# A New Approach to Determining the Network Fractality with Application to Robot-Laser-Hardened Surfaces of Materials

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## Abstract

**:**

## 1. Introduction

## 2. Material Preparation, Methodology and Data Mining

#### 2.1. Material Preparation

- ◦
- Laser beam speed v ∈ [2 mm/s, 5 mm/s];
- ◦
- Surface temperature T ∈ [850 °C, 1400 °C];
- ◦
- Impact angles φ ∈ [45°, 60°, 75°], φ ∈ [105°, 120°, 135°].

#### 2.2. Methodology

_{r}). For each line φ

_{i}, we check how many of the nodes lie on it (n). So, φ

_{i}= n. We determine all pairs (φ

_{i}, n), which results in a linear graph Γ(φ, n). In addition, the linear graph Γ(φ, n) includes nodes T1(φ1, n1), T2(φ2, n2) … Tk(φk, nk). For this graph, we estimate the Hurst exponent H [11]. After this, we can calculate the fractal dimension using the equation D = 2 − H. The Hurst exponent has a value in the interval H ∈ (0, 1); thus, the fractal dimension of complex networks is D ∈ (1, 2). Figure 6 shows the procedure of calculating the fractal dimension of the network.

#### 2.3. Data Mining

- To forecast the fractal dimension of the network of the microstructure of RLH specimens without photos, only non-image input parameters were used.
- In addition to other input characteristics, images were also employed to forecast the fractality of the network of the microstructure of RLH specimens.

## 3. Results, Discussion, GP and CNN Approach

^{2}/n,

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic view of the laser hardening via robot: (

**a**) the laser beam impacts the treated surface perpendicularly (90°); (

**b**) the impact angle modified to the right or left side: φ ∈ [45°, 60°, 75°] and [105°, 120°, 135°], respectively; (

**c**) the impact angle modified forward: φ ∈ [45°, 60°, 75°]; (

**d**) the impact angle is modified downward: φ ∈ [105°, 120°, 135°]; (

**e**) robot laser hardening with overlapping; (

**f**) point robot laser hardening (the laser beam does not move).

**Figure 5.**Three types of self-similar fractal objects: (

**a**) self-similar geometric object; (

**b**) quasi-self-similar natural object; (

**c**) statistically self-affine microstructure of robot-laser-hardened specimen.

**Figure 7.**Convolutional neural network. In the first setting (without images), the global average pooling layer is not connected to fully connected layers, while in the second (with images), it is connected.

**Figure 9.**Microstructure of specimens characterized by a speed treatment of (from

**left**to

**right**) 2 mm/s (S1), 3 mm/s (S2), 4 mm/s (S3), and 5 mm/s (S4) at 1000 °C.

Attributes | # |
---|---|

size of the population of organisms | 600 |

maximum number of generations | 100 |

reproduction probability | 0.5 |

crossover probability | 0.6 |

maximum permissible depth in the creation of the population | 8 |

maximum permissible depth after the operation of crossover of two organisms | 10 |

smallest permissible depth of organisms in generating new organisms | 4 |

tournament size used for selection of organisms | 7 |

S | X1 | X2 | X3 | X4 | Y |
---|---|---|---|---|---|

S1 | 1000 | 2 | 90 | 120,515 | 1.615 |

S2 | 1000 | 3 | 90 | 125,579 | 1.623 |

S3 | 1000 | 4 | 90 | 123,695 | 1.618 |

S4 | 1000 | 5 | 90 | 124,585 | 1.655 |

S5 | 1400 | 2 | 90 | 124,378 | 1.589 |

S6 | 1400 | 3 | 90 | 132,292 | 1.682 |

S7 | 1400 | 4 | 90 | 126,714 | 1.631 |

S8 | 1400 | 5 | 90 | 130,751 | 1.625 |

S9 | 1000 | 2 | 45 | 123,345 | 1.612 |

S10 | 1000 | 3 | 60 | 126,147 | 1.526 |

S11 | 1000 | 4 | 75 | 124,048 | 1.715 |

S12 | 1000 | 2 | 45 | 123,581 | 1.608 |

S13 | 1000 | 3 | 60 | 127,895 | 1.539 |

S14 | 1000 | 4 | 75 | 122,152 | 1.533 |

S15 | 1000 | 2 | 45 | 120,370 | 1.713 |

S16 | 1000 | 3 | 60 | 116,564 | 1.543 |

S17 | 1000 | 4 | 75 | 132,783 | 1.519 |

S18 | 1400 | 0 | 105 | 130,226 | 1.663 |

S19 | 1000 | 0 | 120 | 130,795 | 1.641 |

S20 | 950 | 0 | 135 | 106,468 | 1.585 |

S21 | 850 | 0 | 90 | 120,565 | 1.559 |

S22 | 0 | 0 | 0 | 94,842 | 1.374 |

S | Y | P GP | P CNN |
---|---|---|---|

S1 | 1.615 | 1.58072 | 1.608 |

S2 | 1.623 | 1.62035 | 1.585 |

S3 | 1.618 | 1.61631 | 1.655 |

S4 | 1.655 | 1.61513 | 1.631 |

S5 | 1.589 | 1.61024 | 1.663 |

S6 | 1.682 | 1.69476 | 1.519 |

S7 | 1.631 | 1.61795 | 1.663 |

S8 | 1.625 | 1.61576 | 1.663 |

S9 | 1.612 | 1.60483 | 1.585 |

S10 | 1.526 | 1.53524 | 1.533 |

S11 | 1.715 | 1.61746 | 1.608 |

S12 | 1.608 | 1.60483 | 1.612 |

S13 | 1.539 | 1.53524 | 1.526 |

S14 | 1.533 | 1.61746 | 1.615 |

S15 | 1.713 | 1.60483 | 1.608 |

S16 | 1.543 | 1.53524 | 1.608 |

S17 | 1.519 | 1.61746 | 1.539 |

S18 | 1.663 | 1.65291 | 1.641 |

S19 | 1.641 | 1.64808 | 1.663 |

S20 | 1.585 | 1.58919 | 1.655 |

S21 | 1.559 | 1.55913 | 1.585 |

S22 | 1.374 | 1.37364 | 1.608 |

**Table 4.**Roughness prediction errors (the mean squared error, MSE) in both settings: without and with images.

Number of Hidden Layers | Size of Layers | Activation Function | Without Images | With Images | |
---|---|---|---|---|---|

Train (=Test) | Train | Test | |||

Mean Std | Mean Std | Mean Std | |||

2 hidden layers | 20, 10 | sigmoid | 0.022 ± 0.001 | 0.021 ± 0.002 | 0.022 ± 0.002 |

relu | 0.024 ± 0.002 | 0.018 ± 0.003 | 0.017 ± 0.003 | ||

10, 5 | sigmoid | 0.027 ± 0.001 | 0.028 ± 0.004 | 0.028 ± 0.004 | |

relu | 0.024 ± 0.003 | 0.021 ± 0.004 | 0.022 ± 0.004 | ||

3 hidden layers | 20, 20, 10 | sigmoid | 0.022 ± 0.001 | 0.019 ± 0.002 | 0.021 ± 0.002 |

relu | 0.021 ± 0.002 | 0.015 ± 0.001 | 0.015 ± 0.001 | ||

10, 10, 5 | sigmoid | 0.026 ± 0.001 | 0.026 ± 0.003 | 0.027 ± 0.002 | |

relu | 0.025 ± 0.004 | 0.021 ± 0.005 | 0.021 ± 0.004 |

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**MDPI and ACS Style**

Babič, M.; Marinković, D.
A New Approach to Determining the Network Fractality with Application to Robot-Laser-Hardened Surfaces of Materials. *Fractal Fract.* **2023**, *7*, 710.
https://doi.org/10.3390/fractalfract7100710

**AMA Style**

Babič M, Marinković D.
A New Approach to Determining the Network Fractality with Application to Robot-Laser-Hardened Surfaces of Materials. *Fractal and Fractional*. 2023; 7(10):710.
https://doi.org/10.3390/fractalfract7100710

**Chicago/Turabian Style**

Babič, Matej, and Dragan Marinković.
2023. "A New Approach to Determining the Network Fractality with Application to Robot-Laser-Hardened Surfaces of Materials" *Fractal and Fractional* 7, no. 10: 710.
https://doi.org/10.3390/fractalfract7100710