3.2. Calculation of Fractal Dimension of Cross-Sectional Curve of Joint Surface
As a measure of the irregularity of a complex shape, the fractal dimension reflects the effectiveness of the space occupied by the complex shape. Box-counting N(ε) was refer to the number obtained by covering the measured form with basic units. In the present work, using a scale ε measured N(ε) then the fractal dimension D can be calculated as follows:
Using the coordinate data of the 3D joint surface morphology obtained by the scanning sampling, the cross-sectional curves at different positions in the welding direction and the weld width direction could be obtained. The box-width transformation method [
19,
20] was used to calculate the fractal dimension of the curve. The calculation process is as follows.
The fractal curve was covered by a rectangle with a width of ε, as shown in
Figure 2. The height of the rectangle was determined by the difference between the coordinates in the thickness direction of the highest and lowest points of the fractal curve in the rectangle frame. The rectangle frame was moved step by step to completely cover the fractal curve, and the area of each frame was added to obtain the total area,
Sε. By changing the width of the rectangle to
εi systematically, the corresponding total area,
Sεi, could be obtained. The total area,
Sεi, was divided by
εi2 to obtain the box count:
where
L is the cross-sectional width,
Sε,j is the area of the
j-th rectangular frame with a width of
εi,
Zj is a [(
εi/0.1) + 1] × 1 matrix formed by the coordinate value in the thickness direction with (
εi/0.1) + 1 scanning points covered by the
j-th rectangular frame, and 0.1 is the scanning step.
According to Equation (2), a smaller width and height of the rectangle corresponds to a greater total number of rectangles required to cover the curve and a greater corresponding box count. However, regardless of the measurement method, it is impossible to achieve a width, ε, that tends to zero. Therefore, the measurement scale can only be continuously reduced to achieve a certain accuracy based on the measurement technology. In this study, the minimum width, ε, was taken as the scanning step length. Based on the obtained morphology, this value was far smaller than the joint size, so it met the accuracy requirements.
Under a rotational speed of 1500 rpm and welding speed of 50 mm/min, the fractal dimension (D
C) of 10 curves collected by 10 equal divisions in the direction of welding and the direction of the weld width were calculated using a MATLAB program. The average fractal dimension of 10 curves was taken as the fractal dimension of the joint surface in this welding direction. Taking the cross-sectional curve with a width of
x = 4 mm as an example, the distribution curve of
was plotted, and the slope of the linear region was used as the fractal dimension of the fractal curve.
Figure 3 shows the double logarithm relationship between the box count and the box width of the cross-sectional curve with a width of
x = 4 mm on the welding surface. There was a very strong linear correlation between
and
, indicating that the curve had fractal characteristics (See
Figure 3).
Table 1 lists the calculated fractal dimension of the 10 curves in the welding direction under the welding process parameters for a rotational speed of 1500 rpm and welding speed of 50 mm/min, and the fractal dimension (D
C) in this direction was 1.3141.
In this paper, the curves at
x = 1, 4, and 7 mm were randomly selected for fitting, and the curves at the three positions basically followed a normal distribution (
Figure 4):
where μ and σ represent the average value and standard deviation, respectively.
As shown in
Table 1, when
x = 1 mm, the fractal dimension of the curve was 1.3313, when
x = 4 mm, the fractal dimension of curve was 1.3232, and when
x = 7 mm, the fractal dimension of the curve was 1.3152. The fractal dimensions of the curves at the three positions were very similar.
Figure 4 and
Table 2 show that when
x = 1 mm, μ was 0.0744 and σ was 3.692; when
x = 4 mm, μ was 0.0882 and σ was 3.808; when
x = 7 mm, μ was 0.0945 and σ was 3.976. The average value, μ, and the standard deviation, σ, of the three positions were very similar, which is consistent with the results in
Table 1. This verified that the calculation process of the fractal dimension of the cross-sectional curve of the joint surface was correct.
3.3. Calculation of Fractal Dimension of Curved Surface of Joint Surface
The method of using the fractal dimension to quantitatively characterize the 3D morphology was similar to that of characterizing the fractal dimension of the cross-sectional curve of the joint surface. A rectangle frame with a width of ε was replaced by a square column with a bottom surface of ε × ε (
Figure 5). The height of the square column was determined by the difference between the coordinates in the thickness direction of the highest and lowest points of the scanning surface within the range of ε × ε, and the square column was moved step by step to completely cover the curved surface of the joint. The volumes of the square columns were added to obtain the total volume,
Vε. By systematically changing the width of the bottom of the square column to
εi, the corresponding total volume,
Vε, could be obtained. The total volume
Vε was divided by
εi to obtain the box count:
where
L1 and
L2 are the scanning width in the direction of the welding speed and welding width, respectively,
Vεi,m,n is the volume of the square column at the position of (
m,
n) with a bottom of
εi ×
εi.
Zm,n is the [(
εi/0.1) + 1] × [(
εi/0.1) + 1] order matrix formed by the coordinate values in the thickness direction of the scanning points within the coverage of this square column, and 0.1 is the scanning step length.
Under a rotational speed of 1500 rpm and welding speed of 50 mm/min, the fractal dimension of the joint surface (D
A) was calculated. The above process was conducted in MATLAB, and the
distribution curve was plotted. The slope of the linear part of the curve was taken as the fractal dimension of the joint surface. The calculated results are listed in
Table 3.
Figure 6 shows the double logarithm relationship between box count and box width under the welding process parameters. There was a very high linear correlation between
and
, indicating that the surface morphology had fractal characteristics, and the fractal dimension D
A was 2.438. This calculation result was correct and satisfied the relationship between D
A and D
C, which was D
A ≈ D
C + 1.
3.4. Influence of Welding Process Parameters on Fractal Dimension
To study the influence of the rotational and welding speeds on the fractal dimension of the joint surface, the rotational speed was set to 1000, 1500, and 2000 rpm, and the welding speed was set to 30, 50, and 70 mm/min. The fractal dimension D
C was used as the response surface value. The CCD method was adopted, and certain data were obtained from experiments. A multiple quadratic regression equation was used to fit the functional relationship between the factors and response value. The coding of the factor level is shown in
Table 4. The central composite model in the Design Expert software was used to build the response surface. The CCD of the response surface was obtained by two factor variables. The minimum and maximum coding values of the two factor variables were −1 and 1, respectively, and the middle value was 0. The coding factor level is shown in
Table 4, and the designed matrix and test results are shown in
Table 5.
According to
Table 5 and
Table 6, the independent variables affecting the response surface value of the fractal dimension in the test were the rotational speed
ω and welding speed
ν. For the two variables, the expression of the quadratic linear regression equation “
y” was as follows [
21]:
where
b0 is the intercept term,
bi is the linear term,
bij is the interaction term, and
xi and
xj are designed parameters. In this paper, the response value was the fractal dimension D, which was a function of the rotational speed
ω and the welding speed
ν. Equation (5) can be expressed as follows:
The rotational speed
ω and welding speed
ν of the designed parameters were selected as the
X and
Y coordinate axes, respectively. The following second-order polynomial equation was obtained through multiple regression analysis of the designed matrix and response values in
Table 6:
To better observe the relationship between the welding process parameters and fractal dimension, Equation (7) was fitted to obtain a 3D response surface, as shown in
Figure 7. The response surface could directly reflect the relationship between the welding process parameters and the fractal dimension, and the influence of the welding process parameters on the fractal dimension D
C could thus be obtained. As shown in
Figure 7, when the rotational speed was fixed, the fractal dimension D
C decreased with the increase in the welding speed; when the welding speed was fixed, the fractal dimension D
C decreased with the increase in the rotation speed.
The accuracy of the response surface is the basis for ensuring the effectiveness of the test design and response surface function, and it is necessary for conducting further analysis using the model. The accuracy of the response surface function and the significance of the selected designed parameters can be obtained through an accuracy test of the response surface, which is useful for judging whether the selected designed parameters are reasonable. The response surface model was analyzed using ANONA in the Design Expert software, and the results are shown in
Table 6. In
Table 6, the
F value and probability
P represent the significance of the correlation coefficient. A larger
F value and smaller
P value corresponded to a more significant correlation coefficient.
Table 6 shows that the fitting accuracy of the model was high, and the designed parameters, ω and ν, were very significant, which indicated that the selection of the designed process parameters was reasonable and could reflect the change of the fractal dimension. The interaction between ω and ν was of low significance, indicating that the designed parameters had little correlation. The
F value of ω was 364.86, which was higher than that of ν (198.09). The result indicated that the effect on the fractal dimension D of the rotational speed was more significant than the welding speed. The closer the coefficient of determination (
R2) of the response surface function was to 1, the higher the fitting degree [
14]. The coefficient of determination in this study was 0.9881, indicating that the fitting degree of response surface function was high.
To better observe the fitting accuracy of response surface, the correlation diagram and residual distribution diagram of the experimental and predicted values are given in
Figure 8 and
Figure 9, respectively.
Figure 8 shows that all fractal dimensions D were near the 45° diagonal, and the residual value was basically within the range of 0.005 (
Figure 9), indicating that the predicted value of the fitted response surface function was quite close to the actual value.