[7] J. Tang, X. Geng, D. Li, Y. Shi, J. Tong, H. Xiao, F. Peng, Machine learning-based microstructure prediction during laser sintering of alumina, Scientific Reports 11 (2021) 10724. https://doi.org/10.1038/s41598-021-89816-x

[8] P. Link, J. Bodenstab, L. Penter, S. Ihlenfeldt, Metamodeling of a deep drawing process using conditional Generative Adversarial Networks, IOP Conf. Ser.: Mater. Sci. Eng. 1238 (2022) 12064. https://doi.org/10.1088/1757-899X/1238/1/012064

[9] D.A. Molitor, C. Kubik, M. Becker, R.H. Hetfleisch, F. Lyu, P. Groche, Towards high-performance deep learning models in tool wear classification with generative adversarial networks, Journal of Materials Processing Technology 302 (2022) 117484. https://doi.org/10.1016/j.jmatprotec.2021.117484

[10] J. Luo, J. Huang, H. Li, A case study of conditional deep convolutional generative adversarial networks in machine fault diagnosis, Journal of Intelligent Manufacturing 32 (2021) 407-425. https://doi.org/10.1007/s10845-020-01579-w

[11] M. Li, Z. Siqin, Research on Manufacturing Matrix Data Complementation Method Based on Generative Adversarial Network, in: 2023 4th International Symposium on Computer Engineering and Intelligent Communications (ISCEIC), Nanjing, China, pp. 134-138.

[12] M. Mirza, S. Osindero, Conditional Generative Adversarial Nets, 2014.

[13] A. Radford, L. Metz, S. Chintala, Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks, 2015.

[14] Sahil Sidheekh, Aroof Aimen, Narayanan C Krishnan, On Characterizing GAN Convergence Through Proximal Duality Gap, International Conference on Machine Learning (2021) 9660-9670.

[15] A. Ramdas, N. Trillos, M. Cuturi, On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests, Entropy 19 (2017) 47. https://doi.org/10.3390/e19020047.

[16] R. Fu, J. Chen, S. Zeng, Y. Zhuang, A. Sudjianto, Time Series Simulation by Conditional Generative Adversarial Net, 2019.

[17] J.L. Hodges, The significance probability of the smirnov two-sample test, Ark. Mat. 3 (1958) 469-486. https://doi.org/10.1007/bf02589501
[7] J. Tang, X. Geng, D. Li, Y. Shi, J. Tong, H. Xiao, F. Peng, Machine learning-based microstructure prediction during laser sintering of alumina, Scientific Reports 11 (2021) 10724. https://doi.org/10.1038/s41598-021-89816-x

[8] P. Link, J. Bodenstab, L. Penter, S. Ihlenfeldt, Metamodeling of a deep drawing process using conditional Generative Adversarial Networks, IOP Conf. Ser.: Mater. Sci. Eng. 1238 (2022) 12064. https://doi.org/10.1088/1757-899X/1238/1/012064

[9] D.A. Molitor, C. Kubik, M. Becker, R.H. Hetfleisch, F. Lyu, P. Groche, Towards high-performance deep learning models in tool wear classification with generative adversarial networks, Journal of Materials Processing Technology 302 (2022) 117484. https://doi.org/10.1016/j.jmatprotec.2021.117484

[10] J. Luo, J. Huang, H. Li, A case study of conditional deep convolutional generative adversarial networks in machine fault diagnosis, Journal of Intelligent Manufacturing 32 (2021) 407-425. https://doi.org/10.1007/s10845-020-01579-w

[11] M. Li, Z. Siqin, Research on Manufacturing Matrix Data Complementation Method Based on Generative Adversarial Network, in: 2023 4th International Symposium on Computer Engineering and Intelligent Communications (ISCEIC), Nanjing, China, pp. 134-138.

[12] M. Mirza, S. Osindero, Conditional Generative Adversarial Nets, 2014.

[13] A. Radford, L. Metz, S. Chintala, Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks, 2015.

[14] Sahil Sidheekh, Aroof Aimen, Narayanan C Krishnan, On Characterizing GAN Convergence Through Proximal Duality Gap, International Conference on Machine Learning (2021) 9660-9670.

[15] A. Ramdas, N. Trillos, M. Cuturi, On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests, Entropy 19 (2017) 47. https://doi.org/10.3390/e19020047.

[16] R. Fu, J. Chen, S. Zeng, Y. Zhuang, A. Sudjianto, Time Series Simulation by Conditional Generative Adversarial Net, 2019.

[17] J.L. Hodges, The significance probability of the smirnov two-sample test, Ark. Mat. 3 (1958) 469-486. https://doi.org/10.1007/bf02589501
# Tool path effects on wrinkling in metal sheet incremental roll profiling

BEIGZADEH Ali<fnref n="1" />,ᵃ*, SIMONETTO Enrico<fnref n="1" />,ᵇ, GHIOTTI Andrea<fnref n="1" />,ᶜ, BRUSCHI Stefania<fnref n="1" />,ᵈ

**Keywords:** Incremental Forming, Flexible Roll Bending, Tool Path

**Abstract.** The paper introduces an innovative Incremental Forming (IF) machine, termed "FlexRoll Bending," designed for flexible sheet forming on straight components. Unlike traditional processes, this machine employs two working rollers that can move along three directions and rotate, eliminating the need for specific dies. The study focuses on an "on-edge forming strategy" to manufacture non-uniform cross-section sheet metal parts, presenting a cost-effective alternative to conventional methods. The experimental setup utilizes a 0.8 mm DD11 steel sheet, and a novel toolpath theory is introduced for calculating the movement of the forming rollers. Numerical simulations in LS-DYNA software assess the impact of different toolpaths on the final geometry, revealing successful part manufacturing without defects using a linear toolpath and identifying wrinkling in a two-stage toolpath. The results demonstrate the potential of the FlexRoll Bending process for die-less manufacturing of complex geometry sheet metal parts with low forming forces.

**Introduction**

The rise of Industry 4.0 over the last ten years, along with an increased focus on manufacturing to meet customer needs [1], has highlighted the importance of agile and flexible manufacturing strategies across various sectors. This has become particularly evident in the context of the COVID-19 pandemic [2]. Sheet metal forming processes have commonly found application across a range of industries, including automotive [3], shipbuilding, aerospace industries, shell construction, and power plants, for the fabrication of structural components characterized by enhanced stiffness [4]. However, these procedures usually necessitate specific dies or professional personnel capable of implementing complex duties through manual processes. Dealing with large-sized components poses an extra challenge, as conventional metal forming presses frequently lack the required force or table size for bending operations. Recently, there has been an increasing call for small-batch and single-production runs, contributing to reduced profitability in these operations [5]. In response, manufacturers aim to stay competitive by providing more flexibility, better quality, quicker time-to-market, and lower production costs [6].

Roll forming (RF) is a well-known process for the manufacturing of long-profile components with a uniform cross-section for applications in various industries such as the automotive, aircraft, and building industries [7]. In roll forming, the metal strip is fed through consecutive sets of rotating rollers. Each forming roller stand performs an incremental part of the bend, and the flange is therefore incrementally bent to the desired shape and angle.

To eliminate the uniform cross-section limitation in the RF process, Flexible Roll Forming (FRF) process has been introduced. In this process, some of the rollers stay stationary while others can move perpendicular to the feeding direction and rotate along the vertical axis to produce varying width and depth parts [8, 9].

<figure><img src="image_1395.png" /></figure>

<footnote n="1">¹Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131, Padova, Italy</footnote>

<footnote>ᵃali.beigzadeh@studenti.unipd.it, ᵇenrico.simonetto.1@unipd.it, ᶜandrea.ghiotti@unipd.it,</footnote>

<footnote>ᵈstefania.bruschi@unipd.it</footnote>
# Tool path effects on wrinkling in metal sheet incremental roll profiling

BEIGZADEH Ali<fnref n="1" />,ᵃ*, SIMONETTO Enrico<fnref n="1" />,ᵇ, GHIOTTI Andrea<fnref n="1" />,ᶜ, BRUSCHI Stefania<fnref n="1" />,ᵈ

**Keywords:** Incremental Forming, Flexible Roll Bending, Tool Path

**Abstract.** The paper introduces an innovative Incremental Forming (IF) machine, termed "FlexRoll Bending," designed for flexible sheet forming on straight components. Unlike traditional processes, this machine employs two working rollers that can move along three directions and rotate, eliminating the need for specific dies. The study focuses on an "on-edge forming strategy" to manufacture non-uniform cross-section sheet metal parts, presenting a cost-effective alternative to conventional methods. The experimental setup utilizes a 0.8 mm DD11 steel sheet, and a novel toolpath theory is introduced for calculating the movement of the forming rollers. Numerical simulations in LS-DYNA software assess the impact of different toolpaths on the final geometry, revealing successful part manufacturing without defects using a linear toolpath and identifying wrinkling in a two-stage toolpath. The results demonstrate the potential of the FlexRoll Bending process for die-less manufacturing of complex geometry sheet metal parts with low forming forces.

**Introduction**

The rise of Industry 4.0 over the last ten years, along with an increased focus on manufacturing to meet customer needs [1], has highlighted the importance of agile and flexible manufacturing strategies across various sectors. This has become particularly evident in the context of the COVID-19 pandemic [2]. Sheet metal forming processes have commonly found application across a range of industries, including automotive [3], shipbuilding, aerospace industries, shell construction, and power plants, for the fabrication of structural components characterized by enhanced stiffness [4]. However, these procedures usually necessitate specific dies or professional personnel capable of implementing complex duties through manual processes. Dealing with large-sized components poses an extra challenge, as conventional metal forming presses frequently lack the required force or table size for bending operations. Recently, there has been an increasing call for small-batch and single-production runs, contributing to reduced profitability in these operations [5]. In response, manufacturers aim to stay competitive by providing more flexibility, better quality, quicker time-to-market, and lower production costs [6].

Roll forming (RF) is a well-known process for the manufacturing of long-profile components with a uniform cross-section for applications in various industries such as the automotive, aircraft, and building industries [7]. In roll forming, the metal strip is fed through consecutive sets of rotating rollers. Each forming roller stand performs an incremental part of the bend, and the flange is therefore incrementally bent to the desired shape and angle.

To eliminate the uniform cross-section limitation in the RF process, Flexible Roll Forming (FRF) process has been introduced. In this process, some of the rollers stay stationary while others can move perpendicular to the feeding direction and rotate along the vertical axis to produce varying width and depth parts [8, 9].

<figure><img src="image_1399.png" /></figure>

<footnote n="1">¹Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131, Padova, Italy</footnote>

<footnote>ᵃali.beigzadeh@studenti.unipd.it, ᵇenrico.simonetto.1@unipd.it, ᶜandrea.ghiotti@unipd.it,</footnote>

<footnote>ᵈstefania.bruschi@unipd.it</footnote>
In the profiles manufactured by the FRF process, the main failure mode is flange wrinkling [10]. Wrinkling commonly happens in longitudinal or planar compression situations [9] which commonly occurs in non-uniform cross-section regions.

On the other hand, Incremental Forming (IF) processes are gaining attention as a cost-effective option. Unlike traditional methods, IF doesn't need expensive dies and works well for various part shapes and batch sizes [11]. These processes involve minimal tool-sheet contact, allowing controlled plastic deformation with lower applied forces [12].

Several studies have investigated the application of Incremental Forming (IF) processes for both axis-symmetric and elongated parts. For axis-symmetric parts, metal spinning [13] is considered a promising method for manufacturing large and thick components. The design of the tool path [14] is crucial for ensuring workpiece quality, as the circumferential stress resulting from the reduction of the part's average diameter often leads to wrinkles [15] or other defects. Concerning elongated parts, Voswinckel et al. explored stretch and shrink flanging [16] and evaluated the influence of the tool path on part accuracy [17]. Additional studies have investigated pre-forming flanging procedures for stretch, shrink, and hole-shaped flanges [18]. While many studies have used pin-like tools, others have suggested using a single roller-shaped tool for hemming operations [19], often in conjunction with robotic arms. However, employing this combination to maneuver tools along complex 3D paths [20], despite providing increased flexibility, could face limitations due to the equipment's low stiffness and might not be suitable for thicker workpieces [21].

This paper presents an innovative concept for an Incremental Forming (IF) machine designed specifically for flexible sheet forming operations on straight components, utilizing two working rollers which is called “FlexRoll Bending” process [22, 23]. In this newly introduced IF machine, the mechanics of the metal deformation is similar to RF and FRF, while instead of feeding the stripe and using numerous roller sets, the stripe is restrained, and Roller(s) are able to move along the three directions and rotate in one direction. The strategy of using a pair of simple rollers makes the FlexRoll bending process budget friendly. In the previous research, 90° and 180° bents performed successfully using a pair of rollers. On the other hand, the main purpose of manufacturing of mentioned IF machine is completely die-less manufacturing of complex geometry sheet metal parts with low forming forces. Generation of the toolpath is one of the most important challenges for this IF machine. Controlling the movement of two rollers in three directions increases the complexity of the toolpath needed to generate the final component. Therefore, in the experiments of this research, just one of the forming rollers was used to manufacture the parts.

In this study, a novel approach called "on-edge forming strategy" will be introduced for manufacturing parts non-uniform cross-section part which is more complex compared to previously investigated. The target part has been manufactured by using on edge forming strategy with two different toolpaths on the 0.8 mm DD11 sheet.

DD11 is a hot-rolled steel sheet with very low carbon (<0.12% C) content characterized by its favorable formability properties at ambient temperature, making it widely utilized in cold-forming processes, combined with good weldability. The characteristics of this steel render it well-suited for the subsequent manufacturing of a variety of end products, including stamped, pressed, welded, and drawn parts, as well as tubes, car wheels, agricultural equipment, fasteners, and shelving systems.

## Material

The reference material is a 0.8 mm thick DD11 sheet. To evaluate the behavior of the mentioned material, dog bone samples with a gauge length of 65 ± 0.1 mm and width of 12 ± 0.05 mm were cut from the sheet along the three rolling (0 deg), perpendicular (90 deg), and 45 deg directions based on the ISO 6892 standard. MTS 322TM hydraulic dynamometer with a maximum load
In the profiles manufactured by the FRF process, the main failure mode is flange wrinkling [10]. Wrinkling commonly happens in longitudinal or planar compression situations [9] which commonly occurs in non-uniform cross-section regions.

On the other hand, Incremental Forming (IF) processes are gaining attention as a cost-effective option. Unlike traditional methods, IF doesn't need expensive dies and works well for various part shapes and batch sizes [11]. These processes involve minimal tool-sheet contact, allowing controlled plastic deformation with lower applied forces [12].

Several studies have investigated the application of Incremental Forming (IF) processes for both axis-symmetric and elongated parts. For axis-symmetric parts, metal spinning [13] is considered a promising method for manufacturing large and thick components. The design of the tool path [14] is crucial for ensuring workpiece quality, as the circumferential stress resulting from the reduction of the part's average diameter often leads to wrinkles [15] or other defects. Concerning elongated parts, Voswinckel et al. explored stretch and shrink flanging [16] and evaluated the influence of the tool path on part accuracy [17]. Additional studies have investigated pre-forming flanging procedures for stretch, shrink, and hole-shaped flanges [18]. While many studies have used pin-like tools, others have suggested using a single roller-shaped tool for hemming operations [19], often in conjunction with robotic arms. However, employing this combination to maneuver tools along complex 3D paths [20], despite providing increased flexibility, could face limitations due to the equipment's low stiffness and might not be suitable for thicker workpieces [21].

This paper presents an innovative concept for an Incremental Forming (IF) machine designed specifically for flexible sheet forming operations on straight components, utilizing two working rollers which is called “FlexRoll Bending” process [22, 23]. In this newly introduced IF machine, the mechanics of the metal deformation is similar to RF and FRF, while instead of feeding the stripe and using numerous roller sets, the stripe is restrained, and Roller(s) are able to move along the three directions and rotate in one direction. The strategy of using a pair of simple rollers makes the FlexRoll bending process budget friendly. In the previous research, 90° and 180° bents performed successfully using a pair of rollers. On the other hand, the main purpose of manufacturing of mentioned IF machine is completely die-less manufacturing of complex geometry sheet metal parts with low forming forces. Generation of the toolpath is one of the most important challenges for this IF machine. Controlling the movement of two rollers in three directions increases the complexity of the toolpath needed to generate the final component. Therefore, in the experiments of this research, just one of the forming rollers was used to manufacture the parts.

In this study, a novel approach called "on-edge forming strategy" will be introduced for manufacturing parts non-uniform cross-section part which is more complex compared to previously investigated. The target part has been manufactured by using on edge forming strategy with two different toolpaths on the 0.8 mm DD11 sheet.

DD11 is a hot-rolled steel sheet with very low carbon (<0.12% C) content characterized by its favorable formability properties at ambient temperature, making it widely utilized in cold-forming processes, combined with good weldability. The characteristics of this steel render it well-suited for the subsequent manufacturing of a variety of end products, including stamped, pressed, welded, and drawn parts, as well as tubes, car wheels, agricultural equipment, fasteners, and shelving systems.

## Material

The reference material is a 0.8 mm thick DD11 sheet. To evaluate the behavior of the mentioned material, dog bone samples with a gauge length of 65 ± 0.1 mm and width of 12 ± 0.05 mm were cut from the sheet along the three rolling (0 deg), perpendicular (90 deg), and 45 deg directions based on the ISO 6892 standard. MTS 322TM hydraulic dynamometer with a maximum load
capacity of 50 kN and strain rate of 0.1 s⁻¹ was employed to perform tensile tests. Fig. 1 displays the results of the tensile test along the mentioned directions for the DD11 hot-rolled sheet. Tensile test results demonstrate an average yield strength of 224 MPa, Ultimate Tensile strength of 439 MPa, and Elongation of 21.4%.

<figure><img src="image_1396.png" /><figcaption>Fig. 1. DD11 flow stress curves along the three directions</figcaption></figure>

## Part

The raw material is a 0.8 mm thick DD11 sheet with a length of 500 mm and a width of 150 mm. The sheet metal is bent along the line AA' by movement of the forming roller. In the desired workpiece, the bent angle increases from 0° in BB' section to 60° in CC' section, see Fig. 2. As it illustrated in Fig. 2, AA'C'B' area clamped by the clamping plates.

<figure><img src="image_1397.png" /><figcaption>Fig. 2. Schematic of the Flexible Roll Bending process</figcaption></figure>

## Toolpath theory

For tool path derivation, the deformation in the FlexRoll bending process must be investigated from the geometrical point of view. In this research, to simplify the process, just one of the rollers was used to form the sheet, and the sheet wasn't supported by the support roller. As an additional simplifying assumption, it has been assumed that the thickness of the sheet doesn't change during the process which is deemed acceptable in consideration of the nature of the deformation applied in this research. As the last simplifying assumption, a perfect bending situation happens. Therefore, the bending allowance area doesn't exist. Therefore, *MM'* cross-sectional of the Fig. 2 will be like Fig. 3.
capacity of 50 kN and strain rate of 0.1 s⁻¹ was employed to perform tensile tests. Fig. 1 displays the results of the tensile test along the mentioned directions for the DD11 hot-rolled sheet. Tensile test results demonstrate an average yield strength of 224 MPa, Ultimate Tensile strength of 439 MPa, and Elongation of 21.4%.

<figure><img src="image_1400.png" /><figcaption>Fig. 1. DD11 flow stress curves along the three directions</figcaption></figure>

## Part

The raw material is a 0.8 mm thick DD11 sheet with a length of 500 mm and a width of 150 mm. The sheet metal is bent along the line AA' by movement of the forming roller. In the desired workpiece, the bent angle increases from 0° in BB' section to 60° in CC' section, see Fig. 2. As it illustrated in Fig. 2, AA'C'B' area clamped by the clamping plates.

<figure><img src="image_1401.png" /><figcaption>Fig. 2. Schematic of the Flexible Roll Bending process</figcaption></figure>

## Toolpath theory

For tool path derivation, the deformation in the FlexRoll bending process must be investigated from the geometrical point of view. In this research, to simplify the process, just one of the rollers was used to form the sheet, and the sheet wasn't supported by the support roller. As an additional simplifying assumption, it has been assumed that the thickness of the sheet doesn't change during the process which is deemed acceptable in consideration of the nature of the deformation applied in this research. As the last simplifying assumption, a perfect bending situation happens. Therefore, the bending allowance area doesn't exist. Therefore, *MM'* cross-sectional of the Fig. 2 will be like Fig. 3.
<figure><img src="image_1398.png" /><figcaption>Fig. 3. MM' cross-sectional view (cross-sectional cut of the part with a plan with a normal vector along the y direction)</figcaption></figure>

Hence, if the sheet is subjected to θ bending angle, Δx, and Δz can be driven as a function of θ and, l₁ as following:

$$ \Delta x = l \cdot (1 - \cos \theta) \qquad (1) $$

$$ \Delta z = l \cdot \sin \theta \qquad (2) $$

Due to the inherent dependence between Δx and Δz parameters, any attempt to modify one inherently involves an adjustment in the other parameter in this forming strategy. In order to be more accurate, calculation of the forming tool size effect is necessary. The roundness of the forming roller edge (r) results in the float contact point with the workpiece on the forming roller. The contact point is a function of the θ and roundness radius r, see Fig. 4. By applying the Δx, and Δz the contact point in xz plane shifts from P₁ to P₂ as a result of the forming roller size effect.

<figure><img src="image_1399.png" /><figcaption>Fig. 4. Schematic of the effect of the r on forming roller-sheet contact</figcaption></figure>

In Fig. 4 red arrows show the movement is needed to eliminate the effect of forming roller edge roundness. Difference between the on-work contact (P₂) point and the initial contact point (P₁) along x direction can be calculated as follows:

$$ \Delta x_r = r \sin \theta \qquad (3) $$

and along z-direction:

$$ \Delta z_{r1} = r(1 - \cos \theta) \qquad (4) $$
<figure><img src="image_1402.png" /><figcaption>Fig. 3. **MM'** cross-sectional view (cross-sectional cut of the part with a plan with a normal vector along the y direction)</figcaption></figure>

Hence, if the sheet is subjected to θ bending angle, Δx, and Δz can be driven as a function of θ and, l₁ as following:

$$ \Delta x = l \cdot (1 - \cos \theta) \qquad (1) $$

$$ \Delta z = l \cdot \sin \theta \qquad (2) $$

Due to the inherent dependence between Δx and Δz parameters, any attempt to modify one inherently involves an adjustment in the other parameter in this forming strategy. In order to be more accurate, calculation of the forming tool size effect is necessary. The roundness of the forming roller edge (r) results in the float contact point with the workpiece on the forming roller. The contact point is a function of the θ and roundness radius r, see Fig. 4. By applying the Δx, and Δz the contact point in xz plane shifts from P₁ to P₂ as a result of the forming roller size effect.

<figure><img src="image_1403.png" /><figcaption>Fig. 4. Schematic of the effect of the r on forming roller-sheet contact</figcaption></figure>

In Fig. 4 red arrows show the movement is needed to eliminate the effect of forming roller edge roundness. Difference between the on-work contact (P₂) point and the initial contact point (P₁) along x direction can be calculated as follows:

$$ \Delta x_r = r \sin \theta \qquad (3) $$

and along z-direction:

$$ \Delta z_{r1} = r(1 - \cos \theta) \qquad (4) $$
$$ \Delta z_{r2} = \frac{r(1 - \cos \theta)}{\cos \theta} \qquad (5) $$

Therefore, $\Delta z_r$ is equal to:

$$ \Delta z_r = \Delta z_{r1} + \Delta z_{r2} = \frac{r \sin^2 \theta}{\cos \theta} \qquad (6) $$

Similarly, the radius of the forming roller (R) can result in the float contact point with the workpiece on the forming roller. The float contact point position is a function of R and $\varphi$. By applying the $\Delta z$ is needed to create the $\theta$ angle, the contact point in yz plane, shifts from $Q_1$ to $Q_2$ as a result of the forming roller size effect, see Fig. 5.

<figure><img src="image_1400.png" /><figcaption>Fig. 5. Schematic of the effect of the R on forming roller-sheet contact</figcaption></figure>

Therefore, the difference between the on-work contact point and the initial contact point along y direction can be calculated as following:

$$ \Delta y_R = R \sin \varphi \qquad (7) $$

and along z-direction:

$$ \Delta z_{R1} = R(1 - \cos \varphi) \qquad (8) $$

$$ \Delta z_{R2} = \frac{R(1 - \cos \varphi)}{\cos \varphi} \qquad (9) $$

Therefore, $\Delta z_R$ is equal to:

$$ \Delta z_R = \frac{R \sin^2 \varphi}{\cos \varphi} \qquad (10) $$

Therefore, if the initial position of the forming roller is assumed to be $P_0(x_0, y_0, z_0)$ and its desired final position is assumed to be $P_{f\_ideal}(x_{f\_ideal}, y_{f\_ideal}, z_{f\_ideal})$. Therefore:
$$ \Delta z_{r2} = \frac{r(1 - \cos \theta)}{\cos \theta} \qquad (5) $$

Therefore, $\Delta z_r$ is equal to:

$$ \Delta z_r = \Delta z_{r1} + \Delta z_{r2} = \frac{r \sin^2 \theta}{\cos \theta} \qquad (6) $$

Similarly, the radius of the forming roller (R) can result in the float contact point with the workpiece on the forming roller. The float contact point position is a function of R and $\varphi$. By applying the $\Delta z$ is needed to create the $\theta$ angle, the contact point in yz plane, shifts from $Q_1$ to $Q_2$ as a result of the forming roller size effect, see Fig. 5.

<figure><img src="image_1404.png" /><figcaption>Fig. 5. Schematic of the effect of the R on forming roller-sheet contact</figcaption></figure>

Therefore, the difference between the on-work contact point and the initial contact point along y direction can be calculated as following:

$$ \Delta y_R = R \sin \varphi \qquad (7) $$

and along z-direction:

$$ \Delta z_{R1} = R(1 - \cos \varphi) \qquad (8) $$

$$ \Delta z_{R2} = \frac{R(1 - \cos \varphi)}{\cos \varphi} \qquad (9) $$

Therefore, $\Delta z_R$ is equal to:

$$ \Delta z_R = \frac{R \sin^2 \varphi}{\cos \varphi} \qquad (10) $$

Therefore, if the initial position of the forming roller is assumed to be $P_0(x_0, y_0, z_0)$ and its desired final position is assumed to be $P_{f\_ideal}(x_{f\_ideal}, y_{f\_ideal}, z_{f\_ideal})$. Therefore:
$$ \Delta P_{\text{ideal}} = P_{f_{\text{ideal}}} - P_0 \qquad (11) $$

$$ \Delta x_{\text{ideal}} = x_{f_{\text{ideal}}} - x_0 \qquad (12) $$

$$ \Delta y_{\text{ideal}} = y_{f_{\text{ideal}}} - y_0 \qquad (13) $$

$$ \Delta z_{\text{ideal}} = z_{f_{\text{ideal}}} - z_0 \qquad (14) $$

Therefore, the final real position of the forming roller considering the r and R effect can be calculated as following:

$$ x_{f_{\text{real}}} = x_0 + \Delta x_{\text{ideal}} + \Delta x_r \qquad (15) $$

$$ y_{f_{\text{real}}} = y_0 + \Delta y_{\text{ideal}} + \Delta y_R \qquad (16) $$

$$ z_{f_{\text{real}}} = z_0 + \Delta z_{\text{ideal}} + \Delta z_r + \Delta z_R \qquad (17) $$

## Tool path

For tool path calculation R and r as the constants of the forming roller and l, L, and θ as the parameters related to the geometry of the workpiece is needed which are provided in Table 1.

<table><caption>Table 1: Geometrical parameters of the forming roller and workpiece for tool path generation</caption><thead><tr><td>R (mm)</td><td>r (mm)</td><td>l (mm)</td><td>L (mm)</td><td>θ range</td></tr></thead><tbody><tr><td>175</td><td>5</td><td>50</td><td>500</td><td>0°-60°</td></tr></tbody></table>

The workpiece mentioned in the “part” section has been manufactured and numerically simulated using two different toolpaths following the above-mentioned toolpath theory. I) Part A: using a linear toolpath II) Part B: using a combination of two linear toolpaths

## Numerical Simulation

To evaluate the tool path effect on the final geometry and reach an in-depth knowledge of the process, experiments were simulated in the LS-DYNA™ software environment and solved by the explicit solver. A 0.8 mm thick steel strip with a length of 500 mm and width of 100 mm was modeled as a deformable body and a 175 mm dia. forming roller with a width of 35 mm and edge roundness of 5 mm was modeled as a rigid body. Both the strip and forming roller were meshed by fully integrated shell elements with a 1 mm average mesh size and 5 integration points along the thickness. The stripe was modeled based on the results of the tensile tests results which were reported in the material section, using the Hollomon work hardening model. The dynamic friction coefficient of 0.2 was considered between the forming roller and the stripe. The tool paths were applied based on the formula introduced in the toolpath theory section.

## Results and discussion

Fig. 6 a and b shows the photograph of the part A and B, respectively. It is evident that while part A was manufactured without any wrinkling defects, a wrinkling defect was observed in the vicinity of y = 350 mm in part B. To obtain a thorough comprehension of the wrinkling defect, the parts were scanned using a 7-axis CMM Nikon MCAX 3D laser scanner with a precision of ± 0.01 mm.
$$ \Delta P_{\text{ideal}} = P_{f_{\text{ideal}}} - P_0 \qquad (11) $$

$$ \Delta x_{\text{ideal}} = x_{f_{\text{ideal}}} - x_0 \qquad (12) $$

$$ \Delta y_{\text{ideal}} = y_{f_{\text{ideal}}} - y_0 \qquad (13) $$

$$ \Delta z_{\text{ideal}} = z_{f_{\text{ideal}}} - z_0 \qquad (14) $$

Therefore, the final real position of the forming roller considering the r and R effect can be calculated as following:

$$ x_{f_{\text{real}}} = x_0 + \Delta x_{\text{ideal}} + \Delta x_r \qquad (15) $$

$$ y_{f_{\text{real}}} = y_0 + \Delta y_{\text{ideal}} + \Delta y_R \qquad (16) $$

$$ z_{f_{\text{real}}} = z_0 + \Delta z_{\text{ideal}} + \Delta z_r + \Delta z_R \qquad (17) $$

## Tool path

For tool path calculation R and r as the constants of the forming roller and l, L, and θ as the parameters related to the geometry of the workpiece is needed which are provided in Table 1.

<table><caption>Table 1: Geometrical parameters of the forming roller and workpiece for tool path generation</caption><thead><tr><td>R (mm)</td><td>r (mm)</td><td>l (mm)</td><td>L (mm)</td><td>θ range</td></tr></thead><tbody><tr><td>175</td><td>5</td><td>50</td><td>500</td><td>0°-60°</td></tr></tbody></table>

The workpiece mentioned in the “part” section has been manufactured and numerically simulated using two different toolpaths following the above-mentioned toolpath theory. I) Part A: using a linear toolpath II) Part B: using a combination of two linear toolpaths

## Numerical Simulation

To evaluate the tool path effect on the final geometry and reach an in-depth knowledge of the process, experiments were simulated in the LS-DYNA™ software environment and solved by the explicit solver. A 0.8 mm thick steel strip with a length of 500 mm and width of 100 mm was modeled as a deformable body and a 175 mm dia. forming roller with a width of 35 mm and edge roundness of 5 mm was modeled as a rigid body. Both the strip and forming roller were meshed by fully integrated shell elements with a 1 mm average mesh size and 5 integration points along the thickness. The stripe was modeled based on the results of the tensile tests results which were reported in the material section, using the Hollomon work hardening model. The dynamic friction coefficient of 0.2 was considered between the forming roller and the stripe. The tool paths were applied based on the formula introduced in the toolpath theory section.

## Results and discussion

Fig. 6 a and b shows the photograph of the part A and B, respectively. It is evident that while part A was manufactured without any wrinkling defects, a wrinkling defect was observed in the vicinity of y = 350 mm in part B. To obtain a thorough comprehension of the wrinkling defect, the parts were scanned using a 7-axis CMM Nikon MCAX 3D laser scanner with a precision of ± 0.01 mm.
<figure><img src="image_1401.png" /><figcaption>Fig. 6. The workpiece photograph for (a) part A; (b) part B</figcaption></figure>

The intersection of parts A and B with eleven parallel planes, each at a constant distance of 50 millimeters and with a normal vector along the y-direction, is presented in Fig 7 a and b. On the other hand, the cross sectional view of the ideal target part is presented in Fig 7 (c)

<figure><img src="image_1402.png" /><figcaption>Fig. 7. cross sectional view of the (a) part A; (b) part B (c) ideal target part</figcaption></figure>

In the case of part A, cross sectional cuts revealing the straight flange, except the vicinity of the AA' line in all cross sections. However, in the case of part B, curved flange observed in the cross sections captured in the range of y = 150 mm to y = 350 mm which indicated with blue color in Fig. 7b. The curved flange leads to change the stripe-roller contact point from the round edge to lateral side of the forming roller. In part B, where a two-stage toolpath has been employed in its manufacturing, the tool gets very close to the edge of the sheet in the vicinity of the y = 250 mm and slipping of the sheet on the roller edge and following lateral contact situation which results in wrinkling in vicinity of y = 350 mm. Results of 3d scanning demonstrate that flange angle varies from 4° to 50 and from 3° to 50° for part A and B, respectively. As it mentioned above, flange angle varies from 0° to 60° in the ideal target part.
<figure><img src="image_1405.png" /><figcaption>Fig. 6. The workpiece photograph for (a) part A; (b) part B</figcaption></figure>

The intersection of parts A and B with eleven parallel planes, each at a constant distance of 50 millimeters and with a normal vector along the y-direction, is presented in Fig 7 a and b. On the other hand, the cross sectional view of the ideal target part is presented in Fig 7 (c)

<figure><img src="image_1406.png" /><figcaption>Fig. 7. cross sectional view of the (a) part A; (b) part B (c) ideal target part</figcaption></figure>

In the case of part A, cross sectional cuts revealing the straight flange, except the vicinity of the AA' line in all cross sections. However, in the case of part B, curved flange observed in the cross sections captured in the range of y = 150 mm to y = 350 mm which indicated with blue color in Fig. 7b. The curved flange leads to change the stripe-roller contact point from the round edge to lateral side of the forming roller. In part B, where a two-stage toolpath has been employed in its manufacturing, the tool gets very close to the edge of the sheet in the vicinity of the y = 250 mm and slipping of the sheet on the roller edge and following lateral contact situation which results in wrinkling in vicinity of y = 350 mm. Results of 3d scanning demonstrate that flange angle varies from 4° to 50 and from 3° to 50° for part A and B, respectively. As it mentioned above, flange angle varies from 0° to 60° in the ideal target part.
<figure><img src="image_1403.png" /><figcaption>Fig. 8. Effective plastic strain of (a) part A; (b) part B</figcaption></figure>

Fig. 8 a and b show the results of the numerical simulation of the process for part A and B respectively. Bent angle measurement of cross-sectional cuts shows that flange angle varies in the 5°-55° and 5°-54° range for part A and B, respectively. In the case of Part B, wrinkling occurs in y = 345 mm, while part A manufactured without wrinkling effect. For a more detailed review, the part A can be divided into two sections: I) Both deformation area and strain intensity are ascending with respect to y amount. II) In this section, the deformation area is descending, and strain intensity is ascending with respect to y amount. Near the border of the section I and II secondary deformation has been observed in the contact point of the stripe-forming roller. On the other hand, part B can be divided into three sections: I) Both deformation area and strain intensity are ascending with respect to y amount, same with part A. II) in this section, plastic deformation occurs throughout the entire flange, leading to the curved flange and wrinkling defect. III) In this section, the deformation area is descending and strain intensity is ascending with respect to y amount, same with part A. section II of part A and section III of part B is ideal. Because the deformation area is concentrated and secondary deformation is negligible.

## Conclusions

The paper presented the theory of the tool path for incremental roll profiling. Number of the increments has a profound impact on the geometrical accuracy of the manufactured part. As the number of increments increases, the tool approaches the edge of the sheet, heightening the probability of stripe contact with the lateral surface of the roller which results in curved flange profile and wrinkling defect. Therefore, it's recommended to increase the roller-stripe contact point distance from the flange edge in the case of the increasing of the number of the increments. Presented results shows the capability of the introduced incremental forming process in manufacturing of the complex geometries.

## Acknowledgments

This study was carried out within the PNRR research activities of the consortium iNEST (Interconnected North-Est Innovation Ecosystem) funded by the European Union Next-GenerationEU (Piano Nazionale di Ripresa e Resilienza (PNRR) – Missione 4 Componente 2, Investimento 1.5 – D.D. 1058 23/06/2022, ECS_00000043). This manuscript reflects only the Authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them.

## References

[1] Monostori, László, Botond Kádár, Thomas Bauernhansl, Shinsuke Kondoh, Soundar Kumara, Gunther Reinhart, Olaf Sauer, Gunther Schuh, Wilfried Sihn, and Kenichi Ueda, Cyber-physical systems in manufacturing, Cirp Annals 65, no. 2 (2016): 621-641. https://doi.org/10.1016/j.cirp.2016.06.005
<figure><img src="image_1407.png" /><figcaption>Fig. 8. Effective plastic strain of (a) part A; (b) part B</figcaption></figure>

Fig. 8 a and b show the results of the numerical simulation of the process for part A and B respectively. Bent angle measurement of cross-sectional cuts shows that flange angle varies in the 5°-55° and 5°-54° range for part A and B, respectively. In the case of Part B, wrinkling occurs in y = 345 mm, while part A manufactured without wrinkling effect. For a more detailed review, the part A can be divided into two sections: I) Both deformation area and strain intensity are ascending with respect to y amount. II) In this section, the deformation area is descending, and strain intensity is ascending with respect to y amount. Near the border of the section I and II secondary deformation has been observed in the contact point of the stripe-forming roller. On the other hand, part B can be divided into three sections: I) Both deformation area and strain intensity are ascending with respect to y amount, same with part A. II) in this section, plastic deformation occurs throughout the entire flange, leading to the curved flange and wrinkling defect. III) In this section, the deformation area is descending and strain intensity is ascending with respect to y amount, same with part A. section II of part A and section III of part B is ideal. Because the deformation area is concentrated and secondary deformation is negligible.

## Conclusions

The paper presented the theory of the tool path for incremental roll profiling. Number of the increments has a profound impact on the geometrical accuracy of the manufactured part. As the number of increments increases, the tool approaches the edge of the sheet, heightening the probability of stripe contact with the lateral surface of the roller which results in curved flange profile and wrinkling defect. Therefore, it's recommended to increase the roller-stripe contact point distance from the flange edge in the case of the increasing of the number of the increments. Presented results shows the capability of the introduced incremental forming process in manufacturing of the complex geometries.

## Acknowledgments

This study was carried out within the PNRR research activities of the consortium iNEST (Interconnected North-Est Innovation Ecosystem) funded by the European Union Next-GenerationEU (Piano Nazionale di Ripresa e Resilienza (PNRR) – Missione 4 Componente 2, Investimento 1.5 – D.D. 1058 23/06/2022, ECS_00000043). This manuscript reflects only the Authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them.

## References

[1] Monostori, László, Botond Kádár, Thomas Bauernhansl, Shinsuke Kondoh, Soundar Kumara, Gunther Reinhart, Olaf Sauer, Gunther Schuh, Wilfried Sihn, and Kenichi Ueda, Cyber-physical systems in manufacturing, Cirp Annals 65, no. 2 (2016): 621-641. https://doi.org/10.1016/j.cirp.2016.06.005
[2] Kazancoglu, Ipek, Melisa Ozbiltekin-Pala, Sachin Kumar Mangla, Yigit Kazancoglu, and Fauzia Jabeen, Role of flexibility, agility and responsiveness for sustainable supply chain resilience during COVID-19, *Journal of Cleaner Production* 362 (2022): 132431.
https://doi.org/10.1016/j.jclepro.2022.132431

[3] Ambrogio, G., Luigino Filice, and F. Gagliardi, Formability of lightweight alloys by hot incremental sheet forming, *Materials & Design* 34 (2012): 501-508.
https://doi.org/10.1016/j.matdes.2011.08.024

[4] Dang, Xiaobing, Kai He, Feifei Zhang, Qiyang Zuo, and Ruxu Du, Multi-stage incremental bending to form doubly curved metal plates based on bending limit diagram, *International Journal of Mechanical Sciences* 155 (2019): 19-30.
https://doi.org/10.1016/j.ijmecsci.2019.02.001

[5] Ambrogio, G., L. De Napoli, L. Filice, F. Gagliardi, and M. Muzzupappa, Application of Incremental Forming process for high customised medical product manufacturing, *Journal of materials processing technology* 162 (2005): 156-162.
https://doi.org/10.1016/j.jmatprotec.2005.02.148

[6] Attanasio, Aldo, Elisabetta Ceretti, Claudio Giardini, and Luca Mazzoni, Asymmetric two points incremental forming: improving surface quality and geometric accuracy by tool path optimization, *Journal of materials processing technology* 197, no. 1-3 (2008): 59-67.
https://doi.org/10.1016/j.jmatprotec.2007.05.053

[7] Deole, Aditya D., Matthew R. Barnett, and Matthias Weiss, The numerical prediction of ductile fracture of martensitic steel in roll forming, *International Journal of Solids and Structures* 144 (2018): 20-31.
https://doi.org/10.1016/j.ijsolstr.2018.04.011

[8] Kim, Dongun, MyungHwan Cha, and Yeon Sik Kang, Development of the bus frame by flexible roll forming, *Procedia Engineering* 183 (2017): 11-16.
https://doi.org/10.1016/j.proeng.2017.04.004

[9] Abeyrathna, Buddhika, Bernard Rolfe, Libo Pan, Rui Ge, and Matthias Weiss, Flexible roll forming of an automotive component with variable depth, *Advances in Materials and Processing Technologies* 2, no. 4 (2016): 527-538.
https://doi.org/10.1080/2374068X.2016.1247234

[10] Groche, Peter, Arnd Zettler, Sebastian Berner, and Georg Schneider, Development and verification of a one-step-model for the design of flexible roll formed parts, *International Journal of Material Forming* 4 (2011): 371-377.
https://doi.org/10.1007/s12289-010-0998-3

[11] Jeswiet, J., F. Micari, G. Hirt, A. Bramley, Joost Duflou, and J. Allwood, Asymmetric single point incremental forming of sheet metal, *CIRP annals* 54, no. 2 (2005): 88-114.
https://doi.org/10.1016/S0007-8506(07)60021-3

[12] Patel, Dharmin, and Anishkumar Gandhi, A review article on process parameters affecting Incremental Sheet Forming (ISF), *Materials Today: Proceedings* 63 (2022): 368-375.
https://doi.org/10.1016/j.matpr.2022.03.208

[13] Jackson, Kathryn, and Julian Allwood, The mechanics of incremental sheet forming, *Journal of materials processing technology* 209, no. 3 (2009): 1158-1174.
https://doi.org/10.1016/j.jmatprotec.2008.03.025

[14] Music, O., J. M. Allwood, and K. Kawai, A review of the mechanics of metal spinning, *Journal of materials processing technology* 210, no. 1 (2010): 3-23.
https://doi.org/10.1016/j.jmatprotec.2009.08.021
[2] Kazancoglu, Ipek, Melisa Ozbiltekin-Pala, Sachin Kumar Mangla, Yigit Kazancoglu, and Fauzia Jabeen, Role of flexibility, agility and responsiveness for sustainable supply chain resilience during COVID-19, *Journal of Cleaner Production* 362 (2022): 132431.
https://doi.org/10.1016/j.jclepro.2022.132431

[3] Ambrogio, G., Luigino Filice, and F. Gagliardi, Formability of lightweight alloys by hot incremental sheet forming, *Materials & Design* 34 (2012): 501-508.
https://doi.org/10.1016/j.matdes.2011.08.024

[4] Dang, Xiaobing, Kai He, Feifei Zhang, Qiyang Zuo, and Ruxu Du, Multi-stage incremental bending to form doubly curved metal plates based on bending limit diagram, *International Journal of Mechanical Sciences* 155 (2019): 19-30.
https://doi.org/10.1016/j.ijmecsci.2019.02.001

[5] Ambrogio, G., L. De Napoli, L. Filice, F. Gagliardi, and M. Muzzupappa, Application of Incremental Forming process for high customised medical product manufacturing, *Journal of materials processing technology* 162 (2005): 156-162.
https://doi.org/10.1016/j.jmatprotec.2005.02.148

[6] Attanasio, Aldo, Elisabetta Ceretti, Claudio Giardini, and Luca Mazzoni, Asymmetric two points incremental forming: improving surface quality and geometric accuracy by tool path optimization, *Journal of materials processing technology* 197, no. 1-3 (2008): 59-67.
https://doi.org/10.1016/j.jmatprotec.2007.05.053

[7] Deole, Aditya D., Matthew R. Barnett, and Matthias Weiss, The numerical prediction of ductile fracture of martensitic steel in roll forming, *International Journal of Solids and Structures* 144 (2018): 20-31.
https://doi.org/10.1016/j.ijsolstr.2018.04.011

[8] Kim, Dongun, MyungHwan Cha, and Yeon Sik Kang, Development of the bus frame by flexible roll forming, *Procedia Engineering* 183 (2017): 11-16.
https://doi.org/10.1016/j.proeng.2017.04.004

[9] Abeyrathna, Buddhika, Bernard Rolfe, Libo Pan, Rui Ge, and Matthias Weiss, Flexible roll forming of an automotive component with variable depth, *Advances in Materials and Processing Technologies* 2, no. 4 (2016): 527-538.
https://doi.org/10.1080/2374068X.2016.1247234

[10] Groche, Peter, Arnd Zettler, Sebastian Berner, and Georg Schneider, Development and verification of a one-step-model for the design of flexible roll formed parts, *International Journal of Material Forming* 4 (2011): 371-377.
https://doi.org/10.1007/s12289-010-0998-3

[11] Jeswiet, J., F. Micari, G. Hirt, A. Bramley, Joost Duflou, and J. Allwood, Asymmetric single point incremental forming of sheet metal, *CIRP annals* 54, no. 2 (2005): 88-114.
https://doi.org/10.1016/S0007-8506(07)60021-3

[12] Patel, Dharmin, and Anishkumar Gandhi, A review article on process parameters affecting Incremental Sheet Forming (ISF), *Materials Today: Proceedings* 63 (2022): 368-375.
https://doi.org/10.1016/j.matpr.2022.03.208

[13] Jackson, Kathryn, and Julian Allwood, The mechanics of incremental sheet forming, *Journal of materials processing technology* 209, no. 3 (2009): 1158-1174.
https://doi.org/10.1016/j.jmatprotec.2008.03.025

[14] Music, O., J. M. Allwood, and K. Kawai, A review of the mechanics of metal spinning, *Journal of materials processing technology* 210, no. 1 (2010): 3-23.
https://doi.org/10.1016/j.jmatprotec.2009.08.021
[15] Russo, Iacopo M., Christopher J. Cleaver, and Julian M. Allwood, Seven principles of toolpath design in conventional metal spinning, *Journal of Materials Processing Technology* 294 (2021): 117131. https://doi.org/10.1016/j.jmatprotec.2021.117131

[16] Voswinckel, Holger, Markus Bambach, and Gerhard Hirt, Process limits of stretch and shrink flanging by incremental sheet metal forming, *Key engineering materials* 549 (2013): 45-52. https://doi.org/10.4028/www.scientific.net/KEM.549.45

[17] Voswinckel, Holger, Markus Bambach, and Gerhard Hirt, Improving geometrical accuracy for flanging by incremental sheet metal forming, *International Journal of Material Forming* 8 (2015): 391-399. https://doi.org/10.1007/s12289-014-1182-y

[18] Buranathiti, Thaweepat, Jian Cao, Wei Chen, Lusine Baghdasaryan, and Z. Cedric Xia, Approaches for model validation: methodology and illustration on a sheet metal flanging process, *J. Manuf. Sci. Eng.* 128, no. 2 (2006): 588-597. https://doi.org/10.1115/1.1807852

[19] Le Maoût, Nicolas, S. Thuillier, and P. Y. Manach, Classical and roll-hemming processes of pre-strained metallic sheets, *Experimental mechanics* 50 (2010): 1087-1097. https://doi.org/10.1007/s11340-009-9297-7

[20] Wang, Jian Qiang, Ya Wen Huang, Guang Rui Zhou, Zhao Long Niu, and Zi Wen Cheng, Process Parameters Study on Robot Rope Hemming of the Hood, *Applied Mechanics and Materials* 602 (2014): 950-953. https://doi.org/10.4028/www.scientific.net/AMM.602-605.950

[21] Hu, Xing, Yi Xi Zhao, Shu Hui Li, and Cheng Liu, Numerical Simulation of Dimensional Variations for Roller Hemming, *Advanced Materials Research* 160 (2011): 1601-1605. https://doi.org/10.4028/www.scientific.net/AMR.160-162.1601

[22] Simonetto, Enrico, Andrea Ghiotti, Stefania Bruschi, and Stefano Filippi, Flexible Incremental Roller Flanging process for metal sheets profiles, *Procedia CIRP* 103 (2021): 219-224. https://doi.org/10.1016/j.procir.2021.10.035

[23] Simonetto, Enrico, Andrea Ghiotti, and Stefania Bruschi, Agile manufacturing of complex shaped bent profiles by incremental deformation, *Manufacturing Letters* 36 (2023): 40-43. https://doi.org/10.1016/j.mfglet.2023.01.004
[15] Russo, Iacopo M., Christopher J. Cleaver, and Julian M. Allwood, Seven principles of toolpath design in conventional metal spinning, *Journal of Materials Processing Technology* 294 (2021): 117131. https://doi.org/10.1016/j.jmatprotec.2021.117131

[16] Voswinckel, Holger, Markus Bambach, and Gerhard Hirt, Process limits of stretch and shrink flanging by incremental sheet metal forming, *Key engineering materials* 549 (2013): 45-52. https://doi.org/10.4028/www.scientific.net/KEM.549.45

[17] Voswinckel, Holger, Markus Bambach, and Gerhard Hirt, Improving geometrical accuracy for flanging by incremental sheet metal forming, *International Journal of Material Forming* 8 (2015): 391-399. https://doi.org/10.1007/s12289-014-1182-y

[18] Buranathiti, Thaweepat, Jian Cao, Wei Chen, Lusine Baghdasaryan, and Z. Cedric Xia, Approaches for model validation: methodology and illustration on a sheet metal flanging process, *J. Manuf. Sci. Eng.* 128, no. 2 (2006): 588-597. https://doi.org/10.1115/1.1807852

[19] Le Maoût, Nicolas, S. Thuillier, and P. Y. Manach, Classical and roll-hemming processes of pre-strained metallic sheets, *Experimental mechanics* 50 (2010): 1087-1097. https://doi.org/10.1007/s11340-009-9297-7

[20] Wang, Jian Qiang, Ya Wen Huang, Guang Rui Zhou, Zhao Long Niu, and Zi Wen Cheng, Process Parameters Study on Robot Rope Hemming of the Hood, *Applied Mechanics and Materials* 602 (2014): 950-953. https://doi.org/10.4028/www.scientific.net/AMM.602-605.950

[21] Hu, Xing, Yi Xi Zhao, Shu Hui Li, and Cheng Liu, Numerical Simulation of Dimensional Variations for Roller Hemming, *Advanced Materials Research* 160 (2011): 1601-1605. https://doi.org/10.4028/www.scientific.net/AMR.160-162.1601

[22] Simonetto, Enrico, Andrea Ghiotti, Stefania Bruschi, and Stefano Filippi, Flexible Incremental Roller Flanging process for metal sheets profiles, *Procedia CIRP* 103 (2021): 219-224. https://doi.org/10.1016/j.procir.2021.10.035

[23] Simonetto, Enrico, Andrea Ghiotti, and Stefania Bruschi, Agile manufacturing of complex shaped bent profiles by incremental deformation, *Manufacturing Letters* 36 (2023): 40-43. https://doi.org/10.1016/j.mfglet.2023.01.004
# Investigation of the stress state around the forming zone during the flow form process

CEBIROGLU Bugra¹,ᵃ*, OZASLAN Emre¹,ᵇ*, YETGIN Ali¹,ᶜ and ACAR Bulent¹,ᵈ

¹Repkon Machine and Tools Industry and Trade Inc., Istanbul Turkey

ᵃbugra.cebiroglu@repkon.com.tr, ᵇemre.ozaslan@repkon.com.tr, ᶜali.yetgin@repkon.com.tr,

ᵇbulent.acar@repkon.com.tr

**Keywords:** Flow Forming, Finite Element Modeling, Stress State

**Abstract.** Flow forming process is a cold metal forming process, which has great advantages over conventional forming techniques such as extrusion and tube drawing. Some advantages of the flow forming process are high material utilization, close dimensional tolerances and excellent surface finish. Flow forming finds extensive applications in industries such as aerospace, automotive, and manufacturing, where the demand for lightweight, durable, and intricately shaped components is crucial. Different types of geometries including thin-walled and axisymmetric cylindrical parts can be manufactured by flow form process. In flow forming process, the preform, which is clamped to a rotating mandrel, is formed by the rollers. It involves the incremental shaping of a rotating workpiece by applying axial compressive forces while simultaneously feeding it through a set of rollers. After the process, length of the preform is increased by decreasing the outer diameter. However, one of drawbacks of the process is the complex mechanical behavior around the forming zone. Understanding the material flow and stress state around the forming zone is crucial to perform a successful process. In this study, stress state around the forming zone was investigated numerically. After validating of 3-D explicit finite element model with the measured loads from experiments, a detailed examination of the stress state around the forming zone was performed numerically. Effects of different process parameters such as thickness on the stress state were emphasized. Also, differences between the stress states around the forming zone of the forward and backward flow form processes, which are two different techniques of the flow form process, were stated. It is also shown that different stress states occur at different stages of the process. Finally, the relation between stress state and failure mode was explained.

**Introduction**

In flow forming, rotating workpiece is supported by a mandrel while rollers move through axial direction. The rollers are free to rotate around their own axes. A lubricant is utilized between the workpiece-roller and workpiece-mandrel interface to minimize the shear stress and cool the workpiece. The thickness of the workpiece is reduced and the length is increased during the forming process. There are two main methods of the flow forming process, which are forward and backward. In forward flow forming process, material flow occurs in front of the rollers and in the direction of the roller movement. However, material flows in the opposite direction of the roller movement in backward flow forming process. Forward flow forming is used when no changes to the form on the tail side are requested for the preform. In forward method, the flow direction of the material is restricted by the tailstock. The mechanisms of the forward and backward flow forming methods are illustrated in Figure 1.

<figure><img src="image_1404.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
# Investigation of the stress state around the forming zone during the flow form process

CEBIROGLU Bugra¹,ᵃ*, OZASLAN Emre¹,ᵇ*, YETGIN Ali¹,ᶜ and ACAR Bulent¹,ᵈ

¹Repkon Machine and Tools Industry and Trade Inc., Istanbul Turkey

ᵃbugra.cebiroglu@repkon.com.tr, ᵇemre.ozaslan@repkon.com.tr, ᶜali.yetgin@repkon.com.tr,

ᵇbulent.acar@repkon.com.tr

**Keywords:** Flow Forming, Finite Element Modeling, Stress State

**Abstract.** Flow forming process is a cold metal forming process, which has great advantages over conventional forming techniques such as extrusion and tube drawing. Some advantages of the flow forming process are high material utilization, close dimensional tolerances and excellent surface finish. Flow forming finds extensive applications in industries such as aerospace, automotive, and manufacturing, where the demand for lightweight, durable, and intricately shaped components is crucial. Different types of geometries including thin-walled and axisymmetric cylindrical parts can be manufactured by flow form process. In flow forming process, the preform, which is clamped to a rotating mandrel, is formed by the rollers. It involves the incremental shaping of a rotating workpiece by applying axial compressive forces while simultaneously feeding it through a set of rollers. After the process, length of the preform is increased by decreasing the outer diameter. However, one of drawbacks of the process is the complex mechanical behavior around the forming zone. Understanding the material flow and stress state around the forming zone is crucial to perform a successful process. In this study, stress state around the forming zone was investigated numerically. After validating of 3-D explicit finite element model with the measured loads from experiments, a detailed examination of the stress state around the forming zone was performed numerically. Effects of different process parameters such as thickness on the stress state were emphasized. Also, differences between the stress states around the forming zone of the forward and backward flow form processes, which are two different techniques of the flow form process, were stated. It is also shown that different stress states occur at different stages of the process. Finally, the relation between stress state and failure mode was explained.

**Introduction**

In flow forming, rotating workpiece is supported by a mandrel while rollers move through axial direction. The rollers are free to rotate around their own axes. A lubricant is utilized between the workpiece-roller and workpiece-mandrel interface to minimize the shear stress and cool the workpiece. The thickness of the workpiece is reduced and the length is increased during the forming process. There are two main methods of the flow forming process, which are forward and backward. In forward flow forming process, material flow occurs in front of the rollers and in the direction of the roller movement. However, material flows in the opposite direction of the roller movement in backward flow forming process. Forward flow forming is used when no changes to the form on the tail side are requested for the preform. In forward method, the flow direction of the material is restricted by the tailstock. The mechanisms of the forward and backward flow forming methods are illustrated in Figure 1.

<figure><img src="image_1408.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
<figure><img src="image_1405.png" /><figcaption>Fig. 1. Flow forming process schematic a) forward flow forming process, b) backward flow forming process</figcaption></figure>

Incremental forming with localized contact area causes a complex behavior of the material around the roller contact zone. The deformation is strongly restricted by surrounding material, which causes a complex stress state around the forming zone. Xu et al.[1] performed an analysis to classify the stress state around the forming zone. They did not observe an apparent difference between forward and backward forming in terms of stress state. A non-homogeneous material flow arises resulting from a situation of local deformation between the workpiece and rollers. Because of non-homogeneous material flow, evaluating the workpiece's stress state is essential to its quality during the flow-forming process. S.C. Chang et al.[2] investigated the effects of inhomogeneous material flow on the incrementally thickness reduction for both forward and backward methods in aluminum alloys. However, taking into account the stress state on the preform and the contact pressures between surfaces will yield deeper understanding about this problem. In the study conducted by M.S. Mohebbi and A. Akbarzadeh [3], finite element simulations were carried out to explore the evolution of redundant strains in a single-roller flow forming process. It was observed that the occurrence of reversal straining significantly influences the redundant work in the process.

In this study, a detailed investigation was conducted on the stress state around the forming zone. The comparison was made mainly between forward and backward flow forming processes. In addition, thickness were also considered. Firstly, forces in different directions were compared. The forward and backward forming processes were then compared in terms of stress in various forming zones. In the context of the study, the stress state in the initial stages of the process is under scrutiny. Consequently, an examination is conducted on the impact of forming in the vicinity near the free edge of the tube on the stress state.

## Methodology

Abaqus/Explicit [4], a commercial code was employed to simulate the forming process. The workpiece material in this study was chosen to be S4140. The analyses utilized material properties depicted in Figure 2 generated from JMatPro [5], which varied with strain rate. Even though it is recognized that the material's sensitivity to rate has a limited impact in cold forming, the material was modeled to incorporate rate dependency. An elasto-plastic material model was defined for workpiece, while roller and mandrel were assumed to be rigid.
<figure><img src="image_1409.png" /><figcaption>Fig. 1. Flow forming process schematic a) forward flow forming process, b) backward flow forming process</figcaption></figure>

Incremental forming with localized contact area causes a complex behavior of the material around the roller contact zone. The deformation is strongly restricted by surrounding material, which causes a complex stress state around the forming zone. Xu et al.[1] performed an analysis to classify the stress state around the forming zone. They did not observe an apparent difference between forward and backward forming in terms of stress state. A non-homogeneous material flow arises resulting from a situation of local deformation between the workpiece and rollers. Because of non-homogeneous material flow, evaluating the workpiece's stress state is essential to its quality during the flow-forming process. S.C. Chang et al.[2] investigated the effects of inhomogeneous material flow on the incrementally thickness reduction for both forward and backward methods in aluminum alloys. However, taking into account the stress state on the preform and the contact pressures between surfaces will yield deeper understanding about this problem. In the study conducted by M.S. Mohebbi and A. Akbarzadeh [3], finite element simulations were carried out to explore the evolution of redundant strains in a single-roller flow forming process. It was observed that the occurrence of reversal straining significantly influences the redundant work in the process.

In this study, a detailed investigation was conducted on the stress state around the forming zone. The comparison was made mainly between forward and backward flow forming processes. In addition, thickness were also considered. Firstly, forces in different directions were compared. The forward and backward forming processes were then compared in terms of stress in various forming zones. In the context of the study, the stress state in the initial stages of the process is under scrutiny. Consequently, an examination is conducted on the impact of forming in the vicinity near the free edge of the tube on the stress state.

## Methodology

Abaqus/Explicit [4], a commercial code was employed to simulate the forming process. The workpiece material in this study was chosen to be S4140. The analyses utilized material properties depicted in Figure 2 generated from JMatPro [5], which varied with strain rate. Even though it is recognized that the material's sensitivity to rate has a limited impact in cold forming, the material was modeled to incorporate rate dependency. An elasto-plastic material model was defined for workpiece, while roller and mandrel were assumed to be rigid.
<figure><img src="image_1406.png" /><figcaption>Fig. 2. Stress – Strain curve of the S4140 with strain rate sensitivity.</figcaption></figure>

The finite element setup is depicted in Figure 2. In the analyses, temperature changes that may occur due to plastic deformation and friction have been neglected. A refined mesh was utilized around the forming zone to accurately capture the stress state, while a coarser mesh was employed far from the forming zone to reduce computational time. The linear reduced element type was used to shorten computational time and prevent the shear locking problem, but this element type, which has a one point integration, can cause hourglass, so the enhanced hourglass control was applied. A total of 83500 CD8R elements were used. Mass scaling, a technique to expedite computation by increasing the mass of the workpiece, was implemented in the simulation. Following the analytical solving stage, it was made sure to confirm that the mass scale factor (EMSF indicator) in ABAQUS was low enough to avoid having an effect on the solution. Artificial strain and kinetic energy parameters were checked to ensure that the inertial forces did not influence the solution. Since the purpose of the simulation was to investigate stress state solely, temperature was not included in the simulation. The process parameters and boundary conditions are summarized in Figure 3.

<figure><img src="image_1407.png" /><figcaption>Fig. 2. Finite element model of the flow forming process with three rollers.</figcaption></figure>
<figure><img src="image_1410.png" /><figcaption>Fig. 2. Stress – Strain curve of the S4140 with strain rate sensitivity.</figcaption></figure>

The finite element setup is depicted in Figure 2. In the analyses, temperature changes that may occur due to plastic deformation and friction have been neglected. A refined mesh was utilized around the forming zone to accurately capture the stress state, while a coarser mesh was employed far from the forming zone to reduce computational time. The linear reduced element type was used to shorten computational time and prevent the shear locking problem, but this element type, which has a one point integration, can cause hourglass, so the enhanced hourglass control was applied. A total of 83500 CD8R elements were used. Mass scaling, a technique to expedite computation by increasing the mass of the workpiece, was implemented in the simulation. Following the analytical solving stage, it was made sure to confirm that the mass scale factor (EMSF indicator) in ABAQUS was low enough to avoid having an effect on the solution. Artificial strain and kinetic energy parameters were checked to ensure that the inertial forces did not influence the solution. Since the purpose of the simulation was to investigate stress state solely, temperature was not included in the simulation. The process parameters and boundary conditions are summarized in Figure 3.

<figure><img src="image_1411.png" /><figcaption>Fig. 2. Finite element model of the flow forming process with three rollers.</figcaption></figure>
## Results

Firstly, the forces occurring during the forming process were analyzed and compared with the experimentally measured results. The experimental measurement was performed only for the backward forming, and after verification of the analysis results, the analysis scenarios were continued. The verification results given in Figure 4 belong to the radial direction force data in the backward process performed in Repkon and the results performed in Abaqus/Explicit.

<figure><img src="image_1408.png" /><figcaption>Fig. 3. Comparison of process forces obtained from experiments and analysis</figcaption></figure>

As seen in the Figure 5 the forces obtained from the backward forming analysis were higher than those obtained from forward forming. The larger contact area observed on the bottom surface of the workpiece in backward forming is the reason for the higher forces in backward forming. In the force graph, it can be seen that the load increases rapidly in the first 0.4 seconds (transient), and remains constant during the process (steady state). The steady state region refers to the situation where full contact between the roller and the workpiece is achieved. The presentation includes only the initial 2 seconds of data to provide a detailed view of the initial increasing trend of the load.

<figure><img src="image_1409.png" /><figcaption>Fig. 5. The time dependent force variations and contact pressure distributions in the analyses (B and F refer backward and forward, respectively)</figcaption></figure>

In the radial direction, a compressive stress state was observed on both top and bottom surfaces of the workpiece for the Backward process and these stress distribution are shown in the Figure 6. The contacts of roller/workpiece and mandrel/workpiece created a pure compressive stress in radial direction around the forming zone. This radial compressive stress state plays a crucial role
## Results

Firstly, the forces occurring during the forming process were analyzed and compared with the experimentally measured results. The experimental measurement was performed only for the backward forming, and after verification of the analysis results, the analysis scenarios were continued. The verification results given in Figure 4 belong to the radial direction force data in the backward process performed in Repkon and the results performed in Abaqus/Explicit.

<figure><img src="image_1412.png" /><figcaption>Fig. 3. Comparison of process forces obtained from experiments and analysis</figcaption></figure>

As seen in the Figure 5 the forces obtained from the backward forming analysis were higher than those obtained from forward forming. The larger contact area observed on the bottom surface of the workpiece in backward forming is the reason for the higher forces in backward forming. In the force graph, it can be seen that the load increases rapidly in the first 0.4 seconds (transient), and remains constant during the process (steady state). The steady state region refers to the situation where full contact between the roller and the workpiece is achieved. The presentation includes only the initial 2 seconds of data to provide a detailed view of the initial increasing trend of the load.

<figure><img src="image_1413.png" /><figcaption>Fig. 5. The time dependent force variations and contact pressure distributions in the analyses (B and F refer backward and forward, respectively)</figcaption></figure>

In the radial direction, a compressive stress state was observed on both top and bottom surfaces of the workpiece for the Backward process and these stress distribution are shown in the Figure 6. The contacts of roller/workpiece and mandrel/workpiece created a pure compressive stress in radial direction around the forming zone. This radial compressive stress state plays a crucial role
in enhancing the material deformation characteristics during the forming process. The consistent application of compressive forces by the roller/workpiece and mandrel/workpiece contacts ensures a controlled and uniform shaping of the workpiece in the radial direction. The observed compressive stress on the top and bottom surfaces not only signifies effective material compaction but also contributes to the overall structural integrity and dimensional accuracy of the formed product.

<figure><img src="image_1410.png" /><figcaption>Fig. 6. Radial stress distribution on the top and bottom surface of the workpiece for backward process.</figcaption></figure>

In the circumferential direction, a compressive stress state was inspected on both top and bottom surfaces of the workpiece along the roller path. However, residual tensile stresses appeared behind and in front of the roller on the top surface of the workpiece as shown in Figure 7 below. The mentioned expression of residual stress pertains to the stress that arises within the shaped part of the roller subsequent to the shaping process. Additionally, circumferential tensile stress occurred just in front of the roller on the bottom surface of the workpiece. The magnitude of the tensile stress occurred on the bottom surface in backward forming was more pronounced than in forward forming. Moreover, tensile stress seen on the bottom surface of the workpiece was more noticeable in the thick workpiece.

<figure><img src="image_1411.png" /><figcaption>Fig. 4. Circumferential stress distribution on the top and bottom surface of the workpiece a) backward, b) forward, c) backward – thick workpiece</figcaption></figure>

In the axial direction, a compressive stress state around the roller contact region raised on top surface of the workpiece. However, tensile stress was seen in front of the roller on the bottom surface due to material flow in the circumferential direction. The stress distributions for the forward, backward, and thick workpiece analyses are shown in Figure 8. The axial stress states observed in forward and thick workpiece situations were similar to those in backward forming.
in enhancing the material deformation characteristics during the forming process. The consistent application of compressive forces by the roller/workpiece and mandrel/workpiece contacts ensures a controlled and uniform shaping of the workpiece in the radial direction. The observed compressive stress on the top and bottom surfaces not only signifies effective material compaction but also contributes to the overall structural integrity and dimensional accuracy of the formed product.

<figure><img src="image_1414.png" /><figcaption>Fig. 6. Radial stress distribution on the top and bottom surface of the workpiece for backward process.</figcaption></figure>

In the circumferential direction, a compressive stress state was inspected on both top and bottom surfaces of the workpiece along the roller path. However, residual tensile stresses appeared behind and in front of the roller on the top surface of the workpiece as shown in Figure 7 below. The mentioned expression of residual stress pertains to the stress that arises within the shaped part of the roller subsequent to the shaping process. Additionally, circumferential tensile stress occurred just in front of the roller on the bottom surface of the workpiece. The magnitude of the tensile stress occurred on the bottom surface in backward forming was more pronounced than in forward forming. Moreover, tensile stress seen on the bottom surface of the workpiece was more noticeable in the thick workpiece.

<figure><img src="image_1415.png" /><figcaption>Fig. 4. Circumferential stress distribution on the top and bottom surface of the workpiece a) backward, b) forward, c) backward – thick workpiece</figcaption></figure>

In the axial direction, a compressive stress state around the roller contact region raised on top surface of the workpiece. However, tensile stress was seen in front of the roller on the bottom surface due to material flow in the circumferential direction. The stress distributions for the forward, backward, and thick workpiece analyses are shown in Figure 8. The axial stress states observed in forward and thick workpiece situations were similar to those in backward forming.
<figure><img src="image_1412.png" /><figcaption>Figure 5. Axial stress distribution on the top and bottom surface of the workpiece a) backward, b) forward, c) backward – thick workpiece</figcaption></figure>

Conclusion

Backward and forward flow forming process analysis was performed with three rollers. After the analyses, the stress state on the workpiece in both the backward and forward processes and the contact pressure between the rollers and the workpiece were evaluated accordingly. It is shown that the stress state varies around the forming area. The magnitude of the stress for backward forming was higher than the forward, while the stress state was similar. The contact pressure occurring on top and bottom face of the tube was observed more in the backward flow forming than in the forward. The compressive stress state was observed in the radial direction on the top and bottom surfaces of the workpiece. The compressive stress is higher in the backward flow forming analysis compared to the forward flow forming due to more contact areas on the bottom surfaces of the workpieces. The tensile stress occurring in front of the rollers was also more noticeable in the backward analysis. Tensile stress was observed in front of the rollers in the axial direction due to the material flow.

References

[1]Y. Xu, S.H. Zhang, P. Li, K. Yang, D.B. Shan, Y.Lu, . ‘3D rigid–plastic FEM numerical simulation on Tube Spinning’, Journal of Materials Processing Technology, 113 (2001), pp. 710–713. https://doi.org/10.1016/s0924-0136(01)00644-6

[2] S.C. Chang, C.A. Huang, S.Y. Yu, Y. Chang, W. Han, T.S. Shieh, H.C. Chung, H.T. Yao, G.D. Shyu, H.Y. Hou, C.C. Wang, W.S. Wang, ‘Tube spinnability of AA 2024 and 7075 Aluminum Alloys’, Journal of Materials Processing Technology, 80–81(1998) , pp. 676–682. https://doi.org/10.1016/s0924-0136(98)00174-5

[3] M.S. Mohebbi, A. Akbarzadeh, ‘Experimental study and FEM analysis of redundant strains in flow forming of tubes’, Journal of Materials Processing Technology, 210 (2010), pp. 389–395. https://doi.org/10.1016/j.jmatprotec.2009.09.028

[4] Dassault Systèmes D. Abaqus analysis user’s guide. Tech. rep., Technical Report Abaqus 6.14 Documentation. Simulia Corp; 2016.

[5] Saunders, N., Guo, U. K. Z., Li, X., Miodownik, A. P., & Schillé, J. P.. Using JMatPro to model materials properties and behavior. Jom, 55 (2003) , 60-65.
<figure><img src="image_1416.png" /><figcaption>Figure 5. Axial stress distribution on the top and bottom surface of the workpiece a) backward, b) forward, c) backward – thick workpiece</figcaption></figure>

Conclusion

Backward and forward flow forming process analysis was performed with three rollers. After the analyses, the stress state on the workpiece in both the backward and forward processes and the contact pressure between the rollers and the workpiece were evaluated accordingly. It is shown that the stress state varies around the forming area. The magnitude of the stress for backward forming was higher than the forward, while the stress state was similar. The contact pressure occurring on top and bottom face of the tube was observed more in the backward flow forming than in the forward. The compressive stress state was observed in the radial direction on the top and bottom surfaces of the workpiece. The compressive stress is higher in the backward flow forming analysis compared to the forward flow forming due to more contact areas on the bottom surfaces of the workpieces. The tensile stress occurring in front of the rollers was also more noticeable in the backward analysis. Tensile stress was observed in front of the rollers in the axial direction due to the material flow.

References

[1] Y. Xu, S.H. Zhang, P. Li, K. Yang, D.B. Shan, Y.Lu, '3D rigid-plastic FEM numerical simulation on Tube Spinning', Journal of Materials Processing Technology, 113 (2001), pp. 710-713. https://doi.org/10.1016/s0924-0136(01)00644-6

[2] S.C. Chang, C.A. Huang, S.Y. Yu, Y. Chang, W. Han, T.S. Shieh, H.C. Chung, H.T. Yao, G.D. Shyu, H.Y. Hou, C.C. Wang, W.S. Wang, 'Tube spinnability of AA 2024 and 7075 Aluminum Alloys', Journal of Materials Processing Technology, 80-81(1998), pp. 676-682. https://doi.org/10.1016/s0924-0136(98)00174-5

[3] M.S. Mohebbi, A. Akbarzadeh, 'Experimental study and FEM analysis of redundant strains in flow forming of tubes', Journal of Materials Processing Technology, 210 (2010), pp. 389-395. https://doi.org/10.1016/j.jmatprotec.2009.09.028

[4] Dassault Systèmes D. Abaqus analysis user's guide. Tech. rep., Technical Report Abaqus 6.14 Documentation. Simulia Corp; 2016.

[5] Saunders, N., Guo, U. K. Z., Li, X., Miodownik, A. P., & Schillé, J. P.. Using JMatPro to model materials properties and behavior. Jom, 55 (2003), 60-65.
# Neural network-based estimation and compensation of friction for enhanced deep drawing process control

THIERY Sebastian<fnref n="1" />, a*, EL ABDINE Mazhar Zein<fnref n="1" />, b, HEGER Jens<fnref n="1" />, c and BEN KHALIFA Noomane<fnref n="1" />, 2, d

**Keywords:** Deep Drawing, Material Draw-In, Predictive Modelling, Friction Estimation, Closed-Loop Control, Process Monitoring and Stabilization, Particle Swarm Optimization

**Abstract.** Fluctuating process conditions, such as lubrication, can disturb the production process and lead to faulty components that have cracks or wrinkles. Real-time identification of process parameters can detect deviations in sheet forming operations and enable the process parameters to be adjusted. To increase process robustness, closed-loop control is often used to monitor and influence the material draw-in, which corresponds to the material flow and can be measured by camera systems inside the deep-drawing press. The aim of this work is to develop a control concept that can predict the optimum blank holder force by estimating the coefficient of friction based on the material draw-in of the last stroke. Using a cross-die geometry, it is shown how the material draw-in can be determined experimentally by means of a camera system and numerically by FE simulations. Finally, artificial neural network-based models are trained through simulations and are subsequently tested on a numerical case study in which the coefficient of friction is changed as a disturbance variable and must be compensated for. The widely applicable control concept has the potential to incorporate additional soft sensors, for example to determine material properties, and other target variables, such as the punch force, into the optimization algorithm.

**Introduction**

With increasing component complexity in the automotive industry, the sensitivity of deep drawing operations to process fluctuations increases, which leads to a narrowing of the process window. Due to the high cost pressure, press lines must develop efficient strategies to avoid the occurrence of defects under these challenging conditions. One solution to this is closed-loop control, in which quality-relevant product characteristics are monitored and, if necessary, appropriate actuators are used to make the necessary adjustments to the process parameters. Recently, various approaches were presented for recording component quality by means of optical measurements using camera systems installed in the press and integrating them into closed-loop controls. In addition, there is great potential in data-driven modeling to optimize the manufacturing process or for real-time identification of disturbances during deep drawing.

Deep drawing processes are non-static due to fluctuations, where disturbances can have a negative impact on process stability over time [1]. Short-term disturbances include, for example, variable sheet thickness. Varying material properties, tool wear and tool temperature are examples of long-term disturbances. To that end, the tool warms up at the start of a batch and influences the lubrication conditions as well as the friction behavior between the sheet metal and the tool [2]. Depending on the time characteristics of these disturbances, different concepts can be selected in

<figure><img src="image_1413.png" /></figure>

<footnote n="1">¹ Institute for Production Technology and Systems, Leuphana University Lüneburg, Universitätsallee 1, 21335 Lüneburg, Germany</footnote>

<footnote n="2">² Institute of Material and Process Design, Helmholtz-Zentrum Hereon, Max-Planck-Str. 1, 21502 Geesthacht, Germany</footnote>

<footnote>a sebastian.thiery@leuphana.de, b mazhar.abdine@leuphana.de, c jens.heger@leuphana.de,</footnote>

<footnote>d noomane.ben_khalifa@leuphana.de</footnote>
# Neural network-based estimation and compensation of friction for enhanced deep drawing process control

THIERY Sebastian<fnref n="1" />, a*, EL ABDINE Mazhar Zein<fnref n="1" />, b, HEGER Jens<fnref n="1" />, c and BEN KHALIFA Noomane<fnref n="1" />, 2, d

**Keywords:** Deep Drawing, Material Draw-In, Predictive Modelling, Friction Estimation, Closed-Loop Control, Process Monitoring and Stabilization, Particle Swarm Optimization

**Abstract.** Fluctuating process conditions, such as lubrication, can disturb the production process and lead to faulty components that have cracks or wrinkles. Real-time identification of process parameters can detect deviations in sheet forming operations and enable the process parameters to be adjusted. To increase process robustness, closed-loop control is often used to monitor and influence the material draw-in, which corresponds to the material flow and can be measured by camera systems inside the deep-drawing press. The aim of this work is to develop a control concept that can predict the optimum blank holder force by estimating the coefficient of friction based on the material draw-in of the last stroke. Using a cross-die geometry, it is shown how the material draw-in can be determined experimentally by means of a camera system and numerically by FE simulations. Finally, artificial neural network-based models are trained through simulations and are subsequently tested on a numerical case study in which the coefficient of friction is changed as a disturbance variable and must be compensated for. The widely applicable control concept has the potential to incorporate additional soft sensors, for example to determine material properties, and other target variables, such as the punch force, into the optimization algorithm.

**Introduction**

With increasing component complexity in the automotive industry, the sensitivity of deep drawing operations to process fluctuations increases, which leads to a narrowing of the process window. Due to the high cost pressure, press lines must develop efficient strategies to avoid the occurrence of defects under these challenging conditions. One solution to this is closed-loop control, in which quality-relevant product characteristics are monitored and, if necessary, appropriate actuators are used to make the necessary adjustments to the process parameters. Recently, various approaches were presented for recording component quality by means of optical measurements using camera systems installed in the press and integrating them into closed-loop controls. In addition, there is great potential in data-driven modeling to optimize the manufacturing process or for real-time identification of disturbances during deep drawing.

Deep drawing processes are non-static due to fluctuations, where disturbances can have a negative impact on process stability over time [1]. Short-term disturbances include, for example, variable sheet thickness. Varying material properties, tool wear and tool temperature are examples of long-term disturbances. To that end, the tool warms up at the start of a batch and influences the lubrication conditions as well as the friction behavior between the sheet metal and the tool [2]. Depending on the time characteristics of these disturbances, different concepts can be selected in

<figure><img src="image_1417.png" /></figure>

<footnote n="1">¹ Institute for Production Technology and Systems, Leuphana University Lüneburg, Universitätsallee 1, 21335 Lüneburg, Germany</footnote>

<footnote n="2">² Institute of Material and Process Design, Helmholtz-Zentrum Hereon, Max-Planck-Str. 1, 21502 Geesthacht, Germany</footnote>

<footnote>a sebastian.thiery@leuphana.de, b mazhar.abdine@leuphana.de, c jens.heger@leuphana.de,</footnote>

<footnote>d noomane.ben_khalifa@leuphana.de</footnote>
order to create a closed-loop control to compensate for them. Whereas offline controls are better suited for long-term disturbances, as they measure the component properties between two or more strokes and can make adjustments to the process parameters with a time delay [3], online controls are preferable for short-term disturbances, as they can adjust the process parameters during the stroke. Lo and Yang [4] proposed a concept to detect the minimum side wall thickness and the wrinkle height during the deep drawing process and to continuously adjust the blank holder force (BHF) over the drawing depth by a control algorithm. This concept shows that the choice of sensor technology is linked to the target variable of the control system. Siegert et al. [5] developed a sensor to measure the frictional force between blank and die. They successfully demonstrated that the BHF can be adjusted to achieve a constant frictional force despite varying amounts of lubricant. Force sensors can also be integrated into the punch, as the forming process can be characterized via the punch force [6]. In most cases, however, the material draw-in or material flow is measured to control the deep drawing process even in cases where accessibility of the workpiece in the tool is restricted. For example, Behrens et al. [7] developed a special optical sensor for a contact-free measurement of the material flow.

The potential of machine vision for contactless quality monitoring and component measurement is constantly increasing in deep drawing. One of the first examples of online defect detection in the press shop is the image acquisition system from Gayubo et al. [8], in which the components were inspected at the end of the press line by a camera operated by a robot. A valley detection algorithm was used to identify the cracks. Current developments show that convolutional autoencoders can also be used to detect faulty components, including wrinkles [9]. However, indirect quality features can be defined and visually evaluated instead of directly identifying the defects, such as the material draw-in after the first drawing stage [10]. This offers the advantage of possibly detecting interior cracks or ruptures, which would be otherwise challenging. Moreover, this increases the time-window where corresponding countermeasures can be adopted before the faulty part reaches the end of line inspection. In the case that the flange is cut off before the image can be captured, the material flow can alternatively be measured via the position of the skid-lines [11]. This strategy enables continuous quality monitoring, where intervention limits can be set for the position of the skid-lines, indicating when action is required to prevent part failure [12]. Additionally, camera-based quality monitoring offers advantages for press hardening [13]. Optical systems can also be used as sensors for closed-loop control. Using a kitchen sink as an example, Fisher et al. [14] demonstrated that the draw-in can be controlled using a proportional control algorithm to adjust the BHF. Various control algorithms can be used to stabilize the deep drawing process. A controller designed as an optimal control problem gives good results in the example of a front fender and has an integral behavior [15].

Machine learning methods, in particular employing artificial neural networks (ANNs), can be applied in the context of intelligent process control and process optimization. ANNs are suitable for training models which act as *softsensors*, estimating otherwise unmeasurable product properties and process parameters based on known data [16]. Zhao and Wang [17] trained an ANN for the determination of friction conditions on the basis of the punch force and other process parameters. In a similar way, varying material properties can be identified [18]. An alternative for measuring material properties is the eddy current based material testing, which can be integrated directly into the production process [19]. Mork [20] combined this non-destructive material testing with an ANN to make predictions about the quality of each component and, if necessary, automatically adjust the deep drawing process. Further, ANNs can be employed in combination with an optimization algorithm and aid in identifying the optimum parameters for deep drawing processes. Tai and Lin [21] applied a simulated annealing optimization algorithm to an ANN to search for the optimal clearance between punch and die. This strategy was also used to improve sheet thinning during deep drawing [22]. Particle Swarm Optimization (PSO), belonging to the
order to create a closed-loop control to compensate for them. Whereas offline controls are better suited for long-term disturbances, as they measure the component properties between two or more strokes and can make adjustments to the process parameters with a time delay [3], online controls are preferable for short-term disturbances, as they can adjust the process parameters during the stroke. Lo and Yang [4] proposed a concept to detect the minimum side wall thickness and the wrinkle height during the deep drawing process and to continuously adjust the blank holder force (BHF) over the drawing depth by a control algorithm. This concept shows that the choice of sensor technology is linked to the target variable of the control system. Siegert et al. [5] developed a sensor to measure the frictional force between blank and die. They successfully demonstrated that the BHF can be adjusted to achieve a constant frictional force despite varying amounts of lubricant. Force sensors can also be integrated into the punch, as the forming process can be characterized via the punch force [6]. In most cases, however, the material draw-in or material flow is measured to control the deep drawing process even in cases where accessibility of the workpiece in the tool is restricted. For example, Behrens et al. [7] developed a special optical sensor for a contact-free measurement of the material flow.

The potential of machine vision for contactless quality monitoring and component measurement is constantly increasing in deep drawing. One of the first examples of online defect detection in the press shop is the image acquisition system from Gayubo et al. [8], in which the components were inspected at the end of the press line by a camera operated by a robot. A valley detection algorithm was used to identify the cracks. Current developments show that convolutional autoencoders can also be used to detect faulty components, including wrinkles [9]. However, indirect quality features can be defined and visually evaluated instead of directly identifying the defects, such as the material draw-in after the first drawing stage [10]. This offers the advantage of possibly detecting interior cracks or ruptures, which would be otherwise challenging. Moreover, this increases the time-window where corresponding countermeasures can be adopted before the faulty part reaches the end of line inspection. In the case that the flange is cut off before the image can be captured, the material flow can alternatively be measured via the position of the skid-lines [11]. This strategy enables continuous quality monitoring, where intervention limits can be set for the position of the skid-lines, indicating when action is required to prevent part failure [12]. Additionally, camera-based quality monitoring offers advantages for press hardening [13]. Optical systems can also be used as sensors for closed-loop control. Using a kitchen sink as an example, Fisher et al. [14] demonstrated that the draw-in can be controlled using a proportional control algorithm to adjust the BHF. Various control algorithms can be used to stabilize the deep drawing process. A controller designed as an optimal control problem gives good results in the example of a front fender and has an integral behavior [15].

Machine learning methods, in particular employing artificial neural networks (ANNs), can be applied in the context of intelligent process control and process optimization. ANNs are suitable for training models which act as *softsensors*, estimating otherwise unmeasurable product properties and process parameters based on known data [16]. Zhao and Wang [17] trained an ANN for the determination of friction conditions on the basis of the punch force and other process parameters. In a similar way, varying material properties can be identified [18]. An alternative for measuring material properties is the eddy current based material testing, which can be integrated directly into the production process [19]. Mork [20] combined this non-destructive material testing with an ANN to make predictions about the quality of each component and, if necessary, automatically adjust the deep drawing process. Further, ANNs can be employed in combination with an optimization algorithm and aid in identifying the optimum parameters for deep drawing processes. Tai and Lin [21] applied a simulated annealing optimization algorithm to an ANN to search for the optimal clearance between punch and die. This strategy was also used to improve sheet thinning during deep drawing [22]. Particle Swarm Optimization (PSO), belonging to the
family of meta-heuristic optimization algorithms, is another example of a widely used algorithm. El Mrabti et al. [23] first used a finite element model to generate a database with which they trained an ANN to predict springback. The authors then applied PSO to determine the best settings for the BHF and punch speed to minimize springback.

In this study, a concept for estimating and compensating of friction for enhanced deep drawing process control is presented. First, a camera system is introduced to measure the draw-in during the experimental investigation of the deep drawing process. Next, a finite element model is developed to map the influence of the coefficient of friction on the draw-in. The control concept is then described, which includes a softsensor for estimating the coefficient of friction and a PSO-based optimization algorithm. This concept requires the training of two ANN-based regression models, one to estimate the coefficient of friction and, by extension, the lubrication condition and another for estimating the draw-in. Finally, the effectiveness of the control for stabilizing the deep drawing process is tested on a numerical case study.

## Experimental and numerical setup

Experiments and simulations are used to investigate the behavior of the draw-in with respect to press parameters, material properties and lubrication conditions. The experiments are carried out on a hydraulic press with a drawing cushion. A cross-die shape is selected as the geometry (Fig. 1a). Sensors to measure ram force and BHF are integrated into the press. A SelVision monitoring system provided by the metrology specialists Selmatec Systems GmbH was installed in the press to measure the draw-in. The installed system consists of two cameras mounted in opposite corners of the hydraulic press (Fig. 1b). An interface was consequently set up between the monitoring system and the press control system. This enables continuous recording of the process forces and the tool position for each stroke and their automatic mapping to the images taken by the cameras.

<figure><img src="image_1414.png" /><figcaption>Fig. 1. (a) Cross-die tool in the hydraulic press and (b) close-up of the camera on the right side</figcaption></figure>

Experiments with the parameters listed in Table 1 show that each parameter change has a considerable effect on the draw-in measured by the cameras. On the one hand, the viscosity of the lubricant affects the friction, with lower viscosities shortening the draw-in and provoking material cracks. On the other hand, the BHF also influences friction, with low BHF lengthening the draw-in. The BHF is therefore generally suitable for compensation for changes in lubrication conditions.
family of meta-heuristic optimization algorithms, is another example of a widely used algorithm. El Mrabti et al. [23] first used a finite element model to generate a database with which they trained an ANN to predict springback. The authors then applied PSO to determine the best settings for the BHF and punch speed to minimize springback.

In this study, a concept for estimating and compensating of friction for enhanced deep drawing process control is presented. First, a camera system is introduced to measure the draw-in during the experimental investigation of the deep drawing process. Next, a finite element model is developed to map the influence of the coefficient of friction on the draw-in. The control concept is then described, which includes a softsensor for estimating the coefficient of friction and a PSO-based optimization algorithm. This concept requires the training of two ANN-based regression models, one to estimate the coefficient of friction and, by extension, the lubrication condition and another for estimating the draw-in. Finally, the effectiveness of the control for stabilizing the deep drawing process is tested on a numerical case study.

## Experimental and numerical setup

Experiments and simulations are used to investigate the behavior of the draw-in with respect to press parameters, material properties and lubrication conditions. The experiments are carried out on a hydraulic press with a drawing cushion. A cross-die shape is selected as the geometry (Fig. 1a). Sensors to measure ram force and BHF are integrated into the press. A SelVision monitoring system provided by the metrology specialists Selmatec Systems GmbH was installed in the press to measure the draw-in. The installed system consists of two cameras mounted in opposite corners of the hydraulic press (Fig. 1b). An interface was consequently set up between the monitoring system and the press control system. This enables continuous recording of the process forces and the tool position for each stroke and their automatic mapping to the images taken by the cameras.

<figure><img src="image_1418.png" /><figcaption>Fig. 1. (a) Cross-die tool in the hydraulic press and (b) close-up of the camera on the right side</figcaption></figure>

Experiments with the parameters listed in Table 1 show that each parameter change has a considerable effect on the draw-in measured by the cameras. On the one hand, the viscosity of the lubricant affects the friction, with lower viscosities shortening the draw-in and provoking material cracks. On the other hand, the BHF also influences friction, with low BHF lengthening the draw-in. The BHF is therefore generally suitable for compensation for changes in lubrication conditions.
<table><caption>Table 1. Parameters for the deep drawing experiments</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Material and thickness t [mm]</td><td>DC04 (0.7 – 1.0); DC01 (0.8 – 1.0)</td></tr><tr><td>Blank holder force F<sub>BH</sub> [kN]</td><td>40 – 170 – 300 – 430 – 560 – 690</td></tr><tr><td>Drawing velocity v [mm/s]</td><td>5 – 30</td></tr><tr><td>Lubricant viscosity η [mm<sup>2</sup>/s]</td><td>100 – 400</td></tr><tr><td>Drawing depth h [mm]</td><td>55</td></tr></tbody></table>

To test the measuring accuracy of the monitoring system, selected components were measured using an ATOS 3D-Scanner. Fig. 2a shows the results for the material DC04 with a thickness of 1 mm at a drawing speed of 30 mm/s using the lubricant with a viscosity of 100 mm²/s. The camera system automatically captures the greyscale images after each stroke when the blank holder is at the initial position (Fig. 2b). An edge detection algorithm is used to measure the movement of the workpiece edge in relation to the undeformed blank in pixels. The draw-in is then calculated in millimeters by means of a calibration function. The results confirm that the measured values of the two measuring systems are consistent with one another with an average deviation of 0.2 mm.

<figure><img src="image_1415.png" /><figcaption>Fig. 2. (a) Comparison between ATOS and SelVision and (b) images taken by the camera</figcaption></figure>

A finite element model was developed in Abaqus/explicit to investigate the influence of the coefficient of friction on the draw-in and to create a database for training the ANNs. The drawing velocity is artificially increased to minimize the computational cost, resulting in a total step time of 0.05 s. The sheet metal and tooling have been reduced to a quarter of the original size for efficient data generation. The components of the tool (e.g. punch, die, blank holder) are integrated as rigid surfaces. Friction was defined in the same way for all contacts between tool and workpiece using a constant coefficient of friction, which is varied in the following investigations. The sheet was modeled with an approximate element length of 1.5 mm and C3D8R was selected as the element type. Preliminary tests showed that a number of two elements over the sheet thickness give good agreement with the experimental results in terms of draw-in. The material properties of the four materials used in the experiments were determined by tensile tests according to DIN EN ISO 6892-1 with a gauge length of 80 mm. The properties of DC04 are discussed in more detail as this material was selected for the numerical case study. The yield curve was determined by Swift, Eq. (1), with a yield strength of $\sigma_{f0} = 116.687$ MPa and the anisotropy was defined in accordance with Table 2. The simulation and the experiment match sufficiently well (Fig. 3a).
<table><caption>Table 1. Parameters for the deep drawing experiments</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Material and thickness t [mm]</td><td>DC04 (0.7 – 1.0); DC01 (0.8 – 1.0)</td></tr><tr><td>Blank holder force F<sub>BH</sub> [kN]</td><td>40 – 170 – 300 – 430 – 560 – 690</td></tr><tr><td>Drawing velocity v [mm/s]</td><td>5 – 30</td></tr><tr><td>Lubricant viscosity η [mm<sup>2</sup>/s]</td><td>100 – 400</td></tr><tr><td>Drawing depth h [mm]</td><td>55</td></tr></tbody></table>

To test the measuring accuracy of the monitoring system, selected components were measured using an ATOS 3D-Scanner. Fig. 2a shows the results for the material DC04 with a thickness of 1 mm at a drawing speed of 30 mm/s using the lubricant with a viscosity of 100 mm²/s. The camera system automatically captures the greyscale images after each stroke when the blank holder is at the initial position (Fig. 2b). An edge detection algorithm is used to measure the movement of the workpiece edge in relation to the undeformed blank in pixels. The draw-in is then calculated in millimeters by means of a calibration function. The results confirm that the measured values of the two measuring systems are consistent with one another with an average deviation of 0.2 mm.

<figure><img src="image_1419.png" /><figcaption>Fig. 2. (a) Comparison between ATOS and SelVision and (b) images taken by the camera</figcaption></figure>

A finite element model was developed in Abaqus/explicit to investigate the influence of the coefficient of friction on the draw-in and to create a database for training the ANNs. The drawing velocity is artificially increased to minimize the computational cost, resulting in a total step time of 0.05 s. The sheet metal and tooling have been reduced to a quarter of the original size for efficient data generation. The components of the tool (e.g. punch, die, blank holder) are integrated as rigid surfaces. Friction was defined in the same way for all contacts between tool and workpiece using a constant coefficient of friction, which is varied in the following investigations. The sheet was modeled with an approximate element length of 1.5 mm and C3D8R was selected as the element type. Preliminary tests showed that a number of two elements over the sheet thickness give good agreement with the experimental results in terms of draw-in. The material properties of the four materials used in the experiments were determined by tensile tests according to DIN EN ISO 6892-1 with a gauge length of 80 mm. The properties of DC04 are discussed in more detail as this material was selected for the numerical case study. The yield curve was determined by Swift, Eq. (1), with a yield strength of $\sigma_{f0} = 116.687$ MPa and the anisotropy was defined in accordance with Table 2. The simulation and the experiment match sufficiently well (Fig. 3a).
$$ \sigma_f = 538.374 \text{ MPa} \cdot (0.004 + \varepsilon)^{0.274} \quad (1) $$

<table><caption>Table 2. Normal anisotropy of DC04 sheet metal with $t = 1$ mm</caption><thead><tr><th>r0</th><th>r45</th><th>r90</th><th>r&#773;</th></tr></thead><tbody><tr><td>2.178</td><td>1.696</td><td>2.454</td><td>2.006</td></tr></tbody></table>

<figure><img src="image_1416.png" /><figcaption>Fig. 3. (a) Comparison between simulation and experiment regarding material draw-in and (b) results of the simulation for changing coefficient of friction (DC04, $t = 1$ mm)</figcaption></figure>

The coefficient of friction and the BHF were varied in the simulation for each material according to Table 3. The results make it clear that the draw-in generally decreases with increasing coefficient of friction and increasing BHF (Fig 3b). For a BHF of 40 kN, the draw-in is almost identical regardless of the coefficient of friction. Since a slight wrinkling was observed in the experiments for such a low BHF, the corresponding simulation data of each material were excluded for the model training. All data where a necking was visible in the simulation and the draw-in was less than 10 mm were also excluded. In order to still have sufficient data for training the ANNs, additional simulations were then carried out for DC04 with $t = 1$ mm, varying the BHF in smaller increments of 26 kN over a range of 170 kN to 690 kN for the entire range of the coefficient of friction. Ultimately, a numerical dataset with six features and 247 instances was generated.

<table><caption>Table 3. Parameters for the deep drawing simulations</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Blank holder force F<sub>BH</sub> [kN]</td><td>40 – 170 – 300 – 430 – 560 – 690</td></tr><tr><td>Coefficient if friction μ [1]</td><td>0.06 – 0.07 – 0.08 – ... – 0.13 – 0.14 – 0.15</td></tr></tbody></table>

## Control and modelling approach

A discrete control concept has been developed that intervenes in the deep drawing process between two strokes (Fig. 4). As the material draw-in is a suitable monitoring variable, the control is designed to incorporate a camera system for draw-in detection. The target value of the draw-in is referred to as $l_t$ and the actual measured value is referred to as $l_a$. The draw-in is stabilized by adjusting the BHF $F_{BH}$, which is determined for each stroke via PSO. The PSO implementation was realized through the open-source Python package “pyswarm”, where an objective function is designed to optimize $F_{BH}$ by minimizing the absolute difference between the measured draw-in $l_a$ and the target $l_t$. The algorithm iteratively updates the position of each particle in the bounded
$$ \sigma_f = 538.374 \text{ MPa} \cdot (0.004 + \varepsilon)^{0.274} \quad (1) $$

<table><caption>Table 2. Normal anisotropy of DC04 sheet metal with $t = 1$ mm</caption><thead><tr><th>r0</th><th>r45</th><th>r90</th><th>r&#773;</th></tr></thead><tbody><tr><td>2.178</td><td>1.696</td><td>2.454</td><td>2.006</td></tr></tbody></table>

<figure><img src="image_1420.png" /><figcaption>Fig. 3. (a) Comparison between simulation and experiment regarding material draw-in and (b) results of the simulation for changing coefficient of friction (DC04, $t = 1$ mm)</figcaption></figure>

The coefficient of friction and the BHF were varied in the simulation for each material according to Table 3. The results make it clear that the draw-in generally decreases with increasing coefficient of friction and increasing BHF (Fig 3b). For a BHF of 40 kN, the draw-in is almost identical regardless of the coefficient of friction. Since a slight wrinkling was observed in the experiments for such a low BHF, the corresponding simulation data of each material were excluded for the model training. All data where a necking was visible in the simulation and the draw-in was less than 10 mm were also excluded. In order to still have sufficient data for training the ANNs, additional simulations were then carried out for DC04 with $t = 1$ mm, varying the BHF in smaller increments of 26 kN over a range of 170 kN to 690 kN for the entire range of the coefficient of friction. Ultimately, a numerical dataset with six features and 247 instances was generated.

<table><caption>Table 3. Parameters for the deep drawing simulations</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Blank holder force F<sub>BH</sub> [kN]</td><td>40 – 170 – 300 – 430 – 560 – 690</td></tr><tr><td>Coefficient if friction μ [1]</td><td>0.06 – 0.07 – 0.08 – ... – 0.13 – 0.14 – 0.15</td></tr></tbody></table>

## Control and modelling approach

A discrete control concept has been developed that intervenes in the deep drawing process between two strokes (Fig. 4). As the material draw-in is a suitable monitoring variable, the control is designed to incorporate a camera system for draw-in detection. The target value of the draw-in is referred to as $l_t$ and the actual measured value is referred to as $l_a$. The draw-in is stabilized by adjusting the BHF $F_{BH}$, which is determined for each stroke via PSO. The PSO implementation was realized through the open-source Python package “pyswarm”, where an objective function is designed to optimize $F_{BH}$ by minimizing the absolute difference between the measured draw-in $l_a$ and the target $l_t$. The algorithm iteratively updates the position of each particle in the bounded
search space based on the particle's status and the status of the entire swarm and converges when one of the stopping criteria is met: either through reaching the maximum number of iterations permitted or through a function change smaller than the defined threshold. For this application, the default parameter values were used. The reader is referred to Wang et al. [24] for additional information on the optimization procedure of the PSO algorithm.

The objective function of the optimization algorithm is based on an ANN that predicts the draw-in $l_p$ for various parameter combinations. As the input variables of the objective function include material properties such as sheet thickness $t$, yield strength $\sigma_f$ and anisotropy $\bar{\sigma}$, it can be applied to different cases. Since the focus of this work is on friction, the coefficient of friction is estimated using an ANN-based softsensor, which requires the values of the bank holder force $F_{BH}$ and the draw-in $l_a$ of the last stroke to estimate the coefficient of friction for that stroke. If necessary, other measurements, such as the punch force, can be integrated and additional softsensors can be engaged to estimate the material properties. Moreover, a combination of target variables can be considered in the optimization algorithm. Both ANN-based models are feed forward neural networks trained to estimate a continuous target variable. For efficiently exploring the hyperparameter space, a two-stage approach was performed: in the first stage, optimal settings for the number of hidden layers and the corresponding neurons per layer as well as the non-linear activation function were investigated through a random search, where all possible combinations are shuffled and 50 of which are chosen at random to be tested. In the second stage, the batch size and the learning rate of the adam optimizer were investigated for the best performing parameters from stage 1. Early stopping was also included in the search as to account for computational efficiency. All combinations of the aforementioned parameters to be varied are shown in Table 4. For validation, a four-fold cross-validation approach was implemented to ensure robustness. The model's performance for each hyperparameter set was quantified using the mean squared error across all validation folds, where the loss across all folds was computed and averaged for every combination. After this initial hyperparameter exploration, the best performing combinations were empirically used to train and test different models, this time using an 80/20 data split, stratified according to friction. With the aim to increase model generalizability, L1 and L2 regularization were tested but led to a degradation in performance and were thus not incorporated. A 5 % dropout for the neurons in the hidden layers in combination with early stopping was sufficient so as to prevent critical overfitting. The final models reached R2 values of 0.997 and 0.999 on the training set and reached similar performance on the holdout set of the split.

<table><caption>Table 4. Two-stage hyperparameter exploration combining random and grid search methods</caption><thead><tr><th>Hyperparameter</th><th>Range</th></tr></thead><tbody><tr><td colspan="2"><strong>Stage 1: 50 combinations of</strong></td></tr><tr><td>Number of hidden layers</td><td>[1, 2, 3]</td></tr><tr><td>Neurons per layer</td><td>[4, 8, 16, 32, 64]</td></tr><tr><td>Activation function</td><td>[relu, tanh]</td></tr><tr><td colspan="2"><strong>Stage 2: 9 combinations of</strong></td></tr><tr><td>Learning rate</td><td>[1e-2, 1e-3, 1e-4]</td></tr><tr><td>Batch size</td><td>[8, 16, 32]</td></tr></tbody></table>
search space based on the particle's status and the status of the entire swarm and converges when one of the stopping criteria is met: either through reaching the maximum number of iterations permitted or through a function change smaller than the defined threshold. For this application, the default parameter values were used. The reader is referred to Wang et al. [24] for additional information on the optimization procedure of the PSO algorithm.

The objective function of the optimization algorithm is based on an ANN that predicts the draw-in $l_p$ for various parameter combinations. As the input variables of the objective function include material properties such as sheet thickness $t$, yield strength $\sigma_f$ and anisotropy $\bar{\sigma}$, it can be applied to different cases. Since the focus of this work is on friction, the coefficient of friction is estimated using an ANN-based softsensor, which requires the values of the bank holder force $F_{BH}$ and the draw-in $l_a$ of the last stroke to estimate the coefficient of friction for that stroke. If necessary, other measurements, such as the punch force, can be integrated and additional softsensors can be engaged to estimate the material properties. Moreover, a combination of target variables can be considered in the optimization algorithm. Both ANN-based models are feed forward neural networks trained to estimate a continuous target variable. For efficiently exploring the hyperparameter space, a two-stage approach was performed: in the first stage, optimal settings for the number of hidden layers and the corresponding neurons per layer as well as the non-linear activation function were investigated through a random search, where all possible combinations are shuffled and 50 of which are chosen at random to be tested. In the second stage, the batch size and the learning rate of the adam optimizer were investigated for the best performing parameters from stage 1. Early stopping was also included in the search as to account for computational efficiency. All combinations of the aforementioned parameters to be varied are shown in Table 4. For validation, a four-fold cross-validation approach was implemented to ensure robustness. The model's performance for each hyperparameter set was quantified using the mean squared error across all validation folds, where the loss across all folds was computed and averaged for every combination. After this initial hyperparameter exploration, the best performing combinations were empirically used to train and test different models, this time using an 80/20 data split, stratified according to friction. With the aim to increase model generalizability, L1 and L2 regularization were tested but led to a degradation in performance and were thus not incorporated. A 5 % dropout for the neurons in the hidden layers in combination with early stopping was sufficient so as to prevent critical overfitting. The final models reached R2 values of 0.997 and 0.999 on the training set and reached similar performance on the holdout set of the split.

<table><caption>Table 4. Two-stage hyperparameter exploration combining random and grid search methods</caption><thead><tr><th>Hyperparameter</th><th>Range</th></tr></thead><tbody><tr><td colspan="2"><strong>Stage 1: 50 combinations of</strong></td></tr><tr><td>Number of hidden layers</td><td>[1, 2, 3]</td></tr><tr><td>Neurons per layer</td><td>[4, 8, 16, 32, 64]</td></tr><tr><td>Activation function</td><td>[relu, tanh]</td></tr><tr><td colspan="2"><strong>Stage 2: 9 combinations of</strong></td></tr><tr><td>Learning rate</td><td>[1e-2, 1e-3, 1e-4]</td></tr><tr><td>Batch size</td><td>[8, 16, 32]</td></tr></tbody></table>
<figure><img src="image_1417.png" /><figcaption>Fig. 4. Control concept with ANN-based optimization algorithm and ANN-based softsensor</figcaption></figure>

## Numerical case study

Using the simulation model described before, a case study was developed to test the functionality of the control concept and evaluate it against the uncontrolled process. For the material DC04 with a thickness of 1 mm, 25 parts with a draw-in of 16 mm are to be produced. The coefficient of friction varies between 0.07 and 0.09 and is changed arbitrarily after five parts to simulate a disturbance. In the uncontrolled process, a hold-down force of 300 kN is selected for all strokes. Initially, the draw-in is close to the target value, but larger deviations occur during the course of the case study, which illustrate the negative influence of the disturbances (Fig. 5). When using the control concept, however, the softsensor reacts to the changes of the coefficient of friction. The BHF of the first stroke is set to 300 kN. It takes one stroke for the coefficient of friction to be estimated. The optimization algorithm then adjusts the BHF so that the target value is met (Fig. 6). The fluctuation of the coefficient of friction is the same as in the uncontrolled process. Besides this disturbance, the control can deal with abrupt changes of the target value $l_t$ from 16 to 15 mm. Overall, the case study proves that the concept is suitable for stabilizing the deep drawing process.

<figure><img src="image_1418.png" /><figcaption>Fig. 5. Fluctuating coefficient of friction in the uncontrolled process (DC04, $t = 1$ mm)</figcaption></figure>
<figure><img src="image_1421.png" /><figcaption>Fig. 4. Control concept with ANN-based optimization algorithm and ANN-based softsensor</figcaption></figure>

## Numerical case study

Using the simulation model described before, a case study was developed to test the functionality of the control concept and evaluate it against the uncontrolled process. For the material DC04 with a thickness of 1 mm, 25 parts with a draw-in of 16 mm are to be produced. The coefficient of friction varies between 0.07 and 0.09 and is changed arbitrarily after five parts to simulate a disturbance. In the uncontrolled process, a hold-down force of 300 kN is selected for all strokes. Initially, the draw-in is close to the target value, but larger deviations occur during the course of the case study, which illustrate the negative influence of the disturbances (Fig. 5). When using the control concept, however, the softsensor reacts to the changes of the coefficient of friction. The BHF of the first stroke is set to 300 kN. It takes one stroke for the coefficient of friction to be estimated. The optimization algorithm then adjusts the BHF so that the target value is met (Fig. 6). The fluctuation of the coefficient of friction is the same as in the uncontrolled process. Besides this disturbance, the control can deal with abrupt changes of the target value $l_t$ from 16 to 15 mm. Overall, the case study proves that the concept is suitable for stabilizing the deep drawing process.

<figure><img src="image_1422.png" /><figcaption>Fig. 5. Fluctuating coefficient of friction in the uncontrolled process (DC04, $t = 1$ mm)</figcaption></figure>
<figure><img src="image_1419.png" /><figcaption>Fig. 6. Compensation of friction by adjusting the BHF (DC04, t = 1 mm)</figcaption></figure>

**Summary**

The presented approach, combining real-time monitoring, predictive modeling via ANNs and optimization via PSO demonstrates the potential for significant improvements in the accuracy and stability of the sheet forming operation. The adaptability of this concept allows for the inclusion of more target variables and additional softsensors, which can act as a substitute for physical sensors. This provides a scalable solution for various types of process disturbances, while simultaneously eliminating the need for expensive and difficult to deploy measurement systems.
Finally, the following conclusions can be drawn:

* Camera systems are suitable sensors for measuring the material draw-in after each stroke.

* The draw-in depends, among other things, on the blank holder force (BHF) and the lubrication conditions, such as the coefficient of friction. Since the BHF is known and the draw-in can be measured, the coefficient of friction can be estimated.

* A numerical case study proves that with this approach, the fluctuations of the coefficient of friction are detectable and can be compensated by a prediction of the optimal BHF.

**Acknowledgements**

The authors kindly thank the European Regional Development Fund (EFRE) and the state of Lower Saxony for financing the research project OptimUm.

**References**

[1] B. Endelt, J. Danckert, Iterative Learning and Feedback Control Applied on a Deep Drawing Process, Int. J. Mater. Form. 3 (2010) 25–28. https://doi.org/10.1007/s12289-010-0698-z

[2] J. Heingärtner, D. Bonfanti, D. Harsch, F. Dietrich, P. Hora, Implementation of a tribology-based process control system for deep drawing processes, IOP Conf. Ser.: Mater. Sci. Eng. 418 (2018) 12112. https://doi.org/10.1088/1757-899X/418/1/012112

[3] J.M. Allwood, S.R. Duncan, J. Cao, P. Groche, G. Hirt, B. Kinsey, T. Kuboki, M. Liewald, A. Sterzing, A.E. Tekkaya, Closed-loop control of product properties in metal forming, CIRP Annals. 65 (2016) 573–596. https://doi.org/10.1016/j.cirp.2016.06.002

[4] S.-W. Lo, T.-C. Yang, Closed-loop control of the blank holding force in sheet metal forming with a new embedded-type displacement sensor, Int. J. Adv. Manuf. Technol. 24 (2004) 553–559. https://doi.org/10.1007/s00170-003-1711-1
<figure><img src="image_1423.png" /><figcaption>Fig. 6. Compensation of friction by adjusting the BHF (DC04, t = 1 mm)</figcaption></figure>

**Summary**

The presented approach, combining real-time monitoring, predictive modeling via ANNs and optimization via PSO demonstrates the potential for significant improvements in the accuracy and stability of the sheet forming operation. The adaptability of this concept allows for the inclusion of more target variables and additional softsensors, which can act as a substitute for physical sensors. This provides a scalable solution for various types of process disturbances, while simultaneously eliminating the need for expensive and difficult to deploy measurement systems.
Finally, the following conclusions can be drawn:

* Camera systems are suitable sensors for measuring the material draw-in after each stroke.

* The draw-in depends, among other things, on the blank holder force (BHF) and the lubrication conditions, such as the coefficient of friction. Since the BHF is known and the draw-in can be measured, the coefficient of friction can be estimated.

* A numerical case study proves that with this approach, the fluctuations of the coefficient of friction are detectable and can be compensated by a prediction of the optimal BHF.

**Acknowledgements**

The authors kindly thank the European Regional Development Fund (EFRE) and the state of Lower Saxony for financing the research project OptimUm.

**References**

[1] B. Endelt, J. Danckert, Iterative Learning and Feedback Control Applied on a Deep Drawing Process, Int. J. Mater. Form. 3 (2010) 25–28. https://doi.org/10.1007/s12289-010-0698-z

[2] J. Heingärtner, D. Bonfanti, D. Harsch, F. Dietrich, P. Hora, Implementation of a tribology-based process control system for deep drawing processes, IOP Conf. Ser.: Mater. Sci. Eng. 418 (2018) 12112. https://doi.org/10.1088/1757-899X/418/1/012112

[3] J.M. Allwood, S.R. Duncan, J. Cao, P. Groche, G. Hirt, B. Kinsey, T. Kuboki, M. Liewald, A. Sterzing, A.E. Tekkaya, Closed-loop control of product properties in metal forming, CIRP Annals. 65 (2016) 573–596. https://doi.org/10.1016/j.cirp.2016.06.002

[4] S.-W. Lo, T.-C. Yang, Closed-loop control of the blank holding force in sheet metal forming with a new embedded-type displacement sensor, Int. J. Adv. Manuf. Technol. 24 (2004) 553–559. https://doi.org/10.1007/s00170-003-1711-1
[5] K. Siegert, M. Ziegler, S. Wagner, Closed loop control of the friction force. Deep drawing process, J. Mater. Proc. Technol. 71 (1997) 126-133. https://doi.org/10.1016/S0924-0136(97)00158-1

[6] M. Traversin, R. Kergen, Closed-loop control of the blank-holder force in deep-drawing: finite-element modeling of its effects and advantages, J. Mater. Proc. Technol. 50 (1995) 306-317.
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[7] B.A. Behrens, J.W. Yun, M. Milch, Closed-Loop-Control of the Material Flow in the Deep Drawing Process, AMR. 6-8 (2005) 321-328. https://doi.org/10.4028/www.scientific.net/AMR.6-8.321

[8] F. Gayubo, J.L. Gonzalez, E. de La Fuente, F. Miguel, J.R. Peran, On-line machine vision system for detect split defects in sheet-metal forming processes, in: 18th International Conference on Pattern Recognition (ICPR'06), IEEE, 2006, pp. 723-726.
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[9] J. Heger, G. Desai, M.Z. El Abdine, Anomaly detection in formed sheet metals using convolutional autoencoders, Procedia CIRP. 93 (2020) 1281-1285.
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[10] M. Kraft, U. Bürgel, Novel concept for measurement of global blank draw-in when deep drawing outer skin automotive components, J. Phys.: Conf. Ser. 896 (2017) 12034.
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[11] S. Maier, T. Schmerbeck, A. Liebig, T. Kautz, W. Volk, Potentials for the use of tool-integrated in-line data acquisition systems in press shops, J. Phys.: Conf. Ser. 896 (2017) 12033.
https://doi.org/10.1088/1742-6596/896/1/012033

[12] S. Maier, Inline-Qualitätsprüfung im Presswerk durch intelligente Nachfolgewerkzeuge, Dissertation, Technical University of Munich, TUM.University Press, München, 2018.

[13] A. Pierer, T. Wiener, L. Gjakova, J. Koziorek, Zero-error-production through inline-quality control of presshardened automotive parts by multi-camera systems, IOP Conf. Ser.: Mater. Sci. Eng. 1157 (2021) 12074. https://doi.org/10.1088/1757-899X/1157/1/012074

[14] P. Fischer, J. Heingärtner, Y. Renkci, P. Hora, Experiences with inline feedback control and data acquisition in deep drawing, Proc. Manuf. 15 (2018) 949-954.
https://doi.org/10.1016/j.promfg.2018.07.401

[15] P. Fischer, J. Heingärtner, S. Duncan, P. Hora, On part-to-part feedback optimal control in deep drawing, J. Manuf. Process. 50 (2020) 403-411.
https://doi.org/10.1016/j.jmapro.2019.10.019

[16] W. Homberg, B. Arian, V. Arne, T. Borgert, A. Brosius, P. Groche, C. Hartmann, L. Kersting, R. Laue, J. Martschin, T. Meurer, D. Spies, A.E. Tekkaya, A. Trächtler, W. Volk, F. Wendler, M. Wrobel, Softsensors: key component of property control in forming technology, Prod. Eng. Res. Devel. (2023). https://doi.org/10.1007/s11740-023-01227-1

[17] J. Zhao, F. Wang, Parameter identification by neural network for intelligent deep drawing of axisymmetric workpieces, J. Mater. Proc. Technol. 166 (2005) 387-391.
https://doi.org/10.1016/j.jmatprotec.2004.08.020

[18] K. Manabe, M. Yang, S. Yoshihara, Artificial intelligence identification of process parameters and adaptive control system for deep-drawing process, J. Mater. Proc. Technol. 80-81 (1998) 421-426. https://doi.org/10.1016/S0924-0136(98)00121-6
[5] K. Siegert, M. Ziegler, S. Wagner, Closed loop control of the friction force. Deep drawing process, J. Mater. Proc. Technol. 71 (1997) 126-133. https://doi.org/10.1016/S0924-0136(97)00158-1

[6] M. Traversin, R. Kergen, Closed-loop control of the blank-holder force in deep-drawing: finite-element modeling of its effects and advantages, J. Mater. Proc. Technol. 50 (1995) 306-317.
https://doi.org/10.1016/0924-0136(94)01389-I

[7] B.A. Behrens, J.W. Yun, M. Milch, Closed-Loop-Control of the Material Flow in the Deep Drawing Process, AMR. 6-8 (2005) 321-328. https://doi.org/10.4028/www.scientific.net/AMR.6-8.321

[8] F. Gayubo, J.L. Gonzalez, E. de La Fuente, F. Miguel, J.R. Peran, On-line machine vision system for detect split defects in sheet-metal forming processes, in: 18th International Conference on Pattern Recognition (ICPR'06), IEEE, 2006, pp. 723-726.
https://doi.org/10.1109/ICPR.2006.902

[9] J. Heger, G. Desai, M.Z. El Abdine, Anomaly detection in formed sheet metals using convolutional autoencoders, Procedia CIRP. 93 (2020) 1281-1285.
https://doi.org/10.1016/j.procir.2020.04.106

[10] M. Kraft, U. Bürgel, Novel concept for measurement of global blank draw-in when deep drawing outer skin automotive components, J. Phys.: Conf. Ser. 896 (2017) 12034.
https://doi.org/10.1088/1742-6596/896/1/012034

[11] S. Maier, T. Schmerbeck, A. Liebig, T. Kautz, W. Volk, Potentials for the use of tool-integrated in-line data acquisition systems in press shops, J. Phys.: Conf. Ser. 896 (2017) 12033.
https://doi.org/10.1088/1742-6596/896/1/012033

[12] S. Maier, Inline-Qualitätsprüfung im Presswerk durch intelligente Nachfolgewerkzeuge, Dissertation, Technical University of Munich, TUM.University Press, München, 2018.

[13] A. Pierer, T. Wiener, L. Gjakova, J. Koziorek, Zero-error-production through inline-quality control of presshardened automotive parts by multi-camera systems, IOP Conf. Ser.: Mater. Sci. Eng. 1157 (2021) 12074. https://doi.org/10.1088/1757-899X/1157/1/012074

[14] P. Fischer, J. Heingärtner, Y. Renkci, P. Hora, Experiences with inline feedback control and data acquisition in deep drawing, Proc. Manuf. 15 (2018) 949-954.
https://doi.org/10.1016/j.promfg.2018.07.401

[15] P. Fischer, J. Heingärtner, S. Duncan, P. Hora, On part-to-part feedback optimal control in deep drawing, J. Manuf. Process. 50 (2020) 403-411.
https://doi.org/10.1016/j.jmapro.2019.10.019

[16] W. Homberg, B. Arian, V. Arne, T. Borgert, A. Brosius, P. Groche, C. Hartmann, L. Kersting, R. Laue, J. Martschin, T. Meurer, D. Spies, A.E. Tekkaya, A. Trächtler, W. Volk, F. Wendler, M. Wrobel, Softsensors: key component of property control in forming technology, Prod. Eng. Res. Devel. (2023). https://doi.org/10.1007/s11740-023-01227-1

[17] J. Zhao, F. Wang, Parameter identification by neural network for intelligent deep drawing of axisymmetric workpieces, J. Mater. Proc. Technol. 166 (2005) 387-391.
https://doi.org/10.1016/j.jmatprotec.2004.08.020

[18] K. Manabe, M. Yang, S. Yoshihara, Artificial intelligence identification of process parameters and adaptive control system for deep-drawing process, J. Mater. Proc. Technol. 80-81 (1998) 421-426. https://doi.org/10.1016/S0924-0136(98)00121-6
[19] H. Hoffmann, M.F. Zäh, I. Faass, R. Mork, M. Golle, B. Griesbach, M. Kerschner, Automatic Process Control in Press Shops, KEM. 344 (2007) 881-888.
https://doi.org/10.4028/www.scientific.net/KEM.344.881

[20] R. Mork, Qualitätsbewertung und -regelung für die Fertigung von Karosserieteilen in Presswerken auf Basis Neuronaler Netze, Dissertation, Technical University of Munich, Herbert Utz Verlag, München, 2012.

[21] C.C. Tai, J.C. Lin, The optimisation deep-draw clearance design for deep-draw dies, Int. J. Adv. Manuf. Technol. 14 (1998) 390-398. https://doi.org/10.1007/BF01304617

[22] M. Manoochehri, F. Kolahan, Integration of artificial neural network and simulated annealing algorithm to optimize deep drawing process, Int. J. Adv. Manuf. Technol. 73 (2014) 241-249.
https://doi.org/10.1007/s00170-014-5788-5

[23] I. El Mrabti, A. Touache, A. El Hakimi, A. Chamat, Springback optimization of deep drawing process based on FEM-ANN-PSO strategy, Struct. Multidisc. Optim. 64 (2021) 321-333.
https://doi.org/10.1007/s00158-021-02861-y

[24] D. Wang, D. Tan, L. Liu, Particle swarm optimization algorithm: an overview, Soft Comput. 22 (2018) 387-408. https://doi.org/10.1007/s00500-016-2474-6
[19] H. Hoffmann, M.F. Zäh, I. Faass, R. Mork, M. Golle, B. Griesbach, M. Kerschner, Automatic Process Control in Press Shops, KEM. 344 (2007) 881–888.
https://doi.org/10.4028/www.scientific.net/KEM.344.881

[20] R. Mork, Qualitätsbewertung und -regelung für die Fertigung von Karosserieteilen in Presswerken auf Basis Neuronaler Netze, Dissertation, Technical University of Munich, Herbert Utz Verlag, München, 2012.

[21] C.C. Tai, J.C. Lin, The optimisation deep-draw clearance design for deep-draw dies, Int. J. Adv. Manuf. Technol. 14 (1998) 390–398. https://doi.org/10.1007/BF01304617

[22] M. Manoochehri, F. Kolahan, Integration of artificial neural network and simulated annealing algorithm to optimize deep drawing process, Int. J. Adv. Manuf. Technol. 73 (2014) 241–249.
https://doi.org/10.1007/s00170-014-5788-5

[23] I. El Mrabti, A. Touache, A. El Hakimi, A. Chamat, Springback optimization of deep drawing process based on FEM-ANN-PSO strategy, Struct. Multidisc. Optim. 64 (2021) 321–333.
https://doi.org/10.1007/s00158-021-02861-y

[24] D. Wang, D. Tan, L. Liu, Particle swarm optimization algorithm: an overview, Soft Comput. 22 (2018) 387–408. https://doi.org/10.1007/s00500-016-2474-6
# Study on the numerical prediction quality of material models regarding springback of hollow embossed metallic bipolar half-plates

BECK Maxim¹,ᵃ*, KARADOGAN Celalettin¹, CYRON Patrick¹, ROUVEN RIEDMÜLLER Kim¹ and LIEWALD Mathias¹

¹Institute for Metal Forming Technology, University of Stuttgart, Holzgartenstr. 17, 70174 Stuttgart, Germany

ªmaxim.beck@ifu.uni-stuttgart.de

**Keywords:** Bipolar Plate, Springback Prediction, Forming Simulation

**Abstract.** Due to the ongoing endeavors to reduce CO₂ emissions in the energy sector, particularly through advancements in electric mobility, the demand for thin metallic components is currently increasing. This demand is notably apparent for bipolar plates, which are used in fuel cells. The production of these bipolar plates requires high-precision hollow embossing of thin metallic foils with thicknesses below 100 µm. Even minor variations in this process can result in forming defects and undesirable springback in produced parts. A robust design of manufacturing processes for bipolar plates therefore necessitates the implementation of accurate finite element analysis (FEA), since this is crucial for predicting material behavior during and after the forming process. However, the precision of such simulations is heavily dependent on the material models employed. For this reason, the study presented in this paper specifically focused on the numerical modeling of high-precision hollow embossing of bipolar half-plates. For this purpose, a comprehensive material characterization of a 1.4404 stainless-steel foil with a thickness of 0.1 mm was conducted first. Moreover, an experimental laboratory geometry and a corresponding laboratory tool for the hollow embossing of miniaturized bipolar half-plates was developed in order to enable a comprehensive evaluation of the simulation results. The validation and assessment of widely used material models for metal forming in LS-Dyna, exemplified by MAT_036 Barlat '89 and MAT_133 Barlat 2000, were effectively demonstrated through experimental forming tests. These tests confirmed distinct performance differences between the two material models concerning springback and thinning. Importantly, it was confirmed that the simpler MAT_036 model exhibits high accuracy in predicting both springback and thinning, emphasizing its effectiveness in capturing relevant deformation properties during the hollow embossing process.

**Introduction**

Efficient and environmentally sustainable powertrain systems are essential for advancing the environment friendly electrification of both passenger and commercial transportation. In this context, fuel cell technology is emerging as a compelling alternative to conventional electrical drives, particularly for heavy-duty and long-distance applications. The accompanying rise in demand for fuel cell systems has led to a notable increase in the production volume of metallic bipolar plates made of thin stainless-steel foils having thicknesses below 100 µm. However, the manufacturing of such high-precision metallic bipolar plates using processes such as hollow embossing is significantly challenging. Foils with thicknesses of around 100 µm and less are prone to defects such as microcracks, cracks and wrinkles and are difficult to handle during production [1-4]. In addition, the phenomenon of springback still remains a major issue in achieving dimensional accuracy of components made from thin metallic foils [1, 5-6]. Flow channels as well as the overall geometry must meet very precise dimensional tolerances of a couple micrometers in both vertical and lateral directions [1, 7]. Otherwise, problems may arise during the subsequent joining of the bipolar half-plates, which can lead to misalignment issues affecting the performance

<figure><img src="image_1420.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
# Study on the numerical prediction quality of material models regarding springback of hollow embossed metallic bipolar half-plates

BECK Maxim¹,ᵃ*, KARADOGAN Celalettin¹, CYRON Patrick¹, ROUVEN RIEDMÜLLER Kim¹ and LIEWALD Mathias¹

¹Institute for Metal Forming Technology, University of Stuttgart, Holzgartenstr. 17, 70174 Stuttgart, Germany

ªmaxim.beck@ifu.uni-stuttgart.de

**Keywords:** Bipolar Plate, Springback Prediction, Forming Simulation

**Abstract.** Due to the ongoing endeavors to reduce CO₂ emissions in the energy sector, particularly through advancements in electric mobility, the demand for thin metallic components is currently increasing. This demand is notably apparent for bipolar plates, which are used in fuel cells. The production of these bipolar plates requires high-precision hollow embossing of thin metallic foils with thicknesses below 100 µm. Even minor variations in this process can result in forming defects and undesirable springback in produced parts. A robust design of manufacturing processes for bipolar plates therefore necessitates the implementation of accurate finite element analysis (FEA), since this is crucial for predicting material behavior during and after the forming process. However, the precision of such simulations is heavily dependent on the material models employed. For this reason, the study presented in this paper specifically focused on the numerical modeling of high-precision hollow embossing of bipolar half-plates. For this purpose, a comprehensive material characterization of a 1.4404 stainless-steel foil with a thickness of 0.1 mm was conducted first. Moreover, an experimental laboratory geometry and a corresponding laboratory tool for the hollow embossing of miniaturized bipolar half-plates was developed in order to enable a comprehensive evaluation of the simulation results. The validation and assessment of widely used material models for metal forming in LS-Dyna, exemplified by MAT_036 Barlat '89 and MAT_133 Barlat 2000, were effectively demonstrated through experimental forming tests. These tests confirmed distinct performance differences between the two material models concerning springback and thinning. Importantly, it was confirmed that the simpler MAT_036 model exhibits high accuracy in predicting both springback and thinning, emphasizing its effectiveness in capturing relevant deformation properties during the hollow embossing process.

**Introduction**

Efficient and environmentally sustainable powertrain systems are essential for advancing the environment friendly electrification of both passenger and commercial transportation. In this context, fuel cell technology is emerging as a compelling alternative to conventional electrical drives, particularly for heavy-duty and long-distance applications. The accompanying rise in demand for fuel cell systems has led to a notable increase in the production volume of metallic bipolar plates made of thin stainless-steel foils having thicknesses below 100 µm. However, the manufacturing of such high-precision metallic bipolar plates using processes such as hollow embossing is significantly challenging. Foils with thicknesses of around 100 µm and less are prone to defects such as microcracks, cracks and wrinkles and are difficult to handle during production [1-4]. In addition, the phenomenon of springback still remains a major issue in achieving dimensional accuracy of components made from thin metallic foils [1, 5-6]. Flow channels as well as the overall geometry must meet very precise dimensional tolerances of a couple micrometers in both vertical and lateral directions [1, 7]. Otherwise, problems may arise during the subsequent joining of the bipolar half-plates, which can lead to misalignment issues affecting the performance

<figure><img src="image_1424.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
and efficiency of the entire fuel cell stack. These limitations result in extended production cycle times and increased costs for quality assurance, hindering the economical manufacturing of high quantities of bipolar half-plates until this day [1].

To address these challenges and to optimize forming processes of bipolar half-plates at an early stage of development, nowadays finite element simulations are essential. However, the accuracy of these simulations, which is crucial for obtaining virtual results of complex forming processes, heavily relies on the accurate modeling of material behavior and thus the material characterization procedures [1, 8].

The study presented in this paper aims to evaluate the widely used material models “MAT_3-PARAMETER_BARLAT” (MAT_36) and “MAT_BARLAT YLD2000” (MAT_133) in LS-Dyna regarding the prediction accuracy for hollow-embossed bipolar half-plates made of 1.4404 stainless-steel foil (t₀ = 100 µm). For this purpose, an in-depth material characterization was conducted through tensile and bulge tests, enabling comprehensive material modeling of the stainless-steel used. Building upon the findings of previous studies [1], a suitable laboratory-scale prototype tool was developed, demonstreating key features of an actual bipolar half-plate. This tool made the comprehensive comparison between simulation results and real experiments possible. Here, the thickness distribution within the channels, the overall flatness and the springback behavior of the manufactured part were precisely measured and compared with the FE-model.

## Modeling and numerical study

### Material characterization and modeling.

In the present study, the austenitic stainless-steel 1.4404 with a thickness of t₀ = 100 µm was used as test material. This material is frequently used today for the manufacturing of metallic bipolar plates. For determining required material parameters, quasi-static uni-axial tensile tests according to DIN EN 6892 were carried out on specimen according to DIN 50125 Form H 20 x 80 prepared in 0°, 45° and in 90° rolling directions. Additionally, bulge tests according to ISO 16808 were performed to obtain experimental data for higher true strain values as well as for the biaxial yield stress. For r<sub>b</sub>, a value of 1.0 was chosen according to [9]. For both characterization tests optical measurements and strain evaluation were performed using a GOM ARAMIS system. The material parameters given in Table 1 were obtained by averaging the results of three repetitions in each case as explained in the next paragraph.

<table><caption>Table 1: Material properties of stainless-steel 1.4404, t₀ = 100 µm.</caption><thead><tr><th>σ<sub>0</sub> [MPa]</th><th>σ<sub>45</sub> [MPa]</th><th>σ<sub>90</sub> [MPa]</th><th>σ<sub>b</sub> [MPa]</th><th>r<sub>0</sub> [-]</th><th>r<sub>45</sub> [-]</th><th>r<sub>90</sub> [-]</th><th>r<sub>b</sub> [-]</th></tr></thead><tbody><tr><td>243.45</td><td>250.89</td><td>253.51</td><td>228.42</td><td>0.72</td><td>1.06</td><td>1.22</td><td>1.0</td></tr></tbody></table>

Corresponding averaged true-stress – true-strain curves obtained by tensile and bulge tests are depicted in Fig. 1. Here, the true-stress – true-plastic-strain curve is shown for the rolling direction tensile test as well as for the bulge test. Averaged true-stress – true-strain curves for rolling directions of 45° and 90° were also used to obtain the anisotropy ratios (σ<sub>45</sub>/σ<sub>00</sub>, σ<sub>90</sub>/σ<sub>00</sub>, σ<sub>B</sub>/σ<sub>00</sub>) on the initial yield-locus based on the principle of plastic-energy equivalency.
and efficiency of the entire fuel cell stack. These limitations result in extended production cycle times and increased costs for quality assurance, hindering the economical manufacturing of high quantities of bipolar half-plates until this day [1].

To address these challenges and to optimize forming processes of bipolar half-plates at an early stage of development, nowadays finite element simulations are essential. However, the accuracy of these simulations, which is crucial for obtaining virtual results of complex forming processes, heavily relies on the accurate modeling of material behavior and thus the material characterization procedures [1, 8].

The study presented in this paper aims to evaluate the widely used material models “MAT_3-PARAMETER_BARLAT” (MAT_36) and “MAT_BARLAT YLD2000” (MAT_133) in LS-Dyna regarding the prediction accuracy for hollow-embossed bipolar half-plates made of 1.4404 stainless-steel foil ($t_0 = 100 \mu m$). For this purpose, an in-depth material characterization was conducted through tensile and bulge tests, enabling comprehensive material modeling of the stainless-steel used. Building upon the findings of previous studies [1], a suitable laboratory-scale prototype tool was developed, demonstrating key features of an actual bipolar half-plate. This tool made the comprehensive comparison between simulation results and real experiments possible. Here, the thickness distribution within the channels, the overall flatness and the springback behavior of the manufactured part were precisely measured and compared with the FE-model.

## Modeling and numerical study

### Material characterization and modeling.

In the present study, the austenitic stainless-steel 1.4404 with a thickness of $t_0 = 100 \mu m$ was used as test material. This material is frequently used today for the manufacturing of metallic bipolar plates. For determining required material parameters, quasi-static uni-axial tensile tests according to DIN EN 6892 were carried out on specimen according to DIN 50125 Form H 20 x 80 prepared in 0°, 45° and in 90° rolling directions. Additionally, bulge tests according to ISO 16808 were performed to obtain experimental data for higher true strain values as well as for the biaxial yield stress. For $r_b$, a value of 1.0 was chosen according to [9]. For both characterization tests optical measurements and strain evaluation were performed using a GOM ARAMIS system. The material parameters given in Table 1 were obtained by averaging the results of three repetitions in each case as explained in the next paragraph.

<table><caption>Table 1: Material properties of stainless-steel 1.4404, $t_0 = 100 \mu m$.</caption><thead><tr><th>σ<sub>0</sub> [MPa]</th><th>σ<sub>45</sub> [MPa]</th><th>σ<sub>90</sub> [MPa]</th><th>σ<sub>b</sub> [MPa]</th><th>r<sub>0</sub> [-]</th><th>r<sub>45</sub> [-]</th><th>r<sub>90</sub> [-]</th><th>r<sub>b</sub> [-]</th></tr></thead><tbody><tr><td>243.45</td><td>250.89</td><td>253.51</td><td>228.42</td><td>0.72</td><td>1.06</td><td>1.22</td><td>1.0</td></tr></tbody></table>

Corresponding averaged true-stress – true-strain curves obtained by tensile and bulge tests are depicted in Fig. 1. Here, the true-stress – true-plastic-strain curve is shown for the rolling direction tensile test as well as for the bulge test. Averaged true-stress – true-strain curves for rolling directions of 45° and 90° were also used to obtain the anisotropy ratios ($σ_{45}/σ_{00}$, $σ_{90}/σ_{00}$, $σ_B/σ_{00}$) on the initial yield-locus based on the principle of plastic-energy equivalency.
<figure><img src="image_1421.png" /><figcaption>Fig. 1: (a) True stress - true strain curves obtained by tensile and bulge tests and (b) combined flow curve.</figcaption></figure>

Using the obtained anisotropy ratio for the bulge test, the true-stress – true-plastic-strain diagram of the bulge test was used to obtain an experimental extension for the flow curve using the equivalent plastic deformation energy approach as described in [10]. Finally, a suitable flow curve is obtained, as shown in Fig. 1 b). The final flow curve for use in forming simulations is approximated by the Hockett-Sherby formula according to [11], see Eq. 1.

$$ \sigma_{Hockett-Sherby} = \sigma_{sat} - (\sigma_{sat} - \sigma_{ini}) \cdot e^{-a \cdot \varepsilon_{pl}^{p}} \quad (1) $$

<table><caption>Table 2: Parameters describing the flow-curve approximation according to Hockett-Sherby.</caption><thead><tr><th>σ<sub>sat</sub> [MPa]</th><th>σ<sub>ini</sub> [MPa]</th><th>p [-]</th><th>a [-]</th></tr></thead><tbody><tr><td>993.43</td><td>243.45</td><td>1.12</td><td>4.397</td></tr></tbody></table>

In the LS-Dyna software package, these acquired material data was adopted for the material model MAT_036 - "MAT_3-PARAMETER_BARLAT" by following the methodology outlined in [13], and for MAT_133 - "MAT_BARLAT_YLD2000" according to [9]. The characteristic parameters were entered through the corresponding input interfaces in LS-Dyna.

### Laboratory bipolar half-plate geometry and experimental setup.

Based on common designs and preliminary studies regarding the formability of bipolar plate geometries, a laboratory bipolar half-plate geometry was developed [1]. The design is shown in Fig. 2 which has an active surface area of 84 mm x 50 mm and incorporates key features of bipolar plates such as a flow field, inlet and outlet ports for gaseous media as well as a surrounding sealing bead. This geometry does not possess any specific functional characteristics relevant to the fuel cell, but primarily serves to validate forming simulations and material models and to gain a more comprehensive understanding of the manufacturing process.
<figure><img src="image_1425.png" /><figcaption>Fig. 1: (a) True stress - true strain curves obtained by tensile and bulge tests and (b) combined flow curve.</figcaption></figure>

Using the obtained anisotropy ratio for the bulge test, the true-stress – true-plastic-strain diagram of the bulge test was used to obtain an experimental extension for the flow curve using the equivalent plastic deformation energy approach as described in [10]. Finally, a suitable flow curve is obtained, as shown in Fig. 1 b). The final flow curve for use in forming simulations is approximated by the Hockett-Sherby formula according to [11], see Eq. 1.

$$ \sigma_{Hockett-Sherby} = \sigma_{sat} - (\sigma_{sat} - \sigma_{ini}) \cdot e^{-a \cdot \varepsilon_{pl}^{p}} \quad (1) $$

<table><caption>Table 2: Parameters describing the flow-curve approximation according to Hockett-Sherby.</caption><thead><tr><th>σ<sub>sat</sub> [MPa]</th><th>σ<sub>ini</sub> [MPa]</th><th>p [-]</th><th>a [-]</th></tr></thead><tbody><tr><td>993.43</td><td>243.45</td><td>1.12</td><td>4.397</td></tr></tbody></table>

In the LS-Dyna software package, these acquired material data was adopted for the material model MAT_036 - "MAT_3-PARAMETER_BARLAT" by following the methodology outlined in [13], and for MAT_133 - "MAT_BARLAT_YLD2000" according to [9]. The characteristic parameters were entered through the corresponding input interfaces in LS-Dyna.

### Laboratory bipolar half-plate geometry and experimental setup.

Based on common designs and preliminary studies regarding the formability of bipolar plate geometries, a laboratory bipolar half-plate geometry was developed [1]. The design is shown in Fig. 2 which has an active surface area of 84 mm x 50 mm and incorporates key features of bipolar plates such as a flow field, inlet and outlet ports for gaseous media as well as a surrounding sealing bead. This geometry does not possess any specific functional characteristics relevant to the fuel cell, but primarily serves to validate forming simulations and material models and to gain a more comprehensive understanding of the manufacturing process.
<figure><img src="image_1422.png" /><figcaption>Fig. 2: (a) Top view of developed laboratory geometry showing characteristics of a metallic bipolar half-plate and (b) the designed channel cross section of the bipolar half-plate.</figcaption></figure>

To produce the laboratory specimen shown above, a tool-set for conventional hollow embossing with a massive punch and a die was manufactured. The active surfaces of the upper and lower tool have been offset according to the given foil thickness of $t_0 = 0.1$ mm. Precise alignment of the tool inserts was ensured by appropriate manufacturing accuracy, adherence to tolerance specifications and the incorporation of guiding columns into the tool. No blank holder was used in the laboratory-scale investigations presented here. The configuration of the laboratory tool and its installation in the hydraulic press having a maximum press force of 1600 kN is shown in Fig. 3. A load cell and displacement sensor have been added to the tool frame to measure punch force and stroke.

<figure><img src="image_1423.png" /><figcaption>Fig. 3: (a) Laboratory tool in the press and (b) detailed view of the active tool surface of lower die.</figcaption></figure>

In order to determine the minimum embossing force required for the shown laboratory geometry of a bipolar half-plate, an initial experimental test-series was conducted with variable punch forces. Subsequently, the achieved embossing depth was measured through optical measurements of the flow field geometry using a GOM ATOS system. Here, it was found that at a press force of 120 kN, the embossing depth of 0.25 mm and thus complete forming of the specimen could be achieved. The corresponding optical measurements of the embossed channels and their depths are presented in the following Fig. 4. For simplicity, only sections of the specimen have been considered, and average local depth of channels was evaluated. In this localized analysis, any springback effects have been neglected.
<figure><img src="image_1426.png" /><figcaption>Fig. 2: (a) Top view of developed laboratory geometry showing characteristics of a metallic bipolar half-plate and (b) the designed channel cross section of the bipolar half-plate.</figcaption></figure>

To produce the laboratory specimen shown above, a tool-set for conventional hollow embossing with a massive punch and a die was manufactured. The active surfaces of the upper and lower tool have been offset according to the given foil thickness of $t_0 = 0.1$ mm. Precise alignment of the tool inserts was ensured by appropriate manufacturing accuracy, adherence to tolerance specifications and the incorporation of guiding columns into the tool. No blank holder was used in the laboratory-scale investigations presented here. The configuration of the laboratory tool and its installation in the hydraulic press having a maximum press force of 1600 kN is shown in Fig. 3. A load cell and displacement sensor have been added to the tool frame to measure punch force and stroke.

<figure><img src="image_1427.png" /><figcaption>Fig. 3: (a) Laboratory tool in the press and (b) detailed view of the active tool surface of lower die.</figcaption></figure>

In order to determine the minimum embossing force required for the shown laboratory geometry of a bipolar half-plate, an initial experimental test-series was conducted with variable punch forces. Subsequently, the achieved embossing depth was measured through optical measurements of the flow field geometry using a GOM ATOS system. Here, it was found that at a press force of 120 kN, the embossing depth of 0.25 mm and thus complete forming of the specimen could be achieved. The corresponding optical measurements of the embossed channels and their depths are presented in the following Fig. 4. For simplicity, only sections of the specimen have been considered, and average local depth of channels was evaluated. In this localized analysis, any springback effects have been neglected.
<figure><img src="image_1424.png" /><figcaption>Fig. 4: Comparison of embossing force and embossing depth. Measurements obtained by GOM ATOS.</figcaption></figure>

According to these results, the bipolar half-plates that were produced in further tests were embossed with a constant press force of $F_{max}$ = 120 kN, ensuring comparability of the results. Moreover, no additional lubrication was applied and only the lubricant present on the supplied foils was used during the forming operations.

### FE model setup.

Based on the previously introduced laboratory geometry, forming simulations of the hollow embossing process were set up. These simulations aimed to evaluate the simulation accuracy in relation to the real experiments and to assess the performance of the material models considered. For the forming simulations, the LS-Dyna software package with solver version R12.1 was used. The model, consistent with real experiments, consisted of a blank with size of 84 mm x 50 mm and was meshed with fully integrated shell elements of type "-16" with seven integration points in thickness direction. A constant mesh size of 0.025 mm (edge-length) was applied, allowing the fine radii in the geometry (see Fig. 2 and Fig. 5) to be adequately represented. This meshing strategy resulted in a total of 6.72 million elements in the metallic foil for simulating the hollow embossing process. Due to the fine meshing, distortions of the mesh during the forming process could also be prevented (see Fig. 5).

<figure><img src="image_1425.png" /><figcaption>Fig. 5: Detailed mesh view with thickness distribution: a) Before and b) after hollow embossing.</figcaption></figure>

The active tool surfaces were meshed more finely with element sizes ranging from 0.01 mm to 0.025 mm. The die was fixed, as in the real process, while the punch was allowed to move vertically corresponding to the real process. For the simulation, a conservative friction coefficient of $\mu$ = 0.20 was set in a coulomb friction model based on empirical values from previous studies.

The parameters for mass scaling and contact definition were refined through numerical investigations to optimize computational efficiency without significantly changing the forming simulation results. This way a mass scaling of -2.0 e-07 and "FORMINGام$\_SURFACE\_TO\_SURFACE" contact formulation was used. The final configuration resulted in a computation time of approximately 12 hours using the explicit solver, 32 Intel® Xeon® Gold 6230 cores at 2.10 GHz and approximately 50 Gigabyte of RAM. The calculation of the resulting springback based on the results gained from the forming simulation and required approximately two hours when using an implicit solver. Further details on the boundary conditions
<figure><img src="image_1428.png" /><figcaption>Fig. 4: Comparison of embossing force and embossing depth. Measurements obtained by GOM ATOS.</figcaption></figure>

According to these results, the bipolar half-plates that were produced in further tests were embossed with a constant press force of $F_{max}$ = 120 kN, ensuring comparability of the results. Moreover, no additional lubrication was applied and only the lubricant present on the supplied foils was used during the forming operations.

### FE model setup.

Based on the previously introduced laboratory geometry, forming simulations of the hollow embossing process were set up. These simulations aimed to evaluate the simulation accuracy in relation to the real experiments and to assess the performance of the material models considered. For the forming simulations, the LS-Dyna software package with solver version R12.1 was used. The model, consistent with real experiments, consisted of a blank with size of 84 mm x 50 mm and was meshed with fully integrated shell elements of type "-16" with seven integration points in thickness direction. A constant mesh size of 0.025 mm (edge-length) was applied, allowing the fine radii in the geometry (see Fig. 2 and Fig. 5) to be adequately represented. This meshing strategy resulted in a total of 6.72 million elements in the metallic foil for simulating the hollow embossing process. Due to the fine meshing, distortions of the mesh during the forming process could also be prevented (see Fig. 5).

<figure><img src="image_1429.png" /><figcaption>Fig. 5: Detailed mesh view with thickness distribution: a) Before and b) after hollow embossing.</figcaption></figure>

The active tool surfaces were meshed more finely with element sizes ranging from 0.01 mm to 0.025 mm. The die was fixed, as in the real process, while the punch was allowed to move vertically corresponding to the real process. For the simulation, a conservative friction coefficient of $\mu$ = 0.20 was set in a coulomb friction model based on empirical values from previous studies.

The parameters for mass scaling and contact definition were refined through numerical investigations to optimize computational efficiency without significantly changing the forming simulation results. This way a mass scaling of -2.0 e-07 and "FORMINGام$\_SURFACE\_TO\_SURFACE" contact formulation was used. The final configuration resulted in a computation time of approximately 12 hours using the explicit solver, 32 Intel® Xeon® Gold 6230 cores at 2.10 GHz and approximately 50 Gigabyte of RAM. The calculation of the resulting springback based on the results gained from the forming simulation and required approximately two hours when using an implicit solver. Further details on the boundary conditions
of the springback simulation are provided in the subsequent chapter entitled "Results and discussion".

## Results and discussion

The forming simulations were conducted using the material models MAT_036 and MAT_133 based on the material properties outlined above. For the comparison between simulation and reality, both the thickness distributions within the active flow field geometry as well as the global springback of the entire bipolar half-plate geometry were analyzed. For comparison purposes, statistically averaged component data from real measurements was used.

### Analysis of the thickness distribution.

The thickness distributions obtained for the two material models from the forming simulations are illustrated in Fig. 6. It is evident that material thinning primarily concentrates on the flow field geometry within the inner region of the bipolar half-plate.

<figure><img src="image_1426.png" /><figcaption>Fig. 6: (a) Top view of formed specimen. (b) Thickness distribution in the forming simulation by utilizing MAT_036 and MAT_133 – exemplary comparison along the mid-cross-section A-A.</figcaption></figure>

The average material thinning ranges from approximately 5 to 15% showing individual peaks in areas of flow field direction changes. The maximum thinning was calculated higher at 19.16% when using the material model MAT_036 than with the material model MAT_133, which is limited to 16.37%. When examining the mid-cross-section A-A (see Fig. 6), it becomes clear that significant differences in material thinning can be observed, particularly in the flow field. Such differences are explained in more detail in the analysis of Fig. 7.

For the analysis of the resulting thickness distributions in the real component, the embossed laboratory bipolar half-plates were cut and cross-sections were prepared according to the cutting plane A-A in Fig. 6. In this way, the remaining material thickness after forming could be measured using a microscope and be compared with results from the forming simulation. This publication focuses on the comparison of only a few exemplary channels and not all of the 19 in total due to similar conditions in the respective hollow embossed channels. The corresponding cross-sectional analysis for channels 2, 3, 8 and 10 can be seen in Fig. 7. Seven distinct measurement positions were established for each channel, with four thickness measurements in the radii area and three thickness measurements in the region of flat channel bottoms and channel
of the springback simulation are provided in the subsequent chapter entitled "Results and discussion".

## Results and discussion

The forming simulations were conducted using the material models MAT_036 and MAT_133 based on the material properties outlined above. For the comparison between simulation and reality, both the thickness distributions within the active flow field geometry as well as the global springback of the entire bipolar half-plate geometry were analyzed. For comparison purposes, statistically averaged component data from real measurements was used.

### Analysis of the thickness distribution.

The thickness distributions obtained for the two material models from the forming simulations are illustrated in Fig. 6. It is evident that material thinning primarily concentrates on the flow field geometry within the inner region of the bipolar half-plate.

<figure><img src="image_1430.png" /><figcaption>Fig. 6: (a) Top view of formed specimen. (b) Thickness distribution in the forming simulation by utilizing MAT_036 and MAT_133 – exemplary comparison along the mid-cross-section A-A.</figcaption></figure>

The average material thinning ranges from approximately 5 to 15% showing individual peaks in areas of flow field direction changes. The maximum thinning was calculated higher at 19.16% when using the material model MAT_036 than with the material model MAT_133, which is limited to 16.37%. When examining the mid-cross-section A-A (see Fig. 6), it becomes clear that significant differences in material thinning can be observed, particularly in the flow field. Such differences are explained in more detail in the analysis of Fig. 7.

For the analysis of the resulting thickness distributions in the real component, the embossed laboratory bipolar half-plates were cut and cross-sections were prepared according to the cutting plane A-A in Fig. 6. In this way, the remaining material thickness after forming could be measured using a microscope and be compared with results from the forming simulation. This publication focuses on the comparison of only a few exemplary channels and not all of the 19 in total due to similar conditions in the respective hollow embossed channels. The corresponding cross-sectional analysis for channels 2, 3, 8 and 10 can be seen in Fig. 7. Seven distinct measurement positions were established for each channel, with four thickness measurements in the radii area and three thickness measurements in the region of flat channel bottoms and channel
sidewalls. In line with previous investigations, the mean value from three measurements were used for statistical robustness.

<figure><img src="image_1427.png" /><figcaption>Fig. 7: Distribution of foil thickness at selected channel cross-sections – Real part measurement (blue), simulation with MAT_036 (black) and simulation with MAT_133 (green).</figcaption></figure>

In addition to the real component data (in blue), the simulation results, which correspond to the color scheme from previous Fig.s, are also presented. Here, a maximum thinning can be observed at the measurement positions of the radii, while a greater material thickness is detected in the flat areas of the channel bottom and sidewall. The foil thickness generally decreases from the outer regions in the geometry (Channel 2) towards the center of the component (Channel 10) due to restricted material flow. While outer channels support relative motion of foil in the die leading to higher foil thicknesses, inner channels exhibit lower thicknesses. Here, the material flows out of thickness due to high tension stresses, as expected. As a result, both material models can qualitatively represent the foil thickness distribution within the channels, with higher absolute deviations observed at the measurement positions of the radii. Generally, material model MAT_036 shows lower deviation in the investigations, although the simulation results are less critical than reality. This could possibly be attributed to the limited capability of shell elements to represent changes in foil thickness in thickness direction.

### Analysis of springback behavior.

Springback refers to the geometric alteration that occurs to a component after it is freed from the forming tool's forces at the end of a forming process. To numerically determine the resulting springback, the results of the forming simulation were transferred to a separate simulation. In this process, the occurring strains and stresses of the respective elements were exported in a separate restart-file also known as a "dynain file". In a subsequent step, the resulting springback was determined using an implicit calculation in LS-Dyna. Boundary conditions for the springback simulation were chosen accordingly to the springback measurements in reality. The bipolar half-plate was fixed in a defined state using clay, allowing for a comparison between real data and simulation (see Fig. 8). In the simulation, the corresponding degrees of freedom at the corners of
sidewalls. In line with previous investigations, the mean value from three measurements were used for statistical robustness.

<figure><img src="image_1431.png" /><figcaption>Fig. 7: Distribution of foil thickness at selected channel cross-sections – Real part measurement (blue), simulation with MAT_036 (black) and simulation with MAT_133 (green).</figcaption></figure>

In addition to the real component data (in blue), the simulation results, which correspond to the color scheme from previous Fig.s, are also presented. Here, a maximum thinning can be observed at the measurement positions of the radii, while a greater material thickness is detected in the flat areas of the channel bottom and sidewall. The foil thickness generally decreases from the outer regions in the geometry (Channel 2) towards the center of the component (Channel 10) due to restricted material flow. While outer channels support relative motion of foil in the die leading to higher foil thicknesses, inner channels exhibit lower thicknesses. Here, the material flows out of thickness due to high tension stresses, as expected. As a result, both material models can qualitatively represent the foil thickness distribution within the channels, with higher absolute deviations observed at the measurement positions of the radii. Generally, material model MAT_036 shows lower deviation in the investigations, although the simulation results are less critical than reality. This could possibly be attributed to the limited capability of shell elements to represent changes in foil thickness in thickness direction.

### Analysis of springback behavior.

Springback refers to the geometric alteration that occurs to a component after it is freed from the forming tool's forces at the end of a forming process. To numerically determine the resulting springback, the results of the forming simulation were transferred to a separate simulation. In this process, the occurring strains and stresses of the respective elements were exported in a separate restart-file also known as a "dynain file". In a subsequent step, the resulting springback was determined using an implicit calculation in LS-Dyna. Boundary conditions for the springback simulation were chosen accordingly to the springback measurements in reality. The bipolar half-plate was fixed in a defined state using clay, allowing for a comparison between real data and simulation (see Fig. 8). In the simulation, the corresponding degrees of freedom at the corners of
the plate were constrained. Thus, the occurrence of unstable springback conditions could be prevented. Real component measurements were conducted using GOM ATOS as shown in Fig. 8.

<figure><img src="image_1428.png" /><figcaption>Fig. 8: Springback of the laboratory bipolar half-plate when clamped at four corners (real part) measured with GOM ATOS.</figcaption></figure>

During the investigations, it was observed that all real components exhibited nearly consistent springback behavior in each experimental repetition. Thus, only one bipolar half-plate is considered in the following Fig. 9 for an exemplary analysis. Finally, it can be observed that especially the flange areas exhibit a significant amount of springback, whereas the flow field region shows relatively lower levels of springback in proportion. Similar effects were observed during the simulations. Both material models MAT_036 and MAT_133 were able to represent the springback behavior, with MAT_036 showing a very good match, especially when considering the absolute values of the resulting springback (refer to Fig. 9).

<figure><img src="image_1429.png" /><figcaption>Fig. 9: Comparison of springback simulation results with real component data.</figcaption></figure>

The use of MAT_133 leads to an overestimation of the actual springback under otherwise constant boundary conditions. These discrepancies arise from the variation in deformation due to different material modeling approaches, as already highlighted in Fig. 6 and Fig. 7.

## Summary and Outlook

The study as presented in this paper investigated the potential of LS-Dyna-based shell models and widely used material models “MAT 3-PARAMETER BARLAT” (MAT_036) and “MAT_BARLAT YLD2000” (MAT_133) regarding the accurate prediction of the quality of hollow-embossed bipolar half-plates made of 1.4404 stainless-steel foil. For this an extensive material characterization was conducted previously and a corresponding laboratory-scale prototype tool for hollow embossing of bipolar half-plates was developed. Based on the preceding investigations using simulation results and real part data, the following can be concluded:
the plate were constrained. Thus, the occurrence of unstable springback conditions could be prevented. Real component measurements were conducted using GOM ATOS as shown in Fig. 8.

<figure><img src="image_1432.png" /><figcaption>Fig. 8: Springback of the laboratory bipolar half-plate when clamped at four corners (real part) measured with GOM ATOS.</figcaption></figure>

During the investigations, it was observed that all real components exhibited nearly consistent springback behavior in each experimental repetition. Thus, only one bipolar half-plate is considered in the following Fig. 9 for an exemplary analysis. Finally, it can be observed that especially the flange areas exhibit a significant amount of springback, whereas the flow field region shows relatively lower levels of springback in proportion. Similar effects were observed during the simulations. Both material models MAT_036 and MAT_133 were able to represent the springback behavior, with MAT_036 showing a very good match, especially when considering the absolute values of the resulting springback (refer to Fig. 9).

<figure><img src="image_1433.png" /><figcaption>Fig. 9: Comparison of springback simulation results with real component data.</figcaption></figure>

The use of MAT_133 leads to an overestimation of the actual springback under otherwise constant boundary conditions. These discrepancies arise from the variation in deformation due to different material modeling approaches, as already highlighted in Fig. 6 and Fig. 7.

## Summary and Outlook

The study as presented in this paper investigated the potential of LS-Dyna-based shell models and widely used material models “MAT 3-PARAMETER BARLAT” (MAT_036) and “MAT_BARLAT YLD2000” (MAT_133) regarding the accurate prediction of the quality of hollow-embossed bipolar half-plates made of 1.4404 stainless-steel foil. For this an extensive material characterization was conducted previously and a corresponding laboratory-scale prototype tool for hollow embossing of bipolar half-plates was developed. Based on the preceding investigations using simulation results and real part data, the following can be concluded:
* The distribution of thinning can be predicted in forming simulations with LS-Dyna with minimal deviations. Notable differences arise in the material models, particularly in the areas of radii, with MAT_036 demonstrating better predictive accuracy and a lower relative deviation of max. 7.4 %. Whereas the maximum deviation with MAT_133 is max. 10.9 %.

* The springback can be reliably predicted. Particularly, the material model MAT_36 can reproduce the absolute values with high precision - almost no significant differences in the presented hollow embossing process are notable with deviation between -0.07 mm and +0.11 mm. However, MAT_133 exhibits significant deviations in this aspect, with values between +1.00 mm and +2.88 mm compared to measured real part data.

* Reliable prediction of the resulting quality of hollow embossed bipolar half-plates using common material models in LS-Dyna is possible, although there is further potential for improvement in the accuracy of thickness distribution predictions, requiring additional modeling effort. This can possibly be achieved through more complex shell element formulations.

This comparative analysis has offered valuable insights into the achievable predictive accuracy of the considered common material models in LS-Dyna, thereby enhancing the efficiency of future design processes. Subsequent studies will particularly focus on multi-stage hollow-embossing processes, commonly employed in the production of larger bipolar half-plates. In addition to the simple material models presented here, more intricate material models will be employed. These models can account for kinematic hardening and the Bauschinger effect resulting from cyclic loading and unloading between multiple forming stages. Relevant foundations and approaches have already been introduced in [13].

## Acknowledgements

This work was supported within the "DFG-Fraunhofer Transfer Programme", established by the German Research Foundation (DFG-Project 460294948). Authors feel grateful for the financial support. In addition, the authors would like to thank the Working Group Foil-Forming established at the Institute for Metal Forming Technology, University of Stuttgart for their support.

## References

[1] M. Beck, K. R. Riedmüller, M. Liewald, A. Bertz, M. J. Aslan, D. Carl, Investigation on the influence of geometric parameters on the dimensional accuracy of high-precision embossed metallic bipolar plates, Production at the Leading Edge of Technology WGP (2022) 427-438. http://dx.doi.org/10.1007/978-3-031-18318-8_44

[2] F. Vollertsen, D. Biermann, H. N. Hansen, I.S. Jawahir, K. Kuzman, Size effects in manufacturing of metallic components, CIRP Ann Manuf Technol 58 (2009) 566-587. https://doi.org/10.1016/j.cirp.2009.09.002

[3] D. M. Neto, M. C. Oliveira, J. L. Alves, L. F. Menezes, Numerical study on the formability of metallic bipolar plates for proton exchange membrane (PEM) fuel cells, Metals 9 (2019) 810. https://doi.org/10.3390/met9070810

[4] Q. Hu, D. Zhang, H. Fu, K. Huang, Investigation of stamping process of metallic bipolar plates in PEM fuel cell – numerical simulation and experiments, Int. J. Hydrogen Energy 39 (2014) 13770-13776. http://dx.doi.org/10.1016/j.ijhydene.2014.01.201

[5] A. Bauer, Experimentelle und numerische Untersuchungen zur Analyse der umform- technischen Herstellung metallischer Bipolarplatten, Dissertation, Technische Universität Chemnitz, Chemnitz, 2020.
* The distribution of thinning can be predicted in forming simulations with LS-Dyna with minimal deviations. Notable differences arise in the material models, particularly in the areas of radii, with MAT_036 demonstrating better predictive accuracy and a lower relative deviation of max. 7.4 %. Whereas the maximum deviation with MAT_133 is max. 10.9 %.

* The springback can be reliably predicted. Particularly, the material model MAT_36 can reproduce the absolute values with high precision - almost no significant differences in the presented hollow embossing process are notable with deviation between -0.07 mm and +0.11 mm. However, MAT_133 exhibits significant deviations in this aspect, with values between +1.00 mm and +2.88 mm compared to measured real part data.

* Reliable prediction of the resulting quality of hollow embossed bipolar half-plates using common material models in LS-Dyna is possible, although there is further potential for improvement in the accuracy of thickness distribution predictions, requiring additional modeling effort. This can possibly be achieved through more complex shell element formulations.

This comparative analysis has offered valuable insights into the achievable predictive accuracy of the considered common material models in LS-Dyna, thereby enhancing the efficiency of future design processes. Subsequent studies will particularly focus on multi-stage hollow-embossing processes, commonly employed in the production of larger bipolar half-plates. In addition to the simple material models presented here, more intricate material models will be employed. These models can account for kinematic hardening and the Bauschinger effect resulting from cyclic loading and unloading between multiple forming stages. Relevant foundations and approaches have already been introduced in [13].

## Acknowledgements

This work was supported within the "DFG-Fraunhofer Transfer Programme", established by the German Research Foundation (DFG-Project 460294948). Authors feel grateful for the financial support. In addition, the authors would like to thank the Working Group Foil-Forming established at the Institute for Metal Forming Technology, University of Stuttgart for their support.

## References

[1] M. Beck, K. R. Riedmüller, M. Liewald, A. Bertz, M. J. Aslan, D. Carl, Investigation on the influence of geometric parameters on the dimensional accuracy of high-precision embossed metallic bipolar plates, Production at the Leading Edge of Technology WGP (2022) 427-438. http://dx.doi.org/10.1007/978-3-031-18318-8_44

[2] F. Vollertsen, D. Biermann, H. N. Hansen, I.S. Jawahir, K. Kuzman, Size effects in manufacturing of metallic components, CIRP Ann Manuf Technol 58 (2009) 566-587. https://doi.org/10.1016/j.cirp.2009.09.002

[3] D. M. Neto, M. C. Oliveira, J. L. Alves, L. F. Menezes, Numerical study on the formability of metallic bipolar plates for proton exchange membrane (PEM) fuel cells, Metals 9 (2019) 810. https://doi.org/10.3390/met9070810

[4] Q. Hu, D. Zhang, H. Fu, K. Huang, Investigation of stamping process of metallic bipolar plates in PEM fuel cell – numerical simulation and experiments, Int. J. Hydrogen Energy 39 (2014) 13770-13776. http://dx.doi.org/10.1016/j.ijhydene.2014.01.201

[5] A. Bauer, Experimentelle und numerische Untersuchungen zur Analyse der umform- technischen Herstellung metallischer Bipolarplatten, Dissertation, Technische Universität Chemnitz, Chemnitz, 2020.
[6] F. Adzima, T. Balan, P. Y. Manach, Springback prediction for a mechanical micro connector using CPFEM based numerical simulations, Int J Mater Form 13 (2020) 649-659.
https://doi.org/10.1007/s12289-019-01503-5

[7] S. Porstmann, T. Wannemacher, W. G. Drossel, A comprehensive comparison of state-of-the-art manufacturing methods for fuel cell bipolar plates including anticipated future industry trends, J Manuf Process 60, 366-383. https://doi.org/10.1016/j.jmapro.2020.10.041

[8] C. Pham, F. Adzima, J. Coër, Anti-buckling device for ultra-thin metallic sheets under large and reversed shear strain paths, Experimental Mechanics 57, 593-602.
https://doi.org/10.1007/s11340-017-0256-4

[9] F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, et al., Plane stress yield function for aluminum alloy sheets – part 1: theory, International Journal of Plasticity 19(9) (2003) 1297-1319.
https://doi.org/10.1016/S0749-6419(02)00019-0

[10] A. Mutrux, B. Hochholdinger, P. Hora, A procedure for the evaluation and validation of the hydraulic biaxial experiment, Numisheet (2008).

[11] J. E. Hockett, O. D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures, J Mech Phys Solids 23 (1975) 87-98.
https://doi.org/10.1016/0022-5096(75)90018-6

[12] F. Barlat, K. Lian, Plastic behavior and stretchability of sheet metals. Part 1: A yield function for orthotropic sheets under plane stress conditions, International Journal of Plasticity 5(1) (1989) 51-66.
https://doi.org/10.1016/0749-6419(89)90019-3

[13] C. Karadogan, M. Beck, P. Cyron, K. R. Riedmüller, M. Liewald, Development of a new cyclic shear test setup for characterizing thin metallic foils, IOP Conference Series Materials Science and Engineering (2023), http://dx.doi.org/10.1088/1757-899X/1284/1/012057
[6] F. Adzima, T. Balan, P. Y. Manach, Springback prediction for a mechanical micro connector using CPFEM based numerical simulations, Int J Mater Form 13 (2020) 649-659. https://doi.org/10.1007/s12289-019-01503-5

[7] S. Porstmann, T. Wannemacher, W. G. Drossel, A comprehensive comparison of state-of-the-art manufacturing methods for fuel cell bipolar plates including anticipated future industry trends, J Manuf Process 60, 366-383. https://doi.org/10.1016/j.jmapro.2020.10.041

[8] C. Pham, F. Adzima, J. Coër, Anti-buckling device for ultra-thin metallic sheets under large and reversed shear strain paths, Experimental Mechanics 57, 593-602. https://doi.org/10.1007/s11340-017-0256-4

[9] F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, et al., Plane stress yield function for aluminum alloy sheets – part 1: theory, International Journal of Plasticity 19(9) (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[10] A. Mutrux, B. Hochholdinger, P. Hora, A procedure for the evaluation and validation of the hydraulic biaxial experiment, Numisheet (2008).

[11] J. E. Hockett, O. D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures, J Mech Phys Solids 23 (1975) 87-98. https://doi.org/10.1016/0022-5096(75)90018-6

[12] F. Barlat, K. Lian, Plastic behavior and stretchability of sheet metals. Part 1: A yield function for orthotropic sheets under plane stress conditions, International Journal of Plasticity 5(1) (1989) 51-66. https://doi.org/10.1016/0749-6419(89)90019-3

[13] C. Karadogan, M. Beck, P. Cyron, K. R. Riedmüller, M. Liewald, Development of a new cyclic shear test setup for characterizing thin metallic foils, IOP Conference Series Materials Science and Engineering (2023), http://dx.doi.org/10.1088/1757-899X/1284/1/012057
# Optimization of the heating parameters of a robotized hot incremental forming of high impact polystyrene

DUARTE ROCHA Valentin<fnref n="1" />,ª*, GIRAUD MOREAU Laurence<fnref n="1" />,ᵇ and CHEROUAT Abel<fnref n="2" />,ᶜ

**Keywords:** Robotized Incremental Forming, High Impact Polystyrene, FEA, Heating System, Optimization

**Abstract.** Single point incremental forming (SPIF) is a modern rapid manufacturing technology able to manufacturing complex sheet parts in small quantities. In comparison to conventional deep-drawing process, complex tools can be dispensed in order to reduce tool costs and the time required to achieve the first finished part. This new technology consists of locally and iteratively deforming plastically the sheet material by a punch that is generally hemispherical and of small dimensions, whose trajectory is programmed on a numerically controlled machine or a robot arm. The parts formed are mainly made from metallic materials (steel, aluminum, titanium or copper alloys). Very few studies focus on the incremental forming of thermoplastic parts. Because of the poor formability of thermoplastic parts and the large spring-back at room temperature, the use of heating systems of the work-piece during the forming process is required. In this work, a novel incremental sheet forming assisted by heat transfer was developed to improve the formability of thermoplastic sheet. By means of a series of experimental forming of High Impact Polystyrene pyramid frustum with constant wall angle at different heat temperatures, geometric accuracy was recorded and analyzed. This study aims to evaluate the behavior of polymer material in the hot incremental sheet forming with different pre-heating system using the finite element method.

**Introduction**

Incremental sheet forming is a promising process, well-suited for small batch production or prototyping, with many advantages compared to traditional forming processes such as low cost and higher flexibility. Most studies concern metallic parts or composite parts manufactured using classical three-axis CNC milling machines [1]. Nevertheless, these machines have a limited workspace and can be used to form only simple shape parts. Larger dimension parts and more complex shapes can be produced by industrial robots. Regarding the materials, the use of composite materials is now increasing in many areas (aeronautic, automotive and medical) due to their mechanical properties and environment concerns [2]. There is little to no literature on the incremental forming of high impact polystyrene, although this polymer is widely used in industry. However, this polymeric material having a poor formability at room temperature, a heating system must be considered. The aim of this research is to study experimentally and numerically the robotized hot incremental forming of polymer cone parts. With the help of design of experiment (DOE), the effects of heating system (tool or sheet or combination) were studied to optimize the process.

**Experimental process**

The High Impact Polystyrene part considered in this study is a truncated cone with a wall angle of $α = 30°$, a depth of $h = 35mm$ and a base radius of $R = 81 mm$. The High Impact Polystyrene sheet

<figure><img src="image_1430.png" /></figure>

<footnote n="1">¹UR-Lasmis, UTT, 12 rue Marie Curie CS 42060, 10004 Troyes cedex, France</footnote>

<footnote n="2">²UR-Gamma3, UTT, 12 rue Marie Curie CS 42060, 10004 Troyes cedex, France</footnote>

<footnote>ªvalentin.duarte_rocha@utt.fr, ᵇlaurence.moreau@utt.fr, cabel.cherouat@utt.fr</footnote>