Effect of Flexible 3 D Multipoint Stretch Bending Dies on the Shape Accuracy and the Optimal Design

Aiming at the flexible 3D stretch bending with multipoint dies for the aluminum profile, the numerical simulation analysis of the bending process was carried out by ABAQUS finite element software. In the multipoint stretch bending (MPSB) process, the influence of the number of die units on the springback for the complex section profile was studied. ,e shape error between the forming parts and the target parts was reduced through the method of die surface modification. ,e results showed that the springback of the aluminum profile could reach aminimumwhen the number of die units was 25 under the precondition of saving cost and ensuring the quality of forming parts. In the numerical simulation, the maximum shape error of the forming parts reduced from 18.824mm to 2.456mm; in the test, the maximum shape error reduced from 27.26mm to 6.03mm through the method of die surface modification.


Introduction
In recent years, lightweight structural design has drawn much attention.Reduction of the weight will not only reduce energy consumption, alleviating the increasingly serious energy crisis, but also reduce environmental pollution caused by exhaust emissions, especially in the automotive and high-speed rail industries [1][2][3].Aluminum alloy is widely used in aerospace, rail vehicle manufacturing, and automobile manufacturing as a lightweight material due to its low density, high strength, and ease of recycling [4][5][6].
With the rapid development of the manufacturing industry, people have more and more diversified requirements on various parts and components, while the parts and components formed by traditional methods have been unable to meet people's needs [7].As the 3D bending parts can provide a lighter, stiffer, and more space-saving spatial structure than the conventional 2D parts, the demand for 3D parts is also increasing [8,9].However, the research on the forming method of 3D parts is still in the exploratory stage.
Hermes et al. developed a machine that specializes in 3D forming of profiles and tubes, in which the combined spinning and bending process leads to low bending forces with the possibility of a significant springback reduction [10].Gantner et al. used a free-form bending technology to create a freely definable bending geometry, including transitionless bend-in-bends and spline bends [11].So far, studies about flexible bending are focused on profiles and tubes with rectangular cross section.
In the past decades, stretch bending was the main production method for 2D parts.At present, it is mainly used in high-volume production of the profile and is also widely used in aircraft, high-speed railway, and automobile and shipbuilding industries [12].However, the traditional stretch bending is only suitable for mass production rather than small batch production.On one hand, with the improvement of people's living standards, low-cost and small batch production of many kinds of parts and components will be the main way of industrial manufacturing in the future [13,14].On the other hand, it is difficult for complex cross-sectional profiles to bend in the horizontal direction and vertical direction without unloading condition through the traditional die [15,16].But the use of the combination of the flexible multipoint forming technology and the stretch bending method can be a good solution to the 3D bending profile.According to the characteristics of the flexible 3D bending technology, a new flexible 3D stretch bending (FSB) with the multipoint dies is thus being developed.e 3D stretch bending of the profile was successfully achieved.

Description of the Multipoint Stretch Bending Process.
For one thing, the numerical simulation of flexible discrete die forming is more sophisticated than that of the traditional one due to the discontinuous contacts between the profile and the multipoint dies in the multipoint stretch bending (MPSB) process.As shown in Figure 1, the multipoint dies are made up of multiple die units to reconstruct the die surface by changing the space position in the MPSB.For another, the profile is formed into a 3D part, which requires the profile to be shaped, respectively, in the horizontal direction and vertical direction to bend a spatial contour.
erefore, the MPSB is more diversified than the conventional process due to increased vertical bending of the profile in this process.
A typical MPSB process involves the following stages: (1) Shape adjustment: In the multipoint dies, the position parameters of each flexible die unit are adjusted to obtain the desired envelope surface.(2) Prestretching: Under the action of axial tension, the profile is prestretched to a plastic state, and the prestretch can effectively reduce the springback of the forming parts.(3) Horizontal bending along the x-y plane: e profile is gradually close to the die in the horizontal x-y plane for bending (Figure 2(a)).(4) Vertical bending along the x-z plane: e multipoint stretch bending machine gradually improves the profile in the vertical x-z plane for bending (Figure 2(b)).(5) Poststretching: e profile along the tangential direction makes up stretching after the 3D bending, which can effectively reduce springback of the forming parts.(6) Releasing: e profile is made of a small amount of plastic after the measurement.en, the finished product is put into the warehouse.
In the MPSB process, the shape adjustment is based on the spatial contour of the target parts.In the numerical analysis, the stages from (2) to (6) should be simulated to find out the optimal MPSB process of the profile.

Material Model.
Aluminum alloys are commonly used in the field of high-speed railway structures since they offer good mechanical properties with low weight [17].In the numerical investigations, aluminum alloy 6005A is used, which is 6500 mm in length and about 21 kg in weight.Meanwhile, isotropic, elastoplastic constitutive behavior with isotropic hardening is assumed for the profile material.
e relevant mechanical properties are given as follows: yield strength σ y � 260 MPa, elastic modulus E � 70 GPa, Poisson's coefficient υ � 0.33, and density ρ � 2.71 g/cm 3 .In addition, the preliminary and supplemental tension values are 32.5 mm, and the friction coefficient is 0.1.
Figure 3 shows the nominal stress σ versus nominal strain ε curve obtained by the authors from the tension test, and this curve was used in the finite element simulation.

Models of FE Simulation.
In the MPSB process, the stages from (2) to ( 5) should be quasi-static with the explicit, dynamic version of the ABAQUS/Explicit.After stage (5), we used the ABAQUS/Implicit finite element method into numerical simulation to calculate the springback.As shown in Figure 4, the 1/2 finite element model was used to shorten the calculation time and improve the efficiency because the profile is symmetrical along the middle section.
Figure 5 is the parts diagram in the finite element simulation.In this figure, the geometry side of the profile is shown in Figure 6(c).e die units were defined by the plane in contact with the profile, with a length of 50 mm.eir distribution is determined by the horizontal forming condition of the target parts.Meanwhile, the plane plates with the same shape of the die units were defined by the limit screw to bend in the vertical direction for actually controlling the multipoint dies, with a length of 50 mm.eir distribution is determined by the horizontal forming condition and vertical forming condition of the target parts.And the clamps were used with the same shape of the three-dimensional shell structure as the profile, with a length of 100 mm.
e element C3D8R was used to model the aluminum profile.e C3D8R is a hexahedral solid element with eight nodes and linear displacement interpolation, and it is suitable for large strains and deformations.
In the MPSB process, the element R3D4 was used to model the die units, limit screw, and clamp because it did not undergo deformation.e R3D4 is a bilinear quadrilateral three-dimensional rigid element with four nodes.e use of the rigid body element can effectively shorten the calculation time because it does not calculate the unit level in the simulation.

Multipoint dies
Die units Profile M T M T 2 Advances in Materials Science and Engineering In the nite element simulation, the nite element model is divided into four submodels and six analytic steps, which are de ned as the horizontal bending model, vertical bending model, poststretching model, and springback model, respectively.e horizontal bending model includes the initial analysis step, the prestretching process, and the

Numerical Analysis of the Shape Accuracy in the MPSB
In the MPSB process, the number of die units not only a ects the manufacturing cost of 3D forming parts but also causes its shape accuracy, stress, and strain to change.erefore, in the MPSB process, it is vital to choose the number of die units.Figure 6 shows the parameters of the aluminum pro le in the 3D stretch bending.According to these parameters, the four options of the number of die units m 15, m 20, m 25, and m 30 are chosen to simulate the whole MPSB process in order to determine the most suitable die units of the 3D process in the numerical investigation.

E ect of the 3D Stretch Bending on Shape Accuracy.
Figure 7 shows the maximum shape error between the forming parts and the target parts with di erent numbers of die units after the vertical bending.With the increasing number of die units, the maximum shape error of the forming parts will be reduced.is is mainly due to the fact that the displacement amplitude and clearance value of adjacent dies become smaller with more die units.erefore, the shape accuracy of the forming parts can be e ectively improved.Figure 8 is the graph of the equivalent stress distribution after the 3D stretch bending of di erent numbers of die units.As shown in Figure 8(a), the apparent wrinkle of the aluminum pro le a ects the shape accuracy of the forming   However, with the increasing number of die units, the manufacturing cost of the forming parts will also increase.When the number of die units increases to a certain amount, the pro le will undergo depression (as shown in Figure 8(d)).
erefore, selecting the appropriate number of die units is the precondition for the bending quality of the forming parts.
At the end of the aluminum pro le, that is, at the clamp on the force-applying end of the pro le, the stress-strain eld undergoes a drastic change due to the concentrated load on the clamp.us, it is generally used as the stub bar to handle, or it adopts exible clamping machinery to control the practical production process.

E ect of the Springback on Shape Accuracy.
In the bending process, the defects often a ect the quality of the forming parts, such as springback, wrinkles, and fracture.e inevitable factor is the springback, which is closely related to the entire bending process.e reduction of the   springback can e ectively improve the shape accuracy of the forming parts.erefore, the research on springback is indispensable in the MPSB process.For the study of the pro le springback, we adopt the prede ned eld in ABAQUS/Standard and delete the redundant constraint.It is only observed that the springback phenomenon is caused by the stress-strain eld after the poststretching.
e forming process is more complicated than the traditional bending process because the aluminum pro le is formed along the spatial curve in the MPSB process.erefore, the springback process is divided into horizontal direction along the x-y plane and vertical direction along the x-z plane, as shown in Figure 9, where Δα represents the total springback value, Δα y represents the springback value along the x-y plane, and Δα z represents the springback value along the x-z plane.
Figure 10 shows Δα with the change of twist angle after the MPSB process.As shown in the gure, with the increase of twist angle for the pro le, the springback of the forming parts also increases in the pro le section under the action of the die units.us, controlling the pro le deformation within a certain range can e ectively reduce the springback in the MPSB process.At the same time, the curves in Figure 10 clearly demonstrate that the nal springback value of the aluminum pro le reduces when the number of die units increases.
Figures 11(a) and 11(b) show the horizontal and vertical springback values of the pro le along the length.It can be observed from these results that the horizontal springback value and the vertical springback value also decrease when the number of die units increases.At the same time, we can nd that the springback value in the vertical direction is smaller than that in the horizontal direction.is is because the aluminum pro le has been bent horizontally along the x-y plane before the vertical bending.After the horizontal bending, the stress and strain elds of the aluminum pro le change.
e superposition of the stress and strain elds causes the springback value to be larger in the horizontal direction than that in the vertical direction.
In summary, it can e ectively reduce the springback value of the pro le with the increase of the number of die units.However, with the increase of die units, it will not only increase the manufacturing cost of the forming parts but also make the pro le undergo depression (as shown in Figure 8(d)).erefore, it is very important to choose the suitable number of die units in the MPSB process.

3.3.
e Method Based on Reverse Compensation of the Springback Value.After the MPSB process, the shape error of the pro le is mainly due to the springback along the x-y horizontal plane and the x-z vertical plane.erefore, we used the following method based on reverse compensation of the springback value to modify the die units' surface.Firstly, according to the discrete feature of the multipoint dies, the springback value of the pro le was simulated by ABAQUS.en, the springback value was compensated to each die unit to modify the dies' surface.Finally, the shape error was controlled in the range of the accuracy requirement after multiple iterations.
Figure 12 shows the contrast of the curved surface shape before and after the adjustment in the x-y horizontal plane, where Δθ j i represents the measured springback error, Δη j i represents the displacement of the springback compensation, the superscript j represents the number of iterations, and the subscript i represents the number of shape control points.In addition, the adjustment method along the x-z vertical plane is similar to that along the x-y horizontal plane.Moreover, the initial adjusting method of the die units takes the shape of the target part as the multipoint dies' envelope surface.After the springback compensation, the die units' envelope surface is calculated as follows: 6 Advances in Materials Science and Engineering where α represents the compensation factor of the springback, the value of which is generally [−2.5, −1], and K represents the die units' envelope shape.First, through the ABAQUS nite element simulation, we got the springback values along the x-y horizontal plane and the x-z vertical plane when the number of die units was 25 and then reversed the compensation to each die unit to get the discrete point information.Next, according to this information, the theoretical curves of the horizontal and vertical lines of the forming part were obtained by the fourth-degree polynomial t.
e theoretical curve of the horizontal line along the x-y plane is as follows: Y −0.29477 − 0.00466X + 9.50531 × 10 −5 X 2 − 4.2123 × 10 −9 X 3 + 1.48643 × 10 −12 X 4 . ( e theoretical curve of the vertical line along the x-z plane is as follows: Z −0.17864 − 0.00614X + 9.79094 × 10 −5 X 2 − 6.15386 × 10 −9 X 3 + 1.80589 × 10 −12 X 4 . (3) Moreover, the die units' envelope surface was again adjusted by the theoretical curve.Finally, the ABAQUS nite element software was used to simulate the MPSB process according to the forming parameters after the shape adjustment to obtain the shape error value of the forming parts.e curves in Figure 13 clearly demonstrate that it can effectively reduce the shape error between the forming parts and the target parts after the springback by modifying the die units' envelope surface.e die units' surface modi cation method can reduce the shape error of the forming parts  Advances in Materials Science and Engineering caused by the springback and thus improve the shape accuracy of the forming process.

The Shape Accuracy Test of the Forming Part
In this paper, the exible 3D multipoint stretch bending test was carried out to verify the reliability of the simulation, taking the frame of a high-speed railway car as an example (as shown in Figure 14).e test process was shown as follows: First of all, we used the prototype of exible 3D stretch bending (FSB) equipment to make 3D bending of the aluminum pro le and then scanned the forming pro le by using the NDI large space measuring instrument PRO CMM 3500 optical tracker.Finally, the scanned 3D map of the forming part and the target part was 3D analyzed with the Geomagic Qualify software.Figure 15 is the comparison between the test results and the numerical simulation results.It can be seen that the test results are in agreement with the numerical simulation results.Moreover, the shape error between the forming part and the target part can be signi cantly reduced, and the shape accuracy of the forming part can be improved by using the method of die surface modi cation.

Conclusion
(1) In the MPSB, less die units will reduce the forming accuracy of the pro le, while more die units can reduce the springback of the pro le.But more die units will increase the manufacturing cost of the aluminum pro le at the same time.erefore, it is very important to select the appropriate number of die units in the MPSB process.In this paper, 25 was determined as the ideal number of die units.(2) e x-y horizontal and x-z vertical springback compensations are performed for the complex section pro le.In the numerical simulation, the maximum shape error reduced from 18.824 mm to 2.456 mm; in the test, the maximum shape error reduced from 27.26 mm to 6.03 mm by the method of die surface modi cation.erefore, the method of die surface modi cation is an e ective way to improve the shape accuracy of the forming parts.(3) e 3D MPSB test for the pro le was carried out.e test results showed that they are in agreement with the simulation results.For this purpose, the forming parameters can be optimized by the nite element simulation software to guide the production test and reduce the production cost.

Figure 1 :
Figure 1: Schematic diagram of the multipoint stretch bending machine.

Figure 6 :
Figure 6: e 3D stretch bending parts.(a) Vertical view.(b) Front view.(c) Right view and the geometry size of the pro le.

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Advances in Materials Science and Engineering parts after the vertical bending, when the number of die units is 15.As the number of die units increases, its wrinkle is signi cantly reduced (as shown in Figure8(b)).

Figure 7 :
Figure 7: e maximum shape error between the forming parts and the target parts.

Figure 8 :
Figure 8: e equivalent stress distribution of the pro le after the vertical bending along the x-z plane: (a) the number of die units: m 15, (b) the number of die units: m 20, (c) the number of die units: m 25, and (d) the number of die units: m 30.

Figure 9 :Figure 10 :
Figure 9: e front and back spatial positions of the end face contours.

Figure 11 :Figure 12 :
Figure 11: e springback value along the longitudinal direction of the pro le.(a) e springback value along the x-y plane.(b) e springback value along the x-z plane.

Figure 13 :
Figure 13: e shape error before and after the modi cation.

Figure 14 :
Figure 14: e detection setup and the forming part through the MPSB process.

Figure 15 :
Figure 15: Comparison of the test results and the numerical simulation results.