# A Study on the Genetic Algorithm Optimization of an Asphalt Mixture’s Viscoelastic Parameters Based on a Wheel Tracking Test

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Optimization Objects

#### 2.2. Kriging Surrogate Model

#### 2.3. Optimal Algorithm

- (1)
- According to the constraints in the optimization model, the initial sample points selected randomly using the LHS method are used to establish the initial Kriging “response surface”.
- (2)
- Genetic algorithms are utilized to obtain the optimal solution in this “response surface”.
- (3)
- The true values of the optimization results are calculated using the FEM simulations and this set of true values is added to the initial sample points to renew the Kriging “response surface”.
- (4)
- The second and third steps are looped until the difference between the optimization results SAE of the two adjacent ones is less than 1.

## 3. Materials and Experiment

#### 3.1. Material Properties

#### 3.2. Wheel Tracking Test

#### 3.3. Uniaxial Compression Test

## 4. Results and Discussion

#### 4.1. Finite Element Model Simulation of WTT

#### 4.1.1. Elements Division

#### 4.1.2. Loading Mode and Boundary Conditions

#### 4.2. Implementation of the Proposed Method

#### 4.3. Comparison of Simulation Results

#### 4.4. Sensitivity Analysis of Viscoelastic Parameters

## 5. Conclusions

- The generalized Maxwell model used in the finite element models could accurately reflect the rutting development of asphalt mixtures under high temperatures. The rutting depth variation curves obtained from the finite element models using the optimized parameters were in a high agreement with the WTT curve, showing that the proposed method could determine the viscoelastic parameters of the asphalt mixture for rutting study and prediction.
- Compared with experimental parameters, the optimized parameters could more accurately simulate the trend of the flow-type rutting depths and reflect the rutting resistance of the asphalt mixture compared to parameters from the experiment. In addition, there was a large error in the simulation of the rutting deformation with the viscoelastic parameters obtained under the uniaxial compression loading mode. This might be related to the inability of this loading mode to reflect the shear properties of the material.
- Among the 10 parameters mentioned above, the dynamic modulus corresponding to a larger relaxation time was more sensitive to the mechanical response of the asphalt mixture. The equilibrium modulus had the most significant influence on the growth rate and rutting depth. Therefore, the number of parameters can be appropriately reduced to improve the optimization efficiency in the subsequent parameters’ optimization.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Rutting depth curve: (

**a**) total rutting depth curve over time; (

**b**) rutting depth curve after a loading duration of 1000 s.

**Figure 7.**Finite element model of the wheel tracking test: (

**a**) division of the FEM; (

**b**) loading and boundary conditions of the FEM simulation.

**Figure 12.**Influence of viscoelastic parameters on the simulation results: (

**a**) E

_{5}; (

**b**) E

_{6}; (

**c**) E

_{7}; (

**d**) E

_{∞}.

${\mathit{\tau}}_{\mathit{i}}\left(\mathit{s}\right)$ | ${\mathit{E}}_{\mathit{i}}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ |
---|---|

0.001 | ${E}_{1}$ |

0.01 | ${E}_{2}$ |

0.1 | ${E}_{3}$ |

1 | ${E}_{4}$ |

10 | ${E}_{5}$ |

100 | ${E}_{6}$ |

1000 | ${E}_{7}$ |

Infinity | ${E}_{\infty}$ |

Viscoelastic Parameters | Minimum Value | Maximum Value |
---|---|---|

${E}_{1}$ | 0 | 15,000 MPa |

${E}_{2}$ | 0 | 10,000 MPa |

${E}_{3}$ | 0 | 8000 MPa |

${E}_{4}$ | 0 | 5000 MPa |

${E}_{5}$ | 0 | 3000 MPa |

${E}_{6}$ | 0 | 2000 MPa |

${E}_{7}$ | 0 | 1000 MPa |

${E}_{\infty}$ | 0 | 100 MPa |

${C}_{1}$ | 10 | 30 |

${C}_{2}$ | 10 | 300 |

Items | Penetration (25 °C, 0.1 mm) | Ductility (25 °C, cm) | Softening Point (°C) | Viscosity (135 °C, Pa∙s) |
---|---|---|---|---|

SBS-modified asphalt | 69 | >150 | 81.5 | 2.263 |

Sieve size/mm | 16.0 | 13.2 | 9.5 | 4.75 | 2.36 | 1.18 | 0.6 | 0.3 | 0.15 | 0.075 |

Passing rate/% | 100.0 | 95.0 | 76.5 | 53.0 | 37.0 | 26.5 | 19.0 | 13.5 | 10.0 | 6.0 |

Materials | Relative Rutting Depth (mm) | Rutting Depth (mm) | DS (Times/mm) |
---|---|---|---|

AC-13 | 0.355 | 1.328 | 7875 |

${\mathit{\tau}}_{\mathit{i}}\left(\mathit{s}\right)$ | ${\mathit{E}}_{\mathit{i}}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ |
---|---|

0.001 | 10,445.190 |

0.01 | 7307.682 |

0.1 | 4157.773 |

1 | 1697.304 |

10 | 825.104 |

100 | 158.589 |

1000 | 203.613 |

Infinity | 40.232 |

${\mathit{E}}_{1}$ (MPa) | ${\mathit{E}}_{2}$ (MPa) | ${\mathit{E}}_{3}$ (MPa) | ${\mathit{E}}_{4}$ (MPa) | ${\mathit{E}}_{5}$ (MPa) | ${\mathit{E}}_{6}$ (MPa) | ${\mathit{E}}_{7}$ (MPa) | ${\mathit{E}}_{\mathit{\infty}}$ (MPa) | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | SAE (mm) |
---|---|---|---|---|---|---|---|---|---|---|

14,234.66 | 7921.82 | 4879.95 | 2146.03 | 877.61 | 214.35 | 346.29 | 48.03 | 14.53 | 177.45 | 69.4 |

5714.47 | 9778.50 | 5249.52 | 1326.61 | 1647.28 | 312.91 | 73.13 | 37.44 | 10.49 | 252.92 | 21.0 |

9962.96 | 6257.96 | 3831.73 | 1882.12 | 1630.42 | 275.32 | 203.61 | 49.45 | 21.14 | 181.13 | 190.3 |

11,330.95 | 5723.03 | 4463.77 | 1236.60 | 1051.13 | 322.64 | 62.64 | 49.49 | 21.12 | 197.10 | 28.0 |

6474.03 | 7112.44 | 4693.83 | 2387.69 | 995.12 | 142.90 | 1.71 | 49.54 | 11.33 | 130.31 | 59.2 |

7902.77 | 1769.87 | 4880.03 | 3252.78 | 877.61 | 214.24 | 94.37 | 43.20 | 11.24 | 230.81 | 26.6 |

5452.40 | 5737.01 | 4754.71 | 1066.52 | 1256.93 | 493.06 | 102.91 | 34.63 | 21.01 | 270.39 | 57.6 |

11,707.55 | 7383.45 | 5049.70 | 2427.01 | 1462.02 | 364.57 | 41.73 | 51.06 | 15.31 | 229.41 | 32.1 |

13,779.47 | 6259.29 | 4879.95 | 2145.98 | 1078.85 | 111.75 | 255.83 | 51.37 | 11.24 | 135.03 | 71.1 |

6155.02 | 9163.47 | 3339.41 | 1472.47 | 1579.49 | 430.89 | 398.18 | 25.78 | 15.81 | 290.35 | 186.9 |

7033.77 | 6666.79 | 4879.95 | 2005.77 | 1152.97 | 189.89 | 75.64 | 52.89 | 17.59 | 176.57 | 45.8 |

11,919.99 | 5439.85 | 3735.72 | 2981.71 | 763.80 | 173.23 | 216.89 | 53.08 | 16.85 | 182.77 | 47.6 |

14,437.67 | 8353.78 | 4880.32 | 2126.11 | 877.61 | 214.29 | 89.85 | 54.17 | 10.01 | 133.81 | 82.0 |

7459.34 | 5357.24 | 3782.13 | 1389.79 | 1929.09 | 101.98 | 157.03 | 55.37 | 12.57 | 107.72 | 190.0 |

7658.08 | 7921.82 | 3018.70 | 2327.11 | 1293.32 | 232.09 | 199.00 | 56.18 | 24.75 | 263.28 | 51.6 |

5879.70 | 8965.31 | 4879.95 | 2146.03 | 877.61 | 110.91 | 249.12 | 57.54 | 10.76 | 130.31 | 29.7 |

7943.79 | 6229.67 | 3966.84 | 2617.61 | 835.60 | 168.14 | 86.23 | 59.18 | 19.32 | 157.79 | 104.3 |

9677.71 | 6152.51 | 4879.95 | 2146.03 | 1265.92 | 232.09 | 198.99 | 59.20 | 21.20 | 169.10 | 193.6 |

10,383.75 | 9718.84 | 3134.27 | 1497.97 | 788.03 | 386.57 | 139.89 | 31.41 | 12.44 | 122.27 | 27.9 |

13,369.45 | 9963.37 | 3884.26 | 2757.35 | 637.19 | 357.26 | 279.93 | 43.97 | 13.81 | 294.36 | 149.8 |

14,636.83 | 7818.66 | 4936.38 | 2894.79 | 587.86 | 411.26 | 131.82 | 39.96 | 23.08 | 137.14 | 32.8 |

13,326.93 | 5172.03 | 5783.64 | 1212.75 | 1818.82 | 328.38 | 307.02 | 49.05 | 13.89 | 280.14 | 93.8 |

11,445.29 | 7796.16 | 5774.63 | 2227.54 | 1788.96 | 157.95 | 445.18 | 51.95 | 13.13 | 240.10 | 23.4 |

7344.10 | 7338.86 | 4693.83 | 2389.09 | 690.83 | 335.25 | 93.48 | 22.07 | 12.03 | 232.62 | 22.6 |

${\mathit{E}}_{1}$ (MPa) | ${\mathit{E}}_{2}$ (MPa) | ${\mathit{E}}_{3}$ (MPa) | ${\mathit{E}}_{4}$ (MPa) | ${\mathit{E}}_{5}$ (MPa) | ${\mathit{E}}_{6}$ (MPa) | ${\mathit{E}}_{7}$ (MPa) | ${\mathit{E}}_{\mathit{\infty}}$ (MPa) | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | SAE (mm) |
---|---|---|---|---|---|---|---|---|---|---|

10,133.33 | 7753.27 | 4879.95 | 2147.25 | 877.58 | 214.29 | 255.64 | 41.21 | 11.1 | 129.2 | 6.03 |

Items | Relative Rutting Depth (mm) | DS (Time/min) | Error Rate of Relative Rutting Depth (%) | Error Rate of DS (%) |
---|---|---|---|---|

WTT | 0.355 | 7875 | - | - |

Simulation of OP | 0.366 | 7756 | 3.1 | 1.5 |

Simulation of EP | 0.339 | 6033 | 4.5 | 23.4 |

Items | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{3}$ | ${\mathit{E}}_{4}$ | ${\mathit{E}}_{5}$ | ${\mathit{E}}_{6}$ | ${\mathit{E}}_{7}$ | ${\mathit{E}}_{\mathit{\infty}}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Correlation coefficient | −0.022 | 0.105 | −0.046 | −0.034 | 0.245 | 0.222 | −0.414 | 0.394 | 0.427 | 0.359 |

p-value | 0.731 | 0.099 | 0.475 | 0.597 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 |

Items | ${\mathit{E}}_{5}$ (MPa) | ${\mathit{E}}_{6}$ (MPa) | ${\mathit{E}}_{7}$ (MPa) | ${\mathit{E}}_{\mathit{\infty}}$ (MPa) | SAE (mm) | Error Rate of SAE (%) |
---|---|---|---|---|---|---|

Basic parameters | 877.58 | 214.29 | 255.64 | 41.21 | 6.03 | - |

Increscent ${E}_{5}$ | 1755.15 | 214.29 | 255.64 | 41.21 | 6.66 | 10.45 |

Reductive ${E}_{5}$ | 438.79 | 214.29 | 255.64 | 41.21 | 7.04 | 16.75 |

Increscent ${E}_{6}$ | 877.58 | 428.58 | 255.64 | 41.21 | 7.18 | 19.07 |

Reductive ${E}_{6}$ | 877.58 | 107.15 | 255.64 | 41.21 | 7.08 | 17.41 |

Increscent ${E}_{7}$ | 877.58 | 214.29 | 511.28 | 41.21 | 6.68 | 10.78 |

Reductive ${E}_{7}$ | 877.58 | 214.29 | 127.82 | 41.21 | 9.80 | 62.52 |

Increscent ${E}_{\infty}$ | 877.58 | 214.29 | 255.64 | 82.42 | 101.01 | 1575.12 |

Reductive ${E}_{\infty}$ | 877.58 | 214.29 | 255.64 | 20.60 | 191.58 | 3077.11 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, J.; Zhou, W.; Cao, D.; Zhang, J.
A Study on the Genetic Algorithm Optimization of an Asphalt Mixture’s Viscoelastic Parameters Based on a Wheel Tracking Test. *Infrastructures* **2023**, *8*, 169.
https://doi.org/10.3390/infrastructures8120169

**AMA Style**

Zhang J, Zhou W, Cao D, Zhang J.
A Study on the Genetic Algorithm Optimization of an Asphalt Mixture’s Viscoelastic Parameters Based on a Wheel Tracking Test. *Infrastructures*. 2023; 8(12):169.
https://doi.org/10.3390/infrastructures8120169

**Chicago/Turabian Style**

Zhang, Jinxi, Weiqi Zhou, Dandan Cao, and Jia Zhang.
2023. "A Study on the Genetic Algorithm Optimization of an Asphalt Mixture’s Viscoelastic Parameters Based on a Wheel Tracking Test" *Infrastructures* 8, no. 12: 169.
https://doi.org/10.3390/infrastructures8120169