × 10

^{−2}and 7.43 × 10

^{−3}. This comparison indicates that the searching accuracy of IGSA is far better than GSA. Similar conclusions can be made for the other four algorithms. It can be seen from Table 1 that the mean and standard deviation of f

_{1}obtained from PSOGA are 4.41 × 10

^{−1}and 2.57 × 10

^{−1}, respectively, while corresponding values obtained from IPSO are 3.10 × 10

^{−1}and 1.38 × 10

^{−2}. From the comparison of the above results, it can be found that in the case of multiple local optimal solutions, IPSO has a greater probability of jumping out of the local optimal solution. In order to better demonstrate the generality of the improvement strategy, the following numerical and engineering cases mainly compare GSA, PSO, and SA and the corresponding improved algorithms.

_{1}, whereas the IGSA reached a far smaller value of 0.3099. A similar situation can be found in the other two algorithms.

## 4. Numerical Simulation

^{3}, and the section area of the beam element is 1.717 × 10

^{−3}m

^{2}. The FE model was established by ANSYS, and the beam element was used to simulate the truss members.

_{1}and E

_{2}. The surface represents the kriging model response surface, the scattered points represent the sample points, and the sample points are located on the curved surface. It can be seen from the MSE surface that the maximum error is less than 4 × 10

^{−7}, indicating that the kriging model has high accuracy and can replace the FE model in predicting the structural response.

_{1}obtained from IGSA and GSA are 0.74% and 5.49%. These data indicate that the improved algorithms are better than standard algorithms in model updating. Nevertheless, the updating results of each algorithm still need further discussion.

## 5. Case Study: FE Model Updating of a Cable-Stayed Bridge

#### 5.1. Bridge Description

#### 5.2. In Situ Tests and Experimental Results

#### 5.3. Selection of Updating Parameters

_{1}, D

_{1}), steel box girder (E

_{2}, D

_{2}), bridge pylon (E

_{3}, D

_{3}), pier (E

_{4}, D

_{4}), stay cable (E

_{5}, D

_{5}), and secondary tension of three stay cables (F

_{1}, F

_{2}, F

_{3}). Figure 12 shows the results of the sensitivity analysis. In Figure 12, the MAC1 and MAC2 are the MAC values of the first and second mode shapes, respectively. The f

_{1}to f

_{2}are the first two natural frequencies, respectively.

_{2}, the density of steel box girder D

_{2}, the elastic modulus of bridge pylon E

_{3}, the density of bridge pylon D

_{3}, elastic modulus of stay cable E

_{5}, and the density of stay cable D

_{5}as the updating parameters, and their initial values were 2.1 × 10

^{8}kN/m

^{2}, 78.5 kN/m

^{3}, 3.55 × 10

^{7}kN/m

^{2}, 25 kN/m

^{3}, 2.1 × 10

^{8}kN/m

^{2}, 84.5 kN/m

^{3}, respectively.

#### 5.4. Kriging Model Construction

_{2}, D

_{3}, E

_{5}, and D

_{5}) are set as initial values when calculating the analytical values. Figure 13b shows the kriging model’s mean square error (MSE). The maximum error is smaller than 6 × 10

^{−9}, indicating that the kriging predicted value has very high accuracy and can replace the FE model when calculating the response values.

#### 5.5. Objective Function

#### 5.6. Discussion on the Updating Results

_{1}obtained from SA and ISA are totally different. Except for D

_{2}, an overall increase in the rest of the parameters is apparent. A similar updating rate of elastic modulus for a cable-stayed bridge can be found in reference [42]. This increase indicates that the stiffness and mass distribution of the initial FE model is underestimated. The reason for this could be attributed to the simplifications of the local stiffening ribs, diaphragms, and other local members in the initial FE model. The parameters’ updated values obtained from different algorithms are different, indicating that each algorithm and its improved version converged to a different solution in the design space. It should be mentioned that the changes in the updating parameters only represent the change in the overall stiffness and mass between the FE model and the physical structure. Therefore, the updating parameters can be viewed as equivalent elastic modulus and equivalent density. Nevertheless, the rationality of the updated FE model still needs to be evaluated by analyzing the prediction accuracy.

_{1}obtained from GSA is −6.15%, while that obtained from IGSA is −4.10%. These results indicate that the improved algorithms perform better than standard algorithms in predicting frequency.

^{−2}and 1.14 × 10

^{−1}, respectively. The corresponding standard deviations are 3.65 × 10

^{−2}and 1.29 × 10

^{−1}, respectively. The mean value and standard deviation of IGSA are smaller than those of GSA. The same situation is observed for the other two algorithms. These results indicate that the improved algorithms have higher accuracy and stability.

## 6. Conclusions

- (1)
- The proposed improvement strategy can effectively improve the accuracy and the global convergence of standard metaheuristic algorithms. The random crossover and mutation operations are mainly introduced during later-stage searching and aim to improve the algorithm’s accuracy;
- (2)
- The numerical investigation of the two benchmark functions showed that the improved algorithms had better global convergence and stability than the standard algorithms. The updated truss model with the improved algorithms showed higher prediction accuracy than the standard algorithm;
- (3)
- The discrepancies between the calculated and experimental values of displacement and frequency of the cable-stayed bridge were much smaller after model updating. The updated models with the improved algorithm had smaller relative errors than those obtained with the standard algorithms. In the IGSA algorithm, for example, the relative error in displacement was significantly reduced, with the maximum relative error reduced from 30.94% to 11.33% and the relative error in first-order frequency reduced from 11.03% to 4.21%, indicating that the updated model can better represent the actual structure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Image of the benchmark functions: (

**a**) Ackley’s path function; and (

**b**) the sum of different power function.

**Figure 5.**The first six mode shapes of the truss: (

**a**) first order mode; (

**b**) the second mode; (

**c**) the third mode; (

**d**) the fourth mode; (

**e**) the fifth mode; and (

**f**) the sixth mode.

**Figure 6.**The first-order frequency kriging model and the corresponding MSE: (

**a**) kriging model; and (

**b**) MSE.

**Figure 7.**The Yongjiang Bridge: (

**a**) overview; (

**b**) the elevation layout; (

**c)**the section layout of the steel box girder; and (

**d**) the section layout of the concrete box girder. (Unit: cm).

**Figure 9.**Photos of in situ test: (

**a**) the loading train and (

**b**) the instrumentation and data acquisition system.

**Figure 11.**Measured vibration mode and calculated vibration mode MAC: (

**a**) first order mode and (

**b**) the second mode.

**Figure 13.**Kriging model: (

**a**) first order frequency response surface and (

**b**) MSE error of the first-order frequency.

**Figure 14.**The comparison of the displacement values obtained from experiment (EXP), initial FE model and updated FE models: (

**a**) load case A and B; (

**b**) load case C and D; and (

**c**) load case E and F.

**Figure 15.**The mean and standard deviation of the objective function. (

**a**) Mean; (

**b**) Standard deviation.

**Table 1.**The mean and standard deviation values obtained from different algorithms for benchmark functions.

Algorithm | f_{1} | f_{2} | ||
---|---|---|---|---|

Mean | Standard Deviation | Mean | Standard Deviation | |

IGSA | 5.50 × 10^{−2} | 7.43 × 10^{−3} | 2.72 × 10^{−6} | 9.43 × 10^{−6} |

GSA | 6.71 × 10^{−1} | 1.83 × 10^{−2} | 6.77 × 10^{−6} | 2.53 × 10^{−5} |

IPSO | 3.10 × 10^{−1} | 1.38 × 10^{−2} | 1.89 × 10^{−5} | 2.92 × 10^{−5} |

PSOGA | 4.41 × 10^{−1} | 2.57 × 10^{−1} | 5.43 × 10^{−6} | 2.49 × 10^{−5} |

PSO | 1.25 | 2.51 × 10^{−2} | 6.32 × 10^{−4} | 6.31 × 10^{−5} |

ISA | 6.52 × 10^{−1} | 4.91 × 10^{−3} | 3.46 × 10^{−6} | 3.39 × 10^{−5} |

SA | 8.22 × 10^{−1} | 6.58 × 10^{−3} | 6.51 × 10^{−5} | 2.27 × 10^{−4} |

IAFSA | 5.74 × 10^{−1} | 9.01 × 10^{−3} | 3.15 × 10^{−5} | 3.52 × 10^{−5} |

AFSA | 6.31 × 10^{−1} | 5.92 × 10^{−3} | 9.02 × 10^{−5} | 7.58 × 10^{−5} |

IABC | 5.67 × 10^{−2} | 4.62 × 10^{−3} | 3.00 × 10^{−6} | 9.45 × 10^{−6} |

ABC | 1.79 | 9.72 × 10^{−3} | 9.62 × 10^{−6} | 1.93 × 10^{−5} |

Parameters | E_{1} (100 GPa) | E_{2} (100 GPa) | E_{3} (100 GPa) | Relative Error (%) | |||
---|---|---|---|---|---|---|---|

Damaged Model | 0.7 | 0.7 | 1.3 | - | |||

Initial model | 1 | 1 | 1 | 42.86 | 42.86 | −23.08 | |

Updated model | IGSA | 0.705 | 0.714 | 1.282 | 0.71 | 2.00 | −1.38 |

GSA | 0.738 | 0.682 | 1.280 | 5.43 | −2.57 | −1.54 | |

IPSO | 0.727 | 0.740 | 1.320 | 3.86 | 5.71 | 1.54 | |

PSO | 0.740 | 0.771 | 1.315 | 5.71 | 10.14 | 1.15 | |

ISA | 0.700 | 0.695 | 1.311 | 0.00 | −0.71 | 0.85 | |

SA | 0.703 | 0.673 | 1.342 | 0.43 | −3.86 | 3.23 |

**Table 3.**The comparison of the first six frequencies obtained from the initial FE model and updated FE model.

Frequency | f_{1} (Hz) | f_{2} (Hz) | f_{3} (Hz) | f_{4} (Hz) | f_{5} (Hz) | f_{6} (Hz) | Relative Error (%) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Damage Model | 7.50 | 15.72 | 19.57 | 23.67 | 31.42 | 33.68 | - | ||||||

Initial model | 7.63 | 16.65 | 20.18 | 25.16 | 32.54 | 34.32 | 1.73 | 5.92 | 3.12 | 6.29 | 3.56 | 1.90 | |

Updated model | IGSA | 7.50 | 15.71 | 19.61 | 23.71 | 31.45 | 33.67 | 0.00 | −0.06 | 0.20 | 0.17 | 0.10 | −0.03 |

GSA | 7.50 | 15.72 | 19.61 | 23.71 | 31.46 | 33.68 | 0.00 | 0.00 | 0.20 | 0.17 | 0.13 | 0.00 | |

IPSO | 7.54 | 15.87 | 19.69 | 23.88 | 31.59 | 33.91 | 0.53 | 0.95 | 0.61 | 0.89 | 0.54 | 0.68 | |

PSO | 7.55 | 15.96 | 19.76 | 24.01 | 31.70 | 34.00 | 0.67 | 1.53 | 0.97 | 1.44 | 0.89 | 0.95 | |

ISA | 7.50 | 15.7 | 19.57 | 23.65 | 31.41 | 33.70 | 0.00 | −0.13 | 0.00 | −0.08 | −0.03 | 0.06 | |

SA | 7.50 | 15.71 | 19.54 | 23.61 | 31.37 | 33.74 | 0.00 | −0.06 | −0.15 | −0.25 | −0.16 | 0.18 |

Algorithm | IGSA | GSA | IPSO | PSO | ISA | SA |
---|---|---|---|---|---|---|

Mean | 2.56 × 10^{−4} | 5.20 × 10^{−4} | 1.91 × 10^{−4} | 8.16 × 10^{−4} | 9.64 × 10^{−1} | 9.72 × 10^{−1} |

Standard deviation | 7.61 × 10^{−3} | 1.65 × 10^{−2} | 1.74 × 10^{−3} | 2.00 × 10^{−2} | 1.04 × 10^{−1} | 1.36 × 10^{−1} |

**Table 5.**Calculated value and test value of displacement of each measuring point under each load case.

Test Cases | Test Section | Experimental Value (mm) | Initial FE Model (mm) | Relative Error (%) | ||
---|---|---|---|---|---|---|

Upstream | Downstream | Mean | ||||

A | 1 | 414.00 | 412.60 | 413.30 | 486.21 | 17.64 |

2 | 394.60 | 386.60 | 390.60 | 410.60 | 5.12 | |

B | 1 | 211.00 | 213.00 | 212.00 | 237.14 | 11.86 |

2 | 240.00 | 236.30 | 238.15 | 258.12 | 8.39 | |

C | 3 | 102.00 | 110.00 | 106.00 | 125.73 | 18.61 |

4 | 139.00 | 139.60 | 139.30 | 161.31 | 15.80 | |

D | 2 | 263.00 | 264.30 | 263.65 | 304.01 | 15.31 |

3 | 353.00 | 350.40 | 351.70 | 442.41 | 25.79 | |

E | 2 | 313.00 | 311.80 | 312.40 | 370.15 | 18.49 |

3 | 386.00 | 386.80 | 386.40 | 489.42 | 26.66 | |

F | 1 | 435.00 | 434.50 | 434.75 | 569.27 | 30.94 |

2 | 555.00 | 542.40 | 548.70 | 624.84 | 13.88 |

Vibration Mode | Experimental Value (Hz) | Initial FE Model (Hz) | MAC | Relative Error (%) |
---|---|---|---|---|

1 | 0.390 | 0.347 | 0.936 | −11.03 |

2 | 0.490 | 0.456 | 0.947 | −6.94 |

Parameters | Initial Value | Updated Value | Updating Rate (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IGSA | GSA | IPSO | PSO | ISA | SA | IGSA | GSA | IPSO | PSO | ISA | SA | ||

E_{2} | 3.55 | 4.23 | 4.35 | 4.17 | 3.83 | 3.20 | 4.60 | 19.15 | 22.54 | 17.46 | 7.89 | −9.86 | 29.58 |

D_{2} | 2.50 | 2.78 | 2.65 | 2.72 | 2.59 | 2.72 | 2.60 | 11.20 | 6.00 | 8.80 | 3.60 | 8.80 | 4.00 |

E_{3} | 2.10 | 2.42 | 2.42 | 2.39 | 2.29 | 2.29 | 2.31 | 15.24 | 15.24 | 13.81 | 9.05 | 9.05 | 10.00 |

D_{3} | 7.85 | 7.25 | 8.14 | 7.66 | 7.07 | 7.07 | 7.19 | −7.64 | 3.69 | −2.42 | −9.94 | −9.94 | −8.41 |

E_{5} | 2.00 | 2.52 | 2.46 | 2.47 | 2.50 | 2.48 | 2.55 | 26.00 | 23.00 | 23.50 | 25.00 | 24.00 | 27.50 |

D_{5} | 8.45 | 9.89 | 9.43 | 9.40 | 8.56 | 7.92 | 9.75 | 17.04 | 11.60 | 11.24 | 1.30 | −6.27 | 15.38 |

**Table 8.**Comparison of the displacement relative errors between the initial FE model and updated FE models.

Load Case | A | B | C | D | E | F | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Test Section | 1 | 2 | 1 | 2 | 3 | 4 | 2 | 3 | 2 | 3 | 1 | 2 | |

Initial error (%) | 17.64 | 5.12 | 11.86 | 8.39 | 18.61 | 15.80 | 15.31 | 25.79 | 18.49 | 26.66 | 30.94 | 13.88 | |

Relative error (%) | IGSA | 0.34 | −9.51 | −4.08 | −6.36 | −0.02 | −2.76 | −1.01 | 6.81 | 1.35 | 7.64 | 11.33 | −2.00 |

GSA | 0.01 | −9.96 | −4.48 | −6.84 | −0.14 | −2.66 | −1.49 | 7.10 | 1.85 | 7.94 | 11.75 | −2.48 | |

IPSO | 0.29 | −9.73 | −4.07 | −6.36 | −0.36 | −2.04 | −1.41 | 7.16 | 1.62 | 7.94 | 11.35 | −2.29 | |

PSO | 0.36 | −9.77 | −4.19 | −6.67 | 0.27 | −2.17 | −1.25 | 7.22 | 1.62 | 8.04 | 11.66 | −2.28 | |

ISA | 0.40 | −9.32 | −3.45 | −5.46 | 0.50 | −1.28 | −1.35 | 7.42 | 1.62 | 7.99 | 11.38 | −1.85 | |

SA | 0.58 | −9.99 | −4.27 | −7.05 | −1.00 | −2.81 | −1.37 | 7.28 | 1.69 | 8.23 | 11.64 | −2.56 |

**Table 9.**Comparisons of the frequencies obtained from experiment and the initial FE model and updated FE models.

Frequency | f_{1} (Hz) | f_{2} (Hz) | Relative Error (%) | ||
---|---|---|---|---|---|

Experimental Value | 0.390 | 0.490 | |||

Initial FE model | 0.347 | 0.456 | −11.03 | −6.94 | |

Updating models | IGSA | 0.374 | 0.479 | −4.10 | −2.24 |

GSA | 0.366 | 0.474 | −6.15 | −3.27 | |

IPSO | 0.371 | 0.485 | −4.87 | −1.02 | |

PSO | 0.369 | 0.476 | −5.38 | −2.86 | |

ISA | 0.370 | 0.486 | −5.13 | −0.82 | |

SA | 0.370 | 0.479 | −5.13 | −2.24 |

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## Share and Cite

**MDPI and ACS Style**

Qin, S.; Yuan, Y.; Gan, Y.; Wang, Q.
Improved Metaheuristic Algorithm Based Finite Element Model Updating of a Hybrid Girder Cable-Stayed Railway Bridge. *Buildings* **2022**, *12*, 958.
https://doi.org/10.3390/buildings12070958

**AMA Style**

Qin S, Yuan Y, Gan Y, Wang Q.
Improved Metaheuristic Algorithm Based Finite Element Model Updating of a Hybrid Girder Cable-Stayed Railway Bridge. *Buildings*. 2022; 12(7):958.
https://doi.org/10.3390/buildings12070958

**Chicago/Turabian Style**

Qin, Shiqiang, Yonggang Yuan, Yaowei Gan, and Qiuping Wang.
2022. "Improved Metaheuristic Algorithm Based Finite Element Model Updating of a Hybrid Girder Cable-Stayed Railway Bridge" *Buildings* 12, no. 7: 958.
https://doi.org/10.3390/buildings12070958