1 Introduction

With the development of automotive lightweight technology, steel and aluminum alloy materials are widely used in mixed material body. But the traditional fusion welding process tends to form hard and brittle intermetallic compounds when joining materials of steel and aluminum alloy, which leads to poor joint reliability [1]. To solve this problem, a variety of reliable joining processes for dissimilar materials have been proposed, such as self-piercing riveted technology (SPR) [2], flow drill screwing technology [3], and clinch riveting [4]. Among them, SPR can not only realize the riveting between sheets with the same material properties, but realize the connection between sheets of different materials and different thicknesses. Moreover, the riveting quality and reliability are high, so it is often used for the connection of steel and aluminum materials in the car body [5, 6].

Research on SPR often uses experimental methods to evaluate the quality of riveted joints [7, 8]. Compared with experimental methods, SPR’s finite element simulation provides a more direct way to discover the joint deformation and damage mechanism [9, 10]. But a high-precision SPR joint simulation is crucial to obtain the mechanical performance of the joint. Common mechanical performance evaluations of SPR joints include three working conditions: shearing, cross-drawing, and peeling. Porcaro and Hanssen et al. [11, 12] studied the SPR of 6060-T6 and 6060-T4 aluminum sheets with different thicknesses via the finite element method. A solid finite element model (FEM) was established to simulate the drawing condition. A comparison of simulation and test results shows that the riveted sheet undergoes large plastic deformation or even tearing, resulting in large load force and failure displacement. The main reason for this result is that the model ignores material failure modes. Xue et al. [13] prepared equivalent specimens of sheets with different pre-strain levels by flat rolling process. The effect of considering riveting forming strain on the prediction accuracy was discussed by studying the mechanical performance of SPR joints between DP590 high-strength steel and AC43500-T7 cast aluminum alloy sheets. The prediction accuracy of the pre-strained joint is improved. Therefore, it is very important to consider the material failure model of the riveted sheet and the residual information (stress and strain, etc.) during the rivet forming process [14].

Since the parameters of rivets and dies affect the forming quality and mechanical performance of SPR joints, some scholars have studied the effect of different parameters on different performances to obtain the best structural parameters [15]. Jin et al. [16] carried out SPR of DP590 high-strength steel and 6061T6 aluminum alloy sheets. The influence of process parameters on the riveting forming quality and shear strength were analyzed, such as the height of boss, diameter of die, and length of rivets. The process parameters were optimized by the orthogonal experiment method, and the best combination of rivet length and die parameters was determined. The experimental design optimization method can only find the best combination of set parameters, but it is difficult to find the optimal solution. The calculation efficiency is low.

To improve the optimization efficiency and find the optimal solution, the optimization method based on the static approximate model is applied to the analysis of the forming and tensile mechanical performance of SPR joints. Zhao et al. [17] developed two regression models to predict the joint forming quality and systematically analyzed the influence of rivet and die parameters on joint interlock and remaining thickness, indicating the accuracy of the prediction model. Zhang et al. [18] determined the range of rivet and die parameters through sensitivity analysis, and established the relationship between process parameters and joint forming indicators using the response surface model (RSM). The optimization is carried out by using the genetic algorithm, and the rivet and die parameters of the optimal joint forming parameters are determined. Zhao et al. [19] used an artificial neural network to predict the quality of SPR joints. Combined with a genetic algorithm, the parameters of rivet and die can be quickly determined. Fang et al. [20] proposed a framework integrating machine learning and global sensitivity analysis to optimize the parameters of SPR simulation analysis. Wang et al. [21] established a 3D FEM of SPR. Combined with the multi-criteria decision-making method, the process parameters of SPR joints are optimized, which improves the mechanical performance of the joints. The above research on SPR joints shows that the approximate model is effective in predicting the forming quality of SPR and the mechanical performance of joints. However, as the number and range of joint parameters increase, the changing trend of the joint quality evaluation index will become highly nonlinear. The simulation analysis of the forming and tensile mechanical performance of SPR joints is time-consuming and computationally inefficient. There are many kinds of steel-aluminum dissimilar material connections in the mixed material body, so the optimization method based on the static approximate model will be difficult to efficiently obtain the joint process parameters of the optimal mechanical performance. The sequence approximate optimization method is iteratively calculated by coupling the approximate model and optimization algorithm and eliminates the step of accuracy testing [22]. The sequential approximate optimization method provides an efficient optimization method.

This paper proposes a structure-process-performance integrated optimization method for SPR connection joints of various dissimilar materials to quickly obtain the optimal process parameters of SPR connection. The main content of this paper is as follows: In Section 2, the constitutive relationship of the riveted sheet material and the mechanical performance test of the joint are carried out. In Section 3, the integrated simulation analysis of the forming-tensile mechanical performance of the SPR joint considering the post-forming information is carried out. The SPR joint material-structure-process-performance integrated sequential approximate optimization method is studied in Section 4. Finally, the conclusions are summarized in Section 5.

2 SPR joint test

Based on the part material of the steel-aluminum connection part of a certain car body, the SPR joint is carried out on the top panels made of DC01 steel (DC01) or HC280 steel (HC280) materials, and the upper beam made of 5754-H24 aluminum alloy (5754).

2.1 Material performance test of riveted sheet

2.1.1 Material constitutive model

The Johnson–Cook (J-C) model has a simple form and is often used in engineering [23]. The constitutive relation of the J–C model is shown in Eq. (1).

$${\sigma }_{y}=(A+B{\varepsilon }^{n})\cdot (1+C{\text{ln}}\frac{{\dot{\varepsilon }}_{eff}}{{\dot{\varepsilon }}_{0}})$$
(1)

where \({\sigma }_{y}\) is the flow stress, and its unit is MPa; \({\dot{\varepsilon }}_{eff}\) is the strain rate; \({\dot{\varepsilon }}_{0}\) is the quasi-static strain rate; A, B, C, and n are the constitutive fitting parameters.

During the forming process of the SPR joint, the rivet will pierce the upper sheet. Moreover, both the upper and lower sheets may fail and fracture under tension conditions. The stress state of the sheet is different under different tensile conditions. Stress triaxiality \({\sigma }^{*}\) characterizes various complex stress states of metal materials under loads in different directions [20].

The J-C failure model [24] is shown in Eq. (2).

$${\varepsilon }_{f}^{pl}={D}_{1}+{D}_{2}exp({D}_{3}{\sigma }^{*})$$
(2)

where D1 ~ D3 are failure parameters.

2.1.2 Tensile test of riveted sheet material

The stress triaxiality of the element at the notch is calculated during the deformation process of the notched specimen [25]. According to the test method of GB/T 228.1–2010, the samples with notch radius r of 2 mm, 5 mm, 8 mm, 10 mm, and ∞ are designed. The electronic universal tensile testing machine (rated load 100 kN) is used together with the DIC VIC-2D high-speed full-field strain measurement system. First, speckle processing is performed on the specimen, and then the specimen with speckles is placed at the fixture of the tensile testing machine. The tensile speed of the machine is 5 mm/min. The high-speed camera is turned on simultaneously to collect pictures of the changes in the sample during the stretching process. The tensile testing machine collects the force–time curve, and the full-field strain system collects the displacement and strain versus time. Finally, combining the two can obtain the stress–strain curve and the full-field strain at the speckle position, as shown in Fig. 1. Each group is subjected to three tensile tests, and the data are averaged

  1. 1)

    Material constitutive relationship

Fig. 1
figure 1

Notch test of riveted sheet material

Full size image

The material tensile test of the sample with notch radius ∞ is extracted. The true stress–strain curves of DC01, HC280, and 5754 materials are obtained, as shown in Fig. 2. The basic properties of these three materials are shown in Table 1.

Fig. 2
figure 2

True stress–strain relationship of materials

Full size image
Table 1 Basic properties of materials
Full size table

The J–C model is fitted to the acquired stress–strain curves using the least squares method, as shown in Eq. (3).

$${\sigma }_{DC01}\text{=} \, 170+423.29{\varepsilon }^{0.452}$$
(3)

In the same way, the J-C models of HC280 and 5754 materials are obtained.

$${\sigma }_{HC280}= \text{ } 280+762.19{\varepsilon }^{0.476}$$
(4)
$${\sigma }_{5754}= \text{ } 170+175.3{\varepsilon }^{0.462}$$
(5)
  1. 2)

    Material fracture constitutive model

The stretching process of the notched samples is analyzed through the full-field strain system, and the full-field strain of each sample is obtained, as shown in the strain cloud diagram of Fig. 1. The strain values at the fracture locations of samples of different materials are extracted, as shown in Table 2.

Table 2 Stress triaxiality and failure displacement
Full size table

Combined with the data in Table 2, the J-C failure model is obtained by using the least square method, as shown in Fig. 3.

Fig. 3
figure 3

Failure strains of different stress triaxiality

Full size image

2.2 Forming process and performance test of SPR joint

2.2.1 SPR technology and evaluation method

When performing SPR, it is necessary to determine the type of rivet and mold according to the thickness of the sheet. Considering the SPR of 1 mm DC01 and 1.7 mm 5754, a rivet with a diameter of 5.3 mm and length of 5 mm is selected, as shown in Fig. 4a. The die parameters are shown in Fig. 4b.

Fig. 4
figure 4

Process parameters (dimensions in millimeters)

Full size image

The evaluation methods for the connection performance of SPR joints mainly include the section evaluation method and load evaluation method. The section evaluation method judges the quality of the joint by measuring the geometric dimensions of the main shape features, as shown in Fig. 5. Among them, u is the interlock length, tb is the bottom thickness, and tr is the remaining bottom thickness. The minimum bottom thickness tm is the minimum value of the bottom thickness tb and the remaining thickness tr. The interlock length u and the minimum bottom thickness tm are mainly selected as evaluation indicators [26]. The load evaluation method is to judge the connection performance of the joint by the tensile test.

Fig. 5
figure 5

Feature size of cross-sectional evaluation method

Full size image

2.2.2 SPR sample

Tensile testing of the load evaluation method includes shearing, cross-drawing, and peeling conditions. The geometric dimensions of 1mm DC01 and 1.7 mm 5754 are designed, as shown in Fig. 6a, b, and c, respectively. The upper sheet is made of 1mm DC01 material, and the lower sheet is made of 1.7 mm 5754 material. The hole is where the clamp is secured.

Fig. 6
figure 6

Geometric dimensions of samples (dimensions in millimeters)

Full size image

2.2.3 Forming test of SPR joint

The SPR joint samples are riveted in an automatic riveting machine, in which the equipment model is EP-CTF, and the maximum riveting force is 75 kN. The upper and lower riveting sheets are first placed above the concave die of the SPR machine. The riveting machine then riveted the center of the rivet sheet positioning at a speed of 20 mm/s. The SPR samples of cross-drawing, shear, and peel joints are obtained, as shown in Fig. 7. The riveting force indicator can display the riveting pressure value.

Fig. 7
figure 7

SPR forming test

Full size image

2.3 Mechanical performance test of joint

The cross-drawing, shearing, and peeling tests are carried out on the WSM3100 microcomputer-controlled electronic universal testing machine. The tensile speed is set to 5 mm/min, as shown in Fig. 8a, b, and c.

Fig. 8
figure 8

Tensile test of SPR joints

Full size image

3 Finite element analysis of SPR joint

3.1 Finite element modeling of SPR

SPR forming process of 1mm DC01 and 1.7 mm 5754 material is simulated. To improve the simulation speed of riveted joint forming, a 2D axisymmetric FEM is built using commercial Simufact forming software. The steps of the SPR modeling are as follows:

First, the geometric modeling of the SPR is carried out, including punch, blank holder, rivet, upper and lower riveting sheets, and die. The upper and lower sheets are divided to generate the riveted area and the stretched area, as shown in Fig. 9a.

Fig. 9
figure 9

Simulation model of SPR

Full size image

Then, the material constitutive model and failure model of DC01 and 5754 are assigned to the finite element model. Henrob H4 series rivets are used, and material properties are selected from the material library of software. The tension areas of the upper and lower riveted sheets are set for bonding and binding contact, and the other parts are set for automatic contact. The friction between the parts is simulated using a mixed friction model, including Coulomb friction and shear friction models. The friction coefficients of the Coulomb friction and shear friction models are set to 0.1 and 0.2, respectively. The punch, blank holder, die, and upper and lower sheet stretching areas are defined as rigid body. The pressure of the blank holder is set to 3500N.

Finally, the mesh size of 0.14mm is used for the upper and lower sheet riveting area, and a grid size of 0.05mm is used for the rivets with a secondary roughening algorithm. The element type is quadrilateral. The punch riveting speed is set to 20mm/s. The vertical distance from the punch to the upper sheet is set as the stroke of the punch, and the distance is 6.22 mm to ensure the flatness of the upper surface after riveting. The FEM of SPR joint forming is established, as shown in Fig. 9b.

3.2 Forming quality analysis of SPR

The simulated and tested SPR joint forming evaluation indicators are measured, as shown in Fig. 10. The relative errors of the interlock length and remaining thickness are 8.21% and 2.5%, respectively, as shown in Table 3. It is good consistency between the simulation model and test in terms of SPR joint forming performance.

Fig. 10
figure 10

Forming evaluation index of SPR joint (dimensions in millimeters)

Full size image
Table 3 SPR forming evaluation index
Full size table

3.3 Mechanical performance simulation of SPR joint

3.3.1 Forming analysis of SPR joint

After the 2D model of SPR is formed, the simulation result model is rotated to generate a 3D solid model. The result data of the joint are rotated and mapped, and residual information is retained, such as residual stress and residual strain, as shown in Fig. 11. Then, the tensile mechanical performance of the joints is analyzed to realize the integrated design of the joint forming and mechanical performance simulation considering the residual information.

Fig. 11
figure 11

Forming information mapping of SPR joint

Full size image

Referring to the cross-drawing, shearing, and peeling tensile test, the lower sheets are fixed during the simulation of the tensile mechanical performance of the SPR joint. A constant speed of 5 mm/min is applied to the upper sheet, where the load direction for the cross-drawing and peeling conditions is in the positive Z-axis direction and the shearing condition is in the positive X-axis direction. Due to the possibility of sheet tearing during the stretching process of SPR joints, the J-C material failure model is selected to simulate the sheet tearing combined with Sect. 2.1.2.

3.3.2 Failure mode analysis of joint

Compared with the test results, the simulation results of the SPR joint under three tensile conditions are extracted, as shown in Fig. 12a, b, and c.

Fig. 12
figure 12

Failure mode of SPR joint

Full size image

Figure 12 shows that the failure mode of the SPR joint under the cross-drawing condition is the failure of the upper sheet to come out. This is mainly because the thickness of the upper sheet is thinner and its strength is weaker than the lower sheet. Under the action of axial tensile load along the joint, the upper sheet is concave along the joint, and the bending angle continues to increase. The material of the upper sheet undergoes large plastic deformation at the joint position, and then the upper sheet comes out of the rivets, causing the joint to fail. This is consistent with the cross-draw test results in Fig. 12a.

The failure mode of the SPR joint under the shear condition is the failure of the lower sheet coming out. The main reason is that the load in the shear condition is loaded along the radial direction of the joint, and the rivets continue to tilt, which destroys the interlocking structure of the rivets. Finally, the upper sheet and the rivets come out together. The lower sheet is tearing due to the large interlock value. This is consistent with the shear test results in Fig. 12b.

The form of the SPR joint under the peeling condition is similar to that of the cross-drawing condition, and the upper sheet is pulled out, which leads to the failure of the joint. Since the peeling condition is axial unilateral loading, the riveted area of the lower sheet is bent simultaneously with the upper sheet and the rivet, resulting in larger deformation of the joint area on the tensile side of the upper sheet. Finally, the riveted area of the upper sheet is larger resulting in joint failure. This is consistent with the peel test results in Fig. 12c.

The force–displacement curves of the simulation and test under different tensile conditions are extracted, as shown in Fig. 13. The simulation results are also calculated for different tensile conditions without considering the residual information.

Fig. 13
figure 13

Mechanical performance simulation and test of joint

Full size image

Figure 13 shows that the tensile failure displacement considering the residual stress is larger than that without considering the residual information, and the relative error of the maximum load force is smaller. The variation trend of the simulation curve considering the residual information is consistent with the test. The relative error between the test results of the maximum cross-drawing force, shearing force, and peeling force under different tensile conditions and the simulation considering the residual information is within 10%, and the maximum relative error between the simulation and test of failure displacement is within 10%, as shown in Table 4. In the shear condition, the relative error of the failure displacement without considering the residual information is larger. Therefore, the accuracy of the SPR joint forming-cross drawing/shearing/peeling integrated analysis model considering residual information is higher than the model without considering residual information.

Table 4 Comparison between simulation and test
Full size table

4 Hybrid sequence approximate optimization method of SPR process parameters

When SPR is performed on sheets of different materials and thicknesses, different rivets and molds will affect the quality and reliability of riveted joints. Therefore, it is necessary to optimize the design of the joint process to obtain the optimal rivet and die process parameters.

4.1 Hybrid sequence approximate optimization method

The simulation of forming and mechanical performance of SPR joints involves material nonlinearity, contact nonlinearity, and solution nonlinearity, which are highly nonlinear and require a large amount of calculation. The optimization method based on the approximate model needs to pre-estimate a large number of sample points to improve the calculation accuracy. If the accuracy is not enough, re-sampling is required, which will increase the amount of calculation, and the method cannot guarantee the accuracy of predicting the optimal solution.

Aiming at the insufficiency of the optimization method based on the approximate model, the sequential approximate optimization method can improve the optimization efficiency of the approximate model and enhance global convergence. The specific steps of the sequence approximate optimization method are as follows [22]:

  • Step 1: According to the design variables and design space, the optimal latin hypercube sampling method is used to sample the initial points. The number of initial sample points should not be less than the minimum sample points required to construct an approximate model, as shown in Eq. (6).

    $$N\ge 2{n}_{s}+1$$
    (6)

    where N is the number of sample points; \({n}_{s}\) is the dimension of design space.

  • Step 2: The response values of each sample point are calculated by design of experiment (DOE). The initial number of iterations is set to k = 1. When k ≥ 2, the optimal solution and its response value in Step7 are extracted. Then the sample point and response library are updated;

  • Step 3: The sample points and their response values are extracted. The constraint responses are constructed as RSM, and the objective function response is constructed as radial basis function neural network model (RBFNN);

  • Step 4: The multi-island genetic algorithm is used to globally find the optimal solution of the constructed approximate model and obtain the predicted optimal solution \({{x}_{k}}^{*}\) of the kth iteration and the corresponding predicted response values \(\widetilde{f}({{x}_{k}}^{*})\) and \(\widetilde{g}\left({{x}_{k}}^{*}\right)\);

  • Step 5: The optimal solution \({{x}_{k}}^{*}\) of the kth iteration is substituted into the analytical model to obtain the objective function value \(f({{x}_{k}}^{*})\) and the constraint value \(g({{x}_{k}}^{*})\) of the analytical model of the kth iteration. The results are saved into the optimal solution set.

  • Step 6: When k = 1, go to Step 7 directly. When k > 2, by calculating the relative error of the analytical model response value of the kth iteration and the (k-1)th iteration in the optimization process. By judging whether the relative error satisfies the given convergence criterion \(\varepsilon\) and the analysis model constraint value meets the constraint requirements, as shown in Eq. (7). If it is satisfied, stop the cycle, and the optimal solution obtained in Step 5 is the optimal solution of the analysis model. If not satisfied, go to Step 7.

    $$\begin{array}{c}\left|\frac{f({{x}_{k}}^{*})-f({{x}_{k-1}}^{*})}{f(({{x}_{k-1}}^{*})}\right|\le \varepsilon \\ g\left({{x}_{k}}^{*}\right)\le {g}_{0}\mathrm{ or }g\left({{x}_{k}}^{*}\right)\ge {g}_{0}\end{array}$$
    (7)

    where \(\varepsilon\) is the convergence precision value; \({g}_{0}\) is the constraint reference value.

  • Step 7: The approximate optimal solution \(f({{x}_{k}}^{*})\) and the value of the design variable \({{x}_{k}}^{*}\) are used as a new set of sample points. Then return to Step 2, the new sample points are constructed, and k = k + 1 is set.

4.2 Design of experiment for joint parameters

The material-structure-process parameters of SPR joints are coupled together to explore the mechanical performance of the joints.

The 5754 is connected with DC01 and HC280 respectively in a car body, and the joints of the two materials are connected for DOE. The material parameters x1 (DC01 or HC280 material) of the rivet sheet, process parameters including rivet leg length x2, rivet diameter x3, die diameter x4, die bottom height x5, and die boss height x6, and structural parameters including end distance x7 and the side distance x8 of the rivet sheet are selected as variables, as shown in Fig. 14. The levels of the variables are shown in Table 5. Other setting parameters for analytical solutions are the same as in Sect. 3.1. The performance response of SPR joints is mainly divided into riveting forming indicators (interlock length u, minimum bottom thickness tm, and maximum riveting force Fr) and connection tensile mechanical performance indicators (maximum shear force Fs, maximum draw force Fd and the maximum peel force Fp).

Fig. 14
figure 14

Variable parameters

Full size image
Table 5 Parameter levels
Full size table

When x1 = 1, the steel sheet is 1mm DC01 material, the horizontal numbers 1, 2, and 3 of x2 represent the rivet lengths of 5 mm, 5.2 mm, and 5.3 mm, respectively, and the horizontal numbers 1, 2, 3, and 4 of x4 represent the die diameters are 4.6 mm, 4.8 mm, 5 mm, 5.2 mm; when x1 = 2, the steel sheet is 2.5mm HC280 material, and the horizontal numbers 1, 2, 3 of x2 represent the rivet lengths of 6.8 mm, 7 mm, 7.2 mm respectively, and the horizontal numbers 1, 2, 3, and 4 of x4 represent the die diameters of 4.8 mm, 5 mm, 5.2 mm, and 5.4 mm, respectively.

The DOE of the optimal Latin hypercube method is adopted, and the joint interlock length, remaining thickness, maximum riveting force, maximum cross draw force, maximum shear force, and maximum peel force are selected as responses. The joint forming and mechanical performance integrated simulation analysis of 50 sets of schemes are carried out, and the response values of each sampling point are obtained, as shown in Table 6.

Table 6 Results of DOE
Full size table

4.3 Integrated optimization method of material-structure-process-performance

To quickly find the optimal SPR process parameters and solve the problem of low calculation efficiency caused by the multi-peak and high discreteness of the objectives and constraint functions of the connection process parameters, the structure-process-performance integrated optimization of SPR is carried out using the hybrid sequence approximate optimization method.

Since the shear strength failure is more likely to occur at the body joint, the shear strength Fs is taken as the optimization objective, and the forming quality and mechanical performance of other joints are constrained. The initial values and ranges of the design variables are according to Table 5. The mathematical model for the integrated optimization of material-structure-process-performance of SPR is as follows:

$$\begin{array}{c}\begin{array}{c}\begin{array}{cc}{\text{Find}}& \mathbf{x}\end{array}={\left({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{8}\right)}^{{\text{T}}}\\ \begin{array}{cc}{\text{max}}& {F}_{s}\left(\mathbf{x}\right)\end{array}\end{array}\\ s.t.\left\{\begin{array}{cc}{F}_{d}\left(\mathbf{x}\right)\ge {F}_{d0};& \begin{array}{cc}{F}_{p}\left(\mathbf{x}\right)\ge {F}_{p}0;& {F}_{r}\left(\mathbf{x}\right)\le \end{array}\\ u\left(\mathbf{x}\right)\ge {{\varvec{t}}}_{t0};& {{\varvec{t}}}_{{\text{m}}}(\mathbf{x})\ge {{\varvec{t}}}_{m0}\\ {x}_{1}=0 or 1;& {x}_{L}\le {x}_{i}\le {x}_{U}\end{array}\right.\end{array}{F}_{r0}$$
(8)

where tr0 and tm0 are the interlock length and the minimum bottom thickness of joint forming. Considering that if the minimum thickness of the bottom is too small, it is easy to cause the lower sheet to crack or fall off, the minimum thickness is set to not less than 0.15. Since the larger the interlock length, the higher the degree of interlocking between the rivet and the lower sheet, the interlock length is set to not less than 0.2; Fr0, Fd0, and Fp0 are the maximum riveting force, maximum pulling force, and maximum peeling force of the joint, refer to its performance before optimization, when x1 = 1, Fr0, Fd0, and Fp0 are set to 55 kN, 2.254 kN and 1.248 kN, respectively; when x1 = 2, Fr0, Fd0, and Fp0 are set to 73.78 kN, 3.515 kN and 1.675 kN, respectively; \({x}_{U}\) and \({x}_{L}\) are the upper and lower limits of the design variables, respectively.

According to the steps of the hybrid sequence approximate optimization method in Sect. 4.1 and DOE results in Sect. 4.2, the maximum shear force Fs(x) is used to construct the RBFNN, and the interlock length u, the minimum bottom thickness tm, the maximum riveting force Fr, the maximum cross draw force Fd(x) and the maximum peel force Fp(x) are used to construct the RSM. ε is set to 5%, and other optimization parameter settings are the same as in Sect. 4.1.

4.4 Results and discussion

The optimal SPR process parameters of 5754 and DC01 are obtained through hybrid sequence approximate optimization method after 2 iterations, and the optimal SPR process parameters of 5754 and HC280 are obtained through hybrid sequence approximate optimization method after 4 iterations. The final optimization results are shown in Table 7. The performance indicators of the joints are extracted, as shown in Tables 8 and 9.

Table 7 Values of design variables after optimization
Full size table
Table 8 Performance index of DC01 and 5754 riveting
Full size table
Table 9 Performance index of HC280 and 5754 riveting
Full size table

Tables 8 and 9 show that the maximum relative error between the optimal value and the simulated value of each performance index based on the optimization method is less than 5%, which further illustrates the reliability of the hybrid sequence approximate optimization method in terms of computational accuracy. After the process parameters of DC01 and 5754 riveting are optimized, the maximum cross-draw force, maximum shear force, and maximum peel force increased by 34.53%, 8.92%, and 22.43%, respectively, compared with the initial scheme. After the process parameters of HC280 and 5754 riveting are optimized, the maximum draw force, maximum shear force, and maximum peel force increased by 1.495%, 1.28%, and 6.866%, respectively, compared with the initial scheme. It shows that the hybrid sequence approximate optimization method can carry out the integrated optimization of material-structure-process-performance of SPR joints, and realize the comprehensive improvement of the comprehensive mechanical performance of joints while ensuring accuracy.

5 Conclusion

An integrated optimization method of material-structure-process performance is proposed to optimize the structure-process-performance design of SPR joints of steel/aluminum dissimilar materials. The conclusions are as follows:

  1. (1)

    Quasi-static tensile tests under different stress triaxiality states of 5754, DC01, and HC280 material are carried out. The J-C material constitutive model and failure fracture model considering the stress triaxiality are established. The simulation analysis of joint forming is carried out, and the riveting force and forming quality dimensions are obtained by the simulation and test. Compared with the test results, the relative errors of the simulation results are all within 10%;

  2. (2)

    The integrated simulation analysis of the joint forming and mechanical performances of cross-drawing, shearing, and peeling considering the residual information is carried out. Compared with the test results, the relative error between the test and the simulation of the maximum tensile force under different tensile conditions is within 10%, and the maximum relative error between the failure displacement and the test is within 10%. The failure displacement error of the shear condition without considering the forming information is large, which indicates that the accuracy of the established simulation model is higher; and

  3. (3)

    The optimal design of joint material-structure-process parameters is carried out using the sequence approximate optimization method. Compared with the initial model, after the process parameters of DC01 and 5754 riveting are optimized, the maximum cross draw force, maximum shear force, and maximum peel force are increased by 34.53%, 8.92%, and 22.43%, respectively. After optimizing the process parameters of HC280 and 5754 riveting, the maximum cross-drawing force, maximum shear force, and maximum peel force increased by 1.495%, 1.28%, and 6.866% compared with the initial scheme, respectively.