In this study, GA590 steel sheets with a thickness of 2.3 mm were cut into 150 × 300 mm specimens for welding. A solid wire with a diameter of 1.2 mm conforming to AWS A5.18 ER70S-3 was used. The chemical compositions and mechanical properties of both the base metal and solid wire are presented in Table 1.

To fabricate lap joints with varying weld geometries, joint positions (JP) were arranged in horizontal and vertical configurations, as illustrated in Fig. 1. The welding conditions were defined as shown in Table 2. Two welding modes (WM) were employed: constant voltage (CV) mode (DC, DM500, Daihen Co.) and Cold Metal Transfer (CMT) mode (TPS3200CMT, Fronius Co.). The CMT process enables high-speed and stable arc welding through precise control of welding current and wire feeding¹⁷⁾. For the CV mode, wire feed rates (WFR) of 3.0, 5.0, and 7.0 m/min were used in combination with welding speeds (WS) of 60 and 80 cm/min. In the CMT mode, wire feed rates of 5.0, 7.0, and 9.0 m/min were applied with the same welding speeds. The contact tip to work distance (CTWD) was set to 15 mm, and the work angle was maintained at 45°. A shielding gas mixture of 90% Ar and 10% CO₂ was supplied at a fixed flow rate of 25 L/min.

Fatigue specimens were prepared with reference to the ASTM E466 standard¹⁸⁾, as shown in Fig. 2. To reduce localized stress concentration and ensure repeatability in testing, spacers with the same thickness as the base metal were inserted into the lap joint. Radiographic inspection was conducted on all specimens, and only those free from internal defects such as porosity were selected for fatigue testing.

Fatigue tests were conducted using an Instron 8801 testing machine with a maximum load capacity of 100 kN. A sinusoidal cyclic load was applied for the testing. According to the welding conditions in Table 2, all welded specimens exhibited base metal fracture during tensile-shear testing, confirming a minimum tensile-shear strength of at least 610 MPa. The maximum stress (σ_max) in the fatigue tests was set to begin at 60% of this tensile-shear strength (i.e., 610 MPa), and then decreased in 10% increments down to 20% (122 MPa). The stress ratio (R = σ_min/σ_max) was fixed at 0.1, and the loading frequency was maintained at 20 Hz. While the fatigue endurance limit for conventional S-N curves is typically defined at 2 × 10⁶ cycles, components such as automotive chassis parts have varying fatigue life requirements depending on function and load conditions. Because fatigue testing up to 2 × 10⁶ cycles is time-intensive, the fatigue life in this study was evaluated for cycles exceeding 10⁵.

Fig. 3 presents the S-N curves obtained under the welding conditions listed in Table 2. The x-axis represents the fatigue life (N) until failure, while the y-axis indicates the maximum stress (σ_max) under cyclic loading. Each S-N curve is labeled according to the following condition order: lap joint direction (H: horizontal, V: vertical) / welding mode (DC, CMT) / wire feed rate / welding speed.

Under the same welding mode and stress level, an increase in wire feed rate and a decrease in welding speed tended to result in longer fatigue life. This can be attributed to a greater volume of deposited metal with higher wire feed rates and slower welding speeds, which alleviates stress concentration and improves fatigue life. Although no consistent trend was observed across all welding modes under identical wire feed and welding speed conditions, certain conditions clearly exhibited notable differences in fatigue performance.

Both the geometry and microstructure of the weld zone are known to affect fatigue properties¹⁾. In particular, for single-lap joints, weld geometry has been found to exert a more significant influence on fatigue behavior than microstructural changes⁴⁾. Numerous prior studies have reported that variations in weld toe geometry-especially toe angle-have a substantial impact on fatigue life^5-7). As a result, automotive OEMs monitor geometric weld dimensions such as toe angle, reinforcement height, leg length, weld throat, and penetration depth (as illustrated in Fig. 4) to ensure fatigue performance in chassis components subjected to cyclic loading.

Fig. 5 shows cross-sectional images of welds fabricated under the conditions in Table 2. Fig. 6 presents the measured weld geometric parameters based on the definitions provided in Fig. 4. Because weld bead geometry is influenced by the volume of deposited metal, the x-axis was normalized to deposition rate per unit length (g/cm) to analyze changes in weld geometry. Each graph compares the influence of joint orientation (Horizontal (H), Vertical (V)) and welding mode (DC, CMT). As the deposition rate per unit length increased, toe angle (α), leg length (l), weld throat (d), reinforcement height (r), and penetration depth (p) all showed a tendency to increase linearly. At equivalent deposition levels, differences in weld geometry were observed depending on joint orientation: toe angle, leg length, and weld throat were generally greater in horizontal joints than in vertical ones, while reinforcement height was typically larger in vertical joints. Additionally, in terms of welding mode, DC mode exhibited more pronounced geometric changes in response to deposition rate compared to CMT mode. This is likely due to transitions in droplet transfer behavior caused by wire feed rate variation in DC mode, making weld geometry more sensitive. Moreover, DC welds tended to exhibit deeper penetration than those produced via CMT.

Fig. 7 illustrates the relationship between weld geometry parameters and fatigue life (log(N)), with each graph representing data obtained under different maximum stress levels. All weld geometry parameters exhibited linear correlations with fatigue life. Specifically, an increase in toe angle tended to reduce stress concentration at the toe, thereby enhancing fatigue life. Longer leg lengths contributed to better stress distribution due to increased load-bearing area, and greater weld throat thickness improved structural strength through larger effective cross-sectional area. Reinforcement height helped smooth the stress flow and delay crack initiation, while sufficient penetration improved the structural integrity of the joint, suppressing fatigue crack propagation. Overall, each geometric parameter demonstrated a positive correlation with fatigue life.

The Pearson correlation coefficients between weld geometry factors and fatigue life (log(N)) at different σ_max levels are summarized in Table 3. The correlation strength varied by stress level and geometric factor. Among them, reinforcement height (r) showed the highest correlation with fatigue life, with coefficients exceeding 0.91 under all stress conditions. In contrast, toe angle (α) and weld throat thickness (d) exhibited relatively lower correlation values but still had a statistically significant impact on fatigue life. These results suggest that fatigue life sensitivity to geometric factors depends on the loading condition, and that weld geometry parameters are key design variables for fatigue life prediction.

In this study, five types of regression models were selected and applied as machine learning techniques: Linear Regression, Decision Tree, Support Vector Machine (SVM), Ensemble Regression, and Gaussian Process Regression (GPR)¹⁹⁾.

Linear Regression is the most fundamental regression method that predicts output values based on a linear relationship between input and output variables. It offers advantages such as ease of implementation and computational efficiency. However, its ability to explain complex data structures is limited since it approximates relationships using a single straight line. To overcome this limitation, extended models such as Interaction Linear Regression, Stepwise Linear Regression, and Robust Linear Regression have been developed to improve prediction performance while maintaining the basic structure of linear regression.

The Decision Tree is a non-parametric regression method that repeatedly splits the dataset based on input variables and makes predictions using the outcomes of these branches. It has high interpretability and intuitiveness, as it can incorporate nonlinearities and interactions between variables without explicit equations. However, depending on the depth of the tree, issues such as overfitting or underfitting may arise. To address this, various tree depths-such as Fine Tree, Medium Tree, and Coarse Tree-can be applied and compared for their prediction performance.

SVM is a kernel-based regression method that maps input data into a high-dimensional feature space and constructs a hyperplane with the maximum margin in that space. It maintains high predictive accuracy even for highly nonlinear data, and the choice of kernel function affects the model’s complexity and expressiveness. Kernel functions compute inner products in high-dimensional space to transform nonlinear problems into linear ones, thereby enhancing predictive performance. Commonly used kernel functions include Linear, Quadratic, Cubic, Fine Gaussian, Medium Gaussian, and Coarse Gaussian.

Ensemble Regression combines multiple individual predictive models to achieve greater stability and accuracy than any single model. It is particularly effective for mitigating high variance and overfitting problems. Representative methods include Bagged Trees and Boosted Trees.

GPR is a non-parametric regression method based on Bayesian inference, which models the relationship between input and output as a probabilistic function distribution. Through covariance structures defined by kernel functions, GPR can reflect both local and global characteristics of the data. A notable feature is its ability to provide confidence intervals along with predicted values. Common kernel functions used in GPR include Squared Exponential, Matern 5/2, Exponential, and Rational Quadratic.

In this study, regression models were developed to predict the fatigue life of lap GMAW joints using weld geometry information. Model training and evaluation were conducted using the Regression Learner app in MATLAB R2024b’s Statistics and Machine Learning Toolbox, and the prediction performance of the different regression methods was compared and analyzed.

As shown in Fig. 3, a total of 360 fatigue life data points were used to develop the regression models. Of these, 277 data points (80%) were used as the training set, while 72 data points (20%) were used as the validation set to evaluate the training performance. The input variables consisted of maximum fatigue stress, toe angle, leg length, weld throat, reinforcement height, and penetration depth. The output variable was defined as fatigue life. To reduce the differences in value ranges among the variables and to improve the model’s learning efficiency and predictive performance, Min-Max normalization was applied to the input variables, and a log transformation was applied to the output variable (fatigue life).

The predictive performance of each regression model was evaluated using two metrics: Mean Absolute Percentage Error (MAPE) and the Coefficient of Determination (R²). MAPE quantifies the relative error between predicted and actual values as a percentage, with lower values indicating higher prediction accuracy. It is defined as follows in Equation (1):

Here, y_i is the actual fatigue life, ŷ_i is the predicted fatigue life, and n is the number of data points.

The coefficient of determination, R², represents how well the regression model explains the variance in the dependent variable. Ranging from 0 to 1, higher values indicate better model fit. It is defined as shown in Equation (2):

Here, the numerator is the residual sum of squares (RSS), the denominator is the total sum of squares (TSS), and ȳ is the mean of the actual fatigue life values.

Fig. 8 and Table 4 present the MAPE and R² values of the regression models based on the validation dataset. Overall, linear regression models showed limited predictive performance due to their structural simplicity, yielding relatively low R² and high MAPE compared to other models. In the decision tree family, predictive performance varied significantly with tree depth. The Fine Tree model demonstrated relatively strong performance by effectively capturing nonlinearity and variable interactions through a more granular structure. In contrast, the Coarse Tree model, with a simplified structure, exhibited reduced prediction accuracy. Among the SVM models, prediction performance varied depending on the kernel function. Nonlinear kernels such as Quadratic, Cubic, and Fine Gaussian enabled effective learning of complex boundary conditions in high-dimensional feature spaces, resulting in low prediction errors. However, the Coarse Gaussian kernel, due to its excessively broad bandwidth, failed to capture detailed data structures and only learned general trends, leading to higher prediction errors. In the ensemble regression category, the Bagged Tree model demonstrated stable and consistent performance across the board. On the other hand, the Boosted Tree model, while designed to iteratively reduce errors, showed a tendency for decreased performance due to over-sensitivity to accumulated residuals during training. GPR models exhibited the highest predictive performance among all methods, consistently achieving high R² values and low MAPE. Notably, the Exponential kernel GPR model recorded an R² of 0.9983 and a MAPE of 0.51, marking the best performance in terms of both accuracy and error reduction.

These differences in regression performance reflect the structural characteristics and learning mechanisms of each model. Linear regression models assume a purely linear relationship between inputs and outputs, limiting their ability to capture complex nonlinear patterns and resulting in relatively high prediction errors. While decision trees partially address variable interactions via branching structures, shallow trees can produce discrete and less accurate predictions. In contrast, kernel-based models such as SVM and GPR effectively learn complex boundary conditions in high-dimensional feature spaces, resulting in low prediction errors and reduced variance. In particular, GPR offers the structural advantage of quantifying prediction uncertainty, which contributed to its superior performance in both R² and MAPE.

Residual analysis is a fundamental statistical procedure used to evaluate the fit and reliability of regression models. It provides intuitive insights into how well a model captures the underlying structure of the data. In an ideal regression model, residuals are randomly distributed with constant variance around a mean of zero, exhibiting no systematic patterns with respect to predicted values or input variables. These distributional characteristics of residuals are useful in diagnosing issues such as overfitting or underfitting, the model’s ability to capture nonlinearities, and its effectiveness in learning variable interactions²⁰⁾.

Fig. 9 visualizes the residual distributions for each regression model. In the linear regression category (Fig. 9(a)), a V-shaped pattern was observed, where residuals were larger at low and high predicted values and smaller near the center. This pattern reflects the structural limitations of approximating nonlinear fatigue life data with a single straight line.

In the decision tree category (Fig. 9(b)), the Fine Tree model exhibited a relatively stable residual distribution within ±0.1 across the entire prediction range. In contrast, the Medium Tree and Coarse Tree models displayed discrete clusters of residuals concentrated around specific predicted values. This tendency was especially pronounced in the Coarse Tree model, where the restricted number of decision nodes resulted in repeated output values.

For the SVM models (Fig. 9(c)), the Linear SVM showed a residual pattern similar to that of the linear regression model. However, the Quadratic and Cubic SVM models demonstrated more stable prediction behavior without significant bias or excessive residual variance, indicating that the higher-order kernel structures effectively captured the nonlinear characteristics of the fatigue life data. Among the Gaussian kernel-based models, the Fine and Medium Gaussian SVMs showed stable performance, while the Coarse Gaussian model, due to its overly broad kernel bandwidth, failed to distinguish fine variations in the input data. As a result, it displayed a V-shaped residual pattern similar to that of the Linear SVM.

In the ensemble regression category (Fig. 9(d)), the Boosted Tree model showed a tendency to underpredict, consistently yielding predicted values lower than the actual fatigue life. This behavior may result from the model’s sensitivity to specific regions during iterative weight adjustments to correct previous residuals. On the other hand, the Bagged Tree model produced a more evenly distributed residual pattern, reflecting a stable and unbiased prediction trend. This can be attributed to the bagging algorithm, which trains multiple tree models using bootstrap samples and averages their outputs to effectively reduce variance among individual models.

For the GPR models (Fig. 9(e)), all kernel types exhibited residuals uniformly distributed around a mean of zero, and the discrepancies between predicted and actual values were significantly smaller than those observed in other models. When considering R², MAPE, and residual analysis together, GPR emerged as the most suitable model for fatigue life prediction. In particular, the Exponential kernel-based GPR model showed the best performance, with the lowest MAPE (0.53) and the highest R² value (0.9980). The Matern 5/2 and Rational Quadratic kernel models also demonstrated excellent predictive accuracy.

Fig. 10 presents a comaprison between the actual fatigue life and the predictions obtained using the Exponential GPR model. As presented in Section 3.2, the Exponential GPR model demonstrated the highest prediction accuracy and lowest error among all models. Therefore, it was used to compare predicted and actual fatigue life. Fatigue life was evaluated under selected welding conditions, and the predictions made using the Exponential GPR model showed a strong overall agreement with the experimental results. This indicates that the model effectively captured the complex nonlinear relationships between the weld geometry input variables and fatigue life.

Additionally, Table 5 presents the MAPE values for fatigue life predictions by the Exponential GPR model at various σ_max (MPa) levels. Under the H/DC/5.0/60 condition, the MAPE ranged from 0.05 to 6.14, while for the V/CMT/5.0/60 condition, it ranged from 0.15 to 2.62.