# Predicting the Ultimate Tensile Strength of Friction Stir Welds Using Gaussian Process Regression

*iwb*), Technical University of Munich, Boltzmannstr. 15, 85748 Garching, Germany

^{*}

## Abstract

**:**

## 1. Introduction

- Due to the flexibility in modeling, since GPR is a non-parametric method [9].
- Due to the possibility of quantifying uncertainties for the prediction, which can be a great benefit for safety-critical applications [9].
- Since Verma et al. [8] compared the performance of the GPR to predict the ultimate tensile strength based on the process parameters with the multi-linear regression and support vector machines—the GPR led to the best results.

## 2. Materials and Methods

#### 2.1. Welding Experiments

_{x}, F

_{y}, and F

_{z}, and the spindle torque M

_{z}were recorded with a sampling rate of 9.6 kHz by a dynamometer, which is described in more detail in Krutzlinger et al. [21]. The temperatures at the tool shoulder T

_{S}and the tool probe T

_{P}were measured by thermocouples with a sampling rate of 220 Hz. The temperature measuring system was based on the one described by Costanzi et al. [22]. The accelerations a

_{x}, a

_{y}, and a

_{z}in three spatial directions with a sampling rate of 20 kHz were acquired by an acceleration sensor type 8762A50 of Kistler Instrumente GmbH (Winterthur, Switzerland). The accelerometer was positioned 20 mm away from the immersion point of the welding tool during the experiments. A two-piece tool with a concave shoulder with a radius r

_{S}of 7 mm and a conical probe with a radius r

_{P}of 3 mm was utilized. The probe had a M6 thread and three flats. The tool geometry and the most important dimensions are presented in Figure 2 and Table 2.

_{s}and the tool rotational speed n (r/min rate) were modified. The examined welding speeds were 500 mm/min, 1000 mm/min and 1500 mm/min. As high welding speeds are becoming increasingly important for industrial applications, especially in the context of electromobility [24], welding speeds of up to 1500 mm/min were applied The ratio between the tool rotational speed and the welding speed n/v

_{s}was varied over a wide interval from 1 mm

^{−1}to 7 mm

^{−1}. Furthermore, the tool rotational speed did not exceed 5000 min

^{−1}. Exceeding these boundaries could have damaged the welding tool or the measuring equipment. To generate a sufficient amount of data, the tool rotational speed was adjusted in steps of 200 min

^{−1}within the mentioned boundaries, which resulted in an experimental design totaling 54 experiments. Table A1 in Appendix A shows the process parameters applied in each welding process.

_{m}and the seam underfill u

_{m}, as well as the standard deviations of the flash height S

_{f}, of the seam underfill S

_{u}, and of the weld seam width S

_{w}were calculated. Figure 3c schematically shows the topography along the weld centerline (C). Due to the seam underfill, the topography along the centerline is usually below the sheet surface. The number of local valleys and local peaks along the centerline was counted (n

_{count}) and compared with the theoretical number (n

_{theoret}), which leads to the ratio r

_{arc}:

_{theoret}was calculated using the tool rotational speed n, the welding speed v

_{s}, and the width of the tensile specimen, which was 15 mm:

_{d}of the differences d

_{arc}between the local valleys and the subsequent local peaks were calculated along the 15-mm-long centerline (C) of the tensile specimens. The peak material volume V

_{mp}[25] was determined for area (B) of each tensile specimen. In a previous study it was found that, by employing the peak material volume V

_{mp}, the surface galling of the weld can be quantified [15]. The eight topography indicators utilized to quantify the 15-mm-long weld surface segment on the tensile specimens are summarized in Table 3. These values were later used as input variables for the Gaussian process regression model to predict the ultimate tensile strength based on the weld topography.

_{x}, F

_{y}, F

_{z}, M

_{z}, T

_{P}, T

_{S}, a

_{x}, a

_{y}, a

_{z}) were filtered, cut, and assigned to the corresponding weld segments of the tensile specimens. Afterwards, the following ten statistical values were calculated for each process variable corresponding to the 216 weld segments: arithmetic mean, maximum, minimum, median, root mean square (RMS), variance, kurtosis, skewness, highest amplitude in the frequency spectrum after performing a fast Fourier transform, and the span between the maximum and the minimum signal value of each segment. Thus, a total of 90 different features (nine process variables times ten statistical values) were available for each of the 216 tensile specimens. Some of these values were later provided as inputs for the Gaussian process regression model to predict the ultimate tensile strength based on the process variables.

#### 2.2. Application of the Gaussian Process Regression

_{m}, the mean z-force F

_{z,m}or the tool rotational speed n. The mean function m(

**x**) was set to zero, which is a common choice when modeling with Gaussian processes [9]. As a consequence, the data were normalized to mean zero and to variance one before performing the regression. The data were divided into 80% training data and 20% test data. The distribution of the data between the two data sets was random. Following Rasmussen et al. [9], a five-fold cross-validation was applied.

^{0}and e

^{10}, and afterwards, the model with the lowest MAE for predicting the ultimate tensile strength of the specimens contained in the test data set was applied.

^{®}Core™ i7-6700HQ CPU at 2.60 GHz. The installed RAM was 16 GB.

## 3. Results

#### 3.1. Prediction of the Ultimate Tensile Strength Using Surface Topography Data

_{m}, S

_{f}, u

_{m}, S

_{u}, S

_{w}, r

_{arc}, S

_{d}, V

_{mp}), described in Section 2.1, were used as input variables for the GPR model. The input matrix X thus had eight dimensions d in total. In the first step, the data from all 216 conducted tensile tests were utilized. Some outliers with significantly lower ultimate tensile strengths or very strong flash formation, which occurred especially at very high or very low tool rotational speeds n, were also included in the data. The results are displayed in Table 4. Overall, in the five-fold cross-validation, a mean PCC between the true and the predicted ultimate tensile strength of 0.76 was achieved when the data from all three welding speeds were taken into account. The standard deviation of the five values was 0.17. This result was achieved when using the SM covariance function. The total computation time was 6953 s. This was the time for the training and the testing of the five-fold cross-validation. As described in the previous chapter, for each part of the cross-validation in another nested inner loop, a ten-fold calculation with different initial hyperparameters was performed. The highest mean PCC of 0.87 was achieved at a welding speed of 500 mm/min.

_{s}of 500 mm/min and the Matérn 5/2 covariance function were used to generate Figure 7. A mean PCC of 0.94 and a standard deviation of 0.05 were achieved in this way (see Table 5). In addition to the mean values of the predictions, the 95% confidence intervals are also given for the test samples.

#### 3.2. Predicting Ultimate Tensile Strength Using Process Variables

_{z}, the mean F

_{y}-force, the mean F

_{z}-force and the root mean square (RMS) values of the accelerations in x-, y-, and z-direction. These six input dimensions were determined based on a simple statistical analysis to assess which of the 90 different features have the highest correlation with the ultimate tensile strength.

#### 3.3. Predicting the Ultimate Tensile Strength Using Process Parameters

^{−1}and 3500 min

^{−1}. In Figure 10a, the width of the confidence interval is nearly constant, whereas in Figure 10b, the width of the confidence interval varies in this range. In the ranges of the tool rotational speed that the model has not yet experienced, the width of the confidence interval increases. It seems plausible that predicting the ultimate tensile strength for unknown tool rotational speeds has a higher uncertainty than for tool rotational speeds that have already been tested. The SM covariance function is more flexible than the RBF covariance function and can probably, therefore, learn this uncertainty profile. Additionally, Figure 10 shows that the GPR models are able to reproduce the lower ultimate tensile strength at tool rotational speeds below 1000 min

^{−1}, which was attributed to internal defects, such as tunnel errors.

## 4. Discussion

## 5. Conclusions

- The Gaussian process regression is a powerful approach to non-destructively predict ultimate tensile strength through data evaluation. The uncertainty of the prediction can be quantified, and a confidence interval can be specified within which the ultimate tensile strength is located with a certain probability.
- It is possible to predict the ultimate tensile strength of friction stir welds by evaluating the surface topography through Gaussian process regression. This is especially valid for low welding speeds and when extremely low or high tool rotational speeds are not employed.
- The correlation coefficients for the prediction of the ultimate tensile strength by using the process variables or the process parameters were even higher compared to when using the surface topography data as inputs to the model.
- The differences in the PCCs for the various covariance functions used were low. However, when using the data from all investigated welding speeds, the spectral mixture covariance function according to Wilson et al. [19], always yielded the best results.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Experimental Plan

**Table A1.**Welding experiments performed and mean ultimate tensile strength $\overline{{R}_{m}}$ achieved.

v_{s} = 500 mm/min | v_{s} = 1000 mm/min | v_{s} = 1500 mm/min | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Exp. no. | n min ^{−1} | n/v_{s}mm ^{−1} | $\overline{{\mathit{R}}_{\mathit{m}}}$MPa | Exp. no. | n min ^{−1} | n/v_{s}mm ^{−1} | $\overline{{\mathit{R}}_{\mathit{m}}}$MPa | Exp. no. | n min ^{−1} | n/v_{s}mm ^{−1} | $\overline{{\mathit{R}}_{\mathit{m}}}$MPa | ||

1 | 500 | 1.0 | 127 | 17 | 1100 | 1.1 | 245 | 37 | 1500 | 1.0 | 252 | ||

2 | 700 | 1.4 | 170 | 18 | 1300 | 1.3 | 247 | 38 | 1700 | 1.1 | 255 | ||

3 | 900 | 1.8 | 233 | 19 | 1500 | 1.5 | 248 | 39 | 1900 | 1.3 | 255 | ||

4 | 1100 | 2.2 | 238 | 20 | 1700 | 1.7 | 249 | 40 | 2100 | 1.4 | 254 | ||

5 | 1300 | 2.6 | 239 | 21 | 1900 | 1.9 | 250 | 41 | 2300 | 1.5 | 255 | ||

6 | 1500 | 3.0 | 239 | 22 | 2100 | 2.1 | 250 | 42 | 2500 | 1.7 | 256 | ||

7 | 1700 | 3.4 | 239 | 23 | 2300 | 2.3 | 251 | 43 | 2700 | 1.8 | 257 | ||

8 | 1900 | 3.8 | 238 | 24 | 2500 | 2.5 | 250 | 44 | 2900 | 1.9 | 257 | ||

9 | 2100 | 4.2 | 238 | 25 | 2700 | 2.7 | 249 | 45 | 3100 | 2.1 | 257 | ||

10 | 2300 | 4.6 | 236 | 26 | 2900 | 2.9 | 251 | 46 | 3300 | 2.2 | 255 | ||

11 | 2500 | 5.0 | 236 | 27 | 3100 | 3.1 | 249 | 47 | 3500 | 2.3 | 253 | ||

12 | 2700 | 5.4 | 236 | 28 | 3300 | 3.3 | 246 | 48 | 3700 | 2.5 | 254 | ||

13 | 2900 | 5.8 | 234 | 29 | 3500 | 3.5 | 246 | 49 | 3900 | 2.6 | 253 | ||

14 | 3100 | 6.2 | 234 | 30 | 3700 | 3.7 | 245 | 50 | 4100 | 2.7 | 250 | ||

15 | 3300 | 6.6 | 233 | 31 | 3900 | 3.9 | 244 | 51 | 4300 | 2.9 | 251 | ||

16 | 3500 | 7.0 | 234 | 32 | 4100 | 4.1 | 244 | 52 | 4500 | 3.0 | 253 | ||

Average: | 225 | 33 | 4300 | 4.3 | 247 | 53 | 4700 | 3.1 | 253 | ||||

34 | 4500 | 4.5 | 245 | 54 | 4900 | 3.3 | 248 | ||||||

35 | 4700 | 4.7 | 236 | Average: | 254 | ||||||||

36 | 4900 | 4.9 | 225 | ||||||||||

Average: | 246 |

## Appendix B. Fundamentals of the Gaussian Process Regression

**x**), k(

**x**,

**x′**))

**x**) and the covariance function k(

**x**,

**x′**) of a real process f(

**x**) are defined by [9]:

**x**has a mean, which can be evaluated by the mean function. Furthermore, every two inputs

**x**and

**x’**have a common covariance that can be evaluated by the covariance function. If

**x’**is equal to

**x**, the covariance function returns the variance of

**x**.

**x**is a D-dimensional input vector and y is a scalar output variable. All available data are aggregated in the D $\times $ q matrix X. The GP regression model with noise is given by:

**y**= f(X) +

**ε**

**ε**that follows an independent, identically distributed Gaussian distribution with zero mean and variance ${\sigma}_{\epsilon}^{2}$ [9].

**y**, ${\mathbf{f}}_{\ast}$)

^{T}, also called prior distribution, is defined as:

**y**, ${X}_{\ast}$) is the predicted value of the model from the given data set. This result shows the advantage of applying Gaussian process regression. Instead of a point estimate for the unknown values ${\mathbf{f}}_{\ast}$, the entire probability distribution can be evaluated. Furthermore, if m(X) and m(${X}_{\ast}$) are set to zero, which is very common [9], the prediction depends largely on the selected covariance function (see Equation (A14)).

**θ**. Different covariance functions have different hyperparameters

**θ**[9]. In the following, the covariance functions used in this work are introduced.

_{f}, α, and l.

_{1}, l

_{2}, …, l

_{D}determine the relevance of the corresponding input variable. Input dimensions d with large length-scale l

_{d}imply little variation along the different dimensions d and hence are less important for the predicted outcome ${\mathbf{f}}_{\ast}$. [35]

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**Figure 2.**Tool geometry with the most important dimensions (adapted from Bachmann et al. [23]).

**Figure 3.**(

**a**) Geometry and dimensions of the tensile specimens; (

**b**) Evaluated surface topography; (

**c**) Topography along the centerline of the weld.

**Figure 4.**Comparison of the PCCs means and standard deviations for the different covariance functions and the MLR when using the topography information as input variables and providing the data collected at all three welding speeds. Additionally, the corresponding computation times are given.

**Figure 5.**(

**a**) The lowest and (

**b**) the highest correlation out of the five-fold cross-validation to predict the ultimate tensile strength R

_{m}, taking into account the surface topography data from all three welding speeds v

_{s}and using the SM covariance function.

**Figure 6.**(

**a**) The lowest and (

**b**) the highest correlation out of the five-fold cross-validation to predict the ultimate tensile strength R

_{m}taking into account the surface topography data from all three welding speeds v

_{s}and using the SM covariance function. Additionally, some outliers were removed before training the GPR model in this case.

**Figure 7.**(

**a**) The lowest and (

**b**) the highest correlation out of the five-fold cross-validation to predict the ultimate tensile strength R

_{m}at a welding speed v

_{s}of 500 mm/min when using the surface topography data and the Matérn 5/2 covariance function. The graphs show the mean values of the predictions and the 95 % confidence intervals. In these investigations, some outliers were removed before training the GPR model.

**Figure 8.**(

**a**) The lowest and (

**b**) the highest correlation out of the five-fold cross-validation to predict the ultimate tensile strength R

_{m}, taking into account the process variable data from all three welding speeds v

_{s}and using the SM covariance function.

**Figure 9.**(

**a**) The lowest and (

**b**) the highest correlation out of the five-fold cross-validation to predict the ultimate tensile strength R

_{m}, taking into account the process parameter data from all three welding speeds v

_{s}and using the SM covariance function.

**Figure 10.**Visualization of the GP at a welding speed v

_{s}of 500 mm/min with the tool rotational speed n as the only input dimension for the two covariance functions, (

**a**) RBF and (

**b**) SM.

**Table 1.**Chemical composition of the used material EN AW-6082-T6 in %, which was reported by the selected material supplier, Bikar Metalle GmbH (Bad Berleburg, Germany).

Si | Fe | Cu | Mn | Mg | Cr | Zn | Ti | Others |
---|---|---|---|---|---|---|---|---|

0.90 | 0.42 | 0.10 | 0.44 | 0.70 | 0.03 | 0.13 | 0.03 | max. 0.05 |

Geometry Feature | Value |
---|---|

Probe radius r_{P} | 3 mm |

Shoulder radius r_{S} | 7 mm |

Conical probe angle β | 10° |

Probe length h_{P} | 3.75 mm |

Probe tip radius r_{T} | 10 mm |

Concave shoulder angle γ | 10° |

**Table 3.**Key indicators for quantifying the surface topography of the 15-mm-long weld sections on the tensile specimens.

Surface Feature | Key Indicator 1 | Key Indicator 2 |
---|---|---|

Flash formation | Mean flash height f_{m} | Standard deviation of the flash height S_{f} |

Seam underfill | Mean seam underfill u_{m} | Standard deviation of the seam underfill S_{u} |

Weld seam width | Standard deviation of the weld seam width S_{w} | - |

Arc texture formation | Ratio between the counted and the theoretical number of local valleys and peaks along the weld centerline r_{arc} | Standard deviation of the differences between the local valleys and the subsequent local peaks along the weld centerline S_{d} |

Surface galling | Peak material volume V_{mp} | - |

**Table 4.**Results for predicting the ultimate tensile strength R

_{m}when using the surface topography indicators as inputs for the GPR model.

Welding Speed v_{s} | 500 mm/min | 1000 mm/min | 1500 mm/min | All Data |
---|---|---|---|---|

PCC mean | 0.87 | 0.79 | 0.80 | 0.76 |

PCC standard deviation | 0.15 | 0.23 | 0.05 | 0.17 |

Best covariance function | Add | SM | Mat 5/2 | SM |

Computation time in s | 152 | 1130 | 1017 | 6953 |

**Table 5.**Results for the prediction of the ultimate tensile strength R

_{m}when using the surface topography indicators as input for the GPR model with outliers removed.

Welding Speed v_{s} | 500 mm/min | 1000 mm/min | 1500 mm/min | All Data |
---|---|---|---|---|

PCC mean | 0.94 | 0.93 | 0.83 | 0.96 |

PCC standard deviation | 0.05 | 0.03 | 0.11 | 0.01 |

Best covariance function | Mat 5/2 | Add | Add | SM |

Computation time in s | 28 | 203 | 179 | 4879 |

**Table 6.**Results for the prediction of the ultimate tensile strength R

_{m}when using process variables as input for the GPR model.

Welding Speed v_{s} | 500 mm/min | 1000 mm/min | 1500 mm/min | All Data |
---|---|---|---|---|

PCC mean | 0.99 | 0.91 | 0.93 | 0.99 |

PCC standard deviation | 0.02 | 0.08 | 0.06 | 0.01 |

Best covariance function | Mat 5/2 | RQ | Mat 5/2 | SM |

Computation time in s | 27 | 32 | 30 | 6374 |

**Table 7.**Results for the prediction of the ultimate tensile strength R

_{m}when using process parameters as inputs for the GPR model.

Welding Speed v_{s} | 500 mm/min | 1000 mm/min | 1500 mm/min | All Data |
---|---|---|---|---|

PCC mean | 1.00 | 0.88 | 0.94 | 0.99 |

PCC standard deviation | 0.00 | 0.12 | 0.05 | 0.01 |

Best covariance function | RBF | SM | RQ | SM |

Computation time in s | 625 | 807 | 23 | 5761 |

**Table 8.**Summary of the achieved correlation coefficients between the true and the predicted ultimate tensile strength.

Input Variable | Surface Topography | Process Variables | Process Parameters |
---|---|---|---|

PCC mean | 0.76 | 0.99 | 0.99 |

PCC standard deviation | 0.17 | 0.01 | 0.01 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hartl, R.; Vieltorf, F.; Benker, M.; Zaeh, M.F. Predicting the Ultimate Tensile Strength of Friction Stir Welds Using Gaussian Process Regression. *J. Manuf. Mater. Process.* **2020**, *4*, 75.
https://doi.org/10.3390/jmmp4030075

**AMA Style**

Hartl R, Vieltorf F, Benker M, Zaeh MF. Predicting the Ultimate Tensile Strength of Friction Stir Welds Using Gaussian Process Regression. *Journal of Manufacturing and Materials Processing*. 2020; 4(3):75.
https://doi.org/10.3390/jmmp4030075

**Chicago/Turabian Style**

Hartl, Roman, Fabian Vieltorf, Maximilian Benker, and Michael F. Zaeh. 2020. "Predicting the Ultimate Tensile Strength of Friction Stir Welds Using Gaussian Process Regression" *Journal of Manufacturing and Materials Processing* 4, no. 3: 75.
https://doi.org/10.3390/jmmp4030075