# Estimation of Clinch Joint Characteristics Based on Limited Input Data Using Pre-Trained Metamodels

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Research Questions

## 4. Method

#### 4.1. General Approach

#### 4.2. Numerical Clinching Process

^{−1}causes a compression of the punch-sided material into the die-sided blank based on cold-forming. In this context, Figure 3a depicts determined experimental data compared with the results of the generated FE model. Because the values demonstrate a satisfying agreement, the numerical clinching model can be assumed as valid.

#### 4.3. Design of Experiments

#### 4.4. Metamodeling

#### 4.5. Feature Selection

## 5. Results

#### 5.1. Performance of Machine Learning Algorithms on a Comprehensive Database

#### 5.2. Correlation Analysis

#### 5.3. Performance of Machine Learning Algorithms on Varying Input Data

#### 5.4. Design Equations

## 6. Discussion

## 7. Conclusions

- •
- The linear and polynomial regression models achieved the highest ability to predict the investigated clinch joint properties considering comprehensive and even limited input data.
- •
- The application of a correlation-based feature-selection method enabled the significant decrease of the model complexity based on a systematic reduction of the database. For instance, the accurate estimation of the neck thickness can be achieved by only considering a linear regression model and data regarding the applied die depth and die bottom diameter.
- •
- The experimental evaluation of the generated results confirm a high applicability of the simplified models and design equations for the prediction of clinch joint characteristics even if only a limited number of features are available. Thus, besides the reduction of computational and modeling effort, the results can pave the way to a more versatile application of pretrained regression models on varying tool configurations for a given joining task.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

CoP | Coefficient of Prognosis |

DoE | Design of Experiment |

FE | Finite Element |

GA | Genetic Algorithm |

MLS | Moving Least-Square |

NSGA-II | Non-Dominated Sorting Genetic Algorithm |

PCA | Principal Component Analysis |

RQ | Research Questions |

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**Figure 2.**Illustration of the clinch joining process (

**a**), resulting joint properties (

**b**) and FE model parameters (

**c**).

**Figure 3.**Experimental validation of clinch joint properties (

**a**) and force-displacement shear curve (

**b**).

**Figure 5.**Performance of machine learning algorithms for the estimation of individual clinch joint characteristics.

Input Parameter | Unit | Min.–Max. | Input Parameter | Unit | Min.–Max. |
---|---|---|---|---|---|

Punch | Die | ||||

Diameter d${}_{P}$ | mm | 4.5–6.0 | Diameter d${}_{D}$ | mm | 7.5–8.5 |

Radius r${}_{III}$ | mm | 0.1–0.6 | Depth h${}_{D}$ | mm | 0.8–1.8 |

Side draft angle ${\alpha}_{I}$ | deg | 0.0–4.0 | Groove depth h${}_{DG}$ | mm | 0.5–1.3 |

Face draft angle ${\alpha}_{II}$ | deg | 3.0–8.0 | Bottom diameter d${}_{DB}$ | mm | 3.5–4.8 |

Process | Groove diameter d${}_{DG}$ | mm | 5.6–7.0 | ||

Punch penetration s | % | 70–90 * | Corner radius I r${}_{I}$ | mm | 0.1–0.4 |

Joining velocity | mm s^{−1} | 2 (const.) | Corner radius II r${}_{II}$ | mm | 0.1–0.4 |

Blank holder | Side draft angle ${\alpha}_{III}$ | deg | 0.0–8.0 | ||

Force | N | 785 (const.) | |||

Tool dimensions | const. |

Removed Input Parameter in Each Feature Selection Iteration | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

13 * (1 **) | 12 (2) | 11 (3) | 10 (4) | 9 (5) | 8 (6) | 7 (7) | 6 (8) | 5 (9) | 4 (10) | 3 (11) | 2 (12) | 1 – | |

${\mathit{t}}_{\mathit{NE}}$ | ${\alpha}_{II}$ | ${r}_{II}$ | ${\alpha}_{I}$ | s | ${d}_{P}$ | ${r}_{I}$ | ${d}_{DG}$ | ${d}_{D}$ | ${\alpha}_{III}$ | ${h}_{DG}$ | ${r}_{III}$ | ${d}_{DB}$ | ${h}_{D}$ |

${\mathit{t}}_{\mathit{IL}}$ | ${d}_{D}$ | ${d}_{DG}$ | ${d}_{P}$ | ${r}_{II}$ | ${r}_{I}$ | ${\alpha}_{II}$ | ${\alpha}_{III}$ | ${\alpha}_{I}$ | ${d}_{DB}$ | ${r}_{III}$ | ${h}_{DG}$ | ${h}_{D}$ | s |

${\mathit{F}}_{\mathit{Join}}$ | ${h}_{D}$ | ${d}_{DB}$ | ${h}_{DG}$ | ${r}_{III}$ | ${r}_{I}$ | ${d}_{D}$ | ${r}_{II}$ | ${d}_{DG}$ | ${\alpha}_{III}$ | ${\alpha}_{II}$ | ${\alpha}_{I}$ | s | ${d}_{P}$ |

${\mathit{F}}_{\mathit{Shear}}$ | s *** | ${r}_{II}$ | ${r}_{I}$ | ${\alpha}_{II}$ | ${d}_{DG}$ | ${\alpha}_{I}$ | ${r}_{III}$ | ${d}_{D}$ | ${\alpha}_{III}$ | ${d}_{P}$ | ${d}_{DB}$ | ${h}_{DG}$ | ${h}_{D}$ |

**Table 3.**Overview of the achieved mean CoP values and the minimum required number of input features for the accurate prediction of individual clinch joint characteristics.

Input Data: 330 Samples, 13 Features | Input Data: 330 Samples, N Features | ||||
---|---|---|---|---|---|

Target Variables | ${\overline{\mathit{CoP}}}_{13,\mathit{m}}$ | Metamodel | ${\overline{\mathit{CoP}}}_{\mathit{N},\mathit{m}}$ | Remaining Features N | |

Neck | ${\overline{CoP}}_{13,2}=0.98$ | → | Linear regression | ${\overline{CoP}}_{2,1}=0.89$ | 2 (${d}_{DB}$, ${h}_{D}$) |

Interlock | ${\overline{CoP}}_{13,2}=0.94$ | → | Poly. regression | ${\overline{CoP}}_{6,2}=0.81$ | 6 (${\alpha}_{I}$, ${d}_{DB}$, ${h}_{DG}$, ${r}_{III}$, ${h}_{D}$, s) |

Joining force | ${\overline{CoP}}_{13,2}=0.89$ | → | Linear regression | ${\overline{CoP}}_{5,1}=0.81$ | 5 (${\alpha}_{III}$, ${\alpha}_{II}$, ${\alpha}_{I}$, s, ${d}_{P}$) |

Shear force | ${\overline{CoP}}_{13,2}=0.87$ | → | Poly. regression | ${\overline{CoP}}_{4,2}=0.83$ | 4 (${d}_{P}$, ${d}_{DB}$, ${h}_{DG}$, ${h}_{D}$) |

Regression Functions | Prediction | Exp. Study * | ||
---|---|---|---|---|

${t}_{NE}$ | = | $-0.267{h}_{D}+0.055{d}_{DB}+0.507$ | 0.46 mm | [0.45–0.49] mm |

${t}_{IL}$ | = | $-0.089{r}_{III}-0.001{\alpha}_{I}+0.436{h}_{D}+0.419{h}_{DG}-0.286s$ $-0.295{d}_{DB}-0.173{r}_{III}^{2}-0.005{\alpha}_{I}^{2}-0.088{h}_{D}^{2}+0.021{s}^{2}$ $-0.170{h}_{DG}^{2}+0.043{d}_{DG}^{2}+0.301$ | 0.34 mm | [0.28–0.32] mm |

${F}_{Join}$ | = | $19078{p}_{D}+928{\alpha}_{I}-473{\alpha}_{II}-285{\alpha}_{III}-16888s-54600$ | 29.2 kN | [31.5–35.7] kN |

${F}_{Shear}$ | = | $1611{p}_{D}-267{h}_{D}+192{h}_{DG}+221{d}_{DB}-142{p}_{D}^{2}-80{h}_{D}^{2}$ $+7{h}_{DG}^{2}-6{d}_{DB}^{2}-3430$ | 1895 N | [1902–1946] N |

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**MDPI and ACS Style**

Zirngibl, C.; Schleich, B.; Wartzack, S.
Estimation of Clinch Joint Characteristics Based on Limited Input Data Using Pre-Trained Metamodels. *AI* **2022**, *3*, 990-1006.
https://doi.org/10.3390/ai3040059

**AMA Style**

Zirngibl C, Schleich B, Wartzack S.
Estimation of Clinch Joint Characteristics Based on Limited Input Data Using Pre-Trained Metamodels. *AI*. 2022; 3(4):990-1006.
https://doi.org/10.3390/ai3040059

**Chicago/Turabian Style**

Zirngibl, Christoph, Benjamin Schleich, and Sandro Wartzack.
2022. "Estimation of Clinch Joint Characteristics Based on Limited Input Data Using Pre-Trained Metamodels" *AI* 3, no. 4: 990-1006.
https://doi.org/10.3390/ai3040059