# Process Parameter Selection for Production of Stainless Steel 316L Using Efficient Multi-Objective Bayesian Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Gaussian Process

#### 2.2. Bayesian Optimization: Pareto Front Approximation

#### 2.3. Bayesian Update Procedure: Batch Selection Strategy

## 3. Results

## 4. Discussion

## 5. Conclusions

- The model trained on relatively small batch of data quickly found three points on the Pareto front in just six iterations.
- The highest value of the hardness obtained empirically was $241.3$ HV, corresponding to a VED of $118.5$ J/mm${}^{3}$, with a power of 133 W, a scanning speed of 850 mm/s, and hatch spacing of 66 µm.
- The highest relative density part had a porosity of $0.0007$% and the following parameters: VED $119.72$ J/mm${}^{3}$, power 108 W, scanning speed 465 mm/s, hatch spacing 97 µm.
- The VED that was explored by the algorithm lied in the range of 240–265 J/mm${}^{3}$; the hardness of the produced parts was 224–235 HV, and the porosity was in the range of 0.2–0.37%.
- The recommended processing window corresponded to the parts manufactured with an energy density that lied in the range of 65–280 J/mm${}^{3}$.
- The trained model prescribed the following parameters to ensure quality parts: 58 W, 257 mm/s, 45 µm, with a scan rotation angle of 131 degrees.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AM | Additive manufacturing |

L-PBF | Laser powder bed fusion |

MOBO | Multi-objective Bayesian optimization |

ML | Machine learning |

GP | Gaussian process |

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**Figure 3.**The optical image of the printed samples: (

**a**) sample with energy density $144.13$ J/mm${}^{3}$ with porosity $0.064\%$; (

**b**) sample with energy density 48.23 J/mm${}^{3}$ with porosity $15.60\%$.

**Figure 4.**The scatter plot of (

**a**) porosity (%) and (

**b**) hardness (HV) as a function of VED. The shaded area represents the recommended operating window of 65–280 J/mm${}^{3}$.

**Figure 5.**Performance results. (

**a**) The scatter plot of the hardness versus porosity (Pareto front is colored red); open magenta marks represent candidate configuration, and solid magenta marks denote the suggested configuration after evaluation. (

**b**) The hypervolume improvement plot on the right represents advancements toward the Pareto front.

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

Time (min) | 1 | 4 |

Gas circulation speed (m/s) | 1.5 | 4 |

Laser power (W) | 30 | 175 |

Scan speed (mm/s) | 100 | 3000 |

Hatch distance (µm) | 40 | 120 |

Scan angle (degrees) | 0 | 150 |

Time, min | Gas Feed, m/s | Power, W | Speed, mm/s | Hatch Spacing, µm | Energy Density, J/mm${}^{3}$ | Angle, ° | Hardness, HV |
---|---|---|---|---|---|---|---|

2 | 3.5 | 133.0 | 850.0 | 66.0 | 118.5 | 44.0 | 241.3 |

2 | 3.5 | 154.0 | 335.0 | 90.0 | 255.4 | 48.0 | 240.3 |

2 | 3.5 | 123.0 | 583.0 | 79.0 | 133.5 | 18.0 | 239.3 |

2 | 3.5 | 159.0 | 1128.0 | 75.0 | 93.9 | 50.0 | 237.8 |

2 | 4.0 | 74.3 | 448.6 | 70.0 | 118.3 | 74.9 | 237.0 |

3 | 2.5 | 113.0 | 910.0 | 80.0 | 77.6 | 150.0 | 236.0 |

2 | 3.5 | 101.0 | 505.0 | 87.0 | 114.9 | 5.0 | 235.4 |

3 | 3.5 | 51.5 | 726.9 | 41.0 | 86.4 | 10.7 | 235.1 |

3 | 2.5 | 147.0 | 490.0 | 90.0 | 166.7 | 150.0 | 235.0 |

2 | 3.5 | 114.0 | 454.0 | 67.0 | 187.4 | 41.0 | 235.0 |

Time, min | Gas Feed, m/s | Power, W | Speed, mm/s | Hatch Spacing, µm | Energy Density, J/mm${}^{3}$ | Angle, ° | Porosity, % |
---|---|---|---|---|---|---|---|

2 | 3.5 | 108.0 | 465.0 | 97.0 | 119.72 | 77.0 | 0.0007 |

2 | 3.5 | 163.0 | 616.0 | 71.0 | 186.35 | 51.0 | 0.0013 |

2 | 3.5 | 131.0 | 955.0 | 77.0 | 89.07 | 83.0 | 0.0042 |

2 | 3.5 | 101.0 | 505.0 | 87.0 | 114.94 | 5.0 | 0.0043 |

2 | 3.5 | 159.0 | 1128.0 | 75.0 | 93.97 | 50.0 | 0.0058 |

2 | 3.5 | 146.0 | 606.0 | 88.0 | 136.89 | 8.0 | 0.0067 |

2 | 3.5 | 137.0 | 628.0 | 82.0 | 133.02 | 25.0 | 0.0071 |

2 | 3.5 | 138.0 | 914.0 | 92.0 | 82.06 | 30.0 | 0.0074 |

2 | 3.5 | 133.0 | 850.0 | 66.0 | 118.54 | 44.0 | 0.0075 |

2 | 3.5 | 163.0 | 1268.0 | 74.0 | 86.86 | 35.0 | 0.0083 |

**Table 4.**Process parameters suggested by the DGEMO algorithm. Highlighted rows indicate the Pareto set.

Time, min | Gas Feed, m/s | Power, W | Speed, mm/s | Hatch Spacing, µm | Energy Density, J/mm${}^{3}$ | Angle, ° |
---|---|---|---|---|---|---|

3 | 3.5 | 59.0 | 254.0 | 45.0 | 258.1 | 123.0 |

3 | 3.5 | 58.0 | 257.0 | 47.0 | 240.1 | 131.0 |

3 | 3.5 | 60.0 | 257.0 | 44.0 | 265.3 | 123.0 |

2 | 3.5 | 59.0 | 256.0 | 45.0 | 256.1 | 123.0 |

2 | 3.5 | 59.0 | 258.0 | 45.0 | 254.1 | 124.0 |

2 | 3.5 | 59.0 | 259.0 | 44.0 | 258.9 | 123.0 |

Predicted Hardness (Std.), HV | Actual Hardness, HV | Predicted Porosity (Std.), % | Actual Porosity. % |
---|---|---|---|

248.0 ± 18.0 | 231.0 | 0.510 ± 2.918 | 0.200 |

248.0 ± 17.7 | 235.0 | 0.510 ± 2.854 | 0.310 |

248.0 ± 18.5 | 225.0 | 0.490 ± 2.980 | 0.370 |

248.0 ± 17.9 | 229.0 | 0.520 ± 2.916 | 0.260 |

248.0 ± 18.4 | 224.0 | 0.480 ± 2.930 | 0.200 |

249.0 ± 18.6 | 232.0 | 0.450 ± 2.948 | 0.210 |

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## Share and Cite

**MDPI and ACS Style**

Chepiga, T.; Zhilyaev, P.; Ryabov, A.; Simonov, A.P.; Dubinin, O.N.; Firsov, D.G.; Kuzminova, Y.O.; Evlashin, S.A.
Process Parameter Selection for Production of Stainless Steel 316L Using Efficient Multi-Objective Bayesian Optimization Algorithm. *Materials* **2023**, *16*, 1050.
https://doi.org/10.3390/ma16031050

**AMA Style**

Chepiga T, Zhilyaev P, Ryabov A, Simonov AP, Dubinin ON, Firsov DG, Kuzminova YO, Evlashin SA.
Process Parameter Selection for Production of Stainless Steel 316L Using Efficient Multi-Objective Bayesian Optimization Algorithm. *Materials*. 2023; 16(3):1050.
https://doi.org/10.3390/ma16031050

**Chicago/Turabian Style**

Chepiga, Timur, Petr Zhilyaev, Alexander Ryabov, Alexey P. Simonov, Oleg N. Dubinin, Denis G. Firsov, Yulia O. Kuzminova, and Stanislav A. Evlashin.
2023. "Process Parameter Selection for Production of Stainless Steel 316L Using Efficient Multi-Objective Bayesian Optimization Algorithm" *Materials* 16, no. 3: 1050.
https://doi.org/10.3390/ma16031050