# A Novel Neural Computing Model Applied to Estimate the Dynamic Modulus (DM) of Asphalt Mixtures by the Improved Beetle Antennae Search

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Raw Materials

#### 2.2. Mix Design

#### 2.3. Experimental Tests

#### 2.3.1. Determination of the Dynamic Shear Modulus of the Asphalt Binders

#### 2.3.2. Determination of the DM of the Asphalt Mixtures

#### 2.4. Methods

#### 2.4.1. Backpropagation Neural Network (BPNN)

#### 2.4.2. Improved BAS Algorithm

#### Traditional BAS Algorithm

^{i}

**b**is the step of the beetle at time t. The equation used to update the step size can be determined as follows:

#### Improved BAS with Higher Searching Efficiency

Algorithm 1. Pseudo-code of the MBAS algorithm [84] |

Input Fitness function f(x^i), initial position of the beetle x^0, initial step-size δ^0, maximum iterations n, ratio of antennae length to step-size c, attenuation coefficient of step-size η |

Output: Optimal position x_b, optimal fitness function value f_b. |

FOR I = 1 to n |

Generate random antennae direction b; |

Calculate the antennae length d^i = c × δ^i; |

Calculate the left-hand and right-hand positions x_l and x_r, respectively; |

Calculate the fitness function value f(xl) and f(xr) at the left and right antennae position; |

Calculate the next position x^i; |

Calculate the fitness function value f(x^(i + 1) ) at next position x^(i + 1); |

IF f(x^(i + 1)) < f_b |

THEN Update x_b to x^(i + 1);⋯Update f_b to f(x^(i + 1) ); |

END |

Update step-size δ^(i + 1) using Equations (8) and (9); |

IF |f (x^(i + 1)) − f (x^i)| < μ (fw − fb) |

THEN Update step-size δ^(i + 1) using Levy flight according to Equation (9); |

ELSE Update step-size δ^(i + 1) according to Equation (8) |

END |

i = i + 1; |

END |

## 3. Results and Discussion

#### 3.1. Testing Results and Dataset Description

#### 3.2. Hyperparameter Tuning Results

#### 3.3. Predictive Performance of the Proposed Model

#### 3.4. Importance of the Input Variables

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MEPDG | Mechanistic-Empirical Pavement Design Guide |

BAS | Beetle antennae search |

DM | Dynamic modulus |

AASHTO | American Association of State Highway and Transportation Officials |

PI | Penetration index |

AI | Artificial intelligence |

RC | Reinforced concrete |

BPNN | Backpropagation neural network |

Superpave | SUperior PERforming Asphalt PAVEments |

Va | Air voids |

DSR | Dynamic shear rheometer |

RMSE | Root mean square error |

R | Correlation coefficient |

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**Figure 2.**DM results of the asphalt mixtures as well as the comparison with the Witczak 1-40D model.

Basic Properties | Binder 1 | Binder 2 | Aggregates | |
---|---|---|---|---|

Binder-index properties of asphalt | Penetration @ 25 °C (0.1 mm) | 92.0 | 75.1 | - |

Penetration index (PI) | −1.17 | 0.2 | - | |

Softening point (°C) | 44.0 | 52.5 | - | |

Viscosity @ 135 °C (Pa∙s) | 0.363 | 1.3 | - | |

Physical properties of aggregate | LA abrasion value (%) | - | - | 23.0 |

Aggregate impact value (%) | - | - | 22.0 | |

Water absorption (%) | - | - | 0.14 | |

Combined elongation and flakiness indices (%) | - | - | 28.0 | |

Soundness, magnesium sulfate solution (%) | - | - | 0.4 |

Asphalt Mixtures | Nominal Aggregate Size (mm) | Binder Content (%) | Va (%) |
---|---|---|---|

Asphalt Mixture-1 | 19 | 4.5 | 4.0 |

Asphalt Mixture-2 | 19 | 4.4 | 4.0 |

Asphalt Mixture-3 | 19 | 4.8 | 4.0 |

Asphalt Mixture-4 | 19 | 5.0 | 4.0 |

Asphalt Mixture-5 | 12.5 | 5.0 | 4.0 |

Asphalt Mixture-6 | 12.5 | 5.3 | 4.0 |

Asphalt Mixture-7 | 12.5 | 5.5 | 4.0 |

Asphalt Mixture-8 | 12.5 | 5.4 | 4.0 |

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**MDPI and ACS Style**

Huang, J.; Zhou, M.; Sabri, M.M.S.; Yuan, H.
A Novel Neural Computing Model Applied to Estimate the Dynamic Modulus (DM) of Asphalt Mixtures by the Improved Beetle Antennae Search. *Sustainability* **2022**, *14*, 5938.
https://doi.org/10.3390/su14105938

**AMA Style**

Huang J, Zhou M, Sabri MMS, Yuan H.
A Novel Neural Computing Model Applied to Estimate the Dynamic Modulus (DM) of Asphalt Mixtures by the Improved Beetle Antennae Search. *Sustainability*. 2022; 14(10):5938.
https://doi.org/10.3390/su14105938

**Chicago/Turabian Style**

Huang, Jiandong, Mengmeng Zhou, Mohanad Muayad Sabri Sabri, and Hongwei Yuan.
2022. "A Novel Neural Computing Model Applied to Estimate the Dynamic Modulus (DM) of Asphalt Mixtures by the Improved Beetle Antennae Search" *Sustainability* 14, no. 10: 5938.
https://doi.org/10.3390/su14105938