#### 2.1. Welding Experiments

The experiments were performed on a four-axis milling center adapted for FSW. Each weld had a length of 205 mm and combined two sheets of the aluminum alloy EN AW-6082-T6 in the butt joint configuration. The chemical composition of the used material was specified by the selected supplier Bikar Metalle GmbH (Bad Berleburg, Germany), as listed in

Table 1. The designation “

T6” implies that the material was solution heat-treated and then artificially aged [

20].

Each individual sheet had a dimension of 325 mm × 88.5 mm. The sheet thickness

t was 4 mm, as shown in

Figure 1.

The process forces in three spatial directions

F_{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.

The experiments were performed in position-controlled operation with an immersion depth of 0.1 mm and a tool tilt angle of 2°. The dwell time at the immersion point was one second. The welding speed

v_{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.

Figure 1 displays the areas of removal of the four slices (a)–(d) for the tensile specimens. Since four tensile specimens were taken from each of the 54 manufactured welds, a total of 216 tensile specimens were available. In order not to change the weld seam surface, the tensile specimens were prepared to the correct geometry for the tensile tests after scanning the surface topography of the 216 slices. The dimensions of the tensile specimens are illustrated in

Figure 3a. The topography of the welds was examined using a three-dimensional profilometer VR-3100 (Keyence Deutschland GmbH, Neu-Isenburg, Germany). Thereby, white LEDs projected light from two places onto the weld and the reflected light was measured by a CMOS sensor. The smallest measurable difference in the

z-direction, as shown in

Figure 1, was 1 µm. The sheet surface was always defined as the zero height. The distance between the individual topography points in the

x-

y-plane was approximately 24 µm. Consequently, a total of about 470,000 topography points were generated for the area (A), as shown in

Figure 3a, containing the weld seam on the 15-mm-wide tensile specimens.

The key indicators to quantify the flash formation and the weld seam width were calculated by using area (A), the key indicators for the seam underfill were specified by using area (B), and the arc texture formation was characterized along the weld centerline (C), as shown in

Figure 3b,c.

Figure 3b schematically shows the flash height

f, the seam underfill

u and the weld seam width

w for a section of the weld surface. The weld seam width

w was defined as the distance between the two peaks of the flash formation on the advancing side (AS) and on the retreating side (RS). Due to the distance of the topography points of approximately 24 µm, there were 625 sections of the weld’s topography for each of the 15-mm-wide tensile specimens. From the corresponding 625 values for the flash height

f, the seam underfill

u, and the weld seam width

w, the mean values for the flash height

f_{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}:

The theoretical number

n_{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:

In addition, for each tensile specimen the standard deviation

S_{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.

The signals of the nine different recorded process variables (F_{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.

A period of 7.5 weeks was scheduled between the welding process and the tensile tests. Based on the findings of Brenner et al. [

26], it was assumed that the metallurgical transformations were completed after this period. For the tensile test, a Z050 AllroundLine material testing machine (ZwickRoell GmbH & Co. KG, Ulm, Germany) was utilized. In addition to the 216 tensile specimens for the welds, ten tensile tests were conducted on specimens with the base material for reference. The geometry of the tensile specimens corresponded to the specifications of DIN 50125, Form E [

27]. All tensile tests were performed according to the standards DIN EN ISO 4136 [

28] and DIN EN ISO 6892-1 [

29]. According to the recommendation of the standard DIN EN ISO 6892-1 [

29], the test speed was set to 0.0067 1/s to determine the ultimate tensile strength.

Metallographic specimens were prepared to inspect the welds for internal defects. After taking the samples for metallography from the welded parts, as shown in

Figure 1, they were embedded in an epoxy resin, ground to a fineness of P1200, polished with a 3 µm diamond suspension, and then finely polished with colloidal silica. Finally, the samples were etched with Kroll’s etchant, which is described in Vander Voort [

30].

The values for the eight surface topography indicators, as shown in

Table 3, as well as the ultimate tensile strengths of the 216 tensile specimens, are given in the

Supplementary Materials to the article.

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

For the present work, the Gaussian process regression model, as described in

Appendix B, was applied. The experimental data were stored in a

D ×

q design matrix

X, where

q represented the number of observations and

D corresponded to the total number of features which are, for example, the mean flash height

f_{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.

Five different covariance functions were tested: RBF, RQ, Matérn 5/2, additive GPs, and SM. The covariance functions are described in

Appendix B. For the RBF, the RQ, and the Matérn 5/2 covariance function, the automatic relevance determination (ARD) variant was used (see Equation (A19) in

Appendix B). For the application of the additive GP according to Duvenaud et al. [

18], it was necessary to define the base covariance function and the maximum order up to which calculations should be performed. The RBF covariance function was employed, and the maximum order was set to the maximum number of dimensions

D. For the use of the SM covariance function as described by Wilson et al. [

19], the choice of an integer parameter

Q for the maximum number of mixture components was necessary. All possible models for a

Q from 1 to 20 were calculated and the model with the lowest mean absolute error (MAE) was chosen. In order to learn sensible model parameters, gradient-based optimization of the logarithmic marginal likelihood was performed. To avoid getting stuck in local optima, the process of model fitting was repeated ten times. In each iteration, the initial values of the hyperparameters were randomly set between e

^{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.

The performance measure ultimately utilized to evaluate the deviation between the predicted and the true ultimate tensile strength was the Pearson correlation coefficient (PCC) [

31]. Using the PCC proved to be the best way to compare the results obtained with the different input data for the model and the different covariance functions that were employed. As a benchmark for the GPR, a multi-linear regression (MLR) [

31] was conducted using the same input data, respectively. In the MLR, a division of the data set into 80% training data and 20% test data was implemented as for the GPR by using a fivefold cross-validation.

The computations were performed on the CPU, which was an Intel^{®} Core™ i7-6700HQ CPU at 2.60 GHz. The installed RAM was 16 GB.