Optimization of Multilayer Metal Bellow Hydroforming Process with Response Surface Method and Genetic Algorithm

Abstract

In this paper, an optimization strategy for the hydroforming process of bellows is proposed, based on finite element analysis, design of experiments, response surface methodology, and genetic algorithms. A numerical model of the bellows hydroforming process is developed using the finite element simulation code ABAQUS and validated experimentally. A combination of experimental design, numerical simulations, and regression analysis is employed to establish the mathematical models relating the objectives to the design variables. An analysis of variance (ANOVA) is conducted to evaluate the significance of each individual factor on the response variable. The main and interaction effects of the process parameters on the outer diameter and convolution pitch are illustrated and discussed. Furthermore, the response surface methodology and a Pareto-based multi-objective genetic algorithm (MOGA) are applied to determine optimal solutions within the given optimization criteria. The optimized results show good agreement with the experimental data, demonstrating that the optimization methodology is reliable.

1. Introduction

Metal bellows are a kind of thin-walled shell of revolution with a corrugated profile along the meridian. They are widely used in various fields, such as piping systems, power systems, micro-electromechanical and automotive industries, owing to their exceptional ability to compensate for thermal expansion, contraction, and mechanical vibrations during operation [1,2,3]. Bellows can be classified into single-layer and multilayer structures based on their construction. Compared to single-layer bellows, multilayer bellows (see Figure 1) demonstrate higher pressure resistance, enhanced flexibility and improved fatigue life. These advantages make them ideal for extreme service environments, such as reactor coolant pump seals and liquid rocket engine propellant lines. To meet increasing requirements of high performance and low cost, the hydroforming process has become an advanced and preferred technique for forming multilayer bellows over recent decades [4].

A typical bellows hydroforming process, as illustrated in Figure 2 [5], consists of four main stages:

• During the initial stage, the tube is positioned with both ends sealed, and a series of annular fin dies are fitted equally spaced along its outer surface.
• In the bulging stage, the tube is slightly expanded under internal pressure and secured between the annular fin dies.
• Throughout the folding stage, the tube is shaped into a bellows profile through the combined application of internal pressure and axial compression.
• Finally, during the unloading stage, springback occurs as the internal pressure and axial force are released.

In bellows hydroforming process, defects such as excessive thinning, cracking and insufficient bulging are likely to occur if the process and die parameters are not properly designed. These challenges are especially critical in multilayer bellows, where interlayer slippage during forming intensifies geometric nonlinearity. This will lead to material flows difficulty and stress concentration, thereby increasing the susceptibility to defects such as wrinkling and cracking. To produce defect-free multilayer bellows with precise dimensional accuracy, a proper optimization of both process parameters and die geometry is necessary.

Many researchers have studied the bellows hydroforming process and investigated the influence of process parameters on the product by experiments and finite element (FE) simulations. Faraji et al. [6,7,8] demonstrated that the formed U-shaped bellows are sensitive to internal pressure, axial movement, and die stoke. Lee [9] identified die spacing as the dominant factor affecting the final profile of the U-shaped bellow. Yuan et al. [10] investigated the reinforced S-shaped bellows hydroforming by FE simulation and concluded that internal pressure has the most significant influence on the forming results. These studies collectively indicate that internal pressure and die spacing are the primary influencing factors in the hydroforming of both U-shaped and S-shaped bellows. Several studies have further investigated the profile evolution and springback behaviour of bellows during hydroforming. Liu et al. [11] studied the springback characteristics of single- and bi-layered bellows and observed that the intended U-shaped profile deformed into a tongue-like shape after springback. To compensate for this effect and achieve the desired U-shaped bellows, Liu et al. [12] suggested that the fin die should be designed with a water-drop-shaped profile to account for springback.

To obtain a bellow with the desired shape, both process and die parameters must be taken into account. It remains challenging to derive a universal approach or parameter combination from the existing literature, where studies often focus solely on either process parameters or die parameters. Thus, an optimization algorithm is necessary. Thus, an optimization algorithm is necessary. Typically, optimization approaches such as response surface method (RSM), radial basis function (RBF), kriging (KRG), artificial neural network (ANN) and support vector regression (SVR) are widely adopted for parameter optimization in hydroforming coupled with finite element analysis (FEA). Alaswad et al. [13] studied the effects of loading paths on the hydroforming of a bi-layer T-shaped tube using FE simulation and RSM. The prediction formulas of the formed bellow profile, including convolution height, wall thinning, and wrinkling height were established. A similar methodology was also employed by Raut et al. [14] and Chen et al. [15] to optimize loading paths. The Taguchi method and RSM were applied by Rajaeea et al. [16] and Feng et al. [17] to optimize the process parameters in the hydroforming of stepped tubes and Y-shape tubes, respectively. Additionally, Abbassi et al. [18] adopted an ANN approach coupled with FE simulation for loading path optimization.

Determining the optimal process and die parameters in bellows hydroforming constitutes a multi-objective problem (MOP) involving objectives such as convolution height, pitch, and wall thickness [19]. Several researchers have addressed this using a multi-objective genetic algorithm (MOGA), for its ability to optimize multiple dependent objectives simultaneously in a single run. Bahloul et al. [20] developed an integrated approach combining FE simulation, RSM, and a MOGA to optimize forming process parameters. Sharma et al. [21] utilized artificial neural networks (ANN) and genetic algorithms (GA) for the optimization of tube-to-tubesheet joint forming processes. Han and Kim [22] employed ANN along with a Pareto-based genetic algorithm to enhance the multistage deep drawing process. Darabi et al. [23] adopted both full-factorial and GA methods to identify the optimal combination of bilayer materials that achieve both high formability and minimal weight. Despite these advances, limited research has been conducted on the multi-objective optimization of both process and die parameters specifically in the context of bellows hydroforming.

In this study, a multi-objective optimization framework integrating FE simulation, design of experiments, response surface methodology, and genetic algorithm is proposed for the hydroforming process of bellows. The primary contribution of this work lies in the establishment of an efficient and integrated virtual optimization system specifically designed for bellows hydroforming. This system effectively combines Box–Behnken experimental design with analysis of variance (ANOVA)-based model validation and MOGA optimization. In contrast to prior research that has largely focused on conventional tube hydroforming, this study addresses the distinctive forming mechanisms and process characteristics of metal bellows, thereby enhancing both optimization accuracy and computational efficiency. The paper is structured as follows: Section 2 presents the development of the FE model for the bellows hydroforming process. Section 3 describes the DOE approach and formulates the multi-objective optimization problem. Section 4 details the development of the RSM models, evaluates their significance and reliability using ANOVA, and identifies the optimal solution through MOGA. Experimental validation of the optimization results is also presented.

2. FE Modelling

In this study, a three-layered U-shaped Incoloy 825 bellow is selected. The schematic and dimensions of the bellow are shown in Figure 1 and

Table 1. Specification of the bellow.

Parameters	Values
Outer diameter, D (mm)	106
Inner diameter, d (mm)	85
Convolution height, h (mm)	9.6
Convolution pitch, q (mm)	6.9
Convolution width, a (mm)	4.3
$\mathrm{Radius} \mathrm{of} \mathrm{crest}, r_{out}$(mm)	2.15
$\mathrm{Radius} \mathrm{of} \mathrm{trough}, r_{in}$(mm)	1.3
$\mathrm{Wall} \mathrm{thickness} \mathrm{of} \mathrm{each} \mathrm{layer}, t_0$(mm)	0.3
Number of layers, z	3

. The FE model is developed using ABAQUS software (ABAQUS 6.10) and illustrated in Figure 3. The key modelling techniques are summarized as follows:

• According to the symmetric character of the bellow, a 2D axisymmetric model is used.
• Due to large deformation and dynamic contact, the explicit algorithm is suitable and adopted for the bulging and folding stages, while the implicit algorithm is employed for the springback stage.
• Determination of the internal pressure. The internal pressure p is calculated by Equation (1).

$$p={\displaystyle \frac{2zt{\sigma }_b}{d}}$$

(1)

where z is the number of layers, t is the initial tube thickness, ${\sigma }_b$ is the ultimate tensile strength, and d is the inner diameter.

• Determination of the developed length of a single convolution. The developed length of a single convolution L₀ refers to the length of a complete bellows segment when flattened along its neutral layer. Numerically, it equals the sum of the arc lengths of the crest and trough, plus the lengths of the two straight sections between the crest and trough. The expression for L₀ is given as follows:

  $$L_0=D-d+\left(\pi -2\right)\left(r_{out}+r_{in}\right)-2t$$

  (2)

  where D is the outer diameter, r_out is the radius of crest, and r_in is the radius of trough.

• Determination of spacer block height. In hydroforming, adjacent annular dies are separated by spacer blocks with a height of H_d, and the calculation formula for H_d is as follows:

  $$H_d=L_0-H_m$$

  (3)

  where H_m is the annular die thickness and equals to convolution pitch, as shown in Figure 4.

• During the bulging stage, the internal pressure increases uniformly from 0 to the specified value; during the folding stage, the internal pressure remains constant. The loading path is shown in Figure 5.
• The movement of the dies are controlled by displacement constraints. During the bulging stage, all dies remain stationary. In the folding stage, while one end die is fixed, the opposite end die and the middle dies move axially at a constant velocity until complete closure is achieved (shown in Figure 5).
• The contact between the tube, die, and every two adjacent layers are described with the surface-to-surface contact algorithm. The Coulomb friction coefficients are set as 0.1 for tube-die interfaces and 0.3 for tube-tube contact surfaces.
• The tubes are meshed with the four-node bilinear axisymmetric quadrilateral element (CAX4R), while two-node linear axisymmetric rigidity element RAX2 is used in meshing the rigid dies.
• In the springback model, the workpiece is imported into the Part module. Meanwhile, a predefined field is created in the initial step to import the stress field.

The true stress–strain curves of the Incoloy825 sheet are obtained by uniaxial tensile test. The flow stress is approximated by the Swift law presented in Equation (4) and the mechanical properties are shown in

Table 2. Mechanical properties of the tube material.

Parameters	Values
Young’s modulus, E (GPa)	206
Poisson’s ratio, υ	0.3
Density, ρ (kg/m3)	7850
Yield strength, σs (MPa)	241
Ultimate strength, σb (MPa)	659
Elongation percentage, e (%)	32
Strength coefficient, K (MPa)	1779.21
Strain hardening exponent, n	0.51
Strain constant, ε0	0.061
Normal anisotropic exponent, r	0.68

.

$$\overline{\sigma }=K{\left(\overline{\epsilon }+{\epsilon }_0\right)}^n$$

(4)

where K is the strength coefficient, n is the hardening exponent, and ε₀ is the strain constant.

3. Optimization Scheme

An optimization method is presented and illustrated in Figure 6, comprising the following steps:

• Step 1: Select the design variables and optimization objectives.
• Step 2: Design virtual experiments using the Box–Behnken Design (BBD) method.
• Step 3: Perform numerical simulations according to the BBD matrix and extract the resulting sample data.
• Step 4: Establish a RSM based on the sample data.
• Step 5: Evaluate the accuracy and reliability of the RSM through ANOVA.
• Step 6: determine the optimal solution by applying the MOGA.

3.1. Design of Variables and Objectives

In the bellows hydroforming process, as illustrated in Figure 2 and Figure 4, the internal pressure (p), die angle (θ), die fillet radius (R_d) and die thickness (H_m) are the dominant parameters influencing the formed bellows profile. The upper bound of the internal pressure is determined based on Equation (1), while the lower bound is set to 60% of the calculated pressure. The design of the die angle, fillet radius, and thickness primarily accounts for the effect of springback. After springback, the cross-section of the bellows changes from U-shaped to a tongue-shaped profile, accompanied by an increase in both the convolution pitch and the valley radius. Therefore, compensatory adjustments are incorporated into the die design. Based on empirical experience combined with simulation results, the die angle is set within 0° to 4°, the die fillet radius is selected between 60% and 100% of the trough radius, and the die thickness is chosen in the range of 75% to 100% of the convolution pitch. The specific ranges of the parameters are provided in

Table 3. Ranges of variables in optimization.

Variables	θ (°)	Rd (mm)	Hm (mm)	p (MPa)
Range	[0, 4]	[0.78, 1.3]	[5.16, 6.9]	[8.38, 13.96]

.

The convolution height (relates to the outer diameter) is a paramount installation dimension, while the pitch is the key geometric determinant for performance metrics like compensation capacity, as per standards such as the Standards of the Expansion Joint Manufacture Association (EJMA). During hydroforming, these two parameters are directly defined and constrained by the die geometry, making their variation a direct indicator of process stability and repeatability. Therefore, the outer diameter (D) and the convolution pitch (q) are selected as the response parameters in this study. Due to springback after unloading, which results in a decrease in outer diameter and an increase in convolution pitch, the outer diameter and convolution pitch of the bellows are measured after springback.

According to EJMA standard [24], the manufacturing tolerances for convolution height and convolution pitch of the bellows are specified as ±0.79 mm (corresponding to D = 106 ± 1.58 mm) and ±1.59 mm (corresponding to q = 6.9 ± 1.59 mm), respectively.

3.2. Design of Experiments

The BBD was adopted for virtual simulation experiments of bellows. This design offers efficient estimation of second-order effects while avoiding extreme experimental points, thereby improving numerical stability. Compared to full factorial designs, BBD requires significantly fewer experimental runs while maintaining comparable resolution. In contrast to space-filling designs such as Sobol or Latin Hypercube, BBD is more suitable for building explicit quadratic response surface models. This makes it particularly ideal for multi-parameter optimization studies. In this study, a four-factor, three-level BBD was employed, with the levels of each factor set at −1, 0, and 1. The design variables and their corresponding levels are summarized in

Table 4. The levels of factors for numerical simulations.

Variables	Description	−1	0	1
θ (°)	Die angle	0	2	4
Rd (mm)	Die fillet radius	0.78	1.04	1.30
Hm (mm)	Die thickness	5.16	6.03	6.90
p (MPa)	Internal pressure	8.38	11.17	13.96

. All statistical analyses and response surface plots were performed using MATLAB software (MATLAB R2016b). A total of 29 numerical simulations were conducted, all of which results in formed bellows free from defects such as wrinkling or rupture. The outer diameter ($D$) is determined by measuring the radial distance from the peak of the outermost convolution to the tube centerline, while the convolution pitch ($q$) is obtained from the axial distance between adjacent peaks of the outermost corrugations. The measured results are presented in

Table 5. Design matrix with coded independent design variables and numerical results.

Number	θ (°)	Rd (mm)	Hm (mm)	p (MPa)	$\mathit{D}$ (mm)	$\mathit{q}$ (mm)
1	2	1.30	6.03	8.38	105.23	7.16
2	2	1.04	6.03	11.17	105.50	7.17
3	2	1.04	6.03	11.17	105.50	7.17
4	2	1.04	6.03	11.17	105.50	7.17
5	2	0.78	5.16	11.17	105.23	7.16
6	0	1.30	6.03	11.17	105.75	7.14
7	4	1.30	6.03	11.17	105.25	7.29
8	0	1.04	5.16	11.17	106.26	5.99
9	2	1.04	5.16	8.38	105.63	6.15
10	4	1.04	6.90	11.17	104.75	8.34
11	0	1.04	6.03	8.38	105.43	7.21
12	2	1.04	6.03	11.17	105.50	7.17
13	2	0.78	6.03	13.96	105.96	6.67
14	4	1.04	6.03	8.38	104.97	7.35
15	2	1.30	6.03	13.96	105.91	7.17
16	0	0.78	6.03	11.17	105.56	7.46
17	2	1.04	6.90	8.38	104.64	7.88
18	2	1.04	6.03	11.17	105.50	7.17
19	4	0.78	6.03	11.17	105.20	7.31
20	2	1.30	6.90	11.17	105.02	8.32
21	2	1.04	6.90	13.96	105.39	7.90
22	4	1.04	6.03	13.96	105.80	7.03
23	2	0.78	6.03	8.38	105.08	7.33
24	0	1.04	6.03	13.96	106.22	6.67
25	2	1.30	5.16	11.17	106.12	5.78
26	2	0.78	6.90	11.17	105.04	7.98
27	4	1.04	5.16	11.17	105.69	6.07
28	2	1.04	5.16	13.96	106.36	6.01
29	0	1.04	6.90	11.17	105.32	7.93

.

3.3. Establishment of RSM

In the paper, the second-order polynomial response surface model is used to approximately describe the objective function. This form of model is widely used in modelling metal forming processes due to its ability to capture both main and interaction effects between process variables. A second-order model is preferred over a first-order model to account for potential mild nonlinearities that are commonly present in hydroforming processes. Moreover, the second-order model offers improved predictive capability while maintaining computational efficiency, providing a balanced approach between model accuracy and complexity. The second-order regression polynomial equation is as follows:

$$y=F\left(x\right)={\beta }_0+{\displaystyle \sum {\beta }_i{x}_i+{\displaystyle \sum {\beta }_{ii}{x}_{ii}^2}}+{\displaystyle \sum {\beta }_{ij}{x}_i{x}_j+\epsilon }$$

(5)

where β are polynomial coefficients; ε is minor error; x_i and x_j are the design variables, and y is the response value.

Equation (5) can be expressed in matrix expression as follows:

$$Y=X\beta +\epsilon$$

(6)

where

$$Y=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right],X=\left[\begin{array}{ccccc}1& x_{11}& x_{12}& \dots & x_{1m}\\ 1& x_{21}& x_{22}& \dots & x_{2m}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& x_{n1}& x_{n2}& \dots & x_{nm}\end{array}\right], \beta =\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ ⋮\\ {\beta }_n\end{array}\right], \epsilon =\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ ⋮\\ {\epsilon }_n\end{array}\right]$$

(7)

in which Y is the response vector, X is the matrix of the independent variables, $\beta$ is the vector of the unknown coefficients and $\epsilon$ is the random error vector of the approximation.

3.4. Optimization Using MOGA

In actual forming processes, multiple forming criteria must be met. However, it is challenging to directly determine suitable process parameters. The Pareto-based multi-objective genetic algorithm (MOGA) provides an effective approach for identifying optimized solutions. In multi-objective optimization, the Pareto front represents the optimal trade-off between competing objectives. In a typical two-objective optimization problem, each design objective is represented along a coordinate axis, with the optimization goal being the simultaneous minimization of both objectives. Due to inherent conflicts between the objectives, simultaneous minimization is generally unattainable—optimization of one objective inevitably leads to suboptimal performance in the other. As defined previously, Pareto-optimal solutions represent compromise solutions within the feasible design space that optimally balance these competing objectives [25].

The objective is to determine the optimal combination of forming variables such that the resulting bellow profile, including outer diameter and convolution pitch, conforms as closely as possible to the designed geometry. This optimization problem can be formulated as a constrained nonlinear programming task expressed in the following form:

Minimize

$$F\left(x\right)=\left(f_1\left(x_1\right),f_2\left(x_2\right),\dots ,f_j\left(x_i\right)\right), j=1,2,\hspace{0.17em}\dots \hspace{0.17em},m$$

(8)

Subject to

$$\begin{array}{c}{b}_i^{lower}\le x_i\le b_i^{upper},\hspace{1em}i=1,2,\hspace{0.17em}\dots \hspace{0.17em},n\\ g_k(x_i)\le 0,\hspace{1em}k=1,2,\hspace{0.17em}\dots \hspace{0.17em},p\end{array}$$

(9)

where ${\mathrm{x}}_i$ is the ith design variable; $b_i^{lower}$ and $b_i^{upper}$ represent the lower and upper boundaries of ${\mathrm{x}}_i$, respectively; $f_j\left(x_i\right)$ is the jth objective function of $x_i$ and $g_k\left(x_i\right)$ is the kth constraint function of $x_i$.

The optimization procedure of MOGA comprises the following steps [20]:

• Step 1: Coding. The design variables, represented in real-valued form, are converted into binary numbers.
• Step 2: Fitting. The objective function is constructed using MOGA subject to the constraints given in Equation (10).

$$\begin{array}{l}\begin{array}{c}\begin{array}{c}\end{array}\end{array}0\le \theta \le 4\left(°\right)\\ 0.78\le R_d\le 1.3\left(mm\right)\\ 5.16\le H_m\le 6.9\left(mm\right)\\ 8.38\le p\le 13.96\left(mm\right)\end{array}$$

(10)

• Step 3: Evaluating. The fitness of each individual is evaluated against the objective functions. The Pareto ranking method is applied to assign fitness values.
• Step 4: Grouping. According to the sorting order of the fit value, all the two adjacent individuals are paired into one group.
• Step 5: Crossover and mutation. Within each group, crossover and mutation are car-ried out. The child individuals are generated to replace the parent individuals.
• Step 6: Assembling. The newly generated individuals and the original individuals are combined into one group.
• Step 7: Re-evaluating. Similarly to step 3, the Pareto ranking method is reapplied to determine fitness values with a different objective function.
• Step 8: Selecting. The top half of individuals with higher fitness values are selected as the preferred solutions.
• Step 9: Judging. The termination condition is checked. If met, the MOGA process ends; otherwise, the algorithm returns to Step 4.

4. Results and Discussion

4.1. Factorial Analysis

To evaluate the influence of various factors on the outer diameter (D) and the convolution pitch (q), the main effects of the factors are presented in Figure 7. As shown in Figure 7a, the outer diameter decreases markedly with an increase in die angle (θ) and die thickness (H_m), while it increases significantly with higher internal pressure (p). in contrast, the effect of the die fillet radius (R_d) on the outer diameter is relatively minor. From Figure 7b, it can be observed that the convolution pitch is sensitive to die thickness (H_m), showing a monotonically increasing trend. Other process parameters exhibit negligible influence on the convolution pitch.

The interaction plots for the two responses, outer diameter (D) and convolution pitch (q), are presented in Figure 8. From the interaction plot related to the outer diameter (Figure 8a), it can be observed that the response is significantly influenced by die thickness and internal pressure. The nonparallel lines in the interaction plot between die fillet radius and die thickness indicate a notable interaction effect between these two variables. The largest outer diameter occurs at the maximum die fillet radius and the minimum die thickness. Figure 8b shows the interaction diagram for the convolution pitch. The convolution pitch is strongly affected by die thickness, showing an increasing trend. A combination of greater die thickness and larger die fillet radius leads to a significant increase in convolution pitch. The interaction between die fillet radius and die thickness is also relatively significant. In contrast, the interaction of internal pressure with the other factors exerts negligible effects.

4.2. Complete Quadratic Models

Response surface models for the outer diameter and convolution pitch are developed based on the design variables. Complete quadratic polynomial regression models are established for both the outer diameter and convolution pitch, incorporating all constant, linear, interaction, and quadratic terms, as follows:

$$\begin{array}{l}D=+96.982-0.103\times \theta +9.613\times R_d+1.501\times H_m-0.050\times p-0.067\times \theta \times R_d\\ \hspace{1em}+2.153\times {10}^{-15}\times \theta \times H_m+1.792\times {10}^{-3}\times \theta \times p-1.006\times R_d\times H_m-0.069\times R_d\times p\\ \hspace{1em}+2.060\times {10}^{-3}\times H_m\times p+8.229\times {10}^{-3}\times {\theta }^2-1.085\times R_d^2-0.080\times H_m^2+0.011\times p^2\end{array}$$

(11)

$$\begin{array}{l}q=+16.124-0.503\times \theta -18.387\times R_d-0.638\times H_m+0.037\times p+0.142\times \theta \times R_d\\ \hspace{1em}+0.047\times \theta \times H_m+9.937\times {10}^{-3}\times \theta \times p+1.898\times R_d\times H_m+0.231\times R_d\times p\\ \hspace{1em}+0.017\times H_m\times p+1.469\times {10}^{-4}\times {\theta }^2+1.801\times R_d^2-0.045\times H_m^2-0.020\times p^2\end{array}$$

(12)

The statistical technique of ANOVA is utilized to evaluate the fitness of the models and to identify the main effects of design variables. The total sum of squares SS_T is expressed as follows:

$$S{S}_T={{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^n\left(y_{ij}-\overline{y}\right)}}}^2$$

(13)

where n is the number of replicates within each of k factor levels, y_ij is the jth forming result within factor level I, and $\overline{y}$ is the mean value of the forming results.

The SS_T is a combination of the sum of squares due to error SS_E and the sum of squares due to factors SS_A, given by

$$S{S}_E={{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^n\left(y_{ij}-\overline{y_i}\right)}}}^2$$

(14)

$$S{S}_A=S{S}_T-S{S}_E={{\displaystyle \sum _{i=1}^k{n}_i\left(\overline{y_i}-\overline{y}\right)}}^2$$

(15)

where n_i is the number of the tests of the i-th factor, $\overline{y_i}$ is the mean of results with factor level i.

The mean square (SS_m) is obtained by dividing the SS_A by its associated degrees of freedom (DOF). The F value for each parameter is the mean square of each independent variable divided by the mean square of the residuals. The p-value is the marginal significance level within the hypothesis testing that represents the probability of occurrence of the given event.

The ANOVA results of the outer diameter and the convolution pitch are listed in

Table 6. ANOVA for outer diameter.

Source	Sum of Squares	Dof	Mean Square	F Value	p-Value	Significance
Model	5.18	14	0.37	28.94	<0.0001	**
θ	0.69	1	0.69	54.09	<0.0001	**
Rd	0.12	1	0.12	9.55	0.0080	**
Hm	2.19	1	2.19	171.63	<0.0001	**
p	1.81	1	1.81	141.62	<0.0001	**
θ*Rd	0.0049	1	0.0049	0.38	0.5457
θ*Hm	0	1	0	0	1
θ*p	0.0004	1	0.0004	0.031	0.8621
Rd*Hm	0.21	1	0.21	16.20	0.0013	**
Rd*p	0.01	1	0.01	0.78	0.3913
Hm*p	0.0001	1	0.0001	0.0078	0.9308
${\theta }^2$	0.007	1	0.007	0.55	0.4706
$R_d^2$	0.035	1	0.035	2.73	0.1207
$H_m^2$	0.024	1	0.024	1.88	0.1921
$p^2$	0.047	1	0.047	3.7	0.0749
Residual	0.18	14	0.013
Cor Total	5.36	28

If p < 0.01, mark ** on the significance item; if p < 0.05, mark *.

and

Table 7. ANOVA for convolution pitch.

Source	Sum of Squares	Dof	Mean Square	F Value	p-Value	Significance
Model	12.06	14	0.86	25.05	<0.0001	**
θ	0.084	1	0.084	2.45	0.1399
Rd	0.092	1	0.092	2.67	0.1244
Hm	10.43	1	10.43	303.36	<0.0001	**
p	0.22	1	0.22	6.35	0.0245	*
θ*Rd	0.022	1	0.022	0.63	0.4405
θ*Hm	0.027	1	0.027	0.79	0.3888
θ*p	0.012	1	0.012	0.36	0.5593
Rd*Hm	0.74	1	0.74	21.45	0.0004	**
Rd*p	0.11	1	0.11	3.25	0.0928
Hm*p	0.0066	1	0.0066	0.19	0.6670
${\theta }^2$	2 × 10−6	1	2 × 10−6	6.5 × 10−6	0.9937
$R_d^2$	0.096	1	0.096	2.80	0.1166
$H_m^2$	0.00764	1	0.00764	0.22	0.6445
$p^2$	0.16	1	0.16	4.56	0.0508
Residual	0.48	14	0.034
Cor Total	12.54	28

If p < 0.01, mark ** on the significance item; if p < 0.05, mark *.

, respectively. It can be observed that the “Model p-values” of the outer diameter and the convolution pitch are below 0.0001, indicating that the two models are highly significant. Additionally, terms with p-value greater than 0.05 are considered statistically insignificant. In the case of outer diameter, the significant model terms are θ, R_d, H_m, p and the interaction R_d*H_m. Similarly, for the convolution pitch, H_m, p, and the interaction R_d*H_m are identified as significant terms, as shown in Table 7.

4.3. Reduced Quadratic Models

The second-order polynomial response surface model, as shown in Equation (5), is fitted by the stepwise regression analysis. which is employed to eliminate statistically non-significant terms. Subsequently, based on the identified significant factors, the second-order polynomial models for the outer diameter and the convolution pitch are presented in Equations (16) and (17), respectively.

$$D=+100.457-0.12\times \theta +6.437\times R_d+0.551\times H_m+0.139\times p-1.004\times R_d\times H_m$$

(16)

$$q=+1.222+1.116\times H_m-0.048\times p-0.042\times R_d\times H_m$$

(17)

The ANOVA results for the outer diameter and the convolution pitch are presented in

Table 8. Summary of ANOVA for outer diameter.

Parameter	Value
Standard Deviation	0.12
Mean Value	105.49
C. V. %	0.11
PRESS	0.83
R-squared	0.9377
Adj R-squared	0.9242
Pred R-squared	0.8442
Adeq precision	29.78

and

Table 9. Summary of ANOVA for convolution pitch.

Parameter	Value
Standard Deviation	0.21
Mean Value	7.14
C. V. %	2.95
PRESS	2.00
R-squared	0.9153
Adj R-squared	0.9012
Pred R-squared	0.8401
Adeq precision	31.172

. These results indicate that the response surface models (Equations (16) and (17)) are statistically reasonable.

The distributions of residual error of the regression equations (Equations (16) and (17)) are shown in Figure 9. The relationship between the predicted values and the actual ones are illustrated in Figure 10. Both Figure 9 and Figure 10 indicate that the predicted response values are in good agreement with the actual values. Thus, the second-order response model is suitable for predicting the formed shape in this study.

4.4. Response Surface Analysis

To intuitively reflect the relationship between forming quality and process parameters described by Equations (14) and (15), three-dimensional response surface of the outer diameter and convolution pitch are presented in Figure 11a and Figure 11b, respectively. For this analysis, the die angle (θ) is set to 2°, as the corrugation shape is found to be close to a U-shape in the study by Chen et al. [26].

The influence of die fillet radius (R_d) and die thickness (H_m) on the outer diameter is evaluated at internal pressure (p) values of 8.38 MPa and 13.96 MPa, as shown in Figure 11a and Figure 11b, respectively. It can be observed that the outer diameter increases with an increase in die fillet radius (R_d) and a decrease in die thickness (H_m). Furthermore, a comparison of the two response surfaces indicates that the outer diameter is significantly influenced by the internal pressure (p). The maximum outer diameter occurs at a die fillet radius R_d = 1.3 mm and die thickness H_m = 5.16 mm.

The influence of die fillet radius (R_d) and die thickness (H_m) on convolution pitch was evaluated at internal pressure (p) values of 8.38 MPa and 13.96 MPa, as illustrated in Figure 12a and Figure 12b, respectively. It can be observed that the convolution pitch increases with increasing die thickness (H_m) and decreasing die fillet radius (R_d). Furthermore, the convolution pitch is more significantly influenced by changes in die thickness than by die fillet radius. The maximum convolution pitch occurs at a die fillet radius R_d = 1.3 mm and die thickness H_m = 6.9 mm.

4.5. Multi-Objective Optimization

Four process parameters, namely, die angle (θ), die fillet radius (R_d), die thickness (H_m) and internal pressure (p), were chosen as the design variables, while the outer diameter ($D$) and convolution pitch ($q$) were selected as the optimization objectives. According to the EJMA standard, one objective is to achieve a convolution diameter within a tolerance of ±1.58 mm, and the other is to maintain the convolution pitch within ±1.59 mm, as specified in Figure 1 and Table 1. Based on the preceding analysis, the optimization process for bellows hydroforming can be formulated as follows:

Minimize

$$\begin{array}{l}{F}_1=D_m\left(\theta ,R_d,H_m,p\right)-106\\ F_2=q_m\left(\theta ,R_d,H_m,p\right)-6.9\end{array}$$

(18)

Subject to

$$\begin{array}{l}\begin{array}{c}\begin{array}{c}\end{array}\end{array}0\le \theta \le 4\left(°\right)\\ 0.78\le R_d\le 1.3\left(mm\right)\\ 5.16\le H_m\le 6.9\left(mm\right)\\ 8.38\le p\le 13.96\left(mm\right)\end{array}$$

(19)

In this study, a MOGA-based optimization framework was developed in MATLAB to systematically identify combinations of design variables that satisfy the optimization objectives. The termination criterion is set at 200 generations, a value determined through preliminary convergence tests. These tests indicate that the Pareto front has stabilized beyond this point, with additional generations yielding no significant improvement in solution diversity or accuracy. Following 200 generations of MOGA iteration based on the RSM, the Pareto-optimal solutions for all objectives are plotted in Figure 13, where each point represents a Pareto-optimal solution. To verify the reliability of the optimization method proposed in this study, the design variables corresponding to point A (θ = 1.98°, R_d = 1.28 mm, H_m = 6.90 mm, p = 11.03 MPa) are selected for the experimental validation and FE analysis. However, to accommodate practical process constraints, the die angle (θ), the die fillet radius (R_d), the die thickness (H_m), and the internal pressure (p) are adjusted to 2°, 1.3 mm, 6.9 mm, and 11 MPa, respectively.

The hydroforming process is performed using the apparatus (Aerosun-Tola Expansion Joint Co., Ltd., Nanjing, China) illustrated in Figure 14a. The bellow manufactured under the optimal conditions is shown in Figure 14b. A comparison between the optimized and experimental results under the same optimum conditions is provided in

Table 10. Optimized and experimental results.

Source of the Result	Outer Diameter (mm)	Convolution Pitch (mm)
Optimization	106	6.9
Experiment	105.95	8.08

. For the outer diameter, the optimized result is 106 mm, while the experimental value is 105.95 mm, resulting in a discrepancy of 0.04%. For the convolution pitch, the optimized result is 6.9 mm compared to the experimental value of 8.08 mm, yielding a discrepancy of 17.1%. Notably, both the outer diameter and convolution pitch of the manufactured bellows conform to the manufacturing tolerances specified in the EJMA standards [24]. The 17% error in convolution pitch prediction is primarily caused by the following factors: Accurate convolution pitch prediction is highly sensitive to springback estimation. The bellows’ elastic nature and axial compliance cause highly nonlinear springback behaviour. Additionally, the multi-layer structure introduces greater complexity during forming, leading to intricate deformation mechanisms and challenges in springback prediction. Measurement uncertainties further amplify the discrepancies. Determining convolution pitch requires identifying the centerlines of multiple consecutive crests or troughs, which is sensitive to minor local variations. These variations accumulate and magnify errors in pitch measurement.

To enhance model accuracy, future work will focus on several improvements. Anisotropic constitutive models and kinematic hardening models will be employed to simulate sheet metal behaviour more accurately under complex loading conditions. The springback behaviour of multi-layer bellows will be systematically investigated and compensated through die design modifications or loading path adjustments.

The proposed optimization methodology integrating FE simulation, RSM, GA, and Pareto frontier is also appliable for U-shaped bellows with varying dimensions and materials. However, for bellows with complex cross-sections (e.g., S/Ω-shape), the multiple curvature zones lead to asymmetric material flow during hydroforming and multi-axial springback coupling effects, significantly increasing the complexity of forming prediction and springback control. Therefore, compared with the process optimization of U-shaped bellows, the current method needs to further consider the influence of multi-stage pressure loading and the precision of the mould cavity on the profile and wall thickness uniformity of the bellows in the process optimization for S/Ω-shaped bellows. This will be the key focus area for subsequent research.

5. Conclusions

In this article, an optimization method consisting of FEM, RSM, and MOGA is proposed for the hydroforming process of bellows. The following main conclusions are drawn:

(1)

The outer diameter is primarily influenced by the internal pressure (p), die angle (θ), die thickness (H_m), and die fillet radius (R_d), whereas the convolution pitch is mainly controlled by the die thickness (H_m). An increase in internal pressure (p) enlarges the outer diameter, similarly, a reduction in either the die angle (θ) or die thickness (H_m) also results in a larger outer diameter. In contrast, the convolution pitch increases with greater die thickness (H_m). It is also noteworthy that the interaction between die thickness (H_m) and die fillet radius (R_d) significantly affects both the outer diameter and the convolution pitch.

(2)

Two second-order polynomial equations for predicting the outer diameter and convolution pitch are established using the RSM. The reliability of these equations is verified through the R²-test and ANOVA. Furthermore, the effects of design variable and their interactions in the U-shaped bellows hydroforming are analyzed. The outer diameter reaches its maximum at a die fillet radius (R_d) of 1.3 mm and a die thickness (H_m) of 5.16 mm. Similarly, the convolution pitch is maximized at R_d = 1.3 mm and H_m = 6.9 mm.

(3)

An MOGA-based optimization framework is developed to identify optimal combinations of design variables that meet the optimization objectives. A set of Pareto-optimal solutions is derived. The optimized results are experimentally verified. The application of the proposed procedure demonstrates its effectiveness in identifying feasible process parameters for manufacturing high-quality, defect-free U-shaped bellows.