© D. Zuo et al., Published by EDP Sciences 2026

1 Introduction

Friction stir welding (FSW), as an emerging solid-state joining technology [1], has broad prospects for applications in the connection of structural components such as aircraft skin panels, central wing boxes, and wings. However, Zuo et al. [2] found in the FSW-T joint welding process that the uneven heating of various regions of the T-joint can lead to a gradual decrease in temperature along the width of the weld nugget zone of the T-stiffened plate, resulting in complex structural changes and uneven distribution of residual stresses in the T-joint. Large integral panels of aircraft fuselages belong to the category of columnar panel forming, and CAF is one of the key technologies for the forming of such panels. CAF involves implementing creep and artificial aging simultaneously at a certain temperature to release as many residual stresses in the panel as possible, thereby achieving the forming and strengthening of the integral stiffened plate [3]. Compared to traditional forming methods such as rolling and shot peening, CAF specimens exhibit characteristics such as low residual stresses and good forming performance [4,5], presenting broad prospects in the manufacturing of large integral panels [6], especially in the lightweight of dissimilar alloy welded large integral panels.

However, there is a significant springback in CAF specimens after unloading, which seriously affects their forming accuracy. The introduction of UV can effectively solve the above-mentioned issue by not only reducing and homogenizing residual stresses within the material but also decreasing forming forces and improving forming quality [7]. Yang, Li et al. [8,9] have used UV to reduce residual stresses inside abrasive diamonds and deformation stresses in entropy alloys, enhancing their plastic strain. Liu and colleagues [10] conducted ultrasonic vibration-assisted micro-tension tests on T₂ copper foil, analyzed the fracture morphology and revealed the microscopic mechanism of the acoustic softening effect. To prepare ultrafine-grained Al 6061 alloy strips, Pi et al. [11] proposed a new process of axially ultrasonic vibration-assisted extrusion cutting, which resulted in better surface flatness and smoothness of chip samples compared to traditional cutting processes. For copper T₂ ultrasonic micro-extrusion experiments, Han et al. [12] designed a novel ultrasonic micro-extrusion process with a tool-workpiece composite vibration, finding that this process achieved better micro-extrusion forming characteristics. Furthermore, Zhou and colleagues [13] developed an ultrasonic vibration-assisted forming system and found that when the ultrasonic power reached 1.8 KW, it could reduce the yield stress of copper. Cao et al. [14] introduced an ultrasonic vibration plate device capable of achieving longitudinal full-wave and transverse half-wave (L₂T₁) vibration modes, which reduced the grinding force on Inconel 718 nickel-based superalloy and improved the surface quality. Currently, UV methods mainly focus on research on the same materials, and reports on dissimilar materials are rare. Additionally, Szpunar [15] used a central composite design method to construct a second-order quadratic model to determine the optimal input parameters for optimizing single-point incremental forming processes. Dillip et al. [16] established an optimization model based on multi-objective genetic algorithms (MOGA) to improve the forming rate in machining processes, reduce costs, and minimize product scrap. It can be observed that in any machining process, optimizing process quality through process parameter optimization methods [17] is another effective way to achieve the best and most efficient outputs.

Therefore, this paper innovatively proposes a composite forming method suitable for the UVCAF process of dissimilar FSWed T-stiffened plates. Firstly, a single-objective polynomial response surface model is constructed using the central composite design method (CCD), and the rationality of the model is analyzed using analysis of variance (ANOVA) and residual analysis methods. Secondly, an improved algorithm combining the adaptive multi-objective genetic algorithm (AMOGA) and the entropy weighting method (EWM), named AMOGA-EWM, is used to perform multi-objective optimization on the aforementioned single-objective model. Experimental verification is conducted on self-made UVCAF specialized equipment. Furthermore, a comparison of the macroscopic and microscopic aspects of UVCAF and CAF experimental results is carried out to demonstrate the accuracy and reliability of the multi-objective optimization model. It provides a reference for the high-precision, high-performance integrated manufacturing of lightweight aircraft's large integral panels.

2 Materials and experimental methods

2.1 Specimens and forming device

In this experiment, pre-strained 7055-T6 [18] Al alloy was selected as the skin, with dimensions of 200 × 80 × 1.5 mm, and 2197-T8 Al-Li alloy was used as the stiffener, with dimensions of 200 × 5 × 13 mm. The chemical composition and room temperature properties of the alloys are shown in Tables 1 and 2. The large integral panel of the aircraft fuselage is mainly composed of skin and stiffeners (see Fig. 1a), with the FSWed T-stiffened plate equivalent to a single stiffener specimen cut along the X direction of the columnar panel (see Fig. 1b). Physical images are shown in Figures 1c and 1d. It should be emphasized that when the columnar panel is bent, the stiffeners aligned in the axial direction can be ignored [19].

The UVCAF experiment requires ensuring that the vibration waves are effectively transmitted to the T-stiffened plate inside the thermal environment in the form of longitudinal waves. Therefore, in this experiment, UV was introduced into the forming die directly below the tooling during CAF. The forming die was fixed and installed inside a temperature-controlled chamber with pillar 4, and mould 7 was connected to the external vibration source via special screw 6. The specialized fixture for fixing the vibration is illustrated in Figure 2a, and the vibration is shown in Figure 2b. The ultrasonic vibration device consists of a transducer, an amplifying bar, a vibration transmission bar, and a vibration fixture, as shown in the schematic in Figure 3a, with the forming device depicted in Figure 3b.

Due to the small amplitude generated by the transducer, a 20 kHz ultrasonic frequency can only produce a few microns of amplitude. To increase the mechanical vibration applied to the tooling, an ultrasonic amplitude adjuster is connected between the end face of the transducer and the vibration fixture to achieve the functions of energy focusing and impedance matching [20]. Therefore, this experiment utilizes a self-made three-stage composite amplifying bar with an exponential transition section to enhance the output performance, as shown in the structure and coordinate system in Figure 4.

The dimensions marked in the figure are: l₁ = 27.6 mm, l₂ = 67.4 mm, l₃ = 27.6 mm, l = 122.6 mm, Φ₁ = 60 mm, Φ₂ = 10 mm, Φ₃ = 82.1 mm. It is worth emphasizing that the amplitude adjuster in the above figure is made of TC4 titanium alloy material, where section I and section III are rods with equal circular cross-sections, section II is a rod with an exponential variable circular cross-section, and sections I and II are integral in structure and connected to section III by a special screw.

Fig. 1 Geometry of (a) integral panel and its Simplified (b) T-stiffened plate, and the views of (c) front and (d) back for the mentioned previously T-stiffened plate.

Fig. 2 Illustrations of the (a) fixed vibrator tooling and (b) vibrator.

Fig. 3 Schematic of (a) ultrasonic wave transmission principle and views of (b) forming device.

Fig. 4 Geometry of the amplifying bar.

2.2 Multi-objective optimization of process parameters

2.2.1 Modeling and analysis

During the UVCAF experiment, phenomena such as creep, stress superposition, ultrasonic softening, and work hardening are observed, with a significant impact of UV on the forming quality of the specimens. When subjected to high amplitude vibration, not only does it increase the internal stress of the material, but it also reduces the material's softening effect, hinders its plastic flow ability, and consequently reduces its strength; prolonged vibration leads to deterioration of the viscoplasticity of the aged tensile specimens [21]. Therefore, only appropriate vibration energy can maximally release the residual stresses in the material, enhancing its viscoplastic forming capability. To improve the forming accuracy and performance of the T-stiffened plate, creep temperature x₁, creep time x₂, and amplitude x₃ were selected as design factors, while forming rate Y₁, tensile strength Y₂, and elongation Y₃ were set as optimization objectives. A multi-objective optimization method was employed to obtain the optimal combination of process parameters to enhance the forming rate and mechanical performance of the T-stiffened plates. Initially, the CCD method was used to construct a continuous variable response surface model [22,23], evaluate the impact factors of the objectives and their interactions, define the optimal level ranges, and design the UVCAF experimental results based on the CCD method, as shown in Table 3. It is noteworthy that for ease of calculation, the optimization objectives (Y₁, Y₂, and Y₃) were reciprocally transformed (Y₁', Y₂', and Y₃') to avoid errors caused by magnitude differences among the reciprocals of the objectives. The transformed Y₁' and Y₃' were magnified by 10⁴ times, while Y₂' was magnified by 10⁶ times. Furthermore, a three-factor quadratic regression model was used to fit the optimized process parameters, resulting in a fitted second-order response surface model:

$$\overline{Y}={\overline{b}}_{\text{0}}\text{+}{\sum }_{\text{i=1}}^{\text{k}}{\overline{b}}_{\text{i}}{\text{x}}_{\text{i}}\text{+}{\sum }_{\text{i}<\text{j}}{b}_{\text{ij}}{\text{x}}_{\text{i}}{\text{x}}_{\text{j}}\text{+}{\sum }_{\text{i=1}}^{\text{k}}{\overline{b}}_{\text{ii}}{\text{x}}_{\text{i}}^{\text{2}}.$$(1)

In the equation, ${{\displaystyle \overline{b}}}_0$, ${{\displaystyle \overline{b}}}_{\text{i}}$, ${{\displaystyle \overline{b}}}_{\text{ij}}$, ${{\displaystyle \overline{b}}}_{ii}$ are all constant terms. Let x_i = x_i, x_i² = x_i+k, x_k−1¬x_k = x_k(k+1)/2,${{\displaystyle \overline{b}}}_{\text{i}}$=${{\displaystyle \bar{b}}}_{\text{i}}$, ${{\displaystyle \overline{b}}}_{\text{ij}}$=${{\displaystyle \bar{\text{b}}}}_{\text{i+k}}$, ${{\displaystyle \overline{b}}}_{(\text{k1})\text{k}}$=${{\displaystyle \overline{b}}}_{\text{k}(\text{k+3})\text{/2}}$, n = 0.5k(k+3), where i takes natural numbers from {1,2,..., k}, then formula (1) can be simplified as:

$$\overline{\text{Y}}={\overline{b}}_{\text{0}}\text{+}\underset{\text{i=1}}{\sum ^{\text{n}}}{\overline{b}}_{\text{i}}{\text{x}}_{\text{i}}.$$(2)

In the equation, ${{\displaystyle \overline{b}}}_0$, ${{\displaystyle \overline{b}}}_{\text{i}}$(i = 1,…, n) are undetermined coefficients. To determine these coefficients, the number of experimental trials m needs to satisfy the relationship m ≥ n+1. Let Y^(j) be the response value obtained after m trials, and the response surface function value calculated through equation (2) is denoted as Y^(j), then

$${{\displaystyle \overline{Y}}}^{\left(j\right)}={{\displaystyle \overline{b}}}_0+{\displaystyle {{\displaystyle \sum }}_{i=1}^n}{{\displaystyle \overline{b}}}_i{x_i}^{\left(j\right)}$$(3)

where j takes natural numbers from {1, ..., m}, and the ${{\displaystyle \overline{Y}}}^{\left(j\right)}$calculated through equation (3) is an approximate function of the response value Y^(j). The error value ε_es between the two can be expressed as:

$${\epsilon }_{es}=Y^{\left(j\right)}-{{\displaystyle \overline{b}}}_0-{\displaystyle {{\displaystyle \sum }}_{i=1}^n}{{\displaystyle \overline{b}}}_i{x_i}^{\left(j\right)}.$$(4)

The above formula clarifies the form of the response surface function. However, to solve for the undetermined coefficients in the formula, the least squares principle must be used to minimize the sum of squares of ε_es to determine the most suitable coefficients ${\displaystyle \overline{b}}$ in the formula. First, express formula (1) in matrix form as:

$$\overline{Y}={\overline{b}}_{\text{0}}\text{+}{\text{x}}^{\text{T}}\text{Kx+}{\text{x}}^{\text{T}}\text{L}.$$(5)

In the equation, x^T= (x₁, …, x_k), L^T= (${{\displaystyle \overline{b}}}_1$,…,${{\displaystyle \overline{b}}}_{\text{k}}$), K is a symmetric matrix of order k, as follows:

$$\text{K}=\left[\begin{array}{ccc}\hfill {{\displaystyle \overline{b}}}_{11}\hfill & \hfill \cdots \hfill & \hfill {{\displaystyle \overline{b}}}_{1k}\text{/2}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {{\displaystyle \overline{b}}}_{k1}\text{/2}\hfill & \hfill \cdots \hfill & \hfill {{\displaystyle \overline{b}}}_{kk}\text{/2}\hfill \end{array}\right]$$(6)

Derivation of equation (5) yields:

$${{\displaystyle \overline{Y}}}^{\text{'}}=-L+2Kx.$$(7)

Let equation (7) be equal to 0, the stable point x_s of the quadric surface can be solved, and substitute x_s into equation (3), the predicted response value ${{\displaystyle \overline{\text{Y}}}}_{\text{s}}$ at the stable point can be obtained, the expression is as follows:

$$x_s=L/2K$$(8)

$${{\displaystyle \overline{\text{Y}}}}_s={{\displaystyle \overline{b}}}_0+0.5{x_s}^{T}L.$$(9)

Further apply the least squares algorithm to find the estimated coefficients ${\displaystyle \overline{b}}$ in the matrix K, so that ${\displaystyle \overline{\text{Y}}}$=f(x_k) is optimized. If the matrix x^Tx is non-singular, then only the following equation (5) condition needs to be satisfied by ${\displaystyle \overline{b}}$ to obtain the mathematical model of the second-order response surface:

$$F={\displaystyle {{\displaystyle \sum }}_{j=1}^m}{({\epsilon }_{\mathrm{es}})}^2\to \mathit{min}.$$(10)

Based on the CCD experimental design results in Table 3, the second-order response surface model is fitted using the least squares method to obtain the unknown coefficients in equation (1). Subsequently, regression models for the responses Y₁', Y₂', and Y₃' with respect to x₁, x₂, and x₃ are established, as shown in Table 4.

ANOVA is a statistical model that can analyze the mean differences between multiple sets of data [24]. The test statistic F is defined as the ratio of the sum of squares between groups to the sum of squares within groups. The p-value is the density function value of the F value. The significant relationship between the optimization factors (x₁, x₂, and x₃) and the target variables (Y₁, Y₂, and Y₃), as well as regression model information, is shown in Tables 5 and 6. Generally, the smaller the p-value, the more significant the impact of the factor on the target variable. From Table 5, it can be observed that Y₁ and Y₂ have the relationship p(x₁) < p(x₂) < p(x₃), while Y₃ is arranged in the order p(x₁) < p(x₃) < p(x₂). Therefore, it can be inferred that x₁ and x₂ have a greater impact on Y₁ and Y₂, while x₁ and x₃ have a greater impact on Y₃. Additionally, from Table 6, it is noted that the p-values of the regression terms for Y₁', Y₂', and Y₃' are <0.05, indicating that the constructed response surface model rejects the null hypothesis and the regression model is overall significantly effective. The lack of fit terms for Y₁', Y₂', and Y₃' have p-values (>0.05), showing that the model accepts the null hypothesis and the probability of lack of fit between predicted and observed values does not exist. The correlation coefficients R₂ (>85%) for Y₁', Y₂', and Y₃' are close to the adjusted coefficients, indicating a good overall fit of the model. The adjusted coefficients R_2Adj (>83%) for Y₁', Y₂', and Y₃' imply a high level of confidence in the predictive performance of the model. Furthermore, the analog signal-to-noise ratio is far greater than 4, indicating that the model has sufficient discrimination power. In conclusion, the models constructed for Y₁', Y₂', and Y₃' exhibit high accuracy and reliability.

To further analyze the fitting effect of the model, the goodness of fit of the response surface models for Y₁′, Y₂^′ and Y₃^′ as well as the residual normal probability plots, are shown in Figure 5. From Figure 5a, 5b, and 5c, it can be observed that the predicted values of Y₁′, Y₂^′ and Y₃^′ are distributed on both sides of the 45° angle bisector line with the corresponding true values. Although there is a slight deviation from the ideal scenario where they should overlap and lie on the diagonal line, the deviation is small and can be ignored. Additionally, the residuals of Y₁′, Y₂^′ and Y₃^′ are very close to the fitted regression line, with the most dense and abundant probability density at the center of the residuals, indicating a normal distribution of residuals and a good fit of the regression model. Therefore, this model can be used to predict the changes in the forming accuracy and mechanical properties of T-stiffened plates during the UVCAF process.

Based on the ANOVA results for Y₁, Y₂, and Y₃ in Table 5, it is known that x₁ and x₂ have a more significant impact on Y₁ and Y₂, while x₁ and x₃ have a more significant impact on Y₃. The control of the forming characteristics of the T-stiffened plates by x₃ is more complex, requiring consideration of the variations of x₁ and x₃ on Y₃ at different x₂ conditions. Therefore, to explore the interaction effects between the target variables and factors, contour plots and three-dimensional surface plots of the linear regression models for Y₁', Y₂', and Y₃' are shown in Figures 6–8.

Observing Figure 6a, it is found that Y₁' decreases first and then increases with increasing x₁, decreases with increasing x₂, and reaches a minimum value (located in the deep blue area at 120) when the change intervals of x₁ and x₂ are [175 °C, 195 °C] and [6 h, 15 h], respectively. It can be seen that excessively high creep temperature hinders the improvement of material formability because high temperatures result in coarse material structure, reduced strength, and weakened material plasticity. In the UVCAF process, Y₁' is more significantly influenced by x₁ compared to x₂, as shown in Figure 6b. By using Minitab to capture the optimal forming parameters corresponding to the most significant Y₁' for the T-stiffened plate [178.2 °C, 9.6 h, 13.8 μm], Y₁' is minimized to 118.71, corresponding to Y₁ at 84.25%. To obtain the confidence interval of the points on the regression model, the parameter combination [178.2 °C, 9.6 h, 13.8 μm] is substituted into the response surface model, resulting in a 95% predicted response of Y₁' at 118.85. Comparing the predicted fitted value with the corresponding optimization result shows good consistency, indicating that the regression model Y₁' is highly reliable.

From Figure 7a, it can be observed that when the amplitude x₃ = 12.53 μm, Y₂' shows a decreasing trend initially followed by an increasing trend with increasing x₁ and x₂. When the change intervals of x₁ and x₂ are [143 °C, 175 °C] and [5.5 h, 12.5 h], respectively, Y₂' reaches a minimum value located in the deep blue area (< 2320 MPa), while Y₂ reaches its maximum value. This is because as x₁ or x₂ increases, a large amount of strengthening phases η^′ and θ^′ precipitate within the microstructure, strengthening the alloy matrix, resulting in an increase in material tensile strength. However, when x₁ and x₂ are too large, the material undergoes over-aging, leading to a decrease in the quantity of η^′ or θ^′ phases, an increase in the size of some phases, and a decrease in density, causing Y₂ to decrease.

The interaction effects of x₁ and x₂ have a significant impact on Y₂' of the stiffened plate after UVCAF treatment, as shown in Figure 7b. Using Minitab software, it is determined that when the process parameters are [159.1 °C, 8.9 h, 12.4 μm], Y₂' is minimized and most significant at 2270.10. Substituting these values into the response surface model captures a 95% predicted response of Y₂', with the fitted value at 2271.93. A comparison reveals a high level of consistency between the predicted fitted value and the response optimization result, effectively validating the high reliability of the regression model Y₂'. The contour plots and three-dimensional surface plots of the response surface model for Y₃' are shown in Figure 8.

When x₂= 6 h, x₁ ranges from [150 °C, 175 °C], and x₃ is less than or equal to 8 μm, the minimum value of Y₃' ranges within 1380, as shown in the contour plot in Figure 8a. When x₂ = 8 h, x₁ ranges from [148 °C, 178 °C], and x₃ is not greater than 8 μm, the minimum value range of Y₃' is not greater than 1260, as depicted in the contour plot in Figure 8b. The contour plot in Figure 8c reveals that when x₂ = 10 h, x₁ ranges from [150 °C, 173 °C], and x₃ is not greater than 7.3 μm, the minimum value range of Y₃' does not exceed 1200. Furthermore, from the three-dimensional surface plots in Figures 8a, 8b, and 8c, it is evident that the effects of x₁, x₃, and their interaction on the Y₃' model are significant. When x₂ is 6 h, 8 h, and 10 h, Y₃' shows a decreasing trend followed by an increasing trend with increasing x₁ and x₃, with x₁ having a more significant impact on Y₃' compared to x₃.

Utilizing Mintab analysis tools, it is found that Y₃' reaches its minimum value at 1176.16 when x₁ = 161.4 °C, x₂ = 10 h, and x₃ = 7.1 μm, corresponding to Y₃ at 8.5%. To obtain the confidence interval of the points on the regression model, the process combination [161.4 °C, 10.0 h, 7.1 μm] is input into the variables of the response surface design, capturing a 95% predicted response of the regression model Y₃' at 1187.49. The consistency between the predicted fitted value and the response optimization result validates the high reliability of the regression model Y₃'.

In summary, when x₁ ranges from [140 °C, 185 °C], x₂ ranges from [6 h, 12 h], and x₃ ranges from [8.86 μm, 14.01 μm], the R₂ values of the response surface models for Y₁', Y₂', and Y₃' are all greater than 85%, approaching 100%. This indicates that the above-mentioned models have a high level of accuracy and reliability globally, providing a good predictive effect on the functional relationships. The results obtained by inputting the process parameters corresponding to the response indicators into the response surface models are shown in Table 7. From Figures 6–8 and Table 7, it is observed that when each response indicator is minimized, the forming process parameters are not consistent. Additionally, under the same process parameters, one response indicator may be optimal while the others are not, illustrating a typical optimization problem where multiple response indicators within a given process range interact and conflict, requiring simultaneous optimization to achieve the best results. In essence, the above method demonstrates that the established response surface models can only meet single-target optimization and cannot fulfill the requirements of multi-target optimization.

Fig. 5 Fit degrees of response surface models and residual normal probability plots for (a) Y1′, (b) Y2′ and (c) Y3′.

Fig. 6 (a) Contour and (b) three-dimensional surface plots for the response surface model Y1′.

Fig. 7 (a) Contour and (b) three-dimensional surface plots for the response surface model Y2′.

Fig. 8 Contour and three-dimensional surface plots for the response surface model Y3′ at (a) 6 h, (b) 8 h and (c) 10 h.

2.2.2 Multi-objective optimization via AMOGA-EWM

The essence of multi-objective optimization problems lies in the mathematical relationships formed by multiple objective functions and related equality and inequality constraints, where each objective cannot be optimized simultaneously. To obtain an effective solution set, different weights must be assigned to the objectives [25]. Since the objective functions in Table 4 are all minimization objective functions, a generic functional form with m objectives and n-dimensional design variables can be used to describe the multi-objective optimization problem of the model as follows:

$$\{\begin{array}{c}\hfill \text{min}F(x)={\left[f_1{(x)\mathrm{,f}}_2(x),\cdots f_m(x)\right]}^T\hfill \\ \hfill \mathrm{s.t.}{g}_i(x)\le 0,i=\mathrm{1,2,3}\dots ,p\hfill \\ \hfill h_i(x)=0,j=\mathrm{1,2,3}\dots \mathrm{,q}\hfill \end{array}.$$(11)

In the equation, f₁(x)∈R^m,{i = 1, …, m} represents m-dimensional objective functions, g_i(x),{i = 1, …, p} denotes p inequality constraint functions, h_i(x),{j = 1, …, q} signifies q equality constraint functions, x∈R_n, {x₁, …, x_n} represents n-dimensional design variables, and X = {x|x ∈Rⁿ, g_i(x) ≥ 0, i = 1, …, p, j = 1, …, q} defines the feasible region of equation (11).

The UVCAF process falls within the category of metal material bending processing, with a highly nonlinear and complex process involving multiple target parameters. The relationship between process parameters and the forming quality of T-stiffened plates is nonlinear, time-varying, and highly coupled, making it challenging to establish an accurate mathematical model that describes the relationship between process parameters and T-stiffened plate quality. Therefore, the essence of multi-objective optimization is to search for solutions that closely approach the Pareto optimal solution set [26].

MOGA has become a primary method for solving Pareto front solutions due to its strong pattern recognition, adaptive control, and parameter optimization advantages [27], while EWM ensures the influence of experimental factors on the experimental indicators and the interaction between various experimental indicators [28]. AMOGA-EWM, as one of the favored algorithms for multi-objective optimization, enhances solution diversity through real number encoding and decoding, utilizes gene position mutation operators to obtain excellent individuals, and can achieve a more evenly distributed Pareto optimal solution set with strong stability and adaptability. Therefore, this study will modify the regression models in Table 4 according to equation (11), and then utilize the AMOGA-EWM algorithm to search for the multi-objective response surface models and conduct training until generating satisfactory weight values. The corresponding objective functions and constraints are as follows:

$$\{\begin{array}{c}\hfill \text{min}Y\text{'}=(Y\mathrm{\_1}\text{'}(x),Y\mathrm{\_2}\text{'}(x),Y\mathrm{\_3}\text{'}(x));\hfill \\ \hfill x=\left[x_1{,}_x{,}_x\right]\hfill \\ \hfill s\text{.}t\text{.}140\le x_1\le 185;6\le x_2\le 12;8.86\le x_3\le 14.01\hfill \end{array}$$(12)

Using the AMOGA-EWM algorithm to optimize the regression models Y₁', Y₂', and Y₃', the improved minimum distance selection method is employed to capture the optimal forming process parameters from the Pareto solution set [29]. The flowchart of AMOGA-EWM is shown in Figure 9.

Firstly, input the objective functions and constraints; secondly, form new excellent individuals through crossover and mutation genetic operators; then, eliminate the poorest subpopulations, retaining the best individuals in each genetic iteration; finally, repeat the optimization process calculation according to conditions [30].

In the AMOGA-EWM algorithm, the population size (k) and the number of iterations (n) have a significant impact on obtaining the Pareto optimal solution set [31]. To enhance computational accuracy, the selection of parameters in the multi-objective optimization algorithm needs to be reasonable. In the algorithm operation, the population size is set to k = 300, as a larger population size increases the likelihood of finding the global solution. When iterating n = 100, the overall distribution is highly concentrated, indicating that n = 100 meets the optimization requirements. The comprehensive evaluation model for multi-objective optimization quality is derived as follows:

$$Z=f\text{'}(x)=0.4862\text{'}\text{Y\_1}+0.3617Y\_2\text{'}+0.1521Y\mathrm{\_3}\text{'}$$(13)

Where f'(x) is the reciprocal of ${\displaystyle \bar{\text{Y}}}$= f(x), and the multi-objective optimization design results are shown in Table 8.

Fig. 9 The flowchart of multi-objective optimization with AMOGA-EWM algorithm.

2.2.3 Optimization results and confirmation tests

The optimal processing parameters mentioned above were validated in the UVCAF experiment, and the experimental results are shown in Figure 10. Upon observation of Figure 10, it is noted that the UVCAF experimental result (Y_1exp = 79.6%) aligns well with the AMOGA-EWM computational result (Y_1algo = 80.9%).

Further analysis reveals that the predicted results by the AMOGA-EWM algorithm (Y₁, Y₂, Y₃) are slightly larger by 1.3%, 1.4 MPa, and 0.2% compared to UVCAF, but they still meet engineering practical applications (5%). This indicates that capturing the optimal forming process parameters of the UVCAF experiment using the AMOGA-EWM algorithm is feasible and effective, while also demonstrating the high credibility and good predictive effect of the established response surface regression model.

Fig. 10 Comparison of T-stiffened plate performance under UVCAF and AMOGA-EWM algorithm.

2.3 Experiment design and measurement procedure

To study the effects of UVCAF and CAF under the optimal process parameters on the forming rate and mechanical properties of T-stiffened plates, the plates underwent corresponding processing and tensile tests. The process parameters under different conditions are shown in Table 9. The physical image of the T-stiffened plates after UVCAF processing is shown in Figure 11a. To accurately obtain the curvature radius after springback, a 3D scanner was used to reconstruct the three-dimensional geometric model of the T-stiffened plates, as shown in Figure 11b. Following the “Welded Joint Tensile Test Method” standard [32], the fracture conditions of the tensile specimens under different conditions are shown in Figure 11c, where #1, #2, and #3 correspond to specimens with no treatment, CAF treatment, and UVCAF treatment, respectively. The fractures of all specimens occurred in the nugget zone (NZ), and the fracture surfaces of the three samples were at a 45° angle to the direction of tension. In terms of microscopic characterization, scanning electron microscopy (SEM) was used to observe the fracture morphology, and JEM-2000CX transmission electron microscopy (TEM) was employed to analyze the grain interior and grain boundaries of NZ in the T-stiffened plates.

Fig. 11 Views of (a) T-stiffened plate treated by UVCAF, (b) the forming stiffened plate by 3D scanning and (c) the fracture specimens of T-joints in different working conditions.

3 Experimental results and discussion

3.1 Forming rate and mechanical properties

The forming rate Y₁ and the springback rate (SP) have a complementary relationship [17]. According to equation (14), the calculation formula for Y₁ can be expressed as:

$$\{\begin{array}{c}\hfill Y_1=1-SP\hfill \\ \hfill PP={\mathrm{100}(R}_f-R_0{)\mathrm{/R}}_f\hfill \end{array}$$(14)

Where R₀ = 400 mm. The results of the forming rate and mechanical properties obtained after testing the specimens under different conditions are shown in Figure 12. A comparison reveals that under CAF, the curvature radius of the T-stiffened plates is 539.1 mm, with Y₁ at 74.2%; while under UVCAF, the curvature radius is 502.5 mm, with Y₁ at 79.6%. Compared to the CAF experiment, the Y₁ in the UVCAF experiment increased by 5.4%, indicating a better forming effect. This is because the softening effect caused by UV reduces the steady-state creep strain rate angle, increases the steady-state creep limit, converts more elastic deformation into plastic deformation, and releases residual internal stress more thoroughly, resulting in less springback and a larger Y₁ for the T-stiffened plates.

Furthermore, it can be seen from the graph that after the UVCAF experiment, Y₂ and Y₃ are the highest, being 12.3 MPa and 0.82% larger than after the CAF experiment, respectively, and the smallest without UVCAF. This is because the stiffened plate without UVCAF treatment undergoes grain crushing under the stirring needle action, and the dynamic recrystallization caused by stirring heat makes the grains smaller and more uniform, while there is uneven stress distribution in the NZ, leading to reduced plasticity and strength [33]. In contrast, the stiffened plate after CAF treatment produces certain precipitation phases due to creep aging, enhancing the strength and plasticity at the weld seam [34]. For the stiffened plate treated with UVCAF, the stress superposition caused by UV can soften the material, enabling more elastic deformation to transform into plastic deformation, releasing internal stress more effectively and improving plasticity. From the perspective of fracture mode, due to the combined effects of creep aging and UV, the NZ forms more numerous and smaller dimples, and more equilibrium phases are precipitated on the matrix and grain boundaries, resulting in secondary strengthening of the material [35].

In conclusion, whether it is Y₁, Y₂, or Y₃, the formability control effect of UVCAF under the AMOGA-EWM algorithm is better than that of CAF. It further reveals that under the given experimental process conditions, the AMOGA-EWM optimization algorithm can achieve comprehensive optimization of Y₁, Y₂, and Y₃ after UVCAF, fully validating the effectiveness and feasibility of applying this method to predict composite forming processes.

Fig. 12 Forming rates and mechanical properties of T-stiffened plate under different working conditions.

3.2 Tensile fracture analysis

After the specimens from the CAF and UVCAF experiments were pulled apart, the fracture surfaces were scanned and observed to provide a deeper explanation of the quantitative relationship between the mechanical properties of the T-stiffened plates and their microstructural organization. Figure 13 shows the fracture morphology at different magnifications under SEM for different conditions.

From Figure 13a, it can be seen that the fracture surface of the specimen after the CAF experiment is mainly composed of uneven dimples, with larger pits containing fractured secondary phase particles and accompanied by multiple tearing edges, as well as the presence of varying shear steps (see Fig. 13b), indicating that the primary fracture mode of this alloy is toughness fracture caused by the secondary phase. This is because the second phase distributed in the matrix has a large volume and an extremely irregular shape. Under tensile stress, deformation is relatively concentrated at the second phase, leading to a decrease in local plastic deformation capability, resulting in cracking at the second phase and matrix grain boundaries, forming microcracks that extend along the edges of the second phase grains, leading to the appearance of dimples [36]. Observing Figure 13c, it is found that with the introduction of UV, the number and depth of equiaxed dimples on the fracture surface of the specimen after the UVCAF experiment are increased. From the high-magnification electron micrograph (see Fig. 13d), it is evident that there are residual secondary phase particles at the bottom of the dimples, along with noticeable tearing edges. The increased number of tearing edges and large dimples indicates that the fracture mode of the specimen treated with UVCAF is also a toughness fracture caused by the secondary phase [35,37]. The reason may be that the softening effect generated by UV leads to an increase in the precipitation of strengthening phases in the creep aging specimen, dispersed within the matrix, acting as pinning reinforcements.

Additionally, UV enhances the accumulation of dislocation energy at the grain boundaries, leading to the formation of effective dislocation tangles that hinder dislocation slip, thereby increasing the material strength. It can be observed that the greater and more numerous the deep dimples, the more tearing edges and longer they are, and the smaller the shear step difference, the higher the strength and better the plasticity at a macro level, consistent with the changes in macroscopic tensile properties.

Fig. 13 Tensile fracture morphologies of (a) 500x and (b) 1000x for CAF, and (c) 500x and (d) 1000x for UVCAF.

3.3 TEM analysis

To further investigate the microstructural changes in the NZ of the T-stiffened plates after CAF and UVCAF experiments, Figure 14 displays the bright-field microstructure of the NZ after the CAF and UVCAF experiments. The precipitation of phases within the grains significantly affects dislocation movement. The finer and more uniformly dispersed the precipitates, the stronger the hindrance to dislocation movement, leading to higher alloy strength and hardness. The release sequence of precipitates in 7055 alloy and 2197-T8 alloy is as follows: α (supersaturated solid solution) → GP zone → ή phase (MgZn₂) → η phase (MgZn₂), α → GP zone + δ' → T₁ + δ' + θ', ή phase is metastable and the main source of strengthening the aluminum base. Plate-like η phase is an equilibrium phase, and when it predominates, the alloy strength decreases [38].

As shown in Figure 14a, the alloy material after the CAF experiment contains numerous dislocations, along with a significant amount of ή phase, a small amount of η phase, and a large quantity of oblique intersecting needle-shaped precipitates of the T₁ phase. This is because creep promotes the formation of dislocations in the aluminum matrix and increases the dislocation density in the NZ through creep strain. During the creep aging process, dislocations on either side of the dislocation plane lead to stacking faults, dislocation loops, and tangled dislocations. Stacking faults provide favorable nucleation positions for the non-uniform nucleation of the T₁ phase, and the significant precipitation of T₁ leads to a reduction or even disappearance of the θ' phase. The thermal cycling during the CAF process causes the internal precipitates of the alloy to coarsen gradually. Some metastable ή phases transform into equilibrium phases η. As seen in Figure 14b, the precipitates at the grain boundaries exist in a continuous arrangement, with smaller sizes. With prolonged thermal cycling time, the size of the ή phase at the grain boundaries decreases significantly, but there is some precipitation of the η phase with larger sizes compared to those within the grains, and the presence of a pronounced PFZ near the grain boundaries is not evident. It can be concluded that the main reinforcement of NZ comes from the combination of the T₁ phase and ή phase, leading to enhanced strength and plasticity.

Since creep promotes the formation of dislocations and stacking faults provide nucleation sites for the T₁ phase, while defects such as dislocation loops and tangled dislocations provide superior nucleation regions for the ή phase, specimens subjected to UV creep aging have a small amount of T₁ phase, a large amount of ή phase, and a small amount of equilibrium phase η within the grains, as shown in Figure 14c. Overall, the alloy is still primarily reinforced by the T₁ and ή phases, with ή phase being the most prevalent, indicating that the performance of the NZ after UVCAF is determined by the ή phase. This phenomenon is attributed to the fact that dislocations generated during creep aging are caused by vacancy damage. Excessive vacancies can accelerate the diffusion rate of precipitates in the alloy, speeding up the precipitation behavior. Meanwhile, under the influence of creep aging, the ή phase gradually grows and coarsens. However, under the influence of UV, the ή phase is subdivided into numerous small precipitates, dispersed throughout the matrix, thereby enhancing the alloy performance. Additionally, the stress relaxation effect caused by ultrasonic softening during the creep steady-state stage can also increase the alloy strength [39].

From Figure 14d, it can be seen that with the application of UV, the precipitates at the grain boundaries become smaller, appearing in a discontinuous arrangement, and are uniformly dispersed. The larger precipitates at the grain boundaries transform from a disc shape to elongated plate-like shapes, without any significant presence of a PFZ at the grain boundaries. This is because the grain boundaries have higher dislocation energies, providing superior sites for the nucleation and growth of precipitates. The introduction of UV subdivides the grown precipitates into numerous small grains, dispersing them uniformly within the matrix, thereby improving the alloy's performance. From the above analysis, it is evident that effective UV treatment helps enhance the mechanical properties of creep-aged alloys. In contrast, alloys without UV treatment tend to produce a greater amount of large and coarse η phases in the aluminum matrix after creep aging, leading to a decrease in alloy strength and plasticity.

Fig. 14 TEM images of precipitation phases for (a) grain interior and (b) grain boundary under CAF, for (c) grain interior and (d) grain boundary under UVCAF.

4 Conclusions

This paper reports a forming method that combines ultrasonic vibration and age forming (UVCAF) aimed at achieving high quality of forming dissimilar, friction stir welded 7055-T6/2197-T8 T-stiffened plates. Firstly, a single-objective polynomial response surface model is constructed, and then the AMOGA-EWM algorithm is proposed for the multi-objective model. The optimal process plans obtained from the algorithm are used for UVCAF and CAF experiments. By means of SEM and TEM microscopic methods, the forming control effects of T-stiffened plates under different conditions are compared and analyzed. The main conclusions are as follows:

• By combining variance, fit and residual normal probability plots, it is verified that the single-objective response surface models Y₁′, Y₂′, and Y₃′ constructed based on the CCD method have high accuracy and good fit. When x₁ is [140 °C, 185 °C], x₂ is [6 h, 12 h], and x₃ is [8.86 μm, 14.01 μm], the R₂ values of the three-dimensional response surface models Y₁′, Y₂′, and Y₃′ are all greater than 85%, indicating that the single-objective models have good predictive performance but cannot meet the requirements of multi-objective optimization.

• The UVCAF process demonstrates strong potential for industrial application, particularly in aerospace manufacturing for large-scale, integral panel components (such as wingskins and fuselage panels) with high precision, reduced springback, and improved mechanical properties. For the UVCAF process of T-stiffened plates, an innovative multi-objective optimization quality comprehensive evaluation model based on the AMOGA-EWM algorithm is proposed. The optimal combination of process parameters is determined to be [164.8 °C, 9.2 h, 9.6 μm]. The algorithm-predicted results are only 1.3%, 1.4 MPa, and 0.2% larger than the UVCAF experimental results, demonstrating good consistency. The multi-objective optimization model established for predicting the UVCAF process of T-stiffened plates is reliable.

• The values of Y₁, Y₂, and Y₃ of T-stiffened plates under the UVCAF process are 5.4%, 12.3 MPa, and 0.82% larger than those under the CAF process, indicating that the softening effect produced by UV increases the steady-state creep limit of the material. This leads to more elastic deformation being converted into plastic deformation. More precisely, the specimens treated by UVCAF exhibit less springback, stronger plasticity, and better release of internal residual stresses.

• Microscopic analysis reveals that the fracture mode of specimens under both CAF and UVCAF is a ductile fracture caused by the second phase, with the fracture position located in the NZ. The main reinforcement in the NZ under CAF and UVCAF is the combination of the T₁ phase and the ή phase. However, with the introduction of UV, the ή phase is divided into numerous small precipitated strengthening phases, which further improves the forming accuracy and performance of the CAF specimens.

Acknowledgments

This work was supported by Key Laboratory of Friction Welding Technologies (KLFWT), which is managed by Engineering Department in University of Tokyo, Japan. The authors are also grateful to the CAFUC’s Analysis and Test Center for a test techniques and equipment.

Funding

The China Postdoctoral Science Foundation (NO.2022M722592), the Sichuan Province Engineering Technology Research Center of General Aircraft Maintenance (NO. GAMRC2023ZD03), and the Fundamental Research Funds for the Central Universities (NO.25CAFUC04018).

Data availability statement

All data generated or analyzed during this study are included in this article.

Conflicts of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Author contribution statement

All authors contributed equally to this work.

Additional information

Correspondence and requests for materials should be addressed to Duquan Zuo.

References

All Tables

All Figures

	Fig. 1 Geometry of (a) integral panel and its Simplified (b) T-stiffened plate, and the views of (c) front and (d) back for the mentioned previously T-stiffened plate.
In the text

	Fig. 2 Illustrations of the (a) fixed vibrator tooling and (b) vibrator.
In the text

	Fig. 3 Schematic of (a) ultrasonic wave transmission principle and views of (b) forming device.
In the text

	Fig. 4 Geometry of the amplifying bar.
In the text

	Fig. 5 Fit degrees of response surface models and residual normal probability plots for (a) Y1′, (b) Y2′ and (c) Y3′.
In the text

	Fig. 6 (a) Contour and (b) three-dimensional surface plots for the response surface model Y1′.
In the text

	Fig. 7 (a) Contour and (b) three-dimensional surface plots for the response surface model Y2′.
In the text

	Fig. 8 Contour and three-dimensional surface plots for the response surface model Y3′ at (a) 6 h, (b) 8 h and (c) 10 h.
In the text

	Fig. 9 The flowchart of multi-objective optimization with AMOGA-EWM algorithm.
In the text

	Fig. 10 Comparison of T-stiffened plate performance under UVCAF and AMOGA-EWM algorithm.
In the text

	Fig. 11 Views of (a) T-stiffened plate treated by UVCAF, (b) the forming stiffened plate by 3D scanning and (c) the fracture specimens of T-joints in different working conditions.
In the text

	Fig. 12 Forming rates and mechanical properties of T-stiffened plate under different working conditions.
In the text

	Fig. 13 Tensile fracture morphologies of (a) 500x and (b) 1000x for CAF, and (c) 500x and (d) 1000x for UVCAF.
In the text

	Fig. 14 TEM images of precipitation phases for (a) grain interior and (b) grain boundary under CAF, for (c) grain interior and (d) grain boundary under UVCAF.
In the text