Parameter Identification of the Yoshida-Uemori Hardening Model for Remanufacturing

Abstract

The deformation of plastics during production and service means that retired parts often possess different mechanical states, and this can directly affect not only the properties of remanufactured mechanical parts, but also the design of the remanufacturing process itself. In this paper, we describe the stress-strain relationship for remanufacturing, in particular the cyclic deformation of parts, by using the particle swarm optimization (PSO) method to acquire the Yoshida-Uemori (Y-U) hardening model parameters. To achieve this, tension-compression experimental data of AA7075-O, standard PSO, oscillating second-order PSO (OS-PSO) and variable weight PSO (VW-PSO) were acquired separately. The influence of particle numbers on the inverse analysis efficiency was studied based on standard PSO. Comparing the results of PSO variations showed that: (1) standard PSO is able to avoid local solutions and obtain Y-U model parameters to the same degree of precision as the OS-PSO; (2) by adjusting section weight, the VW-PSO could improve local fitting accuracy and adapt to asymmetric deformation; (3) by reducing particle numbers to a certain extent, the efficiency of analysis can be improved while also maintaining accuracy.

1. Introduction

Remanufacturing is the process of transforming retired products back to their original state and restoring their performance. A sustainable approach to the development and use of materials [1], remanufacturing is considered an essential part of green manufacturing [2]. Currently, remanufacturing research mainly focuses on schemes design, economic applicability and environmental protection [3,4,5]. Few studies have been undertaken on the remanufacturing process itself, which is the basis for remanufacturing implementation. Throughout the process of plastic deformation, which occurs both during production and service, the material properties of parts are changed, including their elastic modulus, yield stress, flow stress, formability, etc. This results in complex material conditions that have the potential to affect the design of remanufacturing processes and determine the mechanical properties of remanufactured parts. Supposing that the retired parts are reformed to their original shapes, this involves a cyclical deformation with symmetrical plastic strain. If the retired parts are reformed into other shapes, the plastic strain is an asymmetric cycle. Research on material deformation behaviors is therefore necessary to aid the design of the remanufacturing process.

A constitutive model is essential when describing material deformation behaviors. In the Y-U hardening model, both the translation and expansion of the bounding surface are considered as the active yield surface only evolves in a kinematic manner. Chongthairungruang [6] has studied the springback effect of different steels, and the Y-U hardening model has been shown to have a comparatively small margin of error compared to this experiment. Based on the Bron-Besson yield function and the Y-U hardening model, the anisotropy hardening behavior of mild steel DC04 and dual phase steel DP500 is described with work-hardening stagnation in the reversed strain path [7]. A sequential response surface built on the Y-U hardening model by Toros et al. [8] showed the same high accuracy in the cyclic stress-strain of several aluminum alloys. In different combinations of yield functions and hardening models, the Y-U hardening model predicts the U-bending springback of DP780 steel with a higher accuracy [9]. When compared to the IH and IH+NKH models, the Y-U hardening model is better at predicting the cyclic behavior and springback of U-shaped channel sections of AA 6022 and HSLA [10]. Similarly, Yan et al. [11] researched the hardening behavior of the QP steel using the Y-U hardening model, with improved fitting accuracy. When considering the Bauschinger effect—the permanent softening and hardening hysteresis effect under tension-compression—the Y-U hardening model is suitable for describing the large cyclic strains that occur during the remanufacturing process. However, while fitting precision is directly determined by the parameters, the inherent characteristics of iterative solutions makes it impossible to use the traditional conjugate gradient algorithm for the identification of Y-U hardening model parameters.

Y-U hardening model parameters are mostly calibrated by the semi-analytical method and the inversion method. However, the semi-analytical method can only determine the material parameters to an acceptable level of accuracy [12,13]. Higher levels of accuracy can be obtained by inverse analysis using intelligent optimization algorithms. Zhu et al. [14] adopted the successive response surface methodology and an adaptive simulated annealing algorithm to obtain the Y-U model parameters of AA 5182-O. Leem et al. [15] employed genetic algorithms to perform the Y-U hardening model calibration for AA 5754-O. Hassan et al. [16] set up a one-element model for simple shear in the finite element software LS-DYNA, and achieved the Y-U model parameters via the curve mapping algorithm, which accurately predicted the springback of the DP600 material. Xu et al. [17] used the simulated annealing algorithm to perform parameter inversion and the accuracy was verified by the high-strength steel V-bend test. Shao et al. [18] used OS-PSO to perform parameter inversion on the Y-U hardening model of aluminum alloy and obtained a fitting accuracy closer to the test curve depending on the nature of the algorithm. The above research shows that the inversion method can obtain parameters with a high level of accuracy but the calculation process is ignored. Moreover, the inversion method performs the overall curve fitting lacking of attention to local precision control.

In this paper, a procedure consisting of variations of the PSO method was adopted in order to inversely determine the parameters of the Y-U model. To illustrate the process of identifying the parameters and to explore the influence of learning factors, the standard PSO and the OS-PSO were studied in comparison. Subsequently, the VW-PSO was used to improve the fitting accuracy for tensile flow stress, Bauschinger effect and permanent softening separately. Additionally, the inversion efficiency with different particle numbers was compared on the standard PSO. As a result of the improved fitting accuracy of the Y-U hardening model, we were able to increase the accuracy of predicting either symmetric or asymmetric cyclic deformation for the remanufacturing process.

2. Methodology

The inversion procedure combines two sections: the graphical section and the PSO section. The graphical section is used to obtain an approximate solution and the PSO section tries to further reduce the residual errors between the results of the experiment and the approximate solution.

The Y-U hardening model is a stress-strain curve that can be obtained through the uniaxial cyclic tension-compression experiment [19]. The model contains seven parameters, namely: the material yield strength Y, the material parameter C that controls the dynamic hardening rate, the material boundary surface parameter cb, the saturation value R_sat, the material parameters k and sb that control the isotropic hardening and the material parameter h that controls the work hardening hysteresis.

The inversion procedure is shown in Figure 1 and is completed as follows:

(1)

Set the parameters’ range and initialize the particle swarm. As with the stress-strain response curve of the tensile-compression experiment, the approximate parameters group for the Y-U hardening model is calculated analytically. Based on these calculations, the parameter range should be narrowed, which will potentially reduce the PSO section iterations. The particle swarm including i groups of parameters should be randomly selected.

(2)

Predict the response curve by simulation. The simulation uses a four-node shell element, with an element size of 1 × 1 mm. The cyclic displacement is loaded in the X-axis direction (the boundary conditions are shown in Figure 2). Import the i groups of parameters for the jth iteration to obtain the predicted response curves.

(3)

Calculate the residual error as fitness value. Determine whether the residual error between the experiment results and the simulation results satisfies the accuracy requirement. If the accuracy is satisfied, skip to step 5. Otherwise, continue to step 4.

(4)

Proceed to the next iteration. Take the i groups of parameters with residual errors as fitting values into the particle swarm iteration, generate the new particle swarm and jump to step 2 for the j + 1st iteration.

(5)

Terminate the iteration with an optimal particle. Obtain optimized parameters group with satisfied residual error for the Y-U hardening model.

2.1. Yoshida-Uemori Hardening Model

The Yoshida-Uemori hardening model adopts combined isotropic-kinematic hardening, containing three different surfaces as yield surfaces, bounding surfaces and hardening stagnation surfaces. While the yield surface f represents kinematic hardening in the bounding surface F, the F is determined by expansion and translation of the hardening stagnation surfaces g. The forms and evolutions of f, F and g are given as follows:

$$f=\frac{3}{2} \left(s-\alpha \right):\left(s-\alpha \right)-Y^2=0,$$

(1a)

$${\dot{\alpha }}_{\ast }=\alpha -\beta =C\left[\left\{\frac{\left(cb\right)+R-Y}{Y}\right\}\left(\sigma -\alpha \right)-{\alpha }_{\ast }\sqrt{\frac{\left(cb\right)+R-Y}{{\overline{\alpha }}_{\ast }}}\right]\dot{\overline{\epsilon }},$$

(1b)

$$F=\frac{3}{2} \left(s-\beta \right):\left(s-\beta \right)-{\left(\left(cb\right)+R\right)}^2=0,$$

(2a)

$$\dot{R}=k\left(R_{sat}-R\right)\dot{\overline{\epsilon }},$$

(2b)

$${\dot{\beta }}^{\prime }=k\left\{\left(\frac{2}{3}\right)\left(sb\right)D^p-{\beta }^{\prime }\dot{\overline{\epsilon }}\right\},$$

(2c)

$$g=\sqrt{\frac{2}{3}}\Vert s-q\Vert -r=0,$$

(3a)

$${\dot{q}}^{\prime }=\left[\frac{3\left({\beta }^{\prime }-q^{\prime }\right):{\dot{\beta }}^{\prime }}{2r^2}-\left(\frac{\dot{r}}{r}\right)\right]\left({\beta }^{\prime }-q^{\prime }\right),$$

(3b)

$$\dot{r}=\left(\frac{3h}{2r}\right)\left({\beta }^{\prime }-q^{\prime }\right):{\dot{\beta }}^{\prime } when \dot{R}>0, \dot{r}=0 if \dot{R}=0,$$

(3c)

where s is Cauchy stress, Y and α are the size and back-stress deviator of yield surface f, β, cb and R are the center, original size and isotropic hardening part of F. $D^p$ is the plastic rate of the deformation tensor. C, R_sat, sb and k are the material parameters. q and r are the center and size of surface g. When β stays on the surface g, the F is supposed to be isotropic hardening.

2.2. Graphical Section

The graphical section, containing four steps, is shown in Figure 3.

(1)

Determine the parameter Y. Fit the data points of the elastic deformation stage to a straight line. Offset 0.2% to intersect with the experiment stress-strain curve and obtain the parameter Y value.

(2)

Determine the parameter cb and k. The voce hardening model is used to describe the tensile flow stress. Fit 50% of the curve with Equation (4) and get the values of cb, k and sb + R_sat.

$${\sigma }_{iso}^{\left(Voce\right)}=cb+\left(sb+R_{sat}\right)\left(1-e^{-k{\epsilon }^p}\right),$$

(4)

where σ is the true stress value and ${\epsilon }^p$ is the plastic strain.

(3)

Determine the parameters sb and R_sat. While uniaxial compression reaches the plastic strain, point (k) represents the fixed isotropic boundary. Point (j) is the intersection of the permanent softening fitting line and the maximum tensile strain. Equation (5) describes the relationship between point (k) and point (j). The parameter sb can be calculated with the known parameter k to achieve the parameter R_sat.

$${\sigma }^{\left(p\right)}=2{\beta }_0=2sb\left(1-e^{-k{\epsilon }^p}\right)$$

(5)

(4)

Determine parameters C and h. C is involved in the stress-strain relationship as follows:

$${\sigma }_B^{\left(t\right)}\approx 2a{e}^{-C{\widehat{\epsilon }}^p},$$

(6)

where ${\sigma }_B^{\left(t\right)}$ is the difference between the permanent softening fitting line and the transient Bauschinger effect; ${\epsilon }^p$ is the maximum tensile strain.

h controls the work hardening hysteresis. Former research [11,12,13,14] indicates that h has no evident influence on fitting accuracy and is generally set to 0.3–0.5.

2.3. PSO Section

PSO is a swarm intelligent stochastic optimization algorithm that solves related optimization problems by simulating the intelligent predation behavior of birds [20]. After each iteration, the particle updates its speed and position with the individual’s best position and the swarm’s best position.

$$v_{id}^{k+1}=\omega v_{id}^k+c_1{r}_1\left(p_{id}^k-x_{id}^k\right)+c_2{r}_2\left(p_{md}^k-x_{id}^k\right),$$

(7)

$$x_{id}^{k+1}=x_{id}^k+v_{id}^{k+1},$$

(8)

where ω is the weight of inertia; d = 1, 2, …, D; i = 1, 2, …, N; k is the current iteration number; v_id is the velocity of the particle, v_id ∈ [-v_max; v_max]; c₁, c₂ are non-negative learning factors; r₁, r₂ are random numbers distributed in the interval [0, 1].

While simulation can predict the stress-strain response curve for each particle, the identification of parameters turns into a curve fitting between experiment and simulation. Establish the corresponding objective function and minimize the residual error:

$$\{\begin{array}{c} \min F\left(X\right)=\sqrt{\sum _{j=1}^M{[g\left(e_j\right)-s_j]}^2}/\sqrt{\sum _{j=1}^M{s}_j^2}\\ \begin{array}{c}\\ s.t.Xϵ\left[X^{lower},X^{upper}\right] \end{array}\end{array},$$

(9)

where F(X) is the objective function and stands for the residual error; g(e_j) is the response stress, where e_j is the strain value on the experimental curve; s_j is the simulated stress value corresponding to e_j; M is the total number of data points; X = [x₁, x₂, x₃, x₄, x₅, x₆, x₇] are variables including x₁ = cb, x₂ = Y, x₃ = sc, x₄ = k, x₅ = R_sat, x₆ = sb and x₇ = h. X^lower and X^upper are the lower and upper limits of variables respectively.

2.4. PSO Variations

In standard PSO, ω = 1 and c₁ = c₂ = 2. The standard PSO maintains a fixed search intensity during the optimization procedure and risks plunging into the local optimum. The OS-PSO is characterized by strengthening the global search in the early stage iteration and strengthening the local search in the later stage iteration in order to avoid falling into the local optimal solution. The evolution equation is as follows:

$$v_{id}^{t+1}=\omega v_{id}^t+c_1{r}_1\left(p_{id}^t-\left(1+ks1\right)\ast x_{id}^t-ks1\ast x_{id}^{t-1}\right)+c_2{r}_2\left(p_{md}^t-\left(1+ks2\right)\ast x_{id}^t-ks2\ast x_{id}^{t-1}\right).$$

(10)

In the early stage of OS-PSO, the second order oscillation increases the particle diversity and thus enhances the global searching ability, resulting in oscillatory convergence:

$$ks1<\frac{2\sqrt{c_1{r}_1}-1}{c_1{r}_1}; ks2<\frac{2\sqrt{c_2{r}_2}-1}{c_2{r}_2}$$

(11)

In the later stage of OS-PSO, progressive convergence possesses better local searching ability by jumping out of the local optimal solution:

$$ks1\ge \frac{2\sqrt{c_1{r}_1}-1}{c_1{r}_1}; ks2\ge \frac{2\sqrt{c_2{r}_2}-1}{c_2{r}_2}$$

(12)

The above PSO variations focus on the overall fitting accuracy control. For the intervals with clear deformation characteristics, improved accuracy will potentially enhance remanufacturing design and prediction, especially for the asymmetric load processing of the retired parts into different products. Therefore, the VW-PSO, by modifying the interval weight in the objective function, is created to increase the fitting accuracy of specific intervals. The redesigned objective function is:

$$\{\begin{array}{c} \min F\left(X\right)=\frac{\sqrt{{\sum }_{j=1}^M{[g\left(e_j\right)-s_j]}^2}}{\sqrt{{\sum }_{j=1}^M{s}_j^2}}j\in \left[1, a\right)\cup \left(b,M\right]\\ \min F\left(X\right)=x\frac{\sqrt{{\sum }_{j=a}^b{[g\left(e_j\right)-s_j]}^2}}{\sqrt{{\sum }_{j=a}^b{s}_j^2}}j\in \left[a, b\right] \end{array},$$

(13)

where [a, b] is the interval to be optimized and x is the interval weight coefficient.

Three PSO variations are comparatively shown in

Table 1. PSO variations comparison.

Algorithm	Algorithm Description
	Weights	Learning Factor	Update Speed	Update Location	Objective Function
Standard PSO	1	c1 = c2 = 2	Equation (7)	x = x + v	Equation (9)
OS-PSO			Equation (10)		Equation (9)
VW-PSO			Equation (7)		Equation (13)

.

In the process of parameter inversion, it is not only necessary to ensure the fitting accuracy but also to control the inversion time in order to improve the inversion efficiency. Finite element simulation is the most time-consuming part in the inversion process. Reducing the simulation times will reduce computation load, resulting in a more efficient inversion. By comparing accuracy and computation load using different particle numbers, the inversion process can be balanced to high levels of accuracy and efficiency.

3. Case Study

3.1. Optimization Process

AA 7075-O was employed for the case study and the tension-compression experiment results are shown in Figure 4 [15]. The sheet specimen was restrained in the normal direction with a self-designed fixture to prevent buckling during compression, while lubrication on both sides of the sheet specimen ensured free tension and compression. Deformation of the gauge segment was captured by CCD cameras on a Sintech 20 G universal material testing machine. The optimization space was set between 50% and 200% of the graphical solution. Specifically, the lower limit was [30, 150, 10, 20, 10] and the upper limit was [120, 700, 90, 200, 80]. Parameter Y, the initial yield stress, was accurately acquired, and parameter h was set to 0.4.

The particle swarm contained 42 sets of randomly selected parameters, for which the corresponding stress-strain curve was explicitly calculated with LS-DYNA. The particle speed range was set to [(−3, 3); (−8, 8); (−3, 3); (−3, 3); (−3, 3)] for the iteration. Iteration for PSO variations was processed three times to prove the stability of the procedure. With standard PSO, the simulated stress-strain response curves and the corresponding parameters gradually converged to the optimal solution, as shown in Figure 5 and Figure 6.

For optimization with VW-PSO, three different intervals (shown in Figure 4), described as tensile flow yield interval (interval I), instantaneous Bauschinger effect interval (interval II) and permanent softening interval (interval III), were separately selected to enhance accuracy, with the weight coefficient equal to 15. These intervals were selected because they are the key sections used to describe the deformation characteristics of materials. The tensile flow yield interval, representing the plastic deformation and hardening with tension load, is important in the plastic forming process, as stamping and deep drawing predict the deformation state after formation. The instantaneous Bauschinger effect interval shows the plastic state entering compression load, and this could be employed to estimate the occurrence of plastic deformation when the retired part is reformed to its original shape. The permanent softening interval reflects the plastic deformation and hardening with compression load, which could be used to predict the deformation state after reforming with a reverse load.

For optimization with standard PSO, various particle numbers, including 10, 20, 30 and 42, were tested in order to study the particle number′s influence on the optimization procedure and results.

3.2. Results

The optimal parameters of standard PSO and OS-PSO are summarized in

Table 2. Optimal parameters with standard PSO and OS-PSO.

Method	cb	Y (MPa)	C	k	Rsat	sb	h	Residual
Graphical method	67.31	80.87	312.23	46.98	110.35	27.99	0.4	13.09%
GA [15]	99.05	79	650.22	29.49	55.06	59.21	0.3	6.84%
Standard PSO	98.27	80.87	584.87	44.05	47.16	37.00	0.4	4.01%
OS-PSO	95.35	80.87	589.50	42.37	55.14	37.61	0.4	4.05%

cb: the material boundary surface parameter; Y: the material yield strength; C: the material parameter; k and sb: the material parameters that control the isotropic hardening; R_sat: the saturation value; h: the material parameter that controls the work hardening hysteresis.

. While the graphical solution left a 13.09% residual error, indicating a large deviation between the experiment and the simulation, both standard PSO and OS-PSO resulted in an improved residual error of around 4%. The genetic algorithm (GA) resulted in a comparatively lower accuracy, with a residual error of 6.84%.

In relation to the different intervals, the VW-PSO method yielded varying optimal parameters, as presented in

Table 3. Optimal parameters with VW-PSO.

Interval	cb	Y (MPa)	C	k	Rsat	sb	h	Residual
interval I	94.61	80.87	718.65	42.04	54.46	46.61	0.4	5.14%
interval II	109.03	80.87	644.45	53.04	37.39	28.53	0.4	4.96%
interval III	106.02	80.87	626.27	34.04	48.02	38.61	0.4	4.15%

. C shows a visible increase in the three sets of optimal parameters in comparison to standard PSO. The results of k, R_sat and sb fluctuated in standard PSO, while cb barely changed. The residual error of each interval was reduced and resulted in an improvement accuracy of around 40%, as shown in

Table 4. Target interval accuracy improvement.

Interval	Standard PSO	Variable Weight PSO	Improvement
interval 1	4.64%	3.09%	33.41%
interval 2	3.48%	1.93%	44.54%
interval 3	2.95%	1.61%	45.42%

. Although VW-PSO focused on the target interval accuracy, the global residual error was ever so slightly increased within an acceptable range.

In terms of the varying particle numbers, standard PSO yielded different optimal parameters, as shown in

Table 5. Optimal parameters with varying particle numbers.

i	cb	Y (MPa)	C	K	Rsat	sb	h	Residual
42	98.27	80.87	584.87	44.05	47.16	37.00	0.4	4.01%
30	97.05	80.87	607.24	46.48	46.53	38.47	0.4	4.05%
20	98.31	80.87	582.83	49.06	50.52	32.83	0.4	4.03%
10	101.96	80.87	517.54	43.29	49.63	33.17	0.4	4.25%

. Compared to standard PSO, VW-PSO with 10 particles resulted in optimal parameters with larger fluctuations. Specifically, C fluctuated within a small range with a particle number above 20, while its value decreased with a particle number of 10. The residual error remained relatively stable at around 4% with a particle number above 20, and increased visibly with a particle number of 10, which indicates that 10 particles is insufficient for the iteration.

3.3. Discussion

The stress-strain response curve optimized with GA and standard PSO shows a different fitting accuracy in each quadrant. As shown in Figure 7, standard PSO described a uniformly distributed residual error while GA depicted a high accuracy in the first quadrant. This could be for two reasons: first, insufficient population diversity during the iteration might have held the result in a local optimal solution; second, the built-in fitness function in LS-DYNA may have been rewritten and turned out to be different from the overall residual error.

In a comparative study of the OS-PSO, standard PSO proved to be able to converge to the optimal solution for the Y-U hardening model, as shown in Figure 8. With a global search in the early stage, OS-PSO presented a relatively fast convergence rate in comparison to standard PSO. Upon reaching the 13th iteration in the middle stage, both methods encountered the same residual error. In the later stage, OS-PSO converged more slowly than standard PSO due to the oscillating search mode. While standard PSO reduced the residual error iteration by iteration, OS-PSO imposed oscillation by trying to jump out of the solution. Finally, standard PSO achieved the same level of residual error as OS-PSO, at around 4%. The optimization procedure and result confirm that standard PSO is appropriate for the identification of Y-U hardening model parameters.

VW-PSO is designed for asymmetric strain envelopes in order to reduce residual error accumulation during cyclic loading. With regard to the remanufacturing process, the strain envelope may not be symmetrical for tension and compression depending on the remanufactured parts. For VW-PSO with an interval of I, the tensile flow yield prediction is shown in Figure 9a, and the overall response stress-strain curve is shown in Figure 9b, which is suitable for processes with large tension and small compression strain envelopes. For VW-PSO with an interval of II, the instantaneous Bauschinger effect prediction is shown in Figure 9c, and the overall response stress-strain curve is shown in Figure 9d, which is suitable for processes with less tension and small compression strain envelopes. For VW-PSO with an interval of III, the permanent softening prediction is shown in Figure 9e, and the overall response stress-strain curve is shown in Figure 9f, which is suitable for processes with less tension and large compression strain envelopes. Interestingly, VW-PSO with an interval of I results in a response stress-strain curve that is quite similar to the GA output, which further implies that the fitness function of built-in LS-DYNA may be slightly modified.

The particle number directly determined the computation load. Simulation times were calculated with Equation (14) and are shown in Figure 10. While the particle number decreased from 42 to 10, the required iteration times increased from 33 to 55 and the simulation times decreased from 1344 to 560. The particle number also had an effect on the convergence rate and the optimal result, as presented in Figure 11. When the particle number decreased within a certain range, the simulation times decreased quickly while maintaining accuracy. Although the simulation times with a particle number of 10 were lowest, we suggest using 20 to 30 particles for the Y-U hardening model parameters optimization in order to take into account iteration times and optimization accuracy.

$$\{\begin{array}{c}\mathrm{Simulation} \mathrm{times}\times \mathrm{Number} \mathrm{of} \mathrm{iterations}\\ \mathrm{Total} \mathrm{time}=T_J\times \mathrm{Number} \mathrm{of} \mathrm{iterations}\times \frac{i}{i^{\prime }}\end{array},$$

(14)

where i is the particle number; T_J is the average time for a simulation; i′ is the achievable simulation number at one time for the iteration.

3.4. Experimental Verification

With the aim of remanufacturing a retired part into its original shape, a bending-leveling experiment was used to verify the Y-U hardening model, with the above parameters obtained by standard PSO. On the three points bending testing machine, the specimen was firstly bent to a specified angle and then straightened, as shown in Figure 12. The distance between the two supporting points was 150 mm, and the press amount for bending was 20 mm. The size of the 1.5 mm-thick AA 7075-O sheet specimen was 300 mm × 50 mm. The experiment was performed three times to ensure stable results.

The residual angles of the sheet specimen after both bending and leveling, as predicted by FE simulation with LS-DYNA using the Y-U model and measured with experiment, are compared in

Table 6. Residual angles of bending-leveling by experiment and simulation.

Bending			Leveling
Bending before rebound	Experiment after rebound	Simulation after rebound	Leveling before rebound	Experiment after rebound	Simulation after rebound
39.6°	19.3°	19.5°	15.9°	0.6°	1.6°

. At the bending stage, the springback result calculated by the Y-U hardening model was accurate, with a prediction error of 0.2°. At the leveling stage, the residual angle predicted with the Y-U hardening model showed an acceptable level of accuracy, with a prediction error of 1°. In general, the Y-U hardening model with determined parameters provided reasonably accurate predictions for the bending-leveling experiment and therefore could potentially be effective in the design of remanufacturing processes.

4. Conclusions

In this paper, the graphical section and the PSO section were combined to improve the accuracy of identifying parameters and the efficiency of the Y-U hardening model, a model that is particularly suitable for the remanufacturing process. This study focused on three aspects: PSO variation, particle number and fitness function. The results are as follows:

• Improved fitting accuracy is achieved by PSO variation. Compared with GA, the overall residual error was reduced by more than 40% with standard PSO and the standard PSO result was verified by the OS-PSO.
• Local interval accuracy is improved with VW-PSO. With an increased weight in the case of the tensile flow stress interval, the instantaneous Bauschinger effect interval and the permanent softening interval; the interval residual error was reduced by 33% to 40% when compared to standard PSO, while also maintaining a stable overall residual error.
• Computation load and iteration times are cut down by reducing the particle number within a certain range, while also maintaining the overall residual error. In this study, the iteration time and computation load were comparatively low with particle numbers of 20 and 30.
• Acceptable accuracy can be achieved for bending-leveling experiments. The springback prediction errors for both the bending and leveling stages were within 1°.

The PSO variations employed in this paper show that is it possible to inversely identify the parameters for the Y-U hardening model with a high level of accuracy and efficiency, which is of great importance in predicting the cyclic stress-strain response of materials. Further work is needed to apply the calibrated Y-U hardening model on remanufacturing processes and to inform the design of flexible remanufacturing processes.