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Abstract

This study proposes a systematic approach for predicting and optimizing the forming quality of self-piercing riveting (SPR) joint between aluminum alloy sheets. First, the finite element (FE) model of the forming of SPR joint was built to monitor the forming process and assess the joint’s forming quality, which was then experimentally validated. Second, based on FE numerical simulation, a parameter study was implemented to explore the action laws of forming process parameters (rivet shank thickness, rivet shank inner diameter, die pip height and die inner diameter) on the joint’s forming quality. Third, taking above forming process parameters as design variables, a hybrid Taguchi–gray relational analysis (GRA) method was employed to conduct a multi-objective optimization of the joint’s forming quality for maximizing interlock value (IV) and bottom thickness (BT) simultaneously, which are two primary evaluation indicators of forming quality of SPR joint. Finally, the multi-objective optimization results were validated. According to the outcomes, the SPR joint achieves increases of IV by 7.41% and BT by 40.63% simultaneously, indicating a good multi-objective optimization of forming quality of SPR joint is obtained. The present study provides an effective new method for predicting and optimizing the forming quality of SPR joint.

Keywords

self-piercing riveting forming quality multi-objective optimization Taguchi gray relational analysis

Introduction

Lightweight design of the vehicle body can effectively promote energy conservation and emission reduction of fuel vehicles or increase the range of electric vehicles, which is an important research direction for current automotive design. As the lightweight potential of vehicle body through structural optimization is increasingly limited, the use of lightweight yet high-strength materials and advanced material joining technologies are key future lightweight directions.¹

Self-piercing riveting (SPR) is a highly promising material joining method for future lightweight vehicle body panels owing to its multiple comprehensive advantages, such as wide material applicability, simple process, high joining efficiency and good joining performance, etc.² Figure 1 illustrates an application example of SPR in a car engine cover, as well as its forming process, including clamping, piercing, flaring and releasing.³ However, the joining performance of the SPR joint, which is a weak part of strength in vehicle body SPR connected structures, is greatly affected by the forming quality of the joint. Therefore, research on the forming quality of SPR joint is currently a hot topic in the field of automotive safety and lightweight research.

Figure 1.

An application example of SPR in a car engine cover.³

In order to clarify the forming mechanism of SRP joint and reveal the factors affecting its forming quality, researchers at home and abroad had carried out a wide range of studies. Wang et al.⁴ established a 3D finite element model for self-piercing riveting of steel and aluminum sheets, and investigated the effects of rivet and die parameters on joint’s forming performance. Karathanasopoulos et al.⁵ analyzed the influence of two geometric parameters, that is, inner radius of the rivet leg and depth of the die central tip, on the forming quality of SPR joints between dissimilar materials. Zhou et al.⁶ systematically investigated the effects of rivet height and sheet stacking sequence on SPR joint’s forming quality between aluminum alloy and high-strength steel sheets through tests. Xu⁷employed ANOVA technique to investigate the impact of rivet length and die type on SPR joint’s undercut, bottom thickness and rivet flaring. Haque⁸ conducted a focused investigation on key parameters affecting the forming quality of SPR joint from a cross-sectional perspective, while also analyzing auxiliary technologies capable of enhancing joint’s forming quality. Deng et al.⁹ explored the impact of die geometry on the forming quality of SPR joint between aluminum alloy and low carbon steel sheets. Kong et al.¹⁰ adopted a hybrid approach, integrating experimental methods with numerical simulations, to evaluate the influence of structural parameters on the forming quality of CFRP/Al SPR joints, etc.

However, a review of above existing relevant literature reveals the following shortcomings in current research: (1) Currently, most research related to the forming process and quality of SRP joint relied on experiments, which were time-consuming, labor-intensive, costly, of poor reproducibility and difficult to clarify the micro-mechanisms of SPR joint forming. At present, there has been a relative lack of effective finite element numerical simulation methods on forming quality of SPR joint, which might not only accurately and efficiently simulate the forming process of SPR joint, but also reveal the influencing factors and corresponding action laws on joint’s forming quality. (2) Most of the existing studies simply studied the independent influence of rivet parameters (such as rivet hardness, rivet length, etc.) or die parameters (such as die depth, die inner diameter, etc.) on the forming quality of SPR joint, currently there have been few studies exploring the effects of rivet parameters and die parameters simultaneously on the forming quality of SPR joint, and even fewer studies discussing the interactive effects between the above two types of parameters.¹¹ (3) At present, there have been relatively few studies on optimization method of forming quality of SPR joint, while traditional optimization methods based on experimental design and surrogate models have much high computational costs, hence how to establish an effective and efficient new optimization method for optimizing the forming quality of SPR joint is worth exploring.

Consequently, the present study proposes a systematic approach for predicting and optimizing the forming quality of SPR joint between aluminum alloy sheets. First, the finite element (FE) model of the SPR joint forming was built to monitor the riveting process and assess the joint’s forming quality, which was then experimentally validated. Second, based on numerical simulation, a param- meter study was implemented to explore the action laws of riveting process parameters (rivet shank inner diameter, rivet shank thickness, die pip height and die inner diameter) on the joint’s forming quality. Third, taking above process parameters as design variables, a hybrid Taguchi–GRA method was employed to conduct a multi-objective optimization of the joint’s forming quality for maximizing interlock value and bottom thickness simultaneously, which are two primary evaluation indicators of forming quality of SPR joint. Finally, the multi-objective optimization results were validated. The present study provides an effective new method for predicting and optimizing the forming quality of SPR joint. Figure 2 demonstrates the flow chart of the present study.

Figure 2.

Flow chart of the present study.

Finite element modeling and verification

Geometrical modeling

In this study, the design software SolidWorks was employed to create the geometrical model of SPR joint forming, including the punch, clamp, die, rivet, top sheet and bottom sheet, which was then imported into Simufact. Forming to monitor the forming process. Figure 3 demonstrates the geometrical model of SPR joint forming and partial structural parameters of rivet, top sheet, bottom sheet and die, which were referred to Liu et al.¹² Specifically, the thicknesses of the upper and lower sheets are 1.2 and 1.5 mm, respectively, the rivet has a height of 5.0 mm and leg outer diameter of 5.3 mm, and the die has an inner diameter of 9.0 mm.

Figure 3.

Geometrical model of SPR joint forming and partial structural parameters.

Material description

The upper and lower sheets were made of aluminum alloy with comprehensive merits of light weight, good ductility and processability, while the rivet was made of boron steel, whose corresponding material parameters and plastic stress–strain curves were referred to Liu et al.¹² and Carandente et al.,¹³ and are listed in Table 1 and demonstrated in Figure 4, respectively. Herein for aluminum alloy, the plastic stress-strain curve under temperature of 200 °C was selected to approximately take into account the thermal softening effect during forming process,^12,13 while this thermal effect on the rivet material was not considered because it has a very limited influence on the mechanical properties of the boron steel.¹³

Table 1.

Material parameters of upper sheet, lower sheet and rivet.^12,13

Material	Elastic modulus (MPa)	Poisson’s ratio
Aluminum alloy	7.00E+04	0.3
Boron steel	2.00E+05	0.3

Figure 4.

Plastic stress-strain curves of aluminum alloy¹² and Boron steel.¹³: (a) aluminum alloy and (b) boron steel.

Finite element modeling

The Simufact. Forming software was used to build the finite element model of SPR joint forming. The die was fixed and the punch was set with a constant vertically downwards velocity of 100 mm/s. The clamping force between the die and the clamp was set to 5.3 kN to ensure consistency with the experimental conditions described in the referenced study.¹² During the self-piercing riveting process, the sheet material and rivet undergo significant plastic deformation, while the punch, clamp and die experience negligible deformation and can therefore be modeled as rigid bodies. The Simufact. Forming incorporates a variety of advanced meshing techniques, of which the Advancing Front Quad method was employed for the upper and lower sheets to handle large deformations, while the Quad tree method was used to aid in creating a more refined mesh at the rivet boundaries, as illustrated in Figure 5.¹⁴ To prevent mesh distortion caused by excessive deformation in the large deformation region, the 2D mesh redistribution was set with a minimum sheet fracture thickness of 0.04 mm.¹⁵ To make a balance between simulation accuracy and computing cost, the mesh sizes for the rivet, top sheet and bottom sheet were 0.08, 0.08 and 1.00 mm, respectively. In addition, the Coulomb friction model was selected to describe the interactions between the contact surfaces, the friction coefficient between the bottom sheet and the die was set to 0.22, while the friction coefficient between other components was set to 0.10.¹²

Figure 5.

FE model of SPR joint forming.

Model validation

Figure 6 demonstrates the simulation results of force-displacement curve of SPR joint forming, that is, riveting force versus riveting displacement, as well as partial deformation modes at middle cross-section (abbreviated as sectional deformation modes) corresponding to different riveting displacements. It can be seen that the forming of SPR joint undergoes a continuous process of piercing into the upper sheet, then piercing throughout the upper sheet and piercing into the lower sheet, plastic deformation and forming an interlocked structure. To assess the final forming quality of SPR joint, two key geometrical parameters in joint’s final sectional deformation diagram, that is, interlock value (IV) and bottom thickness (BT), as shown in Figure 7, were selected as two forming quality indicators.¹³ Obviously, larger IV values are beneficial for improving joint’s joining strength and reliability of, meanwhile larger BT values help avoid small cracks or tears in the lower sheet material caused by rivet compression deformation during the forming process, which could result in a serious reduction in joint’s sealing and corrosion resistance. Thus, lager IV or BT value indicating better joint’s forming quality is preferred.

Figure 6.

Force-displacement curve and deformation modes of SPR joint forming.

Figure 7.

Forming quality indicators of the SPR joint in final sectional deformation diagram.

To verify the accuracy of the built FE model of SPR joint forming and corresponding simulation results, the SPR joint forming experimental result by Liu et al.¹² were used in the present study. Specifically, the experiment with the identical joint and die configuration as our finite element model was employed to conduct experimental verification, and the final sectional deformation modes of joint forming corresponding to simulation and experiment were extracted and compared in Figure 8, and the IV and BT of experiment (IV_exp, BT_exp) and simulation (IV_sim, BT_sim) corresponding to final deformation mode of joint forming were further extracted and compared in Table 2. It can be seen that: (1) The final deformation mode of joint forming of simulation is basically consistent with that of experiment; (2) In addition, the IV and BT of simulation ((IV_sim = 0.53 mm, BT_sim = 0.38 mm) are very close to that of experiment (IV_exp = 0.52 mm, BT_exp = 0.35 mm), with relative errors of 1.92% and 8.57%, respectively, which are all <10% and within an acceptable error range. Therefore, the FE model of SPR joint forming established in this paper is good and reliable.

Figure 8.

Comparison of joint’s final sectional deformation modes between simulation and experiment.¹²

Table 2.

Comparison of joint’s forming quality indicators between simulation and experiment.¹²

Indictors	Experiment	Simulation	Error (%)
IV (mm)	0.52	0.53	1.92%
BT (mm)	0.35	0.38	8.57%

BT: bottom thickness; IV: interlock value.

Parametric study

In order to reveal the forming mechanism of SPR joint, it is necessary to investigate the factors influencing the joint’s forming quality.¹⁶ Previous study had revealed that structural parameters of rivet and die have significant impact on SPR joint’s forming quality.¹² Thus, this study employed a single-factor experimental design based on finite element simulation to examine the impact of rivet parameters, that is, rivet shank thickness (R_t) and rivet shank inner diameter (R_d), and die parameters, that is, die pip height (D_h) and die inner diameter (D_d), on the forming quality of SPR joint. Herein the rivet shank thickness (R_t) refers to the wall thickness of the rivet shank, the rivet shank inner diameter (R_d) defines the internal diameter of the rivet shank, the die pip height (D_h) denotes the protrusion height in the die cavity, while the die inner diameter (D_d) specifies the inner diameter of the die cavity, which jointly govern the deformation behavior and final forming quality of SPR joint.¹⁷

Figure 9 illustrates cross-sectional profiles of rivets and dies with different above parameter values, and for each parameter three levels were considered to conduct single-factor experimental study. Specifically, R_t takes 0.9, 0.95 and 1.0 mm; R_d takes 3.3, 3.5 and 3.7 mm; D_h takes 0, 0.1 and 0.2 mm; D_d takes 8.5, 9.0 and 9.5 mm. Taking the above parameters as design variables, an experimental design table with nine samples, namely S1–S9, was constructed as shown in Table 3. The experimental samples could be divided into four groups: S1, S2, S3; S2, S4, S5; S2, S6, S7 and S2, S8, S9, which could be used to explore the individual effect of R_t, R_d, D_h and D_d, respectively, on the joint’s forming quality.

Figure 9.

Cross-sectional profiles of rivets and dies with different parameters.

Table 3.

Experimental design table.

Joint no.	R _t (mm)	R _d (mm)	D _h (mm)	D _d (mm)
S1	0.90	3.7	0.0	9.0
S2	0.95	3.7	0.0	9.0
S3	1.00	3.7	0.0	9.0
S4	0.95	3.3	0.0	9.0
S5	0.95	3.5	0.0	9.0
S6	0.95	3.7	0.1	9.0
S7	0.95	3.7	0.2	9.0
S8	0.95	3.7	0.0	8.5
S9	0.95	3.7	0.0	9.5

The effect of rivet shank thickness

Figure 10 demonstrates the simulation results of force-displacement curves of forming of SPR joints with different R_t under other parameters unchanged, and Figure 11 further compares the corresponding joint’s final sectional deformation modes and forming quality indicators (IV, BT). It can be seen that: (1) The variation of R_t has little effect on riveting force in the early stage of forming process (displacement ≤4 mm), while it has relatively significant effect in the final stage of forming process (displacement >4 mm). Specifically, under any same riveting displacement, the lager the R_t, the larger the riveting force, indicating the more difficult to complete the final forming process. (2) With the increase of R_t, the IV monotonically decreases while the BT first significantly decreases then slightly increases, indicating decrease of the joint’s overall forming quality. To sum, increasing R_t will lead to an increase in joint’s forming difficulty and a decrease in joint’s overall forming quality.

Figure 10.

Force-displacement curves of SPR joints with different R_t.

Figure 11.

Forming quality indicators and final deformation modes of SPR joints with different R_t.

The effect of rivet shank inner diameter

Figure 12 demonstrates the simulation results of force-displacement curves of forming of SPR joints with different R_d under other parameters unchanged, and Figure 13 further compares the corresponding joint’s final sectional deformation modes and forming quality indicators (IV, BT). It can be seen that: (1) The variation of R_d has little effect on riveting force in the early stage of forming process (displacement ≤4 mm), while it has relatively significant effect in the final stage of forming process (displacement >4 mm). Specifically, under any same riveting displacement, the lager the R_d, the larger the riveting force, indicating the more difficult to complete final forming process. (2) With the increase of R_t, both IV and BT monotonically decrease, indicating decrease of the joint’s overall forming quality. To sum, increasing R_d will also lead to an increase in joint’s forming difficulty and a decrease in joint’s overall forming quality.

Figure 12.

Force-displacement curve of SPR joints with different R_d.

Figure 13.

Forming quality indicators and final deformation modes of SPR joints with different R_d.

The effect of die pip height

Figure 14 demonstrates the simulation results of force-displacement curves of forming of SPR joints with different D_h under other parameters unchanged, and Figure 15 further compares the corresponding joint’s final sectional deformation modes and forming quality indicators (IV, BT). It can be seen that: (1) The variation of D_h has little effect on riveting force in the early and final stage of forming process (displacement ≤2.5 mm and displacement ≥4.5 mm), while has relatively significant effect in the middle stage of forming process (2.5 mm < displacement < 4.5 mm). Specifically, under any same riveting displacement, the lager the D_h, the larger the riveting force, indicating the more difficult to execute middle forming process. (2) With the increase of D_h, the IV monotonically increases while the BT monotonically decreases, indicating uncertainty of change of the joint’s overall forming quality. To sum up, increasing D_h will lead to an increase in joint’s forming difficulty yet an uncertain effect on joint’s overall forming quality.

Figure 14.

Force–displacement curve of SPR joints with different D_h.

Figure 15.

Forming quality indicators and final deformation modes of SPR joints with different D_h.

The effect of die inner diameter

Figure 16 demonstrates the simulation results of force-displacement curves of forming of SPR joints with different die inner diameter (D_d) under other parameters unchanged, and Figure 17 further compares the corresponding joint’s final sectional deformation modes and forming quality indicators (IV, BT). It can be seen that: (1) The variation of D_d has little effect on riveting force in the early stage of forming process (displacement ≤2.5 mm), while it has relatively significant effect in the later stages of forming process (displacement >2.5 mm). Specifically, in the middle stage of forming process (2.5 mm < displacement < 4 mm), the riveting force first increases then decrease with increase of D_d under any same riveting displacement, while in the final stage of forming process (displacement ≥4 mm), the riveting force monotonically decreases with increase of D_d under any same riveting displacement, indicating the easier to complete final forming process. (2) With the increase of D_d, the IV first significantly increases then slightly decreases, while the BT first significantly increases then significantly decreases, indicating first increase then decrease of the joint’s overall forming quality. To sum up, increasing D_d will lead to a decrease in in joint’s forming difficulty yet a fluctuating effect (first increase then decrease) on joint’s overall forming quality.

Figure 16.

Force–displacement curve of SPR joints with different D_d.

Figure 17.

Forming quality indicators and final deformation modes of SPR joints with different D_d.

Multi-objective optimization of forming quality of SPR joint

Optimizing the forming process parameters to improve the forming quality of SPR joint can effectively improve its joining performance and reliability,¹⁸ which thus is worthy of further study. For SPR joint, both IV and BT are “larger-the-better” (LTB) to improve the joint’s forming quality and thus were all taken as optimization objectives. However, as discussed earlier, different rivet and die parameters exert distinct impacts on forming quality indicators (BT, IV), in addition, BT and IV are competitive or even contradictory for some parameter, which goes beyond the capability of traditional single-objective optimization method and calls for multi-objective optimization approach to achieve optimization of the two forming quality indicators simultaneously.¹⁹ Different from time-consuming and labor-intensive surrogate model-based optimization methods adopted by most existing literature, a hybrid Taguchi–GRA method was employed in the present study to conduct the multi-objective optimization of the forming quality of SRP joint.

Taguchi–GRA method

The detailed procedure of the Taguchi–GRA method is described as follows:

Step 1: Taguchi experimental design

The Taguchi method, which adopts orthogonal experimental design and handles the experimental results based on signal-to-noise ratio (SNR), demonstrates notable merits of high efficiency and dramatic simplicity on evaluating factor effects and exploring optimal combination of factor levels for optimizing response.²⁰ In this study, the aforementioned four forming process parameters, that is, rivet shank thickness (R_t), rivet shank inner diameter (R_d), die pip height (D_h) and die inner diameter (D_d) were taken as design factors, and for each of which three distinct levels were set, as shown in Table 4. Without loss of generality, herein the middle level of each factor, marked as R_t2-R_d2-D_h2-D_d2, was taken as the baseline (or original) design. Meanwhile, the two joint’s forming quality indicators, that is, interlock value (IV) and bottom thickness (BT), were taken as two responses.²¹ To systematically investigate the effects of above factors on the two responses, a L₉(3⁴) orthogonal experimental design table was constructed by Minitab, which can efficiently reduce the number of required experiments while ensuring a comprehensive exploration of the factor space. Table 5 demonstrates the specific configuration of the experimental design table, detailing the combination of factor levels for each experimental sample, and the corresponding experimental results of the responses, that is, BT and IV, which were all taken as optimization objectives. These experimental results form the basis for the multi-objective optimization of the joint’s forming quality.

Table 4.

Design variables and corresponding levels.

		Levels
Variables	Symbols	1	2	3
Rivet shank thickness (mm)	R _t	0.9	0.95	1.0
Rivet shank inner diameter (mm)	R _d	3.3	3.5	3.7
Die pip height (mm)	D _h	0	0.1	0.2
Die inner diameter (mm)	D _d	8.5	9.0	9.5

Table 5.

Experimental results and corresponding SNRs.

	Experiment layout				Experimental results
No.	R _t (mm)	R _d (mm)	D _h (mm)	D _d (mm)	IV (mm; LTB)	SNR1 (LTB)	BT (mm; LTB)	SNR2 (LTB)
1	0.90	3.3	0	8.5	0.558	−5.067	0.42	−7.535
2	0.90	3.5	0.1	9.0	0.61	−4.293	0.358	−8.922
3	0.90	3.7	0.2	9.5	0.544	−5.288	0.51	−5.849
4	0.95	3.3	0.1	9.5	0.485	−6.285	0.45	−6.936
5	0.95	3.5	0.2	8.5	0.511	−5.832	0.425	−7.432
6	0.95	3.7	0	9.0	0.53	−5.514	0.38	−8.404
7	1.00	3.3	0.2	9.0	0.479	−6.393	0.37	−8.636
8	1.00	3.5	0	9.5	0.434	−7.250	0.392	−8.134
9	1.00	3.7	0.1	8.5	0.371	−8.613	0.29	−10.752

Step 2: Single-to-noise ratio (SNR)

The Taguchi method utilizes the signal-to-noise ratio (SNR) to assess the performance of the response. This approach effectively considers the magnitude of the average value of the measurement process in relation to its variation.²² Typically, various methods are employed to calculate the signal-to-noise ratio (SNR) based on the corresponding response characteristics. Specifically, if a response exhibits a “larger-the-better” (LTB) characteristic, its SNR can be calculated as²³:

SN R_{LTB} = - 10 \log_{10} (\frac{1}{num} \sum_{i = 1}^{num} \frac{1}{{y_{i}}^{2}})

(1)

Else, if a response demonstrates “smaller-the-better” (STB) characteristic, then its SNR could be calculated as:

SN R_{STB} = - 10 \log_{10} (\frac{1}{num} \sum_{i = 1}^{num} y_{i}^{2})

(2)

where y_i is the experimental result of response y at the ith measurement, num is the total number of measurements per experiment.

In the present study, interlock value (IV) and bottom thickness (BT) were taken as two responses (or objectives) for the multi-objective optimization of SPR joint’s forming quality. As mentioned earlier, both IV and BT are of “larger-the-better” (LTB) characteristics, thus the SNR values for BT and IV, denoted as SNR1 and SNR2, respectively, were calculated using equation (1) and listed as shown in Table 5.

Step 3: Gray relational analysis (GRA)

Typically, the Taguchi method is capable of assessing and comparing the effect of each factor on the response based on SNR and main effects analysis, and further determining the optimal combination of factor levels for a single-objective optimization.²⁴ However, traditional Taguchi method was originally introduced to optimize a single response at a time, which is unsuitable for multi-objective optimization of multiple responses simultaneously. Therefore, GRA was incorporated into Taguchi method to convert multiple responses to a single gray relational grade (GRG), that is, to be maximized. The more detailed procedure of GRA is outlined below:

Gray relational generation

First, the experimental data (from Taguchi experimental design) were preprocessed to (0, 1) using different normalization methods based on response characteristics, which is generally named as gray relational generation. Specifically, for a response with “larger-the-better” (LTB) characteristic²⁴:

y_{i} (k) = \frac{x_{i} (k) - min x_{i} (k)}{max x_{i} (k) - min x_{i} (k)}

(3)

Else, for a response with “smaller-the-better” (STB) characteristic:

y_{i} (k) = \frac{max x_{i} (k) - x_{i} (k)}{max x_{i} (k) - min x_{i} (k)}

(4)

where $x_{i} (k)$ and $y_{i} (k)$ , respectively, denote the original and normalized value of k th response at ith experiment; $max x_{i} (k)$ and $min x_{i} (k)$ , respectively, denote the maximum and minimum value of $x_{i} (k)$ for the k th response among all experiments.

2. Gray relational coefficient (GRC)

Second, the normalized experiment data were utilized to compute the GRC $r_{i} (k)$ , which quantifies the degree of correlation between the current experiment and the desired solution according to the following equation:

r_{i} (k) = \frac{\min_{i} \min_{k} | y_{0} (k) - y_{i} (k) | + δ \max_{i} \max_{k} | y_{0} (k) - y_{i} (k) |}{| y_{0} (k) - y_{i} (k) | + δ \max_{i} \max_{k} | y_{0} (k) - y_{i} (k) |}

(5)

where $y_{0} (k)$ is the ideal normalized value, that is, the maximum normalized SNR for the k th response among all experiments. δ denotes the identification coefficient varying in interval (0, 1), δ = 0.5 was adopted in this paper as generally recommended.

3. Gray relational grade (GRG)

Finally, the gray relational grade (GRG), which quantifies the degree of correlation between the current experiment and the ideal solution, can be calculated by averaging the GRCs obtained from multiple responses:

ε_{i} = (\frac{1}{n}) \sum_{k = 1}^{n} r_{i} (k)

(6)

However, assigning different weights to different responses to calculate a weighted GRG when accounting for differences in response preference or significance:

ε_{i ω} = \sum_{k = 1}^{n} ω_{k} r_{i} (k)

(7)

where ε_i and $ε_{i ω}$ represent the GRG and weighted GRG for i th experiment, ω_k represents the weight of the k th response and n represents the total number of responses.

For multi-objective optimization of the SPR joint’s forming quality in this study, n = 2 as IV and BT were selected as the two responses. Note that the GRG can be considered as an overall indicator response of the SPR joint’s forming quality encompassing IV and BT. Obviously, larger value of ε_i or ε_iw signifying the corresponding combination of factor levels is closer to the optimal condition is preferred.

Step 4: Main effect and interaction effect analyses

With the assistance of GRA, the multi-response optimization of IV and BT for improving SPR joint’s forming quality has been transformed to a single-response optimization of GRG, which now allows the Taguchi analysis to explore the optimal combination of factor levels. Specifically, the average response value of GRG for each level of each factor was first calculated, then the main effect value of each factor on GRG was determined by calculating the difference between the maximum and minimum average response values of GRG among all factor levels. The calculation process can be detailed as follows²⁴:

\bar{f_{p}} = (\frac{1}{N}) \sum_{q = 1}^{N} ε_{pq}

(8)

Δ f = \max (\bar{f_{1}}, \bar{f_{2}} . . ., \bar{f_{M}}) - \min (\bar{f_{1}}, \bar{f_{2}} . . ., \bar{f_{M}})

(9)

where $\bar{f_{p}}$ represents the average GRG value for factor f at p th level, ε_pq represents the GRG of q th experiment with factor f at p th level, N denotes the total number of experiments with factor f at p th level, M represents the total number of levels of factor f and Δf represents the main effect value of factor f.

The average response value of GRG at each factor level, the main effect value of each factor on GRG, and the interaction effect value of any two factors on GRG can be obtained and demonstrated in a factor response table, a main effect plot and an interaction effect plot, respectively. Based on which, the main effect of any one factor and the interaction effect of any two factors on GRG could be figured out, moreover, the optimal combination of factor levels for optimal response value of GRG could be determined. In general, the larger the main effect value of a factor, the bigger the impact of the factor on the response, and vice versa. In contrast, the interaction effect of value of any two factors on response could be measured through interaction effect plots, where parallel lines in an interaction plot indicate no interaction, and the greater the difference in slope between the lines, the higher the degree of interaction.²⁴

Results and discussion

Table 5 presents the SNR ratios (SNR1 of IV and SNR2 of BT) calculated for each experimental result based on equation (1). The response tables for SNR1 of IV and SNR2 of BT were calculated and displayed in Tables 6 and 7, respectively. For a given factor, the mean SNR at a certain level stands for the average SNR of all experiments with this specific factor level, while the Delta denotes the maximum difference of SNR between any two levels, which measures the primary effect of that factor on the SNR. Specifically, factor with higher Delta value exerts relatively larger impact on the SNR (or corresponding response). Thus, according to Table 6, the most influential factor on SNR1 (or IV) is rivet shank thickness (R_t), followed by die inner diameter (D_d), rivet shank inner diameter (R_d) and die pip height (D_h) in sequence. In contrast, according to Table 7, the most influential factor on SNR2 (or BT) is rivet shank thickness (R_t), followed by die inner diameter (D_d), die pip height (D_h) and rivet shank inner diameter (R_d) in sequence. The above rank of factor influence could also be figured out by comparing the slopes of lines in the main effect plots of factors on SNR1 of IV and SNR2 of BT, as shown in Figures 18 and 19, respectively. Nevertheless, a more significant function of main effect plots is to determine the optimal combination of factor levels for each response. Specifically, to maximize SNR1 (namely maximizing IV), the optimal combination of factor levels is R_t1-R_d2-D_h3-D_d2 (namely R_t = 3.3 mm, R_d = 0.95 mm, D_h = 0.2 mm and D_d = 9.0 mm). In contrast, to maximize SNR2 (namely maximizing BT), the optimal combination of factor levels is R_t1–R_d1–D_h3–D_d3 (namely R_t = 3.3 mm, R_d = 0.90 mm, D_h = 0.2 mm and D_d = 9.5 mm).

Table 6.

Response table for SNR1 of IV.

Means	Level	R _t	R _d	D _h	D _d
SNR1 (LTB)	1	−4.883	−5.915	−5.944	−6.504
	2	−5.877	−5.792	−6.397	−5.400
	3	−7.419	−6.472	−5.838	−6.274
	Delta	2.536	0.680	0.559	1.103
	Rank	1	3	4	2
	Total mean	−6.060

Table 7.

Response table for SNR2 of BT.

Means	Level	R _t	R _d	D _h	D _d
SNR2 (LTB)	1	−7.435	−7.702	−8.025	−8.573
	2	−7.591	−8.163	−8.870	−8.654
	3	9.174	−8.335	−7.306	−6.973
	Delta	−1.739	0.633	1.564	1.681
	Rank	1	4	3	2
	Total mean	−8.067

Figure 18.

Main effect plots of factors on SNR1.

Figure 19.

Main effect plots of factors on SNR2.

In addition, to explore the influence of interactions among factors on a single response, the interaction effect plots of factors on SNR1 of IV and SNR2 of BT are shown in Figures 20 and 21, respectively. An interaction plot is a plot of means for each level of a factor with the level of a second factor kept constant, which is capable of assessing the two-way interactions among factors on the responses. Specifically, there is no interaction if the lines are parallel, otherwise the greater the lines deviate from being parallel, the larger the interaction.²⁴ Evidently, no matter for SNR1 or SNR2, the interaction effect is observed between either two of the four factors, indicating the relationship between any one factor and SNR1 or SNR2 depends on the remaining three factors.

Figure 20.

Interaction effect plots of factors on SNR1 of IV.

Figure 21.

Interaction effect plots of factors on SNR2 of BT.

As disclosed above, the optimal combination of factor levels for maximizing IV (R_t1–R_d2–D_h3–D_d2) differs with that for maximizing BT (R_t1–R_d1–D_h3–D_d3), indicating the traditional Taguchi method introduced for optimizing a single response at a time could not figure out an optimal combination of factor levels maximizing IV and BT simultaneously, which thus is unsuitable for handling multi-objective optimization of IV and BT for improving joint’s forming quality. To solve this difficulty, this study employed gray correlation analysis (GRA) to convert the two competing responses (SNR1, SNR2) to a single one, that is, gray relational grade (GRG) that needs to be maximized (LTB). Table 8 listed the gray relational generation of SNR1 and SNR2 based on equation (3), the gray relational coefficient (GRC) of SNR1 and SNR2 based on equation (5), the gray relational grade (GRG) based on equation (6) or (7), as well as the rank of GRG values sorted in descending order. Likewise, the response table for GRG is displayed in Table 9, from which it can be seen that the most influential factor on GRG is rivet shank thickness (R_t), followed by die inner diameter (D_d), die pip height (D_h) and rivet shank inner diameter (R_d) in sequence. The main effect plot for GRG is shown in Figure 22, from which it can be seen that the optimal combination of factor levels for GRG is R_t1–R_d2–D_h3–D_d3 (namely R_t = 0.90 mm, R_d = 3.5 mm, D_h = 0.2 mm and D_d = 9.5 mm). In addition, Figure 23 demonstrates the interaction plot for GRG, likewise the interaction effect is observed between either two of the four factors, indicating the relationship between any one factor and GRG depends on the remaining three factors.

Table 8.

GRC and GRG.

	Experiment levels				Gray relational generation		GRC
No.	R _t	R _d	D _h	D _d	SNR1 (LTB)	SNR2 (LTB)	SNR1 (LTB)	SNR2 (LTB)	GRG (LTB)	Rank
Weight					0.500	0.500	0.500	0.500
Ideal	/	/	/	/	1.000	1.000	1.000	1.000
1	1	1	1	1	0.821	0.65	0.736	0.593	0.665	3
2	1	2	2	2	1.000	0.373	1.000	0.444	0.722	2
3	1	3	3	3	0.770	1.000	0.685	1.000	0.843	1
4	2	1	2	3	0.539	0.778	0.520	0.693	0.607	4
5	2	2	3	1	0.644	0.677	0.584	0.608	0.596	5
6	2	3	1	2	0.717	0.479	0.639	0.490	0.565	6
7	3	1	3	2	0.514	0.432	0.507	0.468	0.488	7
8	3	2	1	3	0.316	0.534	0.422	0.518	0.470	8
9	3	3	2	1	0.000	0.000	0.333	0.333	0.333	9

GRC: gray relational coefficient; GRG: gray relational grade.

Table 9.

Response table for GRG.

Means	Level	R _t	R _d	D _h	D _d
GRG (LTB)	1	0.743	0.587	0.567	0.531
	2	0.589	0.596	0.554	0.592
	3	0.430	0.580	0.642	0.640
	Delta	0.313	0.016	0.088	0.109
	Rank	1	4	3	2
	Total mean	0.588

Figure 22.

Main effect plots of factors on GRG.

Figure 23.

Interaction effect plots of factors on GRG.

Verification of optimization results

Once the optimal combination of factor levels, as well as corresponding parameter values, for better joint’s overall forming quality (IV and BT) was determined, namely R_t1–R_d2–D_h3–D_d3 (R_t = 0.90 mm, R_d = 3.5 mm, D_h = 0.2 mm and D_d = 9.5 mm), the subsequent step is to verify the effectiveness of optimization outcomes. Specifically, the SPR joint’s forming quality using optimized forming process parameters (R_t1–R_d2–D_h3–D_d3) were compared with that using baseline (or original) forming process parameters (R_t2–R_d2–D_h2–D_d2). Figure 24 compares the joint’s sectional deformation modes during forming process before and after optimization, it can be seen that the deformation modes during forming process before and after optimization are quite similar overall, whereas small changes can still be detected. Take the joint’s sectional deformation mode at 0.7% of the simulation time (0.7 t) as an example, it can be seen that the opening amplitude at the end of the rivet leg after optimization is relatively greater than that of original, which might signify the joint’s IV and BT have been improved to a certain extent. In addition, Figure 25 compares the joint’s final sectional deformation modes and forming quality indicators (IV and BT) before and after optimization, Figure 26 more clearly demonstrate the change of joint’s forming quality indicators before and after optimization. It can be seen that compared with that using original forming process parameters, the joint’s IV and BT using optimized forming process parameters gain simultaneous increase by 7.41% and 40.63%, respectively, indicating a good multi-objective optimization of the joint’s forming quality was achieved.

Figure 24.

Comparison of joint’s sectional deformation modes during forming process.

Figure 25.

Comparison of joint’s final sectional deformation mode and forming quality indicators.

Figure 26.

Change of joint’s forming quality indicators before and after optimization.

Conclusion

In this study, a systematic approach was proposed for predicting and optimizing the forming quality of SPR joint between aluminum alloy sheets. The primary findings can be summarized as follows:

The simulation result of final sectional deformation mode of joint forming is basically consistent with that of experiment, and the IV and BT of simulation are very close to that of experiment with relative errors of 1.92% and 8.57%, respectively, which are all <10% and within an acceptable error range. Therefore, the FE model of SPR joint forming established is reliable.

According to the parameter study, increasing R_t or R_d will lead to an increase in joint’s forming difficulty and a decrease in joint’s overall forming quality. In contrast, increasing D_h will lead to an slight increase in joint’s forming difficulty yet an uncertain effect on joint’s overall forming quality, whereas increasing D_d will lead to a decrease in in joint’s forming difficulty yet a fluctuating effect (first increase then decrease) on joint’s overall forming quality.

According to the main effect analysis, there is a difference in the sorting of factor influence between on a single response alone (interlock value or bottom thickness) and on multi-response together (GRG). In addition, evident interaction effects were observed between either two of the four factors on a single response alone and on multi-response together.

Compared with that using original forming process parameters, the joint’s IV and BT using optimized forming process parameters gain simultaneous increase by 7.41% and 40.63%, respectively, indicating a good multi-objective optimization of the joint’s forming quality was achieved, which provides an effective new method for predicting and optimizing the forming quality of SPR joints.

Footnotes

Handling Editor: Shamik Basak

ORCID iDs

Feng Xiong

Jiayi Wei

Shuai Zhang

Funding

The authors disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant no. 52202437), Foundation of State Key Laboratory of Automotive Simulation and Control (no. 20210213), Research and Innovation Team Cultivation Plan of Chongqing University of Technology (2023TDZ013), Graduate Education High Quality Development Project of Chongqing University of Technology (no. CYS240667), Science and Technology Research Project of Henan Province (242102241055), Industry-University-Research Collaborative Innovation Base Project on Automobile Lightweight of “Science and Technology Innovation in Central Plains” (2024KCZY315). The authors would like to express their appreciation for the above fund supports. In addition, the authors have no relevant financial or non-financial interests to disclose.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

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Research and multi-objective optimization of forming quality of SPR joint using Taguchi–gray relational analysis

Abstract

Keywords

Introduction

Finite element modeling and verification

Geometrical modeling

Material description

Finite element modeling

Model validation

Parametric study

The effect of rivet shank thickness

The effect of rivet shank inner diameter

The effect of die pip height

The effect of die inner diameter

Multi-objective optimization of forming quality of SPR joint

Taguchi–GRA method

Step 1: Taguchi experimental design

Step 2: Single-to-noise ratio (SNR)

Step 3: Gray relational analysis (GRA)

Step 4: Main effect and interaction effect analyses

Results and discussion

Verification of optimization results

Conclusion

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

References