[11] C. Liebold and A. Haufe, “process2product simulation: Closing Incompatibilities in Constitutive Modeling and Spatial Discretization with envyo®,” in Dearborn, Michigan, USA, 2018.

[12] Prüfung metallischer Werkstoffe – Zugproben, DIN 50125, DIN Deutsches Institut für Normung e. V., Berlin, 2022.

[13] *Metallische Werkstoffe - Zugversuch: Teil 1: Prüfverfahren bei Raumtemperatur* (ISO 6892-1:2019), DIN EN ISO 6892-1, DIN Deutsches Institut für Normung e. V., Berlin, 2020.

[14] *Metallische Werkstoffe - Biegeversuch* (ISO 7438:2020), DIN EN ISO 7438, DIN Deutsches Institut für Normung e. V., Berlin, 2021.

[15] A. R. Birkert, S. Haage, and M. Straub, *Umformtechnische Herstellung komplexer Karosserieteile: Auslegung von Ziehanlagen*. Berlin, Heidelberg: Springer Vieweg, 2013.

[16] J. E. Hockett and O. D. Sherby, “Large strain deformation of polycrystalline metals at low homologous temperatures,” *Journal of the Mechanics and Physics of Solids*, vol. 23, no. 2, pp. 87–98, 1975. https://doi.org/10.1016/0022-5096(75)90018-6
[11] C. Liebold and A. Haufe, “process2product simulation: Closing Incompatibilities in Constitutive Modeling and Spatial Discretization with envyo®,” in Dearborn, Michigan, USA, 2018.

[12] Prüfung metallischer Werkstoffe – Zugproben, DIN 50125, DIN Deutsches Institut für Normung e. V., Berlin, 2022.

[13] Metallische Werkstoffe - Zugversuch: Teil 1: Prüfverfahren bei Raumtemperatur (ISO 6892-1:2019), DIN EN ISO 6892-1, DIN Deutsches Institut für Normung e. V., Berlin, 2020.

[14] Metallische Werkstoffe - Biegeversuch (ISO 7438:2020), DIN EN ISO 7438, DIN Deutsches Institut für Normung e. V., Berlin, 2021.

[15] A. R. Birkert, S. Haage, and M. Straub, *Umformtechnische Herstellung komplexer Karosserieteile: Auslegung von Ziehanlagen*. Berlin, Heidelberg: Springer Vieweg, 2013.

[16] J. E. Hockett and O. D. Sherby, “Large strain deformation of polycrystalline metals at low homologous temperatures,” *Journal of the Mechanics and Physics of Solids*, vol. 23, no. 2, pp. 87–98, 1975. https://doi.org/10.1016/0022-5096(75)90018-6
# Mesh sensitivity study in the random cellular automata finite element model of dynamic recrystallization

SITKO Mateusz¹,ᵃ*, PAWLIKOWSKI Kacper¹,ᵇ, PERZYNSKI Konrad¹,ᶜ, and
MADEJ Lukasz¹,ᵈ

¹AGH University of Krakow, al. Adama Mickiewicza 30, 30-059 Kraków, Poland

ªmsitko@agh.edu.pl, ᵇkpawliko@agh.edu.pl, ᶜkperzyns@agh.edu.pl, ᵈlmadej@agh.edu.pl

**Keywords:** Random Cellular Automata, Microstructure Evolution, Discontinuous Dynamic Recrystallization

**Abstract.** Predicting microstructure morphology evolution under hot forming conditions and determining final material properties are essential for optimizing metal-forming processes. Cellular Automata (CA) is a widely employed full-field method for modeling microstructure morphology changes during various metal-forming processes. However, at higher temperatures and under conditions of substantial microstructure evolution, the CA method encounters limitations related to computational domain geometry changes. The use of random cellular automata (RCA) offers a more realistic representation of this phenomenon, although it requires additional effort in algorithm optimization for acceptable execution times. This paper contributes to an overarching research effort focused on developing a discontinuous dynamic recrystallization model (DRX) by directly incorporating RCA into the finite element (FE) framework. Different mesh sizes and their impact on the quality of the results are analyzed, and the minimum number of elements that do not degrade the results in the CA model are selected. The investigation aims to enhance the practicality of the proposed model, striking a balance between realistic microstructure representation and computational efficiency.

## Introduction

With the rapid advancements in computer technology, contemporary numerical methods have evolved to analyze the macroscopic response of materials and explicitly track microstructural changes. Multiscale models at the forefront of scientific research employ diverse numerical simulation methods to capture phenomena across various scales. The deformation of metallic materials, a crucial area of study, involves microstructure evolution driven by recrystallization, phase transformation, and texture development. When high-temperature forming operations are considered, dynamic recrystallization (DRX) becomes a major phenomenon affecting microstructure restoration, ultimately impacting the material's final mechanical properties [1].

Dynamic recrystallization is fundamentally comprised of two primary stages: the nucleation and growth of new dislocation-free grains. This process is predominantly steered by the reduction of dislocation density accumulated in the material during plastic deformation under elevated temperatures. Direct experimental observations of DRX during deformation at high temperatures pose significant challenges, leading to the reliance on various numerical models to complement experimental analysis [2,3].

Recently, more often, advanced full-field numerical methods have been employed to support experimental research on that topic [4]. One such method that explicitly captures microstructure evolution during plastic deformation is Cellular Automata (CA) [5]. Originating from a basic model by Goetz and Seetharaman [6], the CA-based models have significantly evolved over the years. Notably, Xiao et al. [7] introduced a DRX model integrating CA with topology deformation to consider the influence of grains' topology changes on DRX kinetics. Gawad and Pietrzyk

<figure><img src="image_2143.png" /></figure>
# Mesh sensitivity study in the random cellular automata finite element model of dynamic recrystallization

SITKO Mateusz<fnref n="1" />,ᵃ*, PAWLIKOWSKI Kacper<fnref n="1" />,ᵇ, PERZYNSKI Konrad<fnref n="1" />,ᶜ, and
MADEJ Lukasz<fnref n="1" />,ᵈ

**Keywords:** Random Cellular Automata, Microstructure Evolution, Discontinuous Dynamic Recrystallization

**Abstract.** Predicting microstructure morphology evolution under hot forming conditions and determining final material properties are essential for optimizing metal-forming processes. Cellular Automata (CA) is a widely employed full-field method for modeling microstructure morphology changes during various metal-forming processes. However, at higher temperatures and under conditions of substantial microstructure evolution, the CA method encounters limitations related to computational domain geometry changes. The use of random cellular automata (RCA) offers a more realistic representation of this phenomenon, although it requires additional effort in algorithm optimization for acceptable execution times. This paper contributes to an overarching research effort focused on developing a discontinuous dynamic recrystallization model (DRX) by directly incorporating RCA into the finite element (FE) framework. Different mesh sizes and their impact on the quality of the results are analyzed, and the minimum number of elements that do not degrade the results in the CA model are selected. The investigation aims to enhance the practicality of the proposed model, striking a balance between realistic microstructure representation and computational efficiency.

**Introduction**

With the rapid advancements in computer technology, contemporary numerical methods have evolved to analyze the macroscopic response of materials and explicitly track microstructural changes. Multiscale models at the forefront of scientific research employ diverse numerical simulation methods to capture phenomena across various scales. The deformation of metallic materials, a crucial area of study, involves microstructure evolution driven by recrystallization, phase transformation, and texture development. When high-temperature forming operations are considered, dynamic recrystallization (DRX) becomes a major phenomenon affecting microstructure restoration, ultimately impacting the material's final mechanical properties [1].

Dynamic recrystallization is fundamentally comprised of two primary stages: the nucleation and growth of new dislocation-free grains. This process is predominantly steered by the reduction of dislocation density accumulated in the material during plastic deformation under elevated temperatures. Direct experimental observations of DRX during deformation at high temperatures pose significant challenges, leading to the reliance on various numerical models to complement experimental analysis [2,3].

Recently, more often, advanced full-field numerical methods have been employed to support experimental research on that topic [4]. One such method that explicitly captures microstructure evolution during plastic deformation is Cellular Automata (CA) [5]. Originating from a basic model by Goetz and Seetharaman [6], the CA-based models have significantly evolved over the years. Notably, Xiao et al. [7] introduced a DRX model integrating CA with topology deformation to consider the influence of grains' topology changes on DRX kinetics. Gawad and Pietrzyk

<figure><img src="image_2134.png" /></figure>

<footnote n="1">¹AGH University of Krakow, al. Adama Mickiewicza 30, 30-059 Kraków, Poland</footnote>

<footnote>ªmsitko@agh.edu.pl, ᵇkpawliko@agh.edu.pl, ᶜkperzyns@agh.edu.pl, ᵈlmadej@agh.edu.pl</footnote>
proposed the Cellular Automata Finite Element (CAFE) [8], offering a simplified approach to account for microstructure deformation in finite element simulations. More complex models combining CA with finite element methods have recently been developed [9]. However, the significant limitation of the classical CA approach is the accurate description of cellular automata computational domain deformation under plastic forming conditions, given its reliance on a regular grid of cells.

Several approaches to overcome this limitation have been proposed in the literature, e.g., the CA space elongation algorithm along a selected axis [10], artificially elongating new recrystallized grains, or the mapping algorithm where the computational space is transferred between two separate coordinate systems [11], reducing the geometric assumptions during simulation. Recently, a more advanced solution has been introduced in [12] based on a direct space deformation concept. Direct coupling between the CA and FE approaches provides many advantages but faces limitations in remeshing operations at higher deformation degrees due to CA space regularity. Therefore, the Authors of the paper presented a novel approach based on the concept of full coupling between the finite element deformation model and the random cellular automata (RCA) model for microstructure evolution [12], that can overcome the mentioned limitations.

However, in the fully coupled RCAFE model, the key aspect is related to the simulation time, which may be excessive, limiting the broader application of the approach. There are several solutions to this issue. It is possible to speed up the RCA model, e.g., by the implementation of an efficient CA neighbor search [13] or by improving the communication procedures between the RCA and FE computing modules. Finally, a reduction of computation effort in the FE module is also possible by a decrease in the number of finite elements during the simulation. However, the latter concept is directly related to the mesh sensitivity issue. Estimating an optimal number of RCA cells in the coupled RCAFE simulations that do not affect the quality of results is the subject of this paper.

## RCAFE DRX simulation

The numerical simulations of the material deformation process are conducted within the classical J2 plasticity FE framework of Abaqus software, while the in-house RCA code carries the microstructure evolutions. Each RCA cell from the microstructure evolution model corresponds to a single FE integration point in this coupled approach. At each time step, the coordinates of integration points, equivalent plastic stress and temperature values, are transferred from the FE to the RCA model. This coupling introduces a potential risk that certain material areas may not be discretized with the correct number of RCA cells, thereby influencing the final results. Proper initial mesh discretization can be established to mitigate this concern, as the RCA model is designed to seamlessly accommodate changes introduced in the FE algorithm.

The simulation in such a coupled RCAFE model generally consists of several steps. As mentioned, at the beginning of each iteration, cells in the RCA space are translated based on corresponding integration point coordinates in the FE model. After that, dislocation density in each cell is evaluated based on the equivalent stress value:

$$ \bar{\rho} = \frac{1}{\rho_0} \left( \frac{\sigma_i}{a_6 G b} \right)^2 \qquad (1) $$

where: G is the shear modulus, $\sigma_i$ is the equivalent stress value, $a_6$ is the model constant, b is the magnitude of the Burgers vector, and $\rho_0$ is the initial dislocation density.

When the critical value of dislocation density is reached, new nuclei are incorporated into RCA space with the nucleation rate described by:
proposed the Cellular Automata Finite Element (CAFE) [8], offering a simplified approach to account for microstructure deformation in finite element simulations. More complex models combining CA with finite element methods have recently been developed [9]. However, the significant limitation of the classical CA approach is the accurate description of cellular automata computational domain deformation under plastic forming conditions, given its reliance on a regular grid of cells.

Several approaches to overcome this limitation have been proposed in the literature, e.g., the CA space elongation algorithm along a selected axis [10], artificially elongating new recrystallized grains, or the mapping algorithm where the computational space is transferred between two separate coordinate systems [11], reducing the geometric assumptions during simulation. Recently, a more advanced solution has been introduced in [12] based on a direct space deformation concept. Direct coupling between the CA and FE approaches provides many advantages but faces limitations in remeshing operations at higher deformation degrees due to CA space regularity. Therefore, the Authors of the paper presented a novel approach based on the concept of full coupling between the finite element deformation model and the random cellular automata (RCA) model for microstructure evolution [12], that can overcome the mentioned limitations.

However, in the fully coupled RCAFE model, the key aspect is related to the simulation time, which may be excessive, limiting the broader application of the approach. There are several solutions to this issue. It is possible to speed up the RCA model, e.g., by the implementation of an efficient CA neighbor search [13] or by improving the communication procedures between the RCA and FE computing modules. Finally, a reduction of computation effort in the FE module is also possible by a decrease in the number of finite elements during the simulation. However, the latter concept is directly related to the mesh sensitivity issue. Estimating an optimal number of RCA cells in the coupled RCAFE simulations that do not affect the quality of results is the subject of this paper.

## RCAFE DRX simulation

The numerical simulations of the material deformation process are conducted within the classical J2 plasticity FE framework of Abaqus software, while the in-house RCA code carries the microstructure evolutions. Each RCA cell from the microstructure evolution model corresponds to a single FE integration point in this coupled approach. At each time step, the coordinates of integration points, equivalent plastic stress and temperature values, are transferred from the FE to the RCA model. This coupling introduces a potential risk that certain material areas may not be discretized with the correct number of RCA cells, thereby influencing the final results. Proper initial mesh discretization can be established to mitigate this concern, as the RCA model is designed to seamlessly accommodate changes introduced in the FE algorithm.

The simulation in such a coupled RCAFE model generally consists of several steps. As mentioned, at the beginning of each iteration, cells in the RCA space are translated based on corresponding integration point coordinates in the FE model. After that, dislocation density in each cell is evaluated based on the equivalent stress value:

$$ \bar{\rho} = \frac{1}{\rho_0} \left( \frac{\sigma_i}{a_6 G b} \right)^2 \qquad (1) $$

where: G is the shear modulus, $\sigma_i$ is the equivalent stress value, $a_6$ is the model constant, b is the magnitude of the Burgers vector, and $\rho_0$ is the initial dislocation density.

When the critical value of dislocation density is reached, new nuclei are incorporated into RCA space with the nucleation rate described by:
$$ \dot{N} = c\dot{\epsilon}\exp\left(-\frac{Q_n}{RT}\right) \qquad (2) $$

where: c is constant, $\dot{\epsilon}$ is the strain rate, $Q_n$ is the activation energy for nucleation, T is the temperature, and R is the universal gas constant.

Subsequently, in each time step, the RCA cell changes its state from unrecrystallized to recrystallized based on its size and growth velocity value calculated as:

$$ V = M_0 \exp\left(-\frac{Q_m}{RT}\right) 2\alpha\tau(\rho_{(k,l)} - \rho_{(i,j)}) \qquad (3) $$

where: $M_0$ is the initial grain boundary mobility, $Q_m$ is the activation energy for grain boundary motion, $\alpha$ is a coefficient from the range (0,1), $\rho_{(k,l)}$ is the dislocation density in the analyzed and neighboring cells, and $\tau$ is the energy required for dislocation movement.

As pointed out, initial investigations with the developed RCAFE solution [14] identified the primary drawback of the coupled model as excessive computational time. While enhancements have been made to the RCA code in terms of designing specific neighborhood types and search algorithms, as documented in [15], the current research is dedicated to further optimizing the algorithmic solutions, focusing on the role of mesh density in the simulation accuracy.

## Procedure

First, a digital material representation model with different levels of mesh discretization was developed based on scanning electron microscopy metallographic investigation. The austenite model alloy Fe30Ni was selected as a case study. The sample for further analysis was extracted from the as-received undeformed material. The electron back-scattered diffraction map with a step size of 0.5µm was used to directly map the investigated microstructure morphology into the digital form, as seen in Fig. 1. An area of 100×100 µm was used for the research.

<figure><img src="image_2144.png" /><figcaption>Figure 1 Procedure for preparation of undeformed material morphology for numerical investigations: a) EBSD map, b) extraction of grain boundaries, c) flood fill algorithm, d) grain growth, e) assignment of grains to FE mesh.</figcaption></figure>

Finite elements used for discretization in the RCAFE model are assigned to particular grains with slightly different material properties reflecting variations in the crystallographic orientation. The number of RCA cells in the cellular automata model directly corresponds to the increasing number of FE elements (from 25×25 to 300×300, Fig. 2) used during the discretization of the digital microstructure model.

Each model was subjected to numerical simulation of deformation under plane strain conditions at 1000°C and a strain rate of 1/s, to reflect the experimental case study.

The radius of the neighborhood in the RCA method was calculated so that, on average, eight cells fall into the neighborhood. With that, the investigation evaluates how mesh sensitivity affects the FE results and subsequent RCA results of discontinuous dynamic recrystallization evolution. The one-way data transfer from the FE to the RCA model is considered at this research stage.
$$ \dot{N} = c\dot{\epsilon}\exp\left(-\frac{Q_n}{RT}\right) \qquad (2) $$

where: c is constant, $\dot{\epsilon}$ is the strain rate, $Q_n$ is the activation energy for nucleation, T is the temperature, and R is the universal gas constant.

Subsequently, in each time step, the RCA cell changes its state from unrecrystallized to recrystallized based on its size and growth velocity value calculated as:

$$ V = M_0 \exp\left(-\frac{Q_m}{RT}\right) 2\alpha\tau(\rho_{(k,l)} - \rho_{(i,j)}) \qquad (3) $$

where: $M_0$ is the initial grain boundary mobility, $Q_m$ is the activation energy for grain boundary motion, $\alpha$ is a coefficient from the range (0,1), $\rho_{(k,l)}$ is the dislocation density in the analyzed and neighboring cells, and $\tau$ is the energy required for dislocation movement.

As pointed out, initial investigations with the developed RCAFE solution [14] identified the primary drawback of the coupled model as excessive computational time. While enhancements have been made to the RCA code in terms of designing specific neighborhood types and search algorithms, as documented in [15], the current research is dedicated to further optimizing the algorithmic solutions, focusing on the role of mesh density in the simulation accuracy.

## Procedure

First, a digital material representation model with different levels of mesh discretization was developed based on scanning electron microscopy metallographic investigation. The austenite model alloy Fe30Ni was selected as a case study. The sample for further analysis was extracted from the as-received undeformed material. The electron back-scattered diffraction map with a step size of 0.5µm was used to directly map the investigated microstructure morphology into the digital form, as seen in Fig. 1. An area of 100×100 µm was used for the research.

<figure><img src="image_2135.png" /><figcaption>Figure 1 Procedure for preparation of undeformed material morphology for numerical investigations: a) EBSD map, b) extraction of grain boundaries, c) flood fill algorithm, d) grain growth, e) assignment of grains to FE mesh.</figcaption></figure>

Finite elements used for discretization in the RCAFE model are assigned to particular grains with slightly different material properties reflecting variations in the crystallographic orientation. The number of RCA cells in the cellular automata model directly corresponds to the increasing number of FE elements (from 25×25 to 300×300, Fig. 2) used during the discretization of the digital microstructure model.

Each model was subjected to numerical simulation of deformation under plane strain conditions at 1000°C and a strain rate of 1/s, to reflect the experimental case study.

The radius of the neighborhood in the RCA method was calculated so that, on average, eight cells fall into the neighborhood. With that, the investigation evaluates how mesh sensitivity affects the FE results and subsequent RCA results of discontinuous dynamic recrystallization evolution. The one-way data transfer from the FE to the RCA model is considered at this research stage.
<figure><img src="image_2145.png" /><figcaption>Figure 2 Initial FE meshes used during the numerical simulation of plane strain compression: a) 25×25, b) 50×50, c) 75×75, d) 100×100, e) 150×150, f) 200×200, g) 250×250, h) 300×300 elements.</figcaption></figure>

## Results

The qualitative and quantitative mesh sensitivity study from a set of FE simulations is presented in Fig. 3, while the corresponding RCA results, showing differences in recrystallization kinetics, are in Fig. 4.

<figure><img src="image_2146.png" /><figcaption>Figure 3 Homogenised stress-strain response and equivalent stress field of the deformed samples (upper left quarter of each sample) for increasing discretization level.</figcaption></figure>

The overall material behavior from the FE simulation is similar despite significant differences in the mesh densities. However, local differences in the equivalent stress fields are clearly visible. In this case, the minimal discretization level at the FE model should be approx. 75×75 elements to provide repetitive results. As the RCA model contains some stochastic components, each variant was then calculated three times to obtain a more comprehensive response. In this case, convergence of RCA model results is obtained for much larger meshes. As shown in Fig. 4, low mesh density simulation variants result in very different responses from the high mesh density models. However, it can be noticed that approximately from the model size of 200×200 averaged recrystallization kinetics converge (Fig. 5). At the same time, final microstructures predicted by smaller meshes have very irregular grain boundary shapes. It is attributed to the fact that new nuclei are formed in very close proximity to each other, and there is no physical space for growth (Fig. 5).
<figure><img src="image_2136.png" /><figcaption>Figure 2 Initial FE meshes used during the numerical simulation of plane strain compression: a) 25×25, b) 50×50, c) 75×75, d) 100×100, e) 150×150, f) 200×200, g) 250×250, h) 300×300 elements.</figcaption></figure>

## Results

The qualitative and quantitative mesh sensitivity study from a set of FE simulations is presented in Fig. 3, while the corresponding RCA results, showing differences in recrystallization kinetics, are in Fig. 4.

<figure><img src="image_2137.png" /><figcaption>Figure 3 Homogenised stress-strain response and equivalent stress field of the deformed samples (upper left quarter of each sample) for increasing discretization level.</figcaption></figure>

The overall material behavior from the FE simulation is similar despite significant differences in the mesh densities. However, local differences in the equivalent stress fields are clearly visible. In this case, the minimal discretization level at the FE model should be approx. 75×75 elements to provide repetitive results. As the RCA model contains some stochastic components, each variant was then calculated three times to obtain a more comprehensive response. In this case, convergence of RCA model results is obtained for much larger meshes. As shown in Fig. 4, low mesh density simulation variants result in very different responses from the high mesh density models. However, it can be noticed that approximately from the model size of 200×200 averaged recrystallization kinetics converge (Fig. 5). At the same time, final microstructures predicted by smaller meshes have very irregular grain boundary shapes. It is attributed to the fact that new nuclei are formed in very close proximity to each other, and there is no physical space for growth (Fig. 5).
<figure><img src="image_2147.png" /><figcaption>Figure 4 Comparison of recrystallization volume fraction for increasing mesh densities a) 25×25, b) 50×50, c) 75×75, d) 100×100, e) 150×150, f) 200×200, g) 250×250, h) 300×300 elements.</figcaption></figure>

<figure><img src="image_2148.png" /><figcaption>Figure 5 Evolution of recrystallization kinetics with corresponding final microstructures morphologies (green/blue initial grains, red/purple recrystallized – visualization with Ovito [16]).</figcaption></figure>

As presented, both the recrystallization kinetics and final microstructure morphologies are comparable for sufficiently high discretization levels. Therefore, in the RCAFE simulations, the mesh quality is determined by the convergence of the RCA part of the model.

## Summary

The work has shown the effects related to mesh quality on RCAFE model predictions. Based on the presented results the following conclusions can be drawn:

- stochastic elements in the RCA model do not introduce significant variations in the simulation results,

- the averaged recrystallization kinetics and the final grain sizes clearly indicate that there is a threshold that provides repetitive results,

- for coarser meshes, grain boundaries of recrystallized grains cannot be adequately reproduced during the simulation,
<figure><img src="image_2138.png" /><figcaption>Figure 4 Comparison of recrystallization volume fraction for increasing mesh densities a) 25×25, b) 50×50, c) 75×75, d) 100×100, e) 150×150, f) 200×200, g) 250×250, h) 300×300 elements.</figcaption></figure>

<figure><img src="image_2139.png" /><figcaption>Figure 5 Evolution of recrystallization kinetics with corresponding final microstructures morphologies (green/blue initial grains, red/purple recrystallized – visualization with Ovito [16]).</figcaption></figure>

As presented, both the recrystallization kinetics and final microstructure morphologies are comparable for sufficiently high discretization levels. Therefore, in the RCAFE simulations, the mesh quality is determined by the convergence of the RCA part of the model.

## Summary

The work has shown the effects related to mesh quality on RCAFE model predictions. Based on the presented results the following conclusions can be drawn:

- stochastic elements in the RCA model do not introduce significant variations in the simulation results,

- the averaged recrystallization kinetics and the final grain sizes clearly indicate that there is a threshold that provides repetitive results,

- for coarser meshes, grain boundaries of recrystallized grains cannot be adequately reproduced during the simulation,
- in the RCAFE simulations, the mesh quality is determined by the convergence of the RCA part of the model.

The next step in RCAFE model development will focus on mesh sensitivity study in a fully coupled model, where the RCA predictions affect the FE simulation in each time step.

## Acknowledgment

The financial assistance of the National Science Centre project No. 2019/35/B/ST8/00046 is acknowledged. We gratefully acknowledge Polish high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2023/016376.

## References

[1] K. Huang, R.E. Logé, A review of dynamic recrystallization phenomena in metallic materials, Mater Des 111 (2016) 548-574. https://doi.org/10.1016/j.matdes.2016.09.012.

[2] M. Bernacki, R.E. Logé, T. Coupez, Level set framework for the finite-element modelling of recrystallization and grain growth in polycrystalline materials, Scr Mater 64 (2011) 525-528. https://doi.org/10.1016/j.scriptamat.2010.11.032

[3] T. Takaki, A. Yamanaka, Y. Tomita, Phase-field modeling for dynamic recrystallization, Advanced Structured Materials 64 (2015) 441-459. https://doi.org/10.1007/978-3-319-19440-0_20

[4] M. Bernacki, Kinetic equations and level-set approach for simulating solid-state microstructure evolutions at the mesoscopic scale: State of the art, limitations, and prospects, Prog Mater Sci 142 (2024) 101224. https://doi.org/10.1016/j.pmatsci.2023.101224

[5] Z.H.U. Huajia, C. Fei, Z. Haiming, C.U.I. Zhenshan, Review on modeling and simulation of microstructure evolution during dynamic recrystallization using cellular automaton method, Sci China Technol Sci m (2019) 1-40.

[6] R.L. Goetz, V. Seetharaman, Modeling dynamic recrystallization using cellular automata, 1997.

[7] N. Xiao, C. Zheng, D. Li, Y. Li, A simulation of dynamic recrystallization by coupling a cellular automaton method with a topology deformation technique, Comput Mater Sci 41 (2008) 366-374. https://doi.org/10.1016/j.commatsci.2007.04.021

[8] J. Gawąd, M. Pietrzyk, Application of cafe multiscale model to description of microstructure development during dynamic recrystallization, 2007.

[9] Ł. Madej, M. Sitko, K. Radwanski, Kuziak, Roman, Validation and predictions of coupled finite element and cellular automata model: Influence of the degree of deformation on static recrystallization kinetics case study, Mater Chem Phys (2016). https://doi.org/10.1016/j.matchemphys.2016.05.040

[10] H. Hallberg, M. Wallin, M. Ristinmaa, Simulation of discontinuous dynamic recrystallization in pure Cu using a probabilistic cellular automaton, Comput Mater Sci 49 (2010) 25-34. https://doi.org/10.1016/j.commatsci.2010.04.012

[11] N. Yazdipour, C.H.J. Davies, P.D. Hodgson, Microstructural modeling of dynamic recrystallization using irregular cellular automata, Comput Mater Sci 44 (2008) 566-576. https://doi.org/10.1016/j.commatsci.2008.04.027

[12] L. Madej, M. Sitko, A. Legwand, K. Perzynski, K. Michalik, Development and evaluation of data transfer protocols in the fully coupled random cellular automata finite element model of dynamic recrystallization, J Comput Sci 26 (2018) 66-77. https://doi.org/10.1016/j.jocs.2018.03.007
- in the RCAFE simulations, the mesh quality is determined by the convergence of the RCA part of the model.

The next step in RCAFE model development will focus on mesh sensitivity study in a fully coupled model, where the RCA predictions affect the FE simulation in each time step.

## Acknowledgment

The financial assistance of the National Science Centre project No. 2019/35/B/ST8/00046 is acknowledged. We gratefully acknowledge Polish high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2023/016376.

## References

[1] K. Huang, R.E. Logé, A review of dynamic recrystallization phenomena in metallic materials, Mater Des 111 (2016) 548-574. https://doi.org/10.1016/j.matdes.2016.09.012.

[2] M. Bernacki, R.E. Logé, T. Coupez, Level set framework for the finite-element modelling of recrystallization and grain growth in polycrystalline materials, Scr Mater 64 (2011) 525-528. https://doi.org/10.1016/j.scriptamat.2010.11.032

[3] T. Takaki, A. Yamanaka, Y. Tomita, Phase-field modeling for dynamic recrystallization, Advanced Structured Materials 64 (2015) 441-459. https://doi.org/10.1007/978-3-319-19440-0_20

[4] M. Bernacki, Kinetic equations and level-set approach for simulating solid-state microstructure evolutions at the mesoscopic scale: State of the art, limitations, and prospects, Prog Mater Sci 142 (2024) 101224. https://doi.org/10.1016/j.pmatsci.2023.101224

[5] Z.H.U. Huajia, C. Fei, Z. Haiming, C.U.I. Zhenshan, Review on modeling and simulation of microstructure evolution during dynamic recrystallization using cellular automaton method, Sci China Technol Sci m (2019) 1-40.

[6] R.L. Goetz, V. Seetharaman, Modeling dynamic recrystallization using cellular automata, 1997.

[7] N. Xiao, C. Zheng, D. Li, Y. Li, A simulation of dynamic recrystallization by coupling a cellular automaton method with a topology deformation technique, Comput Mater Sci 41 (2008) 366-374. https://doi.org/10.1016/j.commatsci.2007.04.021

[8] J. Gawąd, M. Pietrzyk, Application of cafe multiscale model to description of microstructure development during dynamic recrystallization, 2007.

[9] Ł. Madej, M. Sitko, K. Radwanski, Kuziak, Roman, Validation and predictions of coupled finite element and cellular automata model: Influence of the degree of deformation on static recrystallization kinetics case study, Mater Chem Phys (2016). https://doi.org/10.1016/j.matchemphys.2016.05.040

[10] H. Hallberg, M. Wallin, M. Ristinmaa, Simulation of discontinuous dynamic recrystallization in pure Cu using a probabilistic cellular automaton, Comput Mater Sci 49 (2010) 25-34. https://doi.org/10.1016/j.commatsci.2010.04.012

[11] N. Yazdipour, C.H.J. Davies, P.D. Hodgson, Microstructural modeling of dynamic recrystallization using irregular cellular automata, Comput Mater Sci 44 (2008) 566-576. https://doi.org/10.1016/j.commatsci.2008.04.027

[12] L. Madej, M. Sitko, A. Legwand, K. Perzynski, K. Michalik, Development and evaluation of data transfer protocols in the fully coupled random cellular automata finite element model of dynamic recrystallization, J Comput Sci 26 (2018) 66-77. https://doi.org/10.1016/j.jocs.2018.03.007
[13] M. Sitko, M. Czarnecki, K. Pawlikowski, Ł. Madej, Evaluation of the effectiveness of neighbors' selection algorithms in the random cellular automata model of unconstrained grain growth, Materials and Manufacturing Processes (2023) 1-11. https://doi.org/10.1080/10426914.2023.2196753

[14] K. Pawlikowski, M. Sitko, M. Czarnecki, Evaluation of data transfer methods efficiency in the random cellular automata model of dynamic recrystallisation, Materials Research Proceedings 28 (2023) 1559-1564. https://doi.org/10.21741/9781644902479-168

[15] M. Czarnecki, M. Sitko, Ł. Madej, The role of neighborhood density in the random cellular automata model of grain growth, Computer Methods in Material Science 21 (2021). https://doi.org/10.7494/cmms.2021.3.0760

[16] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool, Model Simul Mat Sci Eng 18 (2010) 015012. https://doi.org/10.1088/0965-0393/18/1/015012
[13] M. Sitko, M. Czarnecki, K. Pawlikowski, Ł. Madej, Evaluation of the effectiveness of neighbors' selection algorithms in the random cellular automata model of unconstrained grain growth, Materials and Manufacturing Processes (2023) 1-11. https://doi.org/10.1080/10426914.2023.2196753

[14] K. Pawlikowski, M. Sitko, M. Czarnecki, Evaluation of data transfer methods efficiency in the random cellular automata model of dynamic recrystallisation, Materials Research Proceedings 28 (2023) 1559-1564. https://doi.org/10.21741/9781644902479-168

[15] M. Czarnecki, M. Sitko, Ł. Madej, The role of neighborhood density in the random cellular automata model of grain growth, Computer Methods in Material Science 21 (2021). https://doi.org/10.7494/cmms.2021.3.0760

[16] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool, Model Simul Mat Sci Eng 18 (2010) 015012. https://doi.org/10.1088/0965-0393/18/1/015012
# Comparative study of artificial neural network and physics-informed neural network application in sheet metal forming

MUNZONE Francesco¹,ᵃ*, HAZRATI Javad¹,ᵇ, HAKVOORT Wouter²,ᶜ and
VAN DEN BOOGAARD Ton¹,ᵈ

¹Nonlinear solid mechanics, Faculty of Engineering Technology, University of Twente,
Enschede, Netherlands

²Precision Engineering, Faculty of Engineering Technology, University of Twente, Enschede,
Netherlands

ᵃf.munzone@utwente.nl, ᵇj.hazratimarangalou@utwente.nl, ᶜw.b.j.hakoorkt@utwente.nl,

ᵈa.h.vandenboogaard@utwente.nl

**Keywords:** Surrogate Modeling, Physics-Informed Neural Networks, Metal-Forming

**Abstract.** Accurate prediction of the resultant geometry in sheet metal forming simulation is necessary to achieve zero-defect production. To quantify the effect of process parameters on the final geometry, numerical methods are used to simulate the process outputs for a given set of process variables. Finite element methods are employed in process optimization and design exploration. However, these computationally expensive models are unhelpful for process control applications. Surrogate models allowing fast prediction of resultant geometry or stress distribution can be plausible solutions. In the current study, we propose a sequential surrogate model to fit the stress field as a function of the process variable and the initial spatial coordinates. The framework is composed of two surrogate models. First, an artificial neural network (ANN) evaluates the displacement and the strain. Then, a second surrogate is employed to fit the stress using input strain and displacement. Here, ANN and physics-informed neural networks (PINN) are compared concerning prediction accuracy for the second surrogate model. The PINN is enhanced with the equilibrium equations. The developed method is demonstrated using a v-bending process. The results show that both surrogate models return good approximations, with ANN showing slightly better results.

**Introduction**

Computer-based simulations are widely employed in many engineering fields to characterize and design a process. Numerical solvers such as finite element (FE) simulations, computational fluid dynamics (CFD), and Monte-Carlo simulations are utilized to perform preliminary design exploration and subsequent reliability and maintenance operations during the process life. However, these tasks are computationally expensive and sometimes prohibitive due to the complexity of the problems. An alternative solution involves employing surrogate models to replace these methods. Surrogate models mimic the relation between inputs and outputs data collected from computer-based simulations. The built relation is then used to obtain fast and cheap predictions of unseen data.

Surrogate models are also widely employed in manufacturing for process optimization. In process optimization, surrogate models deliver fast predictions to easily explore the design space and search for optimal process conditions. Similarly, in robust optimization, the features of the final product have to be guaranteed under uncertainties [8]. In global metamodeling, the aim is to build a globally accurate surrogate model within a reasonable computational time [9]. Global metamodels are challenging to achieve due to the curse of dimensionality, i.e., the number of sample points in the design space grows exponentially with the number of parameters. Therefore,

<figure><img src="image_2149.png" /></figure>
# Comparative study of artificial neural network and physics-informed neural network application in sheet metal forming

MUNZONE Francesco¹,ª*, HAZRATI Javad¹,ᵇ, HAKVOORT Wouter²,ᶜ and
VAN DEN BOOGAARD Ton¹,ᵈ

¹Nonlinear solid mechanics, Faculty of Engineering Technology, University of Twente,
Enschede, Netherlands

²Precision Engineering, Faculty of Engineering Technology, University of Twente, Enschede,
Netherlands

ªf.munzone@utwente.nl, ᵇj.hazratimarangalou@utwente.nl, ᶜw.b.j.hakoorkt@utwente.nl,

ᵈa.h.vandenboogaard@utwente.nl

**Keywords:** Surrogate Modeling, Physics-Informed Neural Networks, Metal-Forming

**Abstract.** Accurate prediction of the resultant geometry in sheet metal forming simulation is necessary to achieve zero-defect production. To quantify the effect of process parameters on the final geometry, numerical methods are used to simulate the process outputs for a given set of process variables. Finite element methods are employed in process optimization and design exploration. However, these computationally expensive models are unhelpful for process control applications. Surrogate models allowing fast prediction of resultant geometry or stress distribution can be plausible solutions. In the current study, we propose a sequential surrogate model to fit the stress field as a function of the process variable and the initial spatial coordinates. The framework is composed of two surrogate models. First, an artificial neural network (ANN) evaluates the displacement and the strain. Then, a second surrogate is employed to fit the stress using input strain and displacement. Here, ANN and physics-informed neural networks (PINN) are compared concerning prediction accuracy for the second surrogate model. The PINN is enhanced with the equilibrium equations. The developed method is demonstrated using a v-bending process. The results show that both surrogate models return good approximations, with ANN showing slightly better results.

**Introduction**

Computer-based simulations are widely employed in many engineering fields to characterize and design a process. Numerical solvers such as finite element (FE) simulations, computational fluid dynamics (CFD), and Monte-Carlo simulations are utilized to perform preliminary design exploration and subsequent reliability and maintenance operations during the process life. However, these tasks are computationally expensive and sometimes prohibitive due to the complexity of the problems. An alternative solution involves employing surrogate models to replace these methods. Surrogate models mimic the relation between inputs and outputs data collected from computer-based simulations. The built relation is then used to obtain fast and cheap predictions of unseen data.

Surrogate models are also widely employed in manufacturing for process optimization. In process optimization, surrogate models deliver fast predictions to easily explore the design space and search for optimal process conditions. Similarly, in robust optimization, the features of the final product have to be guaranteed under uncertainties [8]. In global metamodeling, the aim is to build a globally accurate surrogate model within a reasonable computational time [9]. Global metamodels are challenging to achieve due to the curse of dimensionality, i.e., the number of sample points in the design space grows exponentially with the number of parameters. Therefore,

<figure><img src="image_2140.png" /></figure>
selecting the most suited surrogate models, together with a proper sampling plan, is essential to reach appropriate levels of accuracy.

Artificial neural networks (ANN) are surrogate models categorized as regression models. In recent years, scientific interest in ANN has increased due to the rise of machine learning in many engineering fields. Several studies have utilized ANN in elastoplastic deformation problems such as sheet metal forming to obtain accurate predictions. In the work of El Mrabti et al. [1], the authors compare radial basis function (RBF), response surface methodology (RSM), Kriging, and ANN for failure predictions in a deep drawing process. Their results show that ANN performs better than other models in predicting thinning and rupture. Abbassi et al. [2] employed an ANN to predict the material parameters of the Gurson-Tvergaard-Needleman damage model for sheet metal forming. Ghaisari et al. [3] predicted the mechanical properties of ST14 steel using an ANN with nineteen inputs. The model evaluates acceptable mechanical properties on a test dataset. More generally, Gorji et al. [4] used a recurrent neural network (RNN), a class of neural networks that works with time series, to predict anisotropic plane stress plasticity for arbitrary loading paths. The authors proved that RNNs are well-suited for modeling plastic response through supervised learning.

Physics-informed neural network (PINN) is a recently developed surrogate model class. PINNs are neural networks in which the loss function is augmented with governing equations of the system. In the article by Hoffer et al. [5], the authors compare several surrogate models (ANN, PINN, simple Kriging, support vector regression (SVR), K-nearest neighbor regressor (KNNR), and gradient boosting decision tree regressor (GBDTR)) for predicting the outputs of three different use cases using a perfect plasticity model. Haghighat et al. [6] proposed a PINN architecture applied to a linear elastic problem and a nonlinear perfectly plastic problem under the assumption of small deformations. Niu et al. [7] present a novel PINN framework for finite deformation elastoplasticity. A loading, unloading, and reloading cycle is applied to a plate with a hole in the center. The PINN accurately predicts displacement, plastic deformation, and stress state.

In this article, we propose a surrogate model framework to predict the final (deformed) configuration and stress field under plastic deformation from the initial (undeformed) configuration. A comparison between ANN and PINN accuracy is carried out to describe the relationship between total strain and the stress field while satisfying the equilibrium equations in PINN. The data are obtained from the FE simulations of a V-bending process. The sample points are selected using an optimized Latin hypercube (OLH) to achieve a space-filling design of experiments (DOE). The hyperparameters of the surrogate models within the framework are tuned through an automatic tuner. Finally, we tested the proposed framework considering a third of the loading step.

## Artificial and Physics-Informed Neural Networks

Artificial neural networks (ANN) are surrogate models used for classification and regression. Their core structure consists of one or more hidden layers and neurons. The neurons, i.e., the basic units of the layer, receive the outputs of the previous layer as inputs [10]. The activation function $\phi$ applied to the neurons transforms the linear relation between layers into a nonlinear relation. Therefore, activation functions are essential to enforce the nonlinear relationship between inputs and outputs. The node responses $z_l$ for the layer $l$ with $l = \{1, 2, ..., L\}$ can be written as:

$$z_l = \phi_l(b_l + W_l z_{l-1}) \qquad (1)$$

with $W_l$ and $b_l$ the weights matrix and bias vector of the $l$th layer, respectively, which are usually called trainable parameters $\theta$ (Eq. 3). Assuming that for $l = 0$ the layer $z_0$ is the input vector and
selecting the most suited surrogate models, together with a proper sampling plan, is essential to reach appropriate levels of accuracy.

Artificial neural networks (ANN) are surrogate models categorized as regression models. In recent years, scientific interest in ANN has increased due to the rise of machine learning in many engineering fields. Several studies have utilized ANN in elastoplastic deformation problems such as sheet metal forming to obtain accurate predictions. In the work of El Mrabti et al. [1], the authors compare radial basis function (RBF), response surface methodology (RSM), Kriging, and ANN for failure predictions in a deep drawing process. Their results show that ANN performs better than other models in predicting thinning and rupture. Abbassi et al. [2] employed an ANN to predict the material parameters of the Gurson-Tvergaard-Needleman damage model for sheet metal forming. Ghaisari et al. [3] predicted the mechanical properties of ST14 steel using an ANN with nineteen inputs. The model evaluates acceptable mechanical properties on a test dataset. More generally, Gorji et al. [4] used a recurrent neural network (RNN), a class of neural networks that works with time series, to predict anisotropic plane stress plasticity for arbitrary loading paths. The authors proved that RNNs are well-suited for modeling plastic response through supervised learning.

Physics-informed neural network (PINN) is a recently developed surrogate model class. PINNs are neural networks in which the loss function is augmented with governing equations of the system. In the article by Hoffer et al. [5], the authors compare several surrogate models (ANN, PINN, simple Kriging, support vector regression (SVR), K-nearest neighbor regressor (KNNR), and gradient boosting decision tree regressor (GBDTR)) for predicting the outputs of three different use cases using a perfect plasticity model. Haghighat et al. [6] proposed a PINN architecture applied to a linear elastic problem and a nonlinear perfectly plastic problem under the assumption of small deformations. Niu et al. [7] present a novel PINN framework for finite deformation elastoplasticity. A loading, unloading, and reloading cycle is applied to a plate with a hole in the center. The PINN accurately predicts displacement, plastic deformation, and stress state.

In this article, we propose a surrogate model framework to predict the final (deformed) configuration and stress field under plastic deformation from the initial (undeformed) configuration. A comparison between ANN and PINN accuracy is carried out to describe the relationship between total strain and the stress field while satisfying the equilibrium equations in PINN. The data are obtained from the FE simulations of a V-bending process. The sample points are selected using an optimized Latin hypercube (OLH) to achieve a space-filling design of experiments (DOE). The hyperparameters of the surrogate models within the framework are tuned through an automatic tuner. Finally, we tested the proposed framework considering a third of the loading step.

## Artificial and Physics-Informed Neural Networks

Artificial neural networks (ANN) are surrogate models used for classification and regression. Their core structure consists of one or more hidden layers and neurons. The neurons, i.e., the basic units of the layer, receive the outputs of the previous layer as inputs [10]. The activation function $\phi$ applied to the neurons transforms the linear relation between layers into a nonlinear relation. Therefore, activation functions are essential to enforce the nonlinear relationship between inputs and outputs. The node responses $z_l$ for the layer $l$ with $l = \{1, 2, ..., L\}$ can be written as:

$$z_l = \phi_l(b_l + W_l z_{l-1}) \qquad (1)$$

with $W_l$ and $b_l$ the weights matrix and bias vector of the $l$th layer, respectively, which are usually called trainable parameters $\theta$ (Eq. 3). Assuming that for $l = 0$ the layer $z_0$ is the input vector and
for $l = L$ the layer $z_L$ is the output vector, the output layer can be written as the following composition of function:

$$z_L = \hat{y} = z_{L-1}(z_{L-2}(\dots(z_1(z_0)\dots))) \qquad (2)$$

A cost function $\mathcal{L}$, as the average value of a loss $\ell$, defines the performance of a surrogate model. Generally, regression models aim to minimize the cost function via iterative improvement of the trainable parameters (Eq. 3). In ANN, the derivative of the outputs with respect to the inputs is evaluated through the chain rule through the so-called backpropagation algorithm. The ANN exploits the backpropagation algorithm to search iteratively for the trainable parameter that minimizes the cost function.

$$\min_{\theta} \mathcal{L}(\theta) = \frac{1}{N} \sum_{j=1}^{N} \ell(y, \hat{y}) \qquad (3)$$

A common loss function is the quadratic loss $\ell_2$ which, substituted in Eq. 3, returns the mean squared error.

<figure><img src="image_2150.png" /><figcaption>Fig. 1. Schematic representation of the sequential surrogate model; the p vector contains the process variables. In the grey boxes are the inputs and outputs of the first surrogate model, while in the white boxes are the inputs and outputs of the second surrogate model.</figcaption></figure>

Physics-informed neural networks (PINN) are a class of neural networks recently developed by Raissi et al. [11]. PINNs are addressed as surrogate models capable of solving partial differential equations, embedded in the loss function terms. These partial differential equations may involve initial conditions, boundary conditions, and governing equations. Thus, ANN and PINN share almost the same architecture and objective, i.e., minimizing the cost function.

$$\theta = \underset{\theta}{\arg\min} \left[ \omega_F \mathcal{L}_F(\theta) + \omega_B \mathcal{L}_B(\theta) + \omega_d \mathcal{L}_d(\theta) \right] \qquad (4)$$

where $\mathcal{L}_F$, $\mathcal{L}_B$, and $\mathcal{L}_d$ are the cost function concerning the governing equations, the boundary and/or initial conditions, and the target data respectively. The coefficients $\omega_F$, $\omega_B$, $\omega_d$ are corresponding multiplied weights.
for $l = L$ the layer $z_L$ is the output vector, the output layer can be written as the following composition of function:

$$z_L = \hat{y} = z_{L-1}(z_{L-2}(\dots(z_1(z_0)\dots))) \qquad (2)$$

A cost function $\mathcal{L}$, as the average value of a loss $\ell$, defines the performance of a surrogate model. Generally, regression models aim to minimize the cost function via iterative improvement of the trainable parameters (Eq. 3). In ANN, the derivative of the outputs with respect to the inputs is evaluated through the chain rule through the so-called backpropagation algorithm. The ANN exploits the backpropagation algorithm to search iteratively for the trainable parameter that minimizes the cost function.

$$\min_{\theta} \mathcal{L}(\theta) = \frac{1}{N} \sum_{j=1}^{N} \ell(y, \hat{y}) \qquad (3)$$

A common loss function is the quadratic loss $\ell_2$ which, substituted in Eq. 3, returns the mean squared error.

<figure><img src="image_2141.png" /><figcaption>Fig. 1. Schematic representation of the sequential surrogate model; the p vector contains the process variables. In the grey boxes are the inputs and outputs of the first surrogate model, while in the white boxes are the inputs and outputs of the second surrogate model.</figcaption></figure>

Physics-informed neural networks (PINN) are a class of neural networks recently developed by Raissi et al. [11]. PINNs are addressed as surrogate models capable of solving partial differential equations, embedded in the loss function terms. These partial differential equations may involve initial conditions, boundary conditions, and governing equations. Thus, ANN and PINN share almost the same architecture and objective, i.e., minimizing the cost function.

$$\theta = \underset{\theta}{\arg\min} \left[ \omega_F \mathcal{L}_F(\theta) + \omega_B \mathcal{L}_B(\theta) + \omega_d \mathcal{L}_d(\theta) \right] \qquad (4)$$

where $\mathcal{L}_F$, $\mathcal{L}_B$, and $\mathcal{L}_d$ are the cost function concerning the governing equations, the boundary and/or initial conditions, and the target data respectively. The coefficients $\omega_F$, $\omega_B$, $\omega_d$ are corresponding multiplied weights.
In general, neural networks have a flexible architecture. Hyperparameters, such as the number of hidden layers, number of neurons, activation functions, optimizers, and so on, can be chosen arbitrarily. Nevertheless, the possible combinations of hyperparameters entail many appropriate architectures for one problem. Indeed, the hyperparameter tuning procedure involves time-consuming trial and error methods and the operator experience [12]. Despite this, neural networks are widely studied in literature due to their ability to perform implicit sensitivity analysis, dimensionality reduction, and their universal approximator's property [13, 14].

## Sequential surrogate models

The proposed sequential surrogate (SS) model consists of two sequentially connected surrogate models for approximating the response of a finite-deformation elastoplastic problem (Fig. 1). The SS model takes the process variables and the reference configuration coordinates as inputs to generate a stress distribution as a response. Furthermore, the goal is to construct a global metamodel capable of delivering good approximation for unknown samples in the design space. To this end, picking an appropriate design of experiments (DOE) is crucial. The desired DOE must be spread uniformly over the entire domain utilizing the least amount of samples. Indeed, the last condition is necessary when the problem has many input variables to avoid the curse of dimensionality. Optimal Latin hypercube merges a Latin hypercube sampling with the maximin Morris-Mitchell criteria to achieve a space-filling DOE that involves a relatively small number of sample points [15]. Following the guidelines of Jin et al. [16], given six variables, the number of samples for a LH necessary to represent the design space is 112, i.e., 112 FE simulations.

The first surrogate model is an ANN, entitled to evaluate the total strain and displacement. From now on, this surrogate will be referred to as ANNε. The ANNε takes as inputs a set of process variables p and the finite element integration point coordinates of the reference configuration. The latter are used to evaluate the total strain tensor and the displacement at the integration points at the end of the process. In the second segment of the SS model, a comparative study is conducted between ANN and PINN. The goal is to state which surrogate best imitates the relationship relating total strain and the stress tensor while respecting the strong form of the equilibrium equations. We will refer to the models with ANNσ and PINNσ. The second surrogate model uses as inputs the same set of process variables used for the ANNε, extended with the total strain components and the displacement at the integration points at the last increment. Furthermore, from our findings and as demonstrated by Haghighat et al. [6], better fitting of the stress components is achieved if the outputs have dedicated neural networks and, therefore, dedicated trainable parameters. From now on, we will refer to it as a multiple-network structure.

As mentioned, the second surrogate evaluates the stress components through a relation that includes total strain and displacement. In PINNσ the objective is to minimize the cost function as the sum of $L_F$ and $L_d$. The former takes into account the equilibrium equations, which, for a 2D case and under the assumption of plane strain, can be written as follows.

$$ \mathcal{L} = \omega_d |\mathbf{y} - \hat{\mathbf{y}}|^2 + \omega_F \left( \left| \frac{\partial \sigma_{11}}{\partial x_1} + \frac{\partial \sigma_{12}}{\partial x_2} \right|^2 + \left| \frac{\partial \sigma_{12}}{\partial x_1} + \frac{\partial \sigma_{22}}{\partial x_2} \right|^2 \right) \quad (5) $$

For this study, we use a k-fold cross-validation resampling technique to judge the quality of the surrogate models dictated by their accuracy. The dataset used to fit the surrogate is initially shuffled and then divided into k subsets of equal dimensions, in which k − 1 are used for training, and the remaining is used for validation. This procedure is performed for each possible and unique combination of the k groups. Common values used for k are 5, 10, and n, where the latter case represents the leave-one-out cross-validation strategy. A frequent mistake in ANNs is to assess the model generalization ability by monitoring the validation loss. Feature selection should not be performed on data outside the training set for model training, otherwise, it causes a leakage of
In general, neural networks have a flexible architecture. Hyperparameters, such as the number of hidden layers, number of neurons, activation functions, optimizers, and so on, can be chosen arbitrarily. Nevertheless, the possible combinations of hyperparameters entail many appropriate architectures for one problem. Indeed, the hyperparameter tuning procedure involves time-consuming trial and error methods and the operator experience [12]. Despite this, neural networks are widely studied in literature due to their ability to perform implicit sensitivity analysis, dimensionality reduction, and their universal approximator's property [13, 14].

## Sequential surrogate models

The proposed sequential surrogate (SS) model consists of two sequentially connected surrogate models for approximating the response of a finite-deformation elastoplastic problem (Fig. 1). The SS model takes the process variables and the reference configuration coordinates as inputs to generate a stress distribution as a response. Furthermore, the goal is to construct a global metamodel capable of delivering good approximation for unknown samples in the design space. To this end, picking an appropriate design of experiments (DOE) is crucial. The desired DOE must be spread uniformly over the entire domain utilizing the least amount of samples. Indeed, the last condition is necessary when the problem has many input variables to avoid the curse of dimensionality. Optimal Latin hypercube merges a Latin hypercube sampling with the maximin Morris-Mitchell criteria to achieve a space-filling DOE that involves a relatively small number of sample points [15]. Following the guidelines of Jin et al. [16], given six variables, the number of samples for a LH necessary to represent the design space is 112, i.e., 112 FE simulations.

The first surrogate model is an ANN, entitled to evaluate the total strain and displacement. From now on, this surrogate will be referred to as ANNε. The ANNε takes as inputs a set of process variables p and the finite element integration point coordinates of the reference configuration. The latter are used to evaluate the total strain tensor and the displacement at the integration points at the end of the process. In the second segment of the SS model, a comparative study is conducted between ANN and PINN. The goal is to state which surrogate best imitates the relationship relating total strain and the stress tensor while respecting the strong form of the equilibrium equations. We will refer to the models with ANNσ and PINNσ. The second surrogate model uses as inputs the same set of process variables used for the ANNε, extended with the total strain components and the displacement at the integration points at the last increment. Furthermore, from our findings and as demonstrated by Haghighat et al. [6], better fitting of the stress components is achieved if the outputs have dedicated neural networks and, therefore, dedicated trainable parameters. From now on, we will refer to it as a multiple-network structure.

As mentioned, the second surrogate evaluates the stress components through a relation that includes total strain and displacement. In PINNσ the objective is to minimize the cost function as the sum of L<sub>F</sub> and L<sub>d</sub>. The former takes into account the equilibrium equations, which, for a 2D case and under the assumption of plane strain, can be written as follows.

$$ \mathcal{L} = \omega_d |\mathbf{y} - \hat{\mathbf{y}}|^2 + \omega_F \left( \left| \frac{\partial \sigma_{11}}{\partial x_1} + \frac{\partial \sigma_{12}}{\partial x_2} \right|^2 + \left| \frac{\partial \sigma_{12}}{\partial x_1} + \frac{\partial \sigma_{22}}{\partial x_2} \right|^2 \right) \quad (5) $$

For this study, we use a k-fold cross-validation resampling technique to judge the quality of the surrogate models dictated by their accuracy. The dataset used to fit the surrogate is initially shuffled and then divided into k subsets of equal dimensions, in which k − 1 are used for training, and the remaining is used for validation. This procedure is performed for each possible and unique combination of the k groups. Common values used for k are 5, 10, and n, where the latter case represents the leave-one-out cross-validation strategy. A frequent mistake in ANNs is to assess the model generalization ability by monitoring the validation loss. Feature selection should not be performed on data outside the training set for model training, otherwise, it causes a leakage of
information and leads to excessively optimistic predictions. Therefore, training loss and validation loss are utilized to observe if the model is learning in training and if it is flexible enough to predict unseen data. In this regard, a test dataset, i.e., data never seen by the model during training, is built to evaluate the model's accuracy and generalization ability. It should be noted that the overall accuracy of the SS model is estimated through the accuracy in predicting the stress state given the prediction of the ANNε as input. Thus, in testing, the SS model predicts the strain and the stress components given the process parameters and the undeformed configuration.

## V-Bending process

V-bending of metal sheets is characterized by large strain and plastic deformation. The proposed SS model is applied to a V-bending process inspired by the work of Wiebenga et al. [17]. In the article, the authors perform a sensitivity analysis to reduce the number of process variables and select those with the highest influence on the primary angle (α). Consequentially, we choose thickness (t), punch radius ($R_p$), angle of the die (α), and depth (D) as input variables (see Fig. 2). In addition, we include the yield stress ($σ_y$) and the Young modulus (E) of the sheet material as inputs due to their possible effect on the stress state. In Table 1, the process variables and the respective ranges are listed. It is worth mentioning that the angle α is the same for both the punch and the die, and variation in yield stress will correspond to a shift of the stress-strain curve.

<table><caption>Table 1. Process variables used as inputs for the SS model</caption><thead><tr><td>Process variables</td><td>Range</td></tr></thead><tbody><tr><td>Thickness [mm]</td><td>0.49 ≤ t ≤ 0.51</td></tr><tr><td>Radius die [mm]</td><td>0.8 ≤ R<sub>d</sub> ≤ 1.2</td></tr><tr><td>Angle [deg]</td><td>42.5 ≤ α ≤ 47.25</td></tr><tr><td>Depth [mm]</td><td>0.4 ≤ D ≤ 0.6</td></tr><tr><td>Young modulus [GPa]</td><td>190 ≤ E ≤ 210</td></tr><tr><td>Yield stress [MPa]</td><td>280 ≤ σ<sub>y</sub> ≤ 300</td></tr></tbody></table>

To run the FE simulations, the V-bending process is built in Marc Mentat 2022.1, assuming plane strain condition, isotropic hardening, and half of the geometry due to symmetry. The mesh counts 1800 quadrilateral linear elements, split 10 on the thickness and 180 along the length of the sheet.

## Hyperparameters tuning

Tuning the hyperparameters of neural networks is an expensive task typically performed using trial and error methods, relying on the operator's expertise. Another approach involves automatic tuners to optimize an objective function based on testing several sets of hyperparameters. Thus, the hyperparameter searching algorithm is responsible for hyperparameter space exploration and computational efficiency. For example, a random search algorithm will test different hyperparameter sets randomly taken from the hyperparameter space. This method is highly inefficient when many hyperparameters are considered in the optimization process (curse of dimensionality). Bayesian optimization algorithms instead use a Gaussian process to sample successive points in the hyperparameter space.
information and leads to excessively optimistic predictions. Therefore, training loss and validation loss are utilized to observe if the model is learning in training and if it is flexible enough to predict unseen data. In this regard, a test dataset, i.e., data never seen by the model during training, is built to evaluate the model's accuracy and generalization ability. It should be noted that the overall accuracy of the SS model is estimated through the accuracy in predicting the stress state given the prediction of the ANNε as input. Thus, in testing, the SS model predicts the strain and the stress components given the process parameters and the undeformed configuration.

## V-Bending process

V-bending of metal sheets is characterized by large strain and plastic deformation. The proposed SS model is applied to a V-bending process inspired by the work of Wiebenga et al. [17]. In the article, the authors perform a sensitivity analysis to reduce the number of process variables and select those with the highest influence on the primary angle (α). Consequentially, we choose thickness (t), punch radius ($R_p$), angle of the die (α), and depth (D) as input variables (see Fig. 2). In addition, we include the yield stress ($σ_y$) and the Young modulus (E) of the sheet material as inputs due to their possible effect on the stress state. In Table 1, the process variables and the respective ranges are listed. It is worth mentioning that the angle α is the same for both the punch and the die, and variation in yield stress will correspond to a shift of the stress-strain curve.

<table><caption>Table 1. Process variables used as inputs for the SS model</caption><thead><tr><td>Process variables</td><td>Range</td></tr></thead><tbody><tr><td>Thickness [mm]</td><td>0.49 ≤ t ≤ 0.51</td></tr><tr><td>Radius die [mm]</td><td>0.8 ≤ R<sub>d</sub> ≤ 1.2</td></tr><tr><td>Angle [deg]</td><td>42.5 ≤ α ≤ 47.25</td></tr><tr><td>Depth [mm]</td><td>0.4 ≤ D ≤ 0.6</td></tr><tr><td>Young modulus [GPa]</td><td>190 ≤ E ≤ 210</td></tr><tr><td>Yield stress [MPa]</td><td>280 ≤ σ<sub>y</sub> ≤ 300</td></tr></tbody></table>

To run the FE simulations, the V-bending process is built in Marc Mentat 2022.1, assuming plane strain condition, isotropic hardening, and half of the geometry due to symmetry. The mesh counts 1800 quadrilateral linear elements, split 10 on the thickness and 180 along the length of the sheet.

## Hyperparameters tuning

Tuning the hyperparameters of neural networks is an expensive task typically performed using trial and error methods, relying on the operator's expertise. Another approach involves automatic tuners to optimize an objective function based on testing several sets of hyperparameters. Thus, the hyperparameter searching algorithm is responsible for hyperparameter space exploration and computational efficiency. For example, a random search algorithm will test different hyperparameter sets randomly taken from the hyperparameter space. This method is highly inefficient when many hyperparameters are considered in the optimization process (curse of dimensionality). Bayesian optimization algorithms instead use a Gaussian process to sample successive points in the hyperparameter space.
<figure><img src="image_2151.png" /><figcaption>Fig. 2. 2D representation of the V-bending process implemented in the FE solver.</figcaption></figure>

The Bayesian optimization searching algorithm (built in the KerasTuner library and available for Keras and Tensorflow) uses the upper confidence bound as an acquisition function. The hyperparameters selected to tune are the number of hidden layers between 2 and 5, the number of nodes 10 to 50, and the activation functions between rectifier (relu), hyperbolic tangent (tanh), and softplus. The automatic tuner runs the training process for 300 epochs for each hyperparameter set. To prevent overfitting, the searching algorithm aims to minimize the validation loss.

In Fig. 3, five validation losses and the corresponding hyperparameter set, ranked according to the objective function, are depicted. It can be noticed that the three neural networks tend to prefer complex models (higher number of trainable parameters). Concerning the activation function, the ANNε shows better validation using the softplus, while both ANNσ and PINNσ prefer tanh. Based on Fig. 3a, the Bayesian optimization tuner ranks properly the hyperparameter sets for displacement and strain. None of the curves show that the model is overfitting and the lowest validation loss value is reached with hyperparameters set ranked as first. Conversely, in Fig. 3b, the automatic tuner suggests for ANNσ the hyperparameter sets in which the validation loss starts overfitting (around 200 epochs), providing a ranking based only on the minimum value. Hence, the best result is the third option (4 layers, 42 nodes, and tanh), in which the validation loss smoothly decreases over the epochs. Analogously to ANNσ, the first-ranked validation loss of the PINNσ is overfitting due to the model complexity (Fig. 3c). The second set in Fig. 3c, instead, is slowly decreasing and reaches a minimum comparable to the first. It is worth noticing that both chosen hyperparameter sets for the surrogates are similar. Therefore, for the comparative study, we set the number of layers to 4, the number of nodes to 40, and the activation function to tanh.

## Results and discussion

The surrogate models are trained using the hyperparameter found with the Bayesian optimization tuner for 3000 epochs per 5 times due to the 5-fold cross-validation. The OLH is used to generate 112 training simulations and 20 testing simulations, being careful not to have repeated sample points. To compare the results with the FE simulations, a mean field is evaluated.

$$ \bar{\mathbf{y}} = \frac{1}{N_{test}} \sum_{i=1}^{N_{test}} \mathbf{y}_i \qquad (6) $$
<figure><img src="image_2142.png" /><figcaption>Fig. 2. 2D representation of the V-bending process implemented in the FE solver.</figcaption></figure>

The Bayesian optimization searching algorithm (built in the KerasTuner library and available for Keras and Tensorflow) uses the upper confidence bound as an acquisition function. The hyperparameters selected to tune are the number of hidden layers between 2 and 5, the number of nodes 10 to 50, and the activation functions between rectifier (relu), hyperbolic tangent (tanh), and softplus. The automatic tuner runs the training process for 300 epochs for each hyperparameter set. To prevent overfitting, the searching algorithm aims to minimize the validation loss.

In Fig. 3, five validation losses and the corresponding hyperparameter set, ranked according to the objective function, are depicted. It can be noticed that the three neural networks tend to prefer complex models (higher number of trainable parameters). Concerning the activation function, the ANNε shows better validation using the softplus, while both ANNσ and PINNσ prefer tanh. Based on Fig. 3a, the Bayesian optimization tuner ranks properly the hyperparameter sets for displacement and strain. None of the curves show that the model is overfitting and the lowest validation loss value is reached with hyperparameters set ranked as first. Conversely, in Fig. 3b, the automatic tuner suggests for ANNσ the hyperparameter sets in which the validation loss starts overfitting (around 200 epochs), providing a ranking based only on the minimum value. Hence, the best result is the third option (4 layers, 42 nodes, and tanh), in which the validation loss smoothly decreases over the epochs. Analogously to ANNσ, the first-ranked validation loss of the PINNσ is overfitting due to the model complexity (Fig. 3c). The second set in Fig. 3c, instead, is slowly decreasing and reaches a minimum comparable to the first. It is worth noticing that both chosen hyperparameter sets for the surrogates are similar. Therefore, for the comparative study, we set the number of layers to 4, the number of nodes to 40, and the activation function to tanh.

## Results and discussion

The surrogate models are trained using the hyperparameter found with the Bayesian optimization tuner for 3000 epochs per 5 times due to the 5-fold cross-validation. The OLH is used to generate 112 training simulations and 20 testing simulations, being careful not to have repeated sample points. To compare the results with the FE simulations, a mean field is evaluated.

$$ \bar{\mathbf{y}} = \frac{1}{N_{test}} \sum_{i=1}^{N_{test}} \mathbf{y}_i \qquad (6) $$
The error measure chosen is the mean absolute error (MAE) to exhibit the error in terms of *MPa* for stress. As expected, the ANNε approximates both displacement and strain (Fig. 4) well enough. Regarding the strain components, the highest error is located in the bending region under compression, which corresponds to the region in contact with the die, and in proximity of *x* = 0 (symmetry line).

<figure><img src="image_2152.png" /><figcaption>Fig. 3. Validation losses of the best five hyperparameter sets ranked by the automatic tuner: a) ANNε with single network structure, b) ANNσ with multiple network structure, c) PINNσ with multiple network structure.</figcaption></figure>

Fig. 5b and 5c depict the MAE of the stress components for ANNσ and PINNσ concerning the complete loading-unloading cycle. In the bending region and on the tip, the ANNσ fits better the results than PINNσ. In particular close to the tip of the sheet is subject to forging, i.e., the tip is extruded onto the outside, due to extreme cases in which the distance between the die and punch walls is relatively small. Moreover, the predicted σ₁₂ using PINNσ shows an error zone in the middle of the sheet in correspondence with the high gradient region observed in Fig. 5a. It is highly likely that the error in these areas is more emphasized due to the computation of the stress divergence. For example, the stress component σ₁₂ is derived two times with respect to x₁ and x₂ for both equilibrium equations. Thus, the trainable parameters are forced to satisfy both relations. It is worth noting that σ₃₃ error is almost identical for both models as a consequence of the assumption that σ₁₃ = σ₂₃ = 0. In conclusion, ANNε, ANNσ, and PINNσ reproduce a distribution of the quantities similar to FEM, but the last two models are not capable of returning a good fitting of stress components, with MAE in the order of 30 – 50 MPa.
The error measure chosen is the mean absolute error (MAE) to exhibit the error in terms of *MPa* for stress. As expected, the ANNε approximates both displacement and strain (Fig. 4) well enough. Regarding the strain components, the highest error is located in the bending region under compression, which corresponds to the region in contact with the die, and in proximity of *x* = 0 (symmetry line).

<figure><img src="image_2143.png" /><figcaption>Fig. 3. Validation losses of the best five hyperparameter sets ranked by the automatic tuner: a) ANNε with single network structure, b) ANNσ with multiple network structure, c) PINNσ with multiple network structure.</figcaption></figure>

Fig. 5b and 5c depict the MAE of the stress components for ANNσ and PINNσ concerning the complete loading-unloading cycle. In the bending region and on the tip, the ANNσ fits better the results than PINNσ. In particular close to the tip of the sheet is subject to forging, i.e., the tip is extruded onto the outside, due to extreme cases in which the distance between the die and punch walls is relatively small. Moreover, the predicted σ₁₂ using PINNσ shows an error zone in the middle of the sheet in correspondence with the high gradient region observed in Fig. 5a. It is highly likely that the error in these areas is more emphasized due to the computation of the stress divergence. For example, the stress component σ₁₂ is derived two times with respect to x₁ and x₂ for both equilibrium equations. Thus, the trainable parameters are forced to satisfy both relations. It is worth noting that σ₃₃ error is almost identical for both models as a consequence of the assumption that σ₁₃ = σ₂₃ = 0. In conclusion, ANNε, ANNσ, and PINNσ reproduce a distribution of the quantities similar to FEM, but the last two models are not capable of returning a good fitting of stress components, with MAE in the order of 30 – 50 MPa.
<figure><img src="image_2153.png" /><figcaption>Fig. 4. a) Mean field of the strain and b) mean absolute error of the strain for the complete loading-unloading cycle.</figcaption></figure>

<figure><img src="image_2154.png" /><figcaption>Fig. 5. a) Mean field of the stress, b) mean absolute error of ANNσ, and c) mean absolute error of PINNσ for the loading-unloading cycle.</figcaption></figure>
<figure><img src="image_2144.png" /><figcaption>Fig. 4. a) Mean field of the strain and b) mean absolute error of the strain for the complete loading-unloading cycle.</figcaption></figure>

<figure><img src="image_2145.png" /><figcaption>Fig. 5. a) Mean field of the stress, b) mean absolute error of ANNσ, and c) mean absolute error of PINNσ for the loading-unloading cycle.</figcaption></figure>
Processes subject to large and plastic deformation are known to be path-dependent. Predicting the stress field of a complete loading-unloading cycle means that information related to the history of the process is lost. Hence, it can be supposed that the surrogate models should lead to better fitting for small-time steps. Employing the same hyperparameters found with automatic tuners, the surrogates are trained using the information collected at a third of the loading step. Once more, a mean stress field is calculated to compare the predicted errors (Fig. 6a). In this case the predictions are plotted with respect to the maximum errors of ANNσ due to a similar error range. This helps emphasize the areas prone to return worse predictions in PINNσ than ANNσ. The evaluated MAE is almost ten to twenty times lower than the one calculated for the previous case. Both models show good generalization ability on the testing dataset, probably due to the low standard deviation among the test simulations. As expected, both models predict σ₃₃ similarly since it does not contribute to the equilibrium equation. Regarding the other components, the models exhibit an error concentration in correspondence of the contact point of the sheet with the punch and the die. The inferior performance of PINNσ can be ascribed to the combination of high nonlinearity regions and the constraint on the model parameters enforced by the equilibrium equation. Nevertheless, the model returns a satisfactory fit if compared to ANNσ.

Concerning the computational efficiency, the time needed to train the two surrogate models for the same number of epochs is almost the same. In PINNσ the evaluation of the derivatives implies additional computations, which have a negligible impact on the overall efficiency. If time increments are taken into account, it is expected that the computational efficiency of the PINNσ will drop significantly.

<figure><img src="image_2155.png" /><figcaption>Fig. 6. a) Mean field of the stress, b) mean absolute error of ANNσ, and c) PINNσ for a third of the loading step.</figcaption></figure>

## Summary

In this work, we proposed a sequential surrogate model framework to predict the stress field using process variables and initial spatial coordinates as inputs for the surrogate models. Simulations of a V-bending process are run via FE software to collect the necessary data to fit the surrogate models. We have shown the benefit of using automatic tuners as tools for the rapid evaluation of hyperparameters. A comparative study of ANN and PINN has been performed to evaluate their accuracy in predicting stress for the complete loading step and a third of it. In the former case, both surrogates did not fit satisfactorily through the data due to the loss of information related to the path dependency of the plasticity problem. Conversely, the surrogate models delivered better results for a smaller time step. ANN and PINN predictions have comparable results, despite the
Processes subject to large and plastic deformation are known to be path-dependent. Predicting the stress field of a complete loading-unloading cycle means that information related to the history of the process is lost. Hence, it can be supposed that the surrogate models should lead to better fitting for small-time steps. Employing the same hyperparameters found with automatic tuners, the surrogates are trained using the information collected at a third of the loading step. Once more, a mean stress field is calculated to compare the predicted errors (Fig. 6a). In this case the predictions are plotted with respect to the maximum errors of ANNσ due to a similar error range. This helps emphasize the areas prone to return worse predictions in PINNσ than ANNσ. The evaluated MAE is almost ten to twenty times lower than the one calculated for the previous case. Both models show good generalization ability on the testing dataset, probably due to the low standard deviation among the test simulations. As expected, both models predict σ₃₃ similarly since it does not contribute to the equilibrium equation. Regarding the other components, the models exhibit an error concentration in correspondence of the contact point of the sheet with the punch and the die. The inferior performance of PINNσ can be ascribed to the combination of high nonlinearity regions and the constraint on the model parameters enforced by the equilibrium equation. Nevertheless, the model returns a satisfactory fit if compared to ANNσ.

Concerning the computational efficiency, the time needed to train the two surrogate models for the same number of epochs is almost the same. In PINNσ the evaluation of the derivatives implies additional computations, which have a negligible impact on the overall efficiency. If time increments are taken into account, it is expected that the computational efficiency of the PINNσ will drop significantly.

<figure><img src="image_2146.png" /><figcaption>Fig. 6. a) Mean field of the stress, b) mean absolute error of ANNσ, and c) PINNσ for a third of the loading step.</figcaption></figure>

## Summary

In this work, we proposed a sequential surrogate model framework to predict the stress field using process variables and initial spatial coordinates as inputs for the surrogate models. Simulations of a V-bending process are run via FE software to collect the necessary data to fit the surrogate models. We have shown the benefit of using automatic tuners as tools for the rapid evaluation of hyperparameters. A comparative study of ANN and PINN has been performed to evaluate their accuracy in predicting stress for the complete loading step and a third of it. In the former case, both surrogates did not fit satisfactorily through the data due to the loss of information related to the path dependency of the plasticity problem. Conversely, the surrogate models delivered better results for a smaller time step. ANN and PINN predictions have comparable results, despite the
latter showing an inappreciable higher error in the contact regions. Not to mention that PINN fits
very well through data while satisfying the system governing equations. In conclusion, the
subdivision of the loading step in smaller time steps should lead to better predictions. Therefore,
adopting recurrent neural networks (RNN) for sequential data could help to consider path
dependency and the evolution in time of the stress field. Besides, improving the PDE loss by taking
into account time derivatives and constitutive equations may enhance the predictions of PINN.
The former is the subject of our future work.

Acknowledgments

This publication is part of the project Digital Twin (project 2.2) with project number P18-03 of the research programme Perspectief, which is (mainly) financed by the Dutch Research Council (NWO).

References

[1] I. El Mrabti, A. El Hakimi, A. Touache and C. Abderrahim, "A comparative study of surrogate models for predicting process failures during the sheet metal forming process of advanced high-strength steel," *The International Journal of Advanced Manufacturing Technology*, vol. 121, no. 1, pp. 199-214, 2022. https://doi.org/10.1007/s00170-022-09319-5

[2] F. Abbassi, T. Belhadj, S. Mistou and A. Zghal, "Parameter identification of a mechanical ductile damage using Artificial Neural Networks in sheet metal forming," *Material and Design*, vol. 45, pp. 605-615, 2013. https://doi.org/10.1016/j.matdes.2012.09.032

[3] J. Ghaisari, H. Jannesari and M. Vatani, "Artificial neural network predictors for mechanical properties of cold rolling products," *Advance in Engineering Software*, vol. 45, pp. 91-99, 2012. https://doi.org/10.1016/j.advengsoft.2011.09.016

[4] M. B. Gorji, M. Mozaffar, J. H. Heidenreich, J. Cao and D. Mohr, "On the potential of recurrent neural networks for modeling path dependent plasticity," *Journal of the Mechanics and Physics of Solids*, vol. 143, p. 103972, 2020. https://doi.org/10.1016/j.jmps.2020.103972

[5] J. G. Hoffer, B. C. Geiger, P. Ofner and R. Kern, "Mesh-Free Surrogate Models for Structural Mechanic FEM Simulation: A Comparative Study of Approaches," *Applied Sciences*, vol. 11, no. 20, p. 9411, 2021. https://doi.org/10.3390/app11209411

[6] E. Haghighat, M. Raissi, A. Moure, H. Gomez and R. Juanes, "A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics," *Computer Methods in Applied Mechanics and Engineering*, vol. 379, 2021. https://doi.org/10.1016/j.cma.2021.113741

[7] S. Niu, E. Zhang, Y. Bazilevs and V. Srivastava, "Modeling finite-strain plasticity using physics-informed neural network and assessment of the network performance," *J. Mech. Phys. Solids*, vol. 172, p. 105177, 2023. https://doi.org/10.1016/j.jmps.2022.105177

[8] B. de Gooijer, "Surrogate modelling and robust optimization of multi-stage metal forming processes," 2022.

[9] H. Liu, J. Cai and Y. S. Ong, "A Survey of Adaptive Sampling for Global Metamodeling in Support of Simulation-based Complex Engineering Design," *Structural and Multidisciplinary Optimization*, vol. 57, pp. 393-416, 2018. https://doi.org/10.1007/s00158-017-1739-8

[10] K. P. Murphy, Probabilistic Machine Learning: An introduction, MIT Press, 2022.

[11] M. Raissi, P. Perdikaris and G. E. Karniada, "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial
latter showing an inappreciable higher error in the contact regions. Not to mention that PINN fits
very well through data while satisfying the system governing equations. In conclusion, the
subdivision of the loading step in smaller time steps should lead to better predictions. Therefore,
adopting recurrent neural networks (RNN) for sequential data could help to consider path
dependency and the evolution in time of the stress field. Besides, improving the PDE loss by taking
into account time derivatives and constitutive equations may enhance the predictions of PINN.
The former is the subject of our future work.

Acknowledgments

This publication is part of the project Digital Twin (project 2.2) with project number P18-03 of the research programme Perspectief, which is (mainly) financed by the Dutch Research Council (NWO).

References

[1] I. El Mrabti, A. El Hakimi, A. Touache and C. Abderrahim, "A comparative study of surrogate models for predicting process failures during the sheet metal forming process of advanced high-strength steel," *The International Journal of Advanced Manufacturing Technology*, vol. 121, no. 1, pp. 199-214, 2022. https://doi.org/10.1007/s00170-022-09319-5

[2] F. Abbassi, T. Belhadj, S. Mistou and A. Zghal, "Parameter identification of a mechanical ductile damage using Artificial Neural Networks in sheet metal forming," *Material and Design*, vol. 45, pp. 605-615, 2013. https://doi.org/10.1016/j.matdes.2012.09.032

[3] J. Ghaisari, H. Jannesari and M. Vatani, "Artificial neural network predictors for mechanical properties of cold rolling products," *Advance in Engineering Software*, vol. 45, pp. 91-99, 2012. https://doi.org/10.1016/j.advengsoft.2011.09.016

[4] M. B. Gorji, M. Mozaffar, J. H. Heidenreich, J. Cao and D. Mohr, "On the potential of recurrent neural networks for modeling path dependent plasticity," *Journal of the Mechanics and Physics of Solids*, vol. 143, p. 103972, 2020. https://doi.org/10.1016/j.jmps.2020.103972

[5] J. G. Hoffer, B. C. Geiger, P. Ofner and R. Kern, "Mesh-Free Surrogate Models for Structural Mechanic FEM Simulation: A Comparative Study of Approaches," *Applied Sciences*, vol. 11, no. 20, p. 9411, 2021. https://doi.org/10.3390/app11209411

[6] E. Haghighat, M. Raissi, A. Moure, H. Gomez and R. Juanes, "A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics," *Computer Methods in Applied Mechanics and Engineering*, vol. 379, 2021. https://doi.org/10.1016/j.cma.2021.113741

[7] S. Niu, E. Zhang, Y. Bazilevs and V. Srivastava, "Modeling finite-strain plasticity using physics-informed neural network and assessment of the network performance," *J. Mech. Phys. Solids*, vol. 172, p. 105177, 2023. https://doi.org/10.1016/j.jmps.2022.105177

[8] B. de Gooijer, "Surrogate modelling and robust optimization of multi-stage metal forming processes," 2022.

[9] H. Liu, J. Cai and Y. S. Ong, "A Survey of Adaptive Sampling for Global Metamodeling in Support of Simulation-based Complex Engineering Design," *Structural and Multidisciplinary Optimization*, vol. 57, pp. 393-416, 2018. https://doi.org/10.1007/s00158-017-1739-8

[10] K. P. Murphy, Probabilistic Machine Learning: An introduction, MIT Press, 2022.

[11] M. Raissi, P. Perdikaris and G. E. Karniada, "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial
differential equations," *Journal of Computational Physics*, vol. 378, pp. 686-707, 2019.
https://doi.org/10.1016/j.jcp.2018.10.045

[12] S. Razavi, B. A. Tolson and D. H. Burn, "Review of surrogate modeling in water resources," *Water Resources Research*, 2012. https://doi.org/10.1029/2011WR011527

[13] K. Hornik, "Multilayer Feedforward Networks are Universal Approximators," *Neural Networks*, vol. 2, pp. 359-366, 1989. https://doi.org/10.1016/0893-6080(89)90020-8

[14] R. Yondo, E. Andrés and E. Valero, "A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses," *Elsevier*, 2020.

[15] A. J. Forrester and A. J. Kean, "Recent advances in surrogate-based optimization," *Progress in Aerospace Sciences*, vol. 45, no. 1-3, pp. 50-79, 2009.
https://doi.org/10.1016/j.paerosci.2008.11.001

[16] R. Jin, W. Chen and T. W. Simpson, "Comparative studies of metamodeling techniques under multiple modeling criteria," *Structural Multidisciplinary Optimization*, 2001.
https://doi.org/10.2514/6.2000-4801

[17] J. H. Wiebenga, A. H. van den Boogaard and G. Klaseboer, "Sequential robust optimization of a V-bending process using numerical simulations," *Struct Multidisc Optim*, vol. 46, pp. 137-153, 2012. https://doi.org/10.1007/s00158-012-0761-0
differential equations," *Journal of Computational Physics*, vol. 378, pp. 686-707, 2019.
https://doi.org/10.1016/j.jcp.2018.10.045

[12] S. Razavi, B. A. Tolson and D. H. Burn, "Review of surrogate modeling in water resources," *Water Resources Research*, 2012. https://doi.org/10.1029/2011WR011527

[13] K. Hornik, "Multilayer Feedforward Networks are Universal Approximators," *Neural Networks*, vol. 2, pp. 359-366, 1989. https://doi.org/10.1016/0893-6080(89)90020-8

[14] R. Yondo, E. Andrés and E. Valero, "A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses," *Elsevier*, 2020.

[15] A. J. Forrester and A. J. Kean, "Recent advances in surrogate-based optimization," *Progress in Aerospace Sciences*, vol. 45, no. 1-3, pp. 50-79, 2009.
https://doi.org/10.1016/j.paerosci.2008.11.001

[16] R. Jin, W. Chen and T. W. Simpson, "Comparative studies of metamodeling techniques under multiple modeling criteria," *Structural Multidisciplinary Optimization*, 2001.
https://doi.org/10.2514/6.2000-4801

[17] J. H. Wiebenga, A. H. van den Boogaard and G. Klaseboer, "Sequential robust optimization of a V-bending process using numerical simulations," *Struct Multidisc Optim*, vol. 46, pp. 137-153, 2012. https://doi.org/10.1007/s00158-012-0761-0
# Novel approach for data-driven modelling of multi-stage straightening and bending processes

PETERS Henning<fnref n="1" />,ᵃ*, DJAKOW Eugen<fnref n="1" />, ROSTEK Tim<fnref n="2" />,ᵇ, MAZUR Andreas<fnref n="3" />,ᶜ, TRÄCHTLER Ansgar<fnref n="1" />,<fnref n="4" />,ᵈ, HOMBERG Werner<fnref n="2" />,ᵉ and HAMMER Barbara<fnref n="3" />,*f

**Keywords:** Straightening Machine, Punch-Bending Process, Hybrid Modelling

**Abstract.** In multi-stage bending and straightening operations cross-stage and quantity-dependent effects crucially affect the quality of the end product. Using punch-bending units in combination with a mechatronic straightening device can improve the accuracy and repeatability of product features remarkably well. In this work a concept for an innovative hybrid model of a roll straightener in a multi-stage straightening and multi-stage bending process is proposed. This model combines data-driven elements with expert knowledge and aims to minimise residual errors of the roll straightener to reliably decrease the risk of disadvantageous cross-stage and quantity-dependent effects on a subsequent punch-bending process.

**Introduction**

Semi-finished wire products are usually wound onto coils for transport so that they can be forwarded into subsequent processes. One such subsequent process is punch-bending. Punch-bending mainly refers to separating and forming production processes. One or more semi-finished products, for example in the form of strips or wires made of metal, are formed and punched into a finished product on an automatic punch-bending machine in several subsequent steps. The desired change in geometry of a punch-bent part usually occurs by exceeding the yield point or altering the plastic state of a solid body. The quality of complex punch-bending parts depends significantly on the design and realisation of the multi-stage bending and straightening operations used for this purpose [1, 2].

The classical punch-bending process consists of three main stages: The wire coil, the straightener and the punch-bending unit (cf. Fig. 1). The semi-finished product is unwound from the coil and passed through the straightener. A particular challenge of the punch-bending process regarding reproducible part quality is the control of cross-stage or quantity-dependent effects that are induced in the wire in the form of bending and residual stresses as a result of wire production and wire coiling [1, 3]. Due to the alternating arrangement of the straightening rolls, the wire undergoes an elastic-plastic alternating deformation, which minimises the residual stresses and results in a flatness of the wire [1, 3]. The adjustment of the straightening process to counter these effects is carried out in industrial applications based on expert knowledge, which is problematic

<figure><img src="image_2156.png" /></figure>

<footnote n="1">¹Fraunhofer Institute for Mechatronic Systems Design (IEM), Paderborn, Germany</footnote>

<footnote n="2">²Paderborn University, Department of forming and machining technology (LUF), Germany</footnote>

<footnote n="3">³Bielefeld University, Center for Cognitive Interaction Technology (CITEC), Germany</footnote>

<footnote n="4">⁴Heinz Nixdorf Institute (HNI), Paderborn University, Germany</footnote>

<footnote>ªhenning.peters@iem.fraunhofer.de, ᵇtr@luf.uni-paderborn.de,</footnote>

<footnote>ᶜamazur@techfak.uni-bielefeld.de, ᵈansgar.traechtler@iem.fraunhofer.de,</footnote>

<footnote>ᵉwh@luf.uni-paderborn.de, ᵈbhammer@techfak.uni-bielefeld.de</footnote>
# Novel approach for data-driven modelling of multi-stage straightening and bending processes

PETERS Henning<fnref n="1" />,ᵃ*, DJAKOW Eugen<fnref n="1" />, ROSTEK Tim<fnref n="2" />,ᵇ, MAZUR Andreas<fnref n="3" />,ᶜ, TRÄCHTLER Ansgar<fnref n="1" />,<fnref n="4" />,ᵈ, HOMBERG Werner<fnref n="2" />,ᵉ and HAMMER Barbara<fnref n="3" />,*f

**Keywords:** Straightening Machine, Punch-Bending Process, Hybrid Modelling

**Abstract.** In multi-stage bending and straightening operations cross-stage and quantity-dependent effects crucially affect the quality of the end product. Using punch-bending units in combination with a mechatronic straightening device can improve the accuracy and repeatability of product features remarkably well. In this work a concept for an innovative hybrid model of a roll straightener in a multi-stage straightening and multi-stage bending process is proposed. This model combines data-driven elements with expert knowledge and aims to minimise residual errors of the roll straightener to reliably decrease the risk of disadvantageous cross-stage and quantity-dependent effects on a subsequent punch-bending process.

**Introduction**

Semi-finished wire products are usually wound onto coils for transport so that they can be forwarded into subsequent processes. One such subsequent process is punch-bending. Punch-bending mainly refers to separating and forming production processes. One or more semi-finished products, for example in the form of strips or wires made of metal, are formed and punched into a finished product on an automatic punch-bending machine in several subsequent steps. The desired change in geometry of a punch-bent part usually occurs by exceeding the yield point or altering the plastic state of a solid body. The quality of complex punch-bending parts depends significantly on the design and realisation of the multi-stage bending and straightening operations used for this purpose [1, 2].

The classical punch-bending process consists of three main stages: The wire coil, the straightener and the punch-bending unit (cf. Fig. 1). The semi-finished product is unwound from the coil and passed through the straightener. A particular challenge of the punch-bending process regarding reproducible part quality is the control of cross-stage or quantity-dependent effects that are induced in the wire in the form of bending and residual stresses as a result of wire production and wire coiling [1, 3]. Due to the alternating arrangement of the straightening rolls, the wire undergoes an elastic-plastic alternating deformation, which minimises the residual stresses and results in a flatness of the wire [1, 3]. The adjustment of the straightening process to counter these effects is carried out in industrial applications based on expert knowledge, which is problematic

<figure><img src="image_2147.png" /></figure>

<footnote n="1">¹Fraunhofer Institute for Mechatronic Systems Design (IEM), Paderborn, Germany</footnote>

<footnote n="2">²Paderborn University, Department of forming and machining technology (LUF), Germany</footnote>

<footnote n="3">³Bielefeld University, Center for Cognitive Interaction Technology (CITEC), Germany</footnote>

<footnote n="4">⁴Heinz Nixdorf Institute (HNI), Paderborn University, Germany</footnote>

<footnote>ªhenning.peters@iem.fraunhofer.de, ᵇtr@luf.uni-paderborn.de,</footnote>

<footnote>ᶜamazur@techfak.uni-bielefeld.de, ᵈansgar.traechtler@iem.fraunhofer.de,</footnote>

<footnote>ᵉwh@luf.uni-paderborn.de, ᵈbhammer@techfak.uni-bielefeld.de</footnote>
<figure><img src="image_2157.png" /><figcaption>Figure 1: Schematic representation of the straightening and bending process.</figcaption></figure>

due to the complexity of the interactions regarding reproducible straightening quality [1, 4]. Changes in the semi-finished product properties due to the cross-stage and quantity-dependent effects have not yet been captured by measurement technology and therefore lead to a high level of waste [1]. One approach to improve the part quality is to combine a multi-stage mechatronic straightener with multi-stage punch-bending units to create a complete mechatronic system. In order to enable targeted application, a cross-stage and quantity-dependent modelling of the described straightening and bending process and machine system is required [5].

In this article, a concept for a novel hybrid model is presented. The goal of the hybrid model is to minimise the residual errors of a roll straightener that is part of a multi-stage straightening and bending process. Therefore, a customised straightening and punch-bending process setup is presented. The introduction of a hybrid model consisting of a digital twin and a machine learning module to the demonstrator process follows. In combination with a surrogate model, a correction unit is introduced to the hybrid model which shall compute corrective suggestions for the roll straightener to minimise the residual material curvature after the straightening process. Eventually, the importance of interpretable and explainable AI is described in the context of the hybrid model.

## State of the Art

Straightening and Punch-Bending Technology. Electrical connectors and terminals are typical components that are manufactured in large quantities and variants by punch-bending [6]. The high complexity and accuracy requirements of multi-stage punch-bending processes result in a high relevance of interactions with regard to component quality. These can be initiated by fluctuating properties of the semi-finished products, tools or machine tools and can significantly impair the robustness of the process [7].

Previous research work focused on the bending angle of the part to be manufactured as a qualitative parameter [7]. Accordingly, one solution approach of reproducible part quality is to record the same bending angle online in order to correct the process [8, 9]. The corrective intervention then requires a corresponding actuator system using complex correction strategies [10]. An interesting approach in the aforementioned context is the use of so-called self-correcting or mechatronic straighteners. Through mechatronic adjustment, these can achieve a significant improvement in product properties. Since a straightening process is generally used in process sequences for punch-bending strip material, this approach offers great potential in terms of optimising bending processes, provided that the necessary complex knowledge for integration within a multi-stage and quantity-dependent process is available [1, 5].

The evaluation of the complex relationships presented above is currently only possible with the help of expert knowledge and the evaluation of simulation and construction data. Process-accompanying data is also required to enable process monitoring of quality characteristics. There are still deficits in the preparation and preservation as well as the automated evaluation of corresponding information.
<figure><img src="image_2148.png" /><figcaption>Figure 1: Schematic representation of the straightening and bending process.</figcaption></figure>

due to the complexity of the interactions regarding reproducible straightening quality [1, 4]. Changes in the semi-finished product properties due to the cross-stage and quantity-dependent effects have not yet been captured by measurement technology and therefore lead to a high level of waste [1]. One approach to improve the part quality is to combine a multi-stage mechatronic straightener with multi-stage punch-bending units to create a complete mechatronic system. In order to enable targeted application, a cross-stage and quantity-dependent modelling of the described straightening and bending process and machine system is required [5].

In this article, a concept for a novel hybrid model is presented. The goal of the hybrid model is to minimise the residual errors of a roll straightener that is part of a multi-stage straightening and bending process. Therefore, a customised straightening and punch-bending process setup is presented. The introduction of a hybrid model consisting of a digital twin and a machine learning module to the demonstrator process follows. In combination with a surrogate model, a correction unit is introduced to the hybrid model which shall compute corrective suggestions for the roll straightener to minimise the residual material curvature after the straightening process. Eventually, the importance of interpretable and explainable AI is described in the context of the hybrid model.

## State of the Art

Straightening and Punch-Bending Technology. Electrical connectors and terminals are typical components that are manufactured in large quantities and variants by punch-bending [6]. The high complexity and accuracy requirements of multi-stage punch-bending processes result in a high relevance of interactions with regard to component quality. These can be initiated by fluctuating properties of the semi-finished products, tools or machine tools and can significantly impair the robustness of the process [7].

Previous research work focused on the bending angle of the part to be manufactured as a qualitative parameter [7]. Accordingly, one solution approach of reproducible part quality is to record the same bending angle online in order to correct the process [8, 9]. The corrective intervention then requires a corresponding actuator system using complex correction strategies [10]. An interesting approach in the aforementioned context is the use of so-called self-correcting or mechatronic straighteners. Through mechatronic adjustment, these can achieve a significant improvement in product properties. Since a straightening process is generally used in process sequences for punch-bending strip material, this approach offers great potential in terms of optimising bending processes, provided that the necessary complex knowledge for integration within a multi-stage and quantity-dependent process is available [1, 5].

The evaluation of the complex relationships presented above is currently only possible with the help of expert knowledge and the evaluation of simulation and construction data. Process-accompanying data is also required to enable process monitoring of quality characteristics. There are still deficits in the preparation and preservation as well as the automated evaluation of corresponding information.
Modelling Approaches of Straightening and Punch-Bending Processes. In the past, several modelling approaches have been developed to optimise the straightening process, as this is essential for the part quality of the punch-bending process. These models can be divided into analytical and numerical approaches.

Henrich [11] presents an analytical model of the straightening process for sheet metal. Here, the deformation in the straightening process is represented using a bending beam with one loading and one unloading cycle. The elastic and plastic areas of the bending line are determined iteratively using third- and fourth-degree polynomials, which can be used to draw conclusions about the residual stresses, the remaining curvature and the contact forces. Another approach is to determine the contact forces and the remaining curvature by means of the straightener geometry regarding the roll radius and the roll position [12]. Paech [1] presents a simulation program that supports the determination of power requirements for process material forming. The simulation is based on an analytical model that describes the elastic-plastic alternating deformations of the straightening process.

Regarding the numerical approach, the finite element method (FEM) has become firmly established as a tool for process simulation and for analyzing the dynamics of semi-finished products, tools and machines to reduce the extent of cost-intensive preliminary tests. Regarding the straightening process, all required characteristic data such as the bending line, reaction forces or stresses can be determined. The simulation results generally correspond well with the real forming processes if unpredictable disturbances or process fluctuations such as varying material and geometric properties are only present to a small extent [4, 13].

Martinez-de-Pisón et al. [14] investigate the optimisation of the FE model regarding the straightening process using genetic algorithms. The purpose is to reduce the residual stresses after the straightening process. This approach requires a high number of simulations. Grüber [3] uses the FEM to set up a parameter study regarding flatness and minimisation of residual stresses to enable process control. For this purpose, control structures were integrated into the FE simulations to determine the correct roll feed. Regarding the punch-bending process, Mertin [15] presents a numerical analysis of a multi-stage punch-bending process. When using material with spring-back properties, the exact mapping of the material properties plays a decisive role. For this reason, a list of the influencing factors that have to be taken into account was compiled.

The presented model variants have several limitations in terms of process representation. Limitations of analytical models consist of simplifying assumptions of the material behavior to enable a mathematical representation of the complex deformation processes. These are based, for example, on the Bernoulli theory to describe the deformation of a rigid beam under pure bending stress. Furthermore, the hardening behavior is approximated, for example using a linear hardening stress-strain curve. There is also often a one-sided consideration of the isotropic or kinematic hardening model. As a result, deviations from the material properties are possible [3].

Numerical models are limited due to the high computing times, which is unacceptable for inline-capable models or iterative tool adaptions. Regarding the straightening process, these arise since material deformations occur simultaneously at several points and a segment must pass the entire straightener to adequately represent the forming process. Sufficiently fine discretisation is also required to obtain a correct resolution of the plastic deformation. This results in a high number of nodes, lines of freedom and contact points [4, 13].

AI Applications in Straightening and Punch-Bending Processes. The data generated by numerical and analytical models as well as from real punch-bending processes can be used by AI technologies to not only gain insights into the underlying physical relations between resulting product and input parameters but also to improve product quality [16, 17, 18]. For instance, a recent example is given by [17] where a convolutional neural network has been trained to predict the required stroke to bend metal sheets into a desired geometry. They additionally included Swift's law as a
Modelling Approaches of Straightening and Punch-Bending Processes. In the past, several modelling approaches have been developed to optimise the straightening process, as this is essential for the part quality of the punch-bending process. These models can be divided into analytical and numerical approaches.

Henrich [11] presents an analytical model of the straightening process for sheet metal. Here, the deformation in the straightening process is represented using a bending beam with one loading and one unloading cycle. The elastic and plastic areas of the bending line are determined iteratively using third- and fourth-degree polynomials, which can be used to draw conclusions about the residual stresses, the remaining curvature and the contact forces. Another approach is to determine the contact forces and the remaining curvature by means of the straightener geometry regarding the roll radius and the roll position [12]. Paech [1] presents a simulation program that supports the determination of power requirements for process material forming. The simulation is based on an analytical model that describes the elastic-plastic alternating deformations of the straightening process.

Regarding the numerical approach, the finite element method (FEM) has become firmly established as a tool for process simulation and for analyzing the dynamics of semi-finished products, tools and machines to reduce the extent of cost-intensive preliminary tests. Regarding the straightening process, all required characteristic data such as the bending line, reaction forces or stresses can be determined. The simulation results generally correspond well with the real forming processes if unpredictable disturbances or process fluctuations such as varying material and geometric properties are only present to a small extent [4, 13].

Martinez-de-Pisón et al. [14] investigate the optimisation of the FE model regarding the straightening process using genetic algorithms. The purpose is to reduce the residual stresses after the straightening process. This approach requires a high number of simulations. Grüber [3] uses the FEM to set up a parameter study regarding flatness and minimisation of residual stresses to enable process control. For this purpose, control structures were integrated into the FE simulations to determine the correct roll feed. Regarding the punch-bending process, Mertin [15] presents a numerical analysis of a multi-stage punch-bending process. When using material with spring-back properties, the exact mapping of the material properties plays a decisive role. For this reason, a list of the influencing factors that have to be taken into account was compiled.

The presented model variants have several limitations in terms of process representation. Limitations of analytical models consist of simplifying assumptions of the material behavior to enable a mathematical representation of the complex deformation processes. These are based, for example, on the Bernoulli theory to describe the deformation of a rigid beam under pure bending stress. Furthermore, the hardening behavior is approximated, for example using a linear hardening stress-strain curve. There is also often a one-sided consideration of the isotropic or kinematic hardening model. As a result, deviations from the material properties are possible [3].

Numerical models are limited due to the high computing times, which is unacceptable for inline-capable models or iterative tool adaptions. Regarding the straightening process, these arise since material deformations occur simultaneously at several points and a segment must pass the entire straightener to adequately represent the forming process. Sufficiently fine discretisation is also required to obtain a correct resolution of the plastic deformation. This results in a high number of nodes, lines of freedom and contact points [4, 13].

AI Applications in Straightening and Punch-Bending Processes. The data generated by numerical and analytical models as well as from real punch-bending processes can be used by AI technologies to not only gain insights into the underlying physical relations between resulting product and input parameters but also to improve product quality [16, 17, 18]. For instance, a recent example is given by [17] where a convolutional neural network has been trained to predict the required stroke to bend metal sheets into a desired geometry. They additionally included Swift's law as a
regularisation term to the loss function, therefore including domain knowledge into training. Their experiments compare the deviation of the predicted geometry after springback to the target geometry. The results of [17] show that the theory-grounded deep neural network (TG-DNN) outperforms not only support vector machines but also regular DNNs, that do not include Swift's law during training.

The use of a purely data-based model in the form of a "black box" carries the risk of an inadequate representation of the overall process due to the inadequacy of the available data to capture complex interactions in their entirety. For this reason, a pure consideration of knowledge-based or data-based models is insufficient for modelling a complex application such as the straightening and bending process [19]. To compensate for the limitations of the presented analytical, numerical and data-based model types and to combine their strengths, a hybrid model in the sense of [19] is to be developed for combined straightening and punch-bending which takes complex multi-stage and quantity-dependent effects into account. However, some of these effects are difficult or impossible to calculate or to measure. Thus, machine learning (ML) methods are required in addition to the classic approaches of analytical and numerical modelling. To the best of the authors knowledge, currently there is no publication in which a hybrid model was used for modelling the entire straightening and bending process.

## Novel Concept for Data-Driven Multi-Stage Straightening and Bending Processes

The overall aim of the entire research project is to develop an explainable data-driven model for multi-stage punch-bending processes in order to make them usable in the long term, from the start of the process to the forming in the final stage. This should enable the locally limited and reproducible adjustment of process variables, e.g. roller position, number of rollers, bending punch path, tool design, in an environment characterised by a large number of disturbance variables, e.g. material fluctuations and tool wear. Due to the limited scope of this paper, however, the focus lies only on the first step of developing the hybrid model for the entire multi-stage straightening and bending process, which is the straightening process.

In Fig. 2 a schematic representation of the mechatronic, multi-stage straightening and bending process is shown. The process setup is designed for flat wire. In contrast to the straighteners used in the industry with mechanical adjusting screws, this work considers a so-called mechatronic straightener that has been developed in [20]. It consists of seven straightening rolls, in which the four lower rolls are fixed to the structure. The upper three straightening rolls can be adjusted individually in a vertical direction using stepper motors. Load cells are installed on these to record the straightening forces. Three laser triangulation sensors monitor the positioning of the upper rolls. Subsequent to the straightener, a wire buffer is required to enable a transition between the continuous wire feed of the straightener and the sequential wire feed of the punch-bending machine. In the punch-bending machine [10], the flat wire is fastened using a blank holder and then formed using the geometry of the bending punch and the bending die. At the same time, the wire is separated from the rest of the strip. An angle (α) is created in a 50 mm long piece of wire. Inside of the punch-bending unit, the bending angle is determined using an optical measuring system. Furthermore, the punch movement is monitored by an additional laser triangulation sensor.
regularisation term to the loss function, therefore including domain knowledge into training. Their experiments compare the deviation of the predicted geometry after springback to the target geometry. The results of [17] show that the theory-grounded deep neural network (TG-DNN) outperforms not only support vector machines but also regular DNNs, that do not include Swift's law during training.

The use of a purely data-based model in the form of a "black box" carries the risk of an inadequate representation of the overall process due to the inadequacy of the available data to capture complex interactions in their entirety. For this reason, a pure consideration of knowledge-based or data-based models is insufficient for modelling a complex application such as the straightening and bending process [19]. To compensate for the limitations of the presented analytical, numerical and data-based model types and to combine their strengths, a hybrid model in the sense of [19] is to be developed for combined straightening and punch-bending which takes complex multi-stage and quantity-dependent effects into account. However, some of these effects are difficult or impossible to calculate or to measure. Thus, machine learning (ML) methods are required in addition to the classic approaches of analytical and numerical modelling. To the best of the authors knowledge, currently there is no publication in which a hybrid model was used for modelling the entire straightening and bending process.

## Novel Concept for Data-Driven Multi-Stage Straightening and Bending Processes

The overall aim of the entire research project is to develop an explainable data-driven model for multi-stage punch-bending processes in order to make them usable in the long term, from the start of the process to the forming in the final stage. This should enable the locally limited and reproducible adjustment of process variables, e.g. roller position, number of rollers, bending punch path, tool design, in an environment characterised by a large number of disturbance variables, e.g. material fluctuations and tool wear. Due to the limited scope of this paper, however, the focus lies only on the first step of developing the hybrid model for the entire multi-stage straightening and bending process, which is the straightening process.

In Fig. 2 a schematic representation of the mechatronic, multi-stage straightening and bending process is shown. The process setup is designed for flat wire. In contrast to the straighteners used in the industry with mechanical adjusting screws, this work considers a so-called mechatronic straightener that has been developed in [20]. It consists of seven straightening rolls, in which the four lower rolls are fixed to the structure. The upper three straightening rolls can be adjusted individually in a vertical direction using stepper motors. Load cells are installed on these to record the straightening forces. Three laser triangulation sensors monitor the positioning of the upper rolls. Subsequent to the straightener, a wire buffer is required to enable a transition between the continuous wire feed of the straightener and the sequential wire feed of the punch-bending machine. In the punch-bending machine [10], the flat wire is fastened using a blank holder and then formed using the geometry of the bending punch and the bending die. At the same time, the wire is separated from the rest of the strip. An angle (α) is created in a 50 mm long piece of wire. Inside of the punch-bending unit, the bending angle is determined using an optical measuring system. Furthermore, the punch movement is monitored by an additional laser triangulation sensor.
<figure><img src="image_2158.png" /><figcaption>Figure 2: Schematic representation of the multi-stage straightening and bending process.</figcaption></figure>

The multi-stage straightening and bending process generates data that is subordinate to multi-stage and quantity dependent effects in each stage. A hybrid machine learning module (HML-module) is thus connected to the straightening and bending process. It includes a digital twin and a machine learning module (ML-module). Via a PLC, an industrial computer with sensor and actuator interfaces, the process data is forwarded to the digital twin which is to be designed. The digital twin shall manage the generated data and forward it to the ML-module. The ML-module shall process the generated data to optimise the output of the roll straightener by suggesting actionable corrections to the roll positions of the roll straightener. Furthermore, the digital twin receives optimisation suggestions of the machine learning module and decides on whether to feed those back to the PLC. Connecting the data-driven HML-module to the multi-stage straightening and bending process results in the overall concept of the data-driven multi-stage straightening and punch-bending process.

## Hybrid Machine Learning Module

The HML-module (cf. Fig. 3) represents a concept that combines expert knowledge, numerical and analytical models about straightening and bending processes, i.e. domain knowledge, with machine learning. The concept for the ML-module shows how machine learning models could process the features provided by the digital twin in three stages.

### Digital Twin
The digital twin collects and manages data generated by either a simulation, i.e. a numerical model of the straightening and bending process, analytical models, the real process itself or expert knowledge. Using recorded force signals as an example, irrelevant signals are to be filtered out during the start-up phases of the process. During the running process, the force signals should then again be decisive for determining the current material condition in combination with the above-mentioned data sources. Therefore, the data has to be categorised and merged to an information model [21]. Based on the data provided by the digital twin the ML-module proposes corrections to the process. These corrections are then sent back to the digital twin. The digital twin thus acts as a safety measure that validates proposed corrections. One example would be filtering out suggestions that are not within the possible value range of the actuators.
<figure><img src="image_2149.png" /><figcaption>Figure 2: Schematic representation of the multi-stage straightening and bending process.</figcaption></figure>

The multi-stage straightening and bending process generates data that is subordinate to multi-stage and quantity dependent effects in each stage. A hybrid machine learning module (HML-module) is thus connected to the straightening and bending process. It includes a digital twin and a machine learning module (ML-module). Via a PLC, an industrial computer with sensor and actuator interfaces, the process data is forwarded to the digital twin which is to be designed. The digital twin shall manage the generated data and forward it to the ML-module. The ML-module shall process the generated data to optimise the output of the roll straightener by suggesting actionable corrections to the roll positions of the roll straightener. Furthermore, the digital twin receives optimisation suggestions of the machine learning module and decides on whether to feed those back to the PLC. Connecting the data-driven HML-module to the multi-stage straightening and bending process results in the overall concept of the data-driven multi-stage straightening and punch-bending process.

## Hybrid Machine Learning Module

The HML-module (cf. Fig. 3) represents a concept that combines expert knowledge, numerical and analytical models about straightening and bending processes, i.e. domain knowledge, with machine learning. The concept for the ML-module shows how machine learning models could process the features provided by the digital twin in three stages.

### Digital Twin
The digital twin collects and manages data generated by either a simulation, i.e. a numerical model of the straightening and bending process, analytical models, the real process itself or expert knowledge. Using recorded force signals as an example, irrelevant signals are to be filtered out during the start-up phases of the process. During the running process, the force signals should then again be decisive for determining the current material condition in combination with the above-mentioned data sources. Therefore, the data has to be categorised and merged to an information model [21]. Based on the data provided by the digital twin the ML-module proposes corrections to the process. These corrections are then sent back to the digital twin. The digital twin thus acts as a safety measure that validates proposed corrections. One example would be filtering out suggestions that are not within the possible value range of the actuators.
<figure><img src="image_2159.png" /><figcaption>Figure 3: *Structure of the hybrid machine learning module.*</figcaption></figure>

**Representation Learning and Dimensionality Reduction.** The ML-module shall receive force measurements from the digital twin which represent sequential data over thousands of time-steps and describe important material information in detail. To handle this sequential data, an embedding neural network is conceptualized into the ML-module which aims to encode quantity-dependent information about the effects of the unwinding procedure to the material [3].

Structuring the embedding network solely with fully connected layers in the neural network is not recommended, as the force measurements would induce large weight matrices which imply difficult training behavior due to high function complexity [22]. Therefore, architectures like one-dimensional convolutional neural networks (1D-CNNs) [23], Long short-term memory units (LSTMs) [24], or Transformer [25] have to be employed, each of which is able to reduce the force measurements into lower-dimensional meaningful representations, which are referred to as embeddings. For example, 1D-CNNs learn filters that produce lower-dimensional representations by extracting latent features when applied to an input [23]. The neural network in the ML-module shall contain one embedding-network per upper roll in the straightener since different material behavior has to be expected at each roll [1]. Hence, the ML-module contains three embedding-networks and therefore calculates three embeddings per product.

**Surrogate Model.** The goal of the surrogate model is to learn a relation between process data and the residual curvature of the material. This represents a regression problem that can be learned using neural networks or Gaussian process regression [26], for example. The input feature space of the surrogate model consists of the embedding space learnt by the embedding neural networks and the vertical roll positions of the three upper rolls in the roll straightener. Further input features for the surrogate model are conceivable. Those could for instance be the geometry of the forming tools or the horizontal distances between the rolls in the roll straightener. Due to hybrid modelling, there is an inherent possibility for the integration of domain expertise into the training of the surrogate model, thus making the surrogate model a possible grey-box-model. Grey-box-models are reported to come with the benefit of being more interpretable than strict black-box models while producing better results than simple white-box models [17, 27].

**Correction Unit.** Lastly, a correction unit is integrated into the ML-module. For now, it is supposed to propose corrections for the adjustable vertical roll positions of the roll straightener in order to minimise residual material curvature. However, if further process parameters such as the roll
<figure><img src="image_2150.png" /><figcaption>Figure 3: *Structure of the hybrid machine learning module.*</figcaption></figure>

**Representation Learning and Dimensionality Reduction.** The ML-module shall receive force measurements from the digital twin which represent sequential data over thousands of time-steps and describe important material information in detail. To handle this sequential data, an embedding neural network is conceptualized into the ML-module which aims to encode quantity-dependent information about the effects of the unwinding procedure to the material [3].

Structuring the embedding network solely with fully connected layers in the neural network is not recommended, as the force measurements would induce large weight matrices which imply difficult training behavior due to high function complexity [22]. Therefore, architectures like one-dimensional convolutional neural networks (1D-CNNs) [23], Long short-term memory units (LSTMs) [24], or Transformer [25] have to be employed, each of which is able to reduce the force measurements into lower-dimensional meaningful representations, which are referred to as embeddings. For example, 1D-CNNs learn filters that produce lower-dimensional representations by extracting latent features when applied to an input [23]. The neural network in the ML-module shall contain one embedding-network per upper roll in the straightener since different material behavior has to be expected at each roll [1]. Hence, the ML-module contains three embedding-networks and therefore calculates three embeddings per product.

**Surrogate Model.** The goal of the surrogate model is to learn a relation between process data and the residual curvature of the material. This represents a regression problem that can be learned using neural networks or Gaussian process regression [26], for example. The input feature space of the surrogate model consists of the embedding space learnt by the embedding neural networks and the vertical roll positions of the three upper rolls in the roll straightener. Further input features for the surrogate model are conceivable. Those could for instance be the geometry of the forming tools or the horizontal distances between the rolls in the roll straightener. Due to hybrid modelling, there is an inherent possibility for the integration of domain expertise into the training of the surrogate model, thus making the surrogate model a possible grey-box-model. Grey-box-models are reported to come with the benefit of being more interpretable than strict black-box models while producing better results than simple white-box models [17, 27].

**Correction Unit.** Lastly, a correction unit is integrated into the ML-module. For now, it is supposed to propose corrections for the adjustable vertical roll positions of the roll straightener in order to minimise residual material curvature. However, if further process parameters such as the roll
geometry or horizontal roll distances are included, the correction mechanism shall also be applicable to these. For this, the correction unit of the ML-module shall use the learnt relationship between the process parameters and the residual curvature of the surrogate model. Counterfactual explanations [28] allow for the definition of target outcomes of machine learning models, which they realise by altering the original input to the model. In the straightening and bending process, the desired outcome of the surrogate model is zero since a planar material, i.e. material with zero residual curvature, is expected to deteriorate the product quality after punch-bending the least. Eventually, the corrections are to be sent to the digital twin which processes and, if applicable, forwards the suggestions to the actuators of the real process or the simulation. Crucially here is that the surrogate model must learn a *meaningful* relation between the process parameters and the residual curvature. Methodologies from explainable AI can be used to investigate whether the model learned such a meaningful relation.

## Taking Explainable AI into Account

Explainable AI (XAI) deals with the issue of model interpretability. Thereby, model interpretability can be understood as the degree to which a human can understand the cause of a prediction from a machine learning model [29]. Since it is planned to derive process configuration corrections using a machine learning surrogate model, the surrogate model should be investigated on whether it learned a meaningful relationship between the process parameters and the residual material curvature after straightening. A poor relationship would cause the counterfactuals, provided by the correction unit in the HML-module, to yield unsatisfactory corrections for the straightening process. To evaluate the learned relationship of the surrogate model, an empirical evaluation of the counterfactuals on whether they provide satisfactory corrections and an investigation on how the surrogate model ranks its input features in terms of feature importance is planned. A common way to quantify feature importance is to determine the so-called permutation feature importance [28, 30]. It can be calculated by comparing the differences between model errors, which are computed on the original data set and the data set in which the feature of interest is permuted [28]. Another aspect to consider is that the learnt embeddings for the force measurements are not guaranteed to be of good quality. Hence, an investigation of the embedding networks and the embeddings themselves is planned [31, 32, 33]. Embeddings should encode similar features to similar embeddings. In other words, embeddings should be robust towards small changes in the input. If otherwise, there is evidence that the embedding-networks produce non-meaningful embeddings. More explanation methods, which can depend on the chosen network architectures, are conceivable. For instance, if 1D-CNNs were used, post-hoc explanation methods such as activation maximisation [34] can highlight important input features in the force measurements.

## Outlook

Future work involves not only the extension of the feature space of the presented surrogate model but also the development of analytical approaches for individual process stages. Residual stress and curvature determined by analytical models must be passed via the digital twin to the ML-module. As the modelling approach of one stage can differ from the approach for another stage, it is also necessary to investigate how different models can be combined within the digital twin. Similar to the machine learning models for the roll straightener in the ML-module, approaches have to be modelled for the process parameters of the punch-bending unit. Hereby, process parameters can be for example force measurements in the punch-bending unit, the predicted residual curvature after straightening and tool geometries. With that feature space in mind, an analogous surrogate model for the punch-bending stage that predicts the bending angle of the final product is to be designed. Furthermore, a data set has to be created by collecting simulation or real process data. This data can then be used to experimentally evaluate the HML-module. Lastly, the
geometry or horizontal roll distances are included, the correction mechanism shall also be applicable to these. For this, the correction unit of the ML-module shall use the learnt relationship between the process parameters and the residual curvature of the surrogate model. Counterfactual explanations [28] allow for the definition of target outcomes of machine learning models, which they realise by altering the original input to the model. In the straightening and bending process, the desired outcome of the surrogate model is zero since a planar material, i.e. material with zero residual curvature, is expected to deteriorate the product quality after punch-bending the least. Eventually, the corrections are to be sent to the digital twin which processes and, if applicable, forwards the suggestions to the actuators of the real process or the simulation. Crucially here is that the surrogate model must learn a *meaningful* relation between the process parameters and the residual curvature. Methodologies from explainable AI can be used to investigate whether the model learned such a meaningful relation.

## Taking Explainable AI into Account

Explainable AI (XAI) deals with the issue of model interpretability. Thereby, model interpretability can be understood as the degree to which a human can understand the cause of a prediction from a machine learning model [29]. Since it is planned to derive process configuration corrections using a machine learning surrogate model, the surrogate model should be investigated on whether it learned a meaningful relationship between the process parameters and the residual material curvature after straightening. A poor relationship would cause the counterfactuals, provided by the correction unit in the HML-module, to yield unsatisfactory corrections for the straightening process. To evaluate the learned relationship of the surrogate model, an empirical evaluation of the counterfactuals on whether they provide satisfactory corrections and an investigation on how the surrogate model ranks its input features in terms of feature importance is planned. A common way to quantify feature importance is to determine the so-called permutation feature importance [28, 30]. It can be calculated by comparing the differences between model errors, which are computed on the original data set and the data set in which the feature of interest is permuted [28]. Another aspect to consider is that the learnt embeddings for the force measurements are not guaranteed to be of good quality. Hence, an investigation of the embedding networks and the embeddings themselves is planned [31, 32, 33]. Embeddings should encode similar features to similar embeddings. In other words, embeddings should be robust towards small changes in the input. If otherwise, there is evidence that the embedding-networks produce non-meaningful embeddings. More explanation methods, which can depend on the chosen network architectures, are conceivable. For instance, if 1D-CNNs were used, post-hoc explanation methods such as activation maximisation [34] can highlight important input features in the force measurements.

## Outlook

Future work involves not only the extension of the feature space of the presented surrogate model but also the development of analytical approaches for individual process stages. Residual stress and curvature determined by analytical models must be passed via the digital twin to the ML-module. As the modelling approach of one stage can differ from the approach for another stage, it is also necessary to investigate how different models can be combined within the digital twin. Similar to the machine learning models for the roll straightener in the ML-module, approaches have to be modelled for the process parameters of the punch-bending unit. Hereby, process parameters can be for example force measurements in the punch-bending unit, the predicted residual curvature after straightening and tool geometries. With that feature space in mind, an analogous surrogate model for the punch-bending stage that predicts the bending angle of the final product is to be designed. Furthermore, a data set has to be created by collecting simulation or real process data. This data can then be used to experimentally evaluate the HML-module. Lastly, the
developed process representation is to be evaluated in terms of identifying and analysing the effects and interactions that occur within the process. This should enable an understanding of the complex multi-stage forming process as well as the design of such processes and the tools required for them.

**Summary**

In this paper, a novel concept for data-driven, multi-stage straightening and bending processes using the straightener as an example was developed. After an extensive overview of existing modelling approaches for straightening and bending processes, the said concept and its interfaces were presented. It combines a HML-module which integrates a digital twin and machine learning with a multi-stage straightening and bending process. Thereby, the digital twin collects and manages process data from various sources and forwards it to a ML-module. It furthermore filters out irrelevant input data from the process as well as possible incorrect actuator corrections by the ML-module. The ML-module contains embeddings networks, a surrogate model and a correction unit. Hereby, the aim of the embedding networks is to learn meaningful lower-dimensional representations for the force-measurements that encode quantity-dependent effects. The surrogate model shall learn a meaningful relationship between said embeddings, roll positions and residual curvatures after straightening. The correction unit shall use the surrogate model to adapt the actuators based on the residual curvature. In the end, methods of explainable AI were discussed which are essential to verify the ML-module on whether it learned a meaningful relation between process and target parameters, as well as methods that allow for an evaluation of the embedding space.

**Acknowledgements**

The authors gratefully acknowledge the funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for the project "Data-Driven Modelling of Metal Bending Processes" funded in the frame of the DFG priority program 2422 "Data-driven process modelling in metal forming technology".

**References**

[1] M. Paech, Advanced semi-automatic straightening technology, Wire Journal International, vol. 41, no. 7 (2008), 74-79.

[2] K. Richter, F. Reuther, R. Müller, D. Landgrebe, Investigating the influence of bending parameters on the springback behavior of ultra-high strength spring strips, Materials Science Forum, vol. 918 (2018), 125-133. https://doi.org/10.4028/www.scientific.net/MSF.918.125

[3] M. Grüber, Konzepte zur Steuerung des Richtwalzprozesses bei variierenden Richtguteigenschaften, Dissertation, RWTH Aachen Universität, Verlagshaus Mainz GmbH Aachen, 2019.

[4] A. Amor, M. Rachik, H. Sfar, "Combination of finite-element and semi-analytical models for sheet metal leveling simulation," Key Engineering Materials, vol. 473 (2011), 182-189. https://doi.org/10.4028/www.scientific.net/KEM.473.182

[5] A. Trächtler, W. Homberg, Entwicklung eines intelligenten Einrichtassistenzsystems für Richtapparate, um die Prozessgrenzen der Verarbeitung hochfester Drahtwerkstoffe zu erweitern, Forschung für die Praxis P 1273 / IGF-Nr. 19799 N, 2021.

[6] G. Knoblauch, Steckverbinder: Systemkonzepte und Technologien, Expert Verlag, Esslingen, 2002.

[7] N. Auserhoff, Integrierte Produkt- und Prozessgestaltung für die Herstellung von Blechbiegeteilen, Shaker, 1999.
developed process representation is to be evaluated in terms of identifying and analysing the effects and interactions that occur within the process. This should enable an understanding of the complex multi-stage forming process as well as the design of such processes and the tools required for them.

## Summary

In this paper, a novel concept for data-driven, multi-stage straightening and bending processes using the straightener as an example was developed. After an extensive overview of existing modelling approaches for straightening and bending processes, the said concept and its interfaces were presented. It combines a HML-module which integrates a digital twin and machine learning with a multi-stage straightening and bending process. Thereby, the digital twin collects and manages process data from various sources and forwards it to a ML-module. It furthermore filters out irrelevant input data from the process as well as possible incorrect actuator corrections by the ML-module. The ML-module contains embeddings networks, a surrogate model and a correction unit. Hereby, the aim of the embedding networks is to learn meaningful lower-dimensional representations for the force-measurements that encode quantity-dependent effects. The surrogate model shall learn a meaningful relationship between said embeddings, roll positions and residual curvatures after straightening. The correction unit shall use the surrogate model to adapt the actuators based on the residual curvature. In the end, methods of explainable AI were discussed which are essential to verify the ML-module on whether it learned a meaningful relation between process and target parameters, as well as methods that allow for an evaluation of the embedding space.

## Acknowledgements

The authors gratefully acknowledge the funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for the project "Data-Driven Modelling of Metal Bending Processes" funded in the frame of the DFG priority program 2422 "Data-driven process modelling in metal forming technology".

## References

[1] M. Paech, Advanced semi-automatic straightening technology, Wire Journal International, vol. 41, no. 7 (2008), 74-79.

[2] K. Richter, F. Reuther, R. Müller, D. Landgrebe, Investigating the influence of bending parameters on the springback behavior of ultra-high strength spring strips, Materials Science Forum, vol. 918 (2018), 125-133. https://doi.org/10.4028/www.scientific.net/MSF.918.125

[3] M. Grüber, Konzepte zur Steuerung des Richtwalzprozesses bei variierenden Richtguteigenschaften, Dissertation, RWTH Aachen Universität, Verlagshaus Mainz GmbH Aachen, 2019.

[4] A. Amor, M. Rachik, H. Sfar, "Combination of finite-element and semi-analytical models for sheet metal leveling simulation," Key Engineering Materials, vol. 473 (2011), 182-189. https://doi.org/10.4028/www.scientific.net/KEM.473.182

[5] A. Trächtler, W. Homberg, Entwicklung eines intelligenten Einrichtassistenzsystems für Richtapparate, um die Prozessgrenzen der Verarbeitung hochfester Drahtwerkstoffe zu erweitern, Forschung für die Praxis P 1273 / IGF-Nr. 19799 N, 2021.

[6] G. Knoblauch, Steckverbinder: Systemkonzepte und Technologien, Expert Verlag, Esslingen, 2002.

[7] N. Auserhoff, Integrierte Produkt- und Prozessgestaltung für die Herstellung von Blechbiegeteilen, Shaker, 1999.
[8] S. Chatti, Optimierung der Fertigungsgenauigkeit beim Profilbiegen. Shaker, 1998.

[9] K. Siegert, Blechumformung: Verfahren, Werkzeuge und Maschinen. Springer-Verlag, 2015.

[10] U.-H. Damerow, Korrigierende Eingriffsmöglichkeiten in den Herstellungsprozess von Stanzbiegeteilen, Dissertation (D 466), Universität Paderborn, Shaker Verlag Aachen, 2017.

[11] L.-S. Henrich, Theoretische und experimentelle Untersuchungen zum Richtwalzen von Blechen, Dissertation, Universität-Gesamthochschule Siegen, TIM-Bericht Nr. T06-08.94, 1994.

[12] V. Shinkin, Arithmetical method of calculation of power parameters of 2n-roller straightening machine under flattening of steel sheet, CIS Iron and Steel Review, vol. 14 (2017), 22-27. http://dx.doi.org/10.17580/cisisr.2017.02.05

[13] L. Steinwender, A. Kainz, K. Krimpelstätter, K. Zeman, Numerical simulation of tension losses and reaction forces in tension levellers, Special Edition: 10th International Conference on Technology of Plasticity, ICTP (2011), 343-348.

[14] F. Martínez-de Pisón, R. Lostado, A. Pernia, R. Fernandez, Optimising tension levelling process by means of genetic algorithms and finite element method, Ironmaking & Steelpmaking, vol. 38, no. 1 (2011), 45-52. https://doi.org/10.1179/030192310X12700328926029

[15] C. F. Mertin, Experimentelle und numerische Analyse von mehrstufigen Stanzbiegeprozessen für Bauteile mit federnden Eigenschaften, Dissertation, RWTH Aachen Universität, Verlagshaus Mainz GmbH, Aachen, 2019.

[16] A. Forcellese, F. Gabrielli, R. Ruffini, Effect of the training set size on springback control by neural network in an air bending process, Journal of Materials Processing Technology, vol. 80 (1998), 493-500. https://doi.org/10.1016/S0924-0136(98)00122-8

[17] S. Liu, Y. Xia, Z. Shi, H. Yu, Z. Li, J. Lin, Deep learning in sheet metal bending with a novel theory-guided deep neural network, IEEE/CAA Journal of Automatica Sinica, vol. 8, no. 3 (2021), 565-581, 2021. https://doi.org/10.1109/JAS.2021.1003871

[18] J. Havinga, T. Van den Boogaard, F. Dallinger, P. Hora, Feedforward control of sheet bending based on force measurements, Journal of Manufacturing Processes, vol. 31 (2018), 260-272. https://doi.org/10.1016/j.jmapro.2017.10.011

[19] A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova, V. Kumar, Theory-guided data science: A new paradigm for scientific discovery from data, IEEE Transactions on knowledge and data engineering, vol. 29, no. 10 (2017), 2318-2331. https://doi.org/10.1109/TKDE.2017.2720168

[20] L. Bathelt, M. Scurk, E. Djakow, C. Henke, A. Trächtler, Novel straightening-machine design with integrated force measurement for straightening of high-strength flat wire, Sensors, vol. 23, no. 22: 9091 (2023). https://doi.org/10.3390/s23229091

[21] J. Li, Z. Wang, S. Zhang, Y. Lin, L. Jiang, J. Tan, Task incremental learning-driven Digital-Twin predictive modeling for customized metal forming product manufacturing process, Robotics and Computer-Integrated Manufacturing, Volume 85 (2023), 1-16. https://doi.org/10.1016/j.rcim.2023.102647

[22] X. Hu, L. Chu, J. Pei, W. Liu, J. Bian, Model complexity of deep learning: A survey, Knowledge and Information Systems, vol. 63 (2021), 2585-2619. https://doi.org/10.48550/arXiv.2103.05127

[23] Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature, vol. 521, no. 7553 (2015), 436-444. https://doi.org/10.1038/nature14539
[8] S. Chatti, Optimierung der Fertigungsgenauigkeit beim Profilbiegen. Shaker, 1998.

[9] K. Siegert, Blechumformung: Verfahren, Werkzeuge und Maschinen. Springer-Verlag, 2015.

[10] U.-H. Damerow, Korrigierende Eingriffsmöglichkeiten in den Herstellungsprozess von Stanzbiegeteilen, Dissertation (D 466), Universität Paderborn, Shaker Verlag Aachen, 2017.

[11] L.-S. Henrich, Theoretische und experimentelle Untersuchungen zum Richtwalzen von Blechen, Dissertation, Universität-Gesamthochschule Siegen, TIM-Bericht Nr. T06-08.94, 1994.

[12] V. Shinkin, Arithmetical method of calculation of power parameters of 2n-roller straightening machine under flattening of steel sheet, CIS Iron and Steel Review, vol. 14 (2017), 22-27. http://dx.doi.org/10.17580/cisisr.2017.02.05

[13] L. Steinwender, A. Kainz, K. Krimpelstätter, K. Zeman, Numerical simulation of tension losses and reaction forces in tension levellers, Special Edition: 10th International Conference on Technology of Plasticity, ICTP (2011), 343-348.

[14] F. Martínez-de Pisón, R. Lostado, A. Pernia, R. Fernandez, Optimising tension levelling process by means of genetic algorithms and finite element method, Ironmaking & Steelmaking, vol. 38, no. 1 (2011), 45-52. https://doi.org/10.1179/030192310X12700328926029

[15] C. F. Mertin, Experimentelle und numerische Analyse von mehrstufigen Stanzbiegeprozessen für Bauteile mit federnden Eigenschaften, Dissertation, RWTH Aachen Universität, Verlagshaus Mainz GmbH, Aachen, 2019.

[16] A. Forcellese, F. Gabrielli, R. Ruffini, Effect of the training set size on springback control by neural network in an air bending process, Journal of Materials Processing Technology, vol. 80 (1998), 493-500. https://doi.org/10.1016/S0924-0136(98)00122-8

[17] S. Liu, Y. Xia, Z. Shi, H. Yu, Z. Li, J. Lin, Deep learning in sheet metal bending with a novel theory-guided deep neural network, IEEE/CAA Journal of Automatica Sinica, vol. 8, no. 3 (2021), 565-581, 2021. https://doi.org/10.1109/JAS.2021.1003871

[18] J. Havinga, T. Van den Boogaard, F. Dallinger, P. Hora, Feedforward control of sheet bending based on force measurements, Journal of Manufacturing Processes, vol. 31 (2018), 260-272. https://doi.org/10.1016/j.jmapro.2017.10.011

[19] A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova, V. Kumar, Theory-guided data science: A new paradigm for scientific discovery from data, IEEE Transactions on knowledge and data engineering, vol. 29, no. 10 (2017), 2318-2331. https://doi.org/10.1109/TKDE.2017.2720168

[20] L. Bathelt, M. Scurk, E. Djakow, C. Henke, A. Trächtler, Novel straightening-machine design with integrated force measurement for straightening of high-strength flat wire, Sensors, vol. 23, no. 22: 9091 (2023). https://doi.org/10.3390/s23229091

[21] J. Li, Z. Wang, S. Zhang, Y. Lin, L. Jiang, J. Tan, Task incremental learning-driven Digital-Twin predictive modeling for customized metal forming product manufacturing process, Robotics and Computer-Integrated Manufacturing, Volume 85 (2023), 1-16. https://doi.org/10.1016/j.rcim.2023.102647

[22] X. Hu, L. Chu, J. Pei, W. Liu, J. Bian, Model complexity of deep learning: A survey, Knowledge and Information Systems, vol. 63 (2021), 2585-2619. https://doi.org/10.48550/arXiv.2103.05127

[23] Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature, vol. 521, no. 7553 (2015), 436-444. https://doi.org/10.1038/nature14539
[24] S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural computation, vol. 9, no. 8 (1997), 1735-1780. https://doi.org/10.1162/neco.1997.9.8.1735

[25] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin, Attention is all you need, Advances in neural information processing systems, vol. 30 (2017). https://doi.org/10.48550/arXiv.1706.03762

[26] C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning, The MIT Press, 2005. https://doi.org/10.7551/mitpress/3206.001.0001

[27] E. Pintelas, I. E. Livieris, P. Pintelas, A grey-box ensemble model exploiting black-box accuracy and white-box intrinsic interpretability, Algorithms, vol. 13, no. 1 (2020). https://doi.org/10.3390/a13010017

[28] C. Molnar, Interpretable machine learning, (2020). Information on https://christophm.github.io/interpretable-ml-book

[29] T. Miller, Explanation in artificial intelligence: Insights from the social sciences, Artificial intelligence, vol. 267 (2019), 1-38. https://doi.org/10.1016/j.artint.2018.07.007

[30] A. Fisher, C. Rudin, F. Dominici, All models are wrong, but many are useful: Learning a variable's importance by studying an entire class of prediction models simultaneously. J. Mach. Learn. Res., vol. 20, no. 177 (2019), 1-81. https://doi.org/10.48550/arXiv.1801.01489

[31] T. Rojat, R. Puget, D. Filliat, J. Del Ser, R. Gelin, N. Díaz-Rodríguez, "Explainable artificial intelligence (xai) on timeseries data: A survey," arXiv preprint arXiv:2104.00950 (2021). https://doi.org/10.48550/arXiv.2104.00950

[32] A. Ali, T. Schnake, O. Eberle, G. Montavon, K.-R. Müller, L. Wolf, Xai for transformers: Better explanations through conservative propagation, in International Conference on Machine Learning. PMLR (2022), 435-451. https://doi.org/10.48550/arXiv.2202.07304

[33] Z. Qin, F. Yu, C. Liu, X. Chen, How convolutional neural network see the world-A survey of convolutional neural network visualization methods, arXiv preprint arXiv:1804.11191 (2018). https://doi.org/10.3934/mfc.2018008

[34] K. Simonyan, A. Vedaldi, A. Zisserman, Deep inside convolutional networks: Visualising image classification models and saliency maps, arXiv preprint arXiv:1312.6034, 2013. https://doi.org/10.48550/arXiv.1312.6034
[24] S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural computation, vol. 9, no. 8 (1997), 1735-1780. https://doi.org/10.1162/neco.1997.9.8.1735

[25] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin, Attention is all you need, Advances in neural information processing systems, vol. 30 (2017). https://doi.org/10.48550/arXiv.1706.03762

[26] C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning, The MIT Press, 2005. https://doi.org/10.7551/mitpress/3206.001.0001

[27] E. Pintelas, I. E. Livieris, P. Pintelas, A grey-box ensemble model exploiting black-box accuracy and white-box intrinsic interpretability, Algorithms, vol. 13, no. 1 (2020). https://doi.org/10.3390/a13010017

[28] C. Molnar, Interpretable machine learning, (2020). Information on https://christophm.github.io/interpretable-ml-book

[29] T. Miller, Explanation in artificial intelligence: Insights from the social sciences, Artificial intelligence, vol. 267 (2019), 1-38. https://doi.org/10.1016/j.artint.2018.07.007

[30] A. Fisher, C. Rudin, F. Dominici, All models are wrong, but many are useful: Learning a variable's importance by studying an entire class of prediction models simultaneously. J. Mach. Learn. Res., vol. 20, no. 177 (2019), 1-81. https://doi.org/10.48550/arXiv.1801.01489

[31] T. Rojat, R. Puget, D. Filliat, J. Del Ser, R. Gelin, N. Díaz-Rodríguez, "Explainable artificial intelligence (xai) on timeseries data: A survey," arXiv preprint arXiv:2104.00950 (2021). https://doi.org/10.48550/arXiv.2104.00950

[32] A. Ali, T. Schnake, O. Eberle, G. Montavon, K.-R. Müller, L. Wolf, Xai for transformers: Better explanations through conservative propagation, in International Conference on Machine Learning. PMLR (2022), 435-451. https://doi.org/10.48550/arXiv.2202.07304

[33] Z. Qin, F. Yu, C. Liu, X. Chen, How convolutional neural network see the world-A survey of convolutional neural network visualization methods, arXiv preprint arXiv:1804.11191 (2018). https://doi.org/10.3934/mfc.2018008

[34] K. Simonyan, A. Vedaldi, A. Zisserman, Deep inside convolutional networks: Visualising image classification models and saliency maps, arXiv preprint arXiv:1312.6034, 2013. https://doi.org/10.48550/arXiv.1312.6034
# Fast prediction of the material displacement in open die forging using neural networks

JAGTAP Nikhil Vijay¹,ᵃ*, REINISCH Niklas¹,ᵇ and BAILLY David¹,ᶜ

¹Institute of Metal Forming, RWTH Aachen University, Intzestraße 10, 52072 Aachen, Germany

ᵃnikhil.jagtap@ibf.rwth-aachen.de, ᵇniklas.reinisch@ibf.rwth-aachen.de,

ᶜdavid.bailly@ibf.rwth-aachen.de

**Keywords:** Open Die Forging, Neural Networks, Data-Driven Modeling, FEA

**Abstract.** This paper presents a data-driven approach to predict the material displacement in open die forging using neural networks. Training data for different process parameters and workpiece geometries is generated using finite element simulations. A neural network architecture is designed that takes the process parameters and the coordinates of a point in the geometry as inputs and outputs the displacement of that point after the deformation. This is systematically implemented for open die forging, using relevant process information. The neural network model is trained and tested on various FEA-simulations for different process parameters and shows good accuracy and generalization. The model is also able to simulate multiple strokes of a single pass in a fast and efficient way. It is demonstrated how the neural network model can enable building a digital material shadow of open die forging processes. The advantages and limitations of the approach are then further discussed.

**Introduction**

A complete open die forging process is comprised of incrementally forming the workpiece in between saddles using multiple strokes. This can not only produce desired geometries but also improve material properties. The cast ingots that are the input to the forging often suffer from inherent undesirable microstructure including inhomogeneous, often large, grain size and orientation. Furthermore, segregation as well as cavities and pores could be the result of the casting process leading to insufficient quality of the casted workpiece. It is possible to remove these defects and produce finer microstructure using open die forging.

The flexibility offered by open die forging is exploited to manufacture a variety of semi-finished products like disks, rings, rods, shafts, hollow shafts, flanged shafts, crankshafts, and housings. However, these applications involve multiple manufacturing steps after forging to produce the final product. The workpiece geometry and the properties are transformed in each of this step. Integrating the relevant parameters like material properties between all these production steps is crucial for enabling knowledge-based decision-making in between the steps, quality assurance and long-term predicting e.g. the life of the final product. The large volume of data generated during each production step can be systematically shared and transformed throughout the process chain in a Digital Material Shadow (DMS) [1]. Even though, a lot of the material defects are located in the core region of a casted ingot, functional areas in most of the mentioned products are located mainly on the surface. Hence, the equivalent strain and geometry evolution outside the core is also essential for creating meaningful DMS. Additionally, the ability to simulate in short time can enable to trace every workpiece, thus capturing the deviations and inaccuracies in the individual process. This is especially important for highly loaded products like drive shafts.

Various methods have been used to calculate the strain and temperature evolution along with other parameters by simulating a forging process. Finite element analysis (FEA) is used for most numerical simulations e.g. for the macroscale analysis of the forging process in 2D and 3D [2]. FEA-simulations can be accurate but take larger amount of time and computation power to be able

<figure><img src="image_2160.png" /></figure>
# Fast prediction of the material displacement in open die forging using neural networks

JAGTAP Nikhil Vijay¹,ᵃ*, REINISCH Niklas¹,ᵇ and BAILLY David¹,ᶜ

¹Institute of Metal Forming, RWTH Aachen University, Intzestraße 10, 52072 Aachen, Germany

ᵃnikhil.jagtap@ibf.rwth-aachen.de, ᵇniklas.reinisch@ibf.rwth-aachen.de,

ᶜdavid.bailly@ibf.rwth-aachen.de

**Keywords:** Open Die Forging, Neural Networks, Data-Driven Modeling, FEA

**Abstract.** This paper presents a data-driven approach to predict the material displacement in open die forging using neural networks. Training data for different process parameters and workpiece geometries is generated using finite element simulations. A neural network architecture is designed that takes the process parameters and the coordinates of a point in the geometry as inputs and outputs the displacement of that point after the deformation. This is systematically implemented for open die forging, using relevant process information. The neural network model is trained and tested on various FEA-simulations for different process parameters and shows good accuracy and generalization. The model is also able to simulate multiple strokes of a single pass in a fast and efficient way. It is demonstrated how the neural network model can enable building a digital material shadow of open die forging processes. The advantages and limitations of the approach are then further discussed.

**Introduction**

A complete open die forging process is comprised of incrementally forming the workpiece in between saddles using multiple strokes. This can not only produce desired geometries but also improve material properties. The cast ingots that are the input to the forging often suffer from inherent undesirable microstructure including inhomogeneous, often large, grain size and orientation. Furthermore, segregation as well as cavities and pores could be the result of the casting process leading to insufficient quality of the casted workpiece. It is possible to remove these defects and produce finer microstructure using open die forging.

The flexibility offered by open die forging is exploited to manufacture a variety of semi-finished products like disks, rings, rods, shafts, hollow shafts, flanged shafts, crankshafts, and housings. However, these applications involve multiple manufacturing steps after forging to produce the final product. The workpiece geometry and the properties are transformed in each of this step. Integrating the relevant parameters like material properties between all these production steps is crucial for enabling knowledge-based decision-making in between the steps, quality assurance and long-term predicting e.g. the life of the final product. The large volume of data generated during each production step can be systematically shared and transformed throughout the process chain in a Digital Material Shadow (DMS) [1]. Even though, a lot of the material defects are located in the core region of a casted ingot, functional areas in most of the mentioned products are located mainly on the surface. Hence, the equivalent strain and geometry evolution outside the core is also essential for creating meaningful DMS. Additionally, the ability to simulate in short time can enable to trace every workpiece, thus capturing the deviations and inaccuracies in the individual process. This is especially important for highly loaded products like drive shafts.

Various methods have been used to calculate the strain and temperature evolution along with other parameters by simulating a forging process. Finite element analysis (FEA) is used for most numerical simulations e.g. for the macroscale analysis of the forging process in 2D and 3D [2]. FEA-simulations can be accurate but take larger amount of time and computation power to be able

<figure><img src="image_2151.png" /></figure>
to do so. One current alternative is the use of state-of-the-art fast models that calculate important workpiece parameters quickly. Beyond others, Recker et al. [3] presented a combined approach that can calculate equivalent strain as well as temperature and austenite grain size in the core of the open die forging workpiece. For equivalent strain, the model use Siemens approximation [4] whereas for grain size, the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is used [5]. The change in the global geometry of the workpiece can be calculated using e.g. the spread equations given by Tomlinson and Stringer [6] to superimpose these results. These models provide fast results contrary to the FEA-simulations and thus find applications in optimization and online monitoring [7]. Here, the expected process results can be visualized and enhanced by online process adaption. The results are only limited to the core fiber (Fig. 1), which is an imaginary longitudinal line passing through the center of the workpiece.

Besides full black box approaches, advanced data driven methods also offer ways to integrate the knowledge of the known physics with systematically collected data. By doing this the data efficiency of the models can be increased, which is especially useful for use cases with limited amount of data like open die forging. In this study, neural networks are used, which can be understood as highly nonlinear function approximator inspired by biological neurons [8]. According to the universal approximation theorem, neural networks can be capable of approximating any function [9]. Knapp [10] used neural networks to predict the spread in open die forging which is essential for optimal process control. Experimental data was used to train and test this model. Lee [11] used convolutional neural networks to design the preform geometry of axisymmetric closed-die forgings to improve strain distribution. Chan [12] used FEA to train a neural network for predicting parameters like forming load and effective stress for critical design parameters to evaluate the quality of the punch die design.

In this paper, the application of neural networks is studied for simulating the local displacement in 2D cross-section of the forging workpiece. Since various process results like equivalent strain can be inferred from the local displacement, this should enable the creation of a DMS in the future. Various process parameters like bite ratio (bite length b/ height h) and height reduction are used to define a process in open die forging and they directly influence the local deformation. These process parameters thus form an input to the neural networks suiting for a variety of pass schedules. Finally, the accuracy of the neural network model is compared with the FEA results.

## Data Collection and Feature Study

In a systematic implementation of machine learning using neural networks, the solution is highly dependent on the model architecture as well as the data the model is trained with. Both of these components can be modeled to convey the physical aspects of the manufacturing process [13]. The insights from the process can be embedded in the data through the data collection methodology and feature study. The data should represent the complete domain without any bias. The domain can be defined by identifying the key variables in open die forging that affect the displacement. Hence, a short discussion of the main influences on the displacement follows. The open die forging process consists of multiple forging passes, where the workpiece is typically rotated around the longitudinal axis mostly based on the target geometry. During each pass, the workpiece is incrementally forged usually from one end of the workpiece to the other through multiple strokes. Each stroke deforms the workpiece locally in between the saddles. All the strokes in a given pass are characterized by process parameters like bite ratio and height reduction. All the passes and their corresponding process parameters are listed in the pass schedule.

According to Knapp [10], the spread in the workpiece is influenced by the ratio of height to width (here referred to as the aspect ratio $\alpha$ of the lateral cross-section),

$$ \alpha_0 = \frac{h_0}{w_0} \qquad (1) $$
to do so. One current alternative is the use of state-of-the-art fast models that calculate important workpiece parameters quickly. Beyond others, Recker et al. [3] presented a combined approach that can calculate equivalent strain as well as temperature and austenite grain size in the core of the open die forging workpiece. For equivalent strain, the model use Siemens approximation [4] whereas for grain size, the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is used [5]. The change in the global geometry of the workpiece can be calculated using e.g. the spread equations given by Tomlinson and Stringer [6] to superimpose these results. These models provide fast results contrary to the FEA-simulations and thus find applications in optimization and online monitoring [7]. Here, the expected process results can be visualized and enhanced by online process adaption. The results are only limited to the core fiber (Fig. 1), which is an imaginary longitudinal line passing through the center of the workpiece.

Besides full black box approaches, advanced data driven methods also offer ways to integrate the knowledge of the known physics with systematically collected data. By doing this the data efficiency of the models can be increased, which is especially useful for use cases with limited amount of data like open die forging. In this study, neural networks are used, which can be understood as highly nonlinear function approximator inspired by biological neurons [8]. According to the universal approximation theorem, neural networks can be capable of approximating any function [9]. Knapp [10] used neural networks to predict the spread in open die forging which is essential for optimal process control. Experimental data was used to train and test this model. Lee [11] used convolutional neural networks to design the preform geometry of axisymmetric closed-die forgings to improve strain distribution. Chan [12] used FEA to train a neural network for predicting parameters like forming load and effective stress for critical design parameters to evaluate the quality of the punch die design.

In this paper, the application of neural networks is studied for simulating the local displacement in 2D cross-section of the forging workpiece. Since various process results like equivalent strain can be inferred from the local displacement, this should enable the creation of a DMS in the future. Various process parameters like bite ratio (bite length b/ height h) and height reduction are used to define a process in open die forging and they directly influence the local deformation. These process parameters thus form an input to the neural networks suiting for a variety of pass schedules. Finally, the accuracy of the neural network model is compared with the FEA results.

## Data Collection and Feature Study

In a systematic implementation of machine learning using neural networks, the solution is highly dependent on the model architecture as well as the data the model is trained with. Both of these components can be modeled to convey the physical aspects of the manufacturing process [13]. The insights from the process can be embedded in the data through the data collection methodology and feature study. The data should represent the complete domain without any bias. The domain can be defined by identifying the key variables in open die forging that affect the displacement. Hence, a short discussion of the main influences on the displacement follows. The open die forging process consists of multiple forging passes, where the workpiece is typically rotated around the longitudinal axis mostly based on the target geometry. During each pass, the workpiece is incrementally forged usually from one end of the workpiece to the other through multiple strokes. Each stroke deforms the workpiece locally in between the saddles. All the strokes in a given pass are characterized by process parameters like bite ratio and height reduction. All the passes and their corresponding process parameters are listed in the pass schedule.

According to Knapp [10], the spread in the workpiece is influenced by the ratio of height to width (here referred to as the aspect ratio $\alpha$ of the lateral cross-section),

$$ \alpha_0 = \frac{h_0}{w_0} \qquad (1) $$
the relative bite length (or bite ratio $x_b$),

$$x_b = \frac{b}{h_0} \qquad (2)$$

relative deformation (or height reduction $\varepsilon_h$),

$$\varepsilon_h = \frac{h_0 - h_1}{h_0} \qquad (3)$$

temperature and steel grade. As shown in Fig. 1, a long workpiece with rectangular cross-section with initial width $w_0$, height $h_0$ and length $l_0$ is considered. The longitudinal direction of the workpiece is along z-axis and the press moves along y-axis. The core fibre passes through the core of the workpiece in z-direction. The material spreads freely in the x-direction during each stroke, thus changing the aspect ratio of the rectangular cross-section.

<figure><img src="image_2161.png" /><figcaption>Fig. 1. A typical open die forging workpiece and saddle in side view (a) and front view (b) along with the respective coordinate systems.</figcaption></figure>

These three parameters, aspect ratio, bite ratio and height reduction given in Eq. 1, Eq. 2 and Eq. 3 form the variables of the study due to their influence on the spread as well as process description. The domain of each parameter or the *variable space* is defined in Table 1. These ranges are most commonly observed in practical forging processes. Each point in this variable space refers to a single *process* that is characterized by corresponding parameters.

<table><caption>Table 1. Important process parameters and their corresponding ranges that form the variable space.</caption><thead><tr><td>Variables</td><td>Range [-]</td></tr></thead><tbody><tr><td>α<sub>0</sub> - aspect ratio</td><td>0.9 to 1.5</td></tr><tr><td>x<sub>b</sub> - bite ratio</td><td>0.3 to 0.9</td></tr><tr><td>ε<sub>h</sub> - height reduction</td><td>5% to 26%</td></tr></tbody></table>

To begin with, two datasets are gathered using FEA for the complete variable space defined in the Table 1. The first, respectively the training dataset, is used for training and validation of the
the relative bite length (or bite ratio $x_b$),

$$x_b = \frac{b}{h_0} \qquad (2)$$

relative deformation (or height reduction $\varepsilon_h$),

$$\varepsilon_h = \frac{h_0 - h_1}{h_0} \qquad (3)$$

temperature and steel grade. As shown in Fig. 1, a long workpiece with rectangular cross-section with initial width $w_0$, height $h_0$ and length $l_0$ is considered. The longitudinal direction of the workpiece is along z-axis and the press moves along y-axis. The core fibre passes through the core of the workpiece in z-direction. The material spreads freely in the x-direction during each stroke, thus changing the aspect ratio of the rectangular cross-section.

<figure><img src="image_2152.png" /><figcaption>Fig. 1. A typical open die forging workpiece and saddle in side view (a) and front view (b) along with the respective coordinate systems.</figcaption></figure>

These three parameters, aspect ratio, bite ratio and height reduction given in Eq. 1, Eq. 2 and Eq. 3 form the variables of the study due to their influence on the spread as well as process description. The domain of each parameter or the *variable space* is defined in Table 1. These ranges are most commonly observed in practical forging processes. Each point in this variable space refers to a single *process* that is characterized by corresponding parameters.

<table><caption>Table 1. Important process parameters and their corresponding ranges that form the variable space.</caption><thead><tr><td>Variables</td><td>Range [-]</td></tr></thead><tbody><tr><td>α<sub>0</sub> - aspect ratio</td><td>0.9 to 1.5</td></tr><tr><td>x<sub>b</sub> - bite ratio</td><td>0.3 to 0.9</td></tr><tr><td>ε<sub>h</sub> - height reduction</td><td>5% to 26%</td></tr></tbody></table>

To begin with, two datasets are gathered using FEA for the complete variable space defined in the Table 1. The first, respectively the training dataset, is used for training and validation of the
neural networks. The sampling of the processes from the variable space was kept random yet ensuring sufficient representation of each variable using *latin hypercube sampling* (LHS). Altogether, 200 processes were sampled from the variable space for the training dataset. The second, respectively the test dataset is a collection of 54 processes from the variable space that are not part of the training dataset. The performance of the trained model is determined based on how accurate it predicts on the unseen test data. The accuracy of the model on this unseen dataset indicates the ability to generalize the underlying patterns in the data.

Each process sampled above is simulated for the corresponding values of aspect ratio, bite ratio and height reduction in a FEA software, *Simufact Forging 15.0*. Addressing the requirement of high accuracy data, a template simulation was setup in Simufact. The parameters that affect the quality and the correctness of the results are defined in this template. Mesh convergence was performed to select suitable mesh size optimized to the application. Symmetry planes were not used to allow more data points that can be augmented by mirroring. For simplicity, the material grade and starting temperature are fixed in this study. A commonly used alloy steel 42CrMo4 (1.7225) is chosen in this study. The furnace or starting temperature is set to 1100°C. The height and width of the workpiece depends on the aspect ratio, however for the square aspect ratio ($α_0$ = 1) the height of the workpiece is 100 mm. Each simulation includes three forging strokes to reach a steady deformation state, since the first two strokes exhibit deviating deformation due to their proximity to the ingots end. The simulation time increments are automatically determined by Simufact and the results of those at the end of each sub step, like at the end of the feed, end of the stroke and end of the backstroke, are saved.

A software program is developed in MATLAB, which creates copies of the template and writes the corresponding process parameters from the variable space to a new simulation. Later, it runs the simulation on the designated machine and the results are stored. A data handling program was written to manage this distributed database while extracting, transforming and loading the data on demand so that it can be used by neural networks for respectively training, validation and testing.

## Displacement Prediction Model based on Neural Networks

A process inspired architecture of the model, that incorporates features arising from the forging process, can bear meaningful results. Thus, the architecture is systematically developed by first studying the data. This method is highly iterative and the final model is a result of multiple adaptions in the data, model architecture and the hyper parameters in the training process. After each adaption, the network is tested for certain criteria derived from their importance to the forging process like error in the surface nodes. These criteria are mentioned in the next chapter. The following analysis of the features and model architecture refer to the final model.

This study is limited to a 2D plane in the direction of the press and passing through the core fiber (plane A-A in Fig. 1), which in these simulations is the yz-plane. Nevertheless, within the data collection, 3D FEA was used and the 2D cross-sections are extracted from 3D results. Fig. 2 shows the displacement in each y and z-directions for the third stroke in an example simulation ($α_0$=1.4, $x_b$=0.75, $ε_h$=17%). The deformation zone can be distinguished from the rest of the workpiece in this figure. Some nodes in the center of the deformation zone are not displaced in the z-direction, mainly due to friction. However, those outside show almost constant displacement indicating the increase in length in both the positive and negative z-direction. On the contrary, there is no displacement in y-direction in this region outside the deformation zone and a large movement of nodes due to the movement of press in y-direction.

The center of deformation changes based on the stroke number, bite ratio and height reduction. All the deformation zones can be positioned uniformly by moving the origin of the coordinate system to the left edge of the saddle. This way, all the nodes under the press will lie in positive z-direction starting from the origin (Fig. 1 and Fig. 2), enabling the model a reliable guess of the general material flow.
neural networks. The sampling of the processes from the variable space was kept random yet ensuring sufficient representation of each variable using *latin hypercube sampling* (LHS). Altogether, 200 processes were sampled from the variable space for the training dataset. The second, respectively the test dataset is a collection of 54 processes from the variable space that are not part of the training dataset. The performance of the trained model is determined based on how accurate it predicts on the unseen test data. The accuracy of the model on this unseen dataset indicates the ability to generalize the underlying patterns in the data.

Each process sampled above is simulated for the corresponding values of aspect ratio, bite ratio and height reduction in a FEA software, *Simufact Forging 15.0*. Addressing the requirement of high accuracy data, a template simulation was setup in Simufact. The parameters that affect the quality and the correctness of the results are defined in this template. Mesh convergence was performed to select suitable mesh size optimized to the application. Symmetry planes were not used to allow more data points that can be augmented by mirroring. For simplicity, the material grade and starting temperature are fixed in this study. A commonly used alloy steel 42CrMo4 (1.7225) is chosen in this study. The furnace or starting temperature is set to 1100°C. The height and width of the workpiece depends on the aspect ratio, however for the square aspect ratio ($α_0$ = 1) the height of the workpiece is 100 mm. Each simulation includes three forging strokes to reach a steady deformation state, since the first two strokes exhibit deviating deformation due to their proximity to the ingots end. The simulation time increments are automatically determined by Simufact and the results of those at the end of each sub step, like at the end of the feed, end of the stroke and end of the backstroke, are saved.

A software program is developed in MATLAB, which creates copies of the template and writes the corresponding process parameters from the variable space to a new simulation. Later, it runs the simulation on the designated machine and the results are stored. A data handling program was written to manage this distributed database while extracting, transforming and loading the data on demand so that it can be used by neural networks for respectively training, validation and testing.

## Displacement Prediction Model based on Neural Networks

A process inspired architecture of the model, that incorporates features arising from the forging process, can bear meaningful results. Thus, the architecture is systematically developed by first studying the data. This method is highly iterative and the final model is a result of multiple adaptions in the data, model architecture and the hyper parameters in the training process. After each adaption, the network is tested for certain criteria derived from their importance to the forging process like error in the surface nodes. These criteria are mentioned in the next chapter. The following analysis of the features and model architecture refer to the final model.

This study is limited to a 2D plane in the direction of the press and passing through the core fiber (plane A-A in Fig. 1), which in these simulations is the yz-plane. Nevertheless, within the data collection, 3D FEA was used and the 2D cross-sections are extracted from 3D results. Fig. 2 shows the displacement in each y and z-directions for the third stroke in an example simulation ($α_0=1.4$, $x_b=0.75$, $ε_h=17\%$). The deformation zone can be distinguished from the rest of the workpiece in this figure. Some nodes in the center of the deformation zone are not displaced in the z-direction, mainly due to friction. However, those outside show almost constant displacement indicating the increase in length in both the positive and negative z-direction. On the contrary, there is no displacement in y-direction in this region outside the deformation zone and a large movement of nodes due to the movement of press in y-direction.

The center of deformation changes based on the stroke number, bite ratio and height reduction. All the deformation zones can be positioned uniformly by moving the origin of the coordinate system to the left edge of the saddle. This way, all the nodes under the press will lie in positive z-direction starting from the origin (Fig. 1 and Fig. 2), enabling the model a reliable guess of the general material flow.
<figure><img src="image_2162.png" /><figcaption>Fig. 2. An example of a forging stroke ($\alpha_0=1.4$, $x_b=0.75$, $\epsilon_h=17\%$) and resulting displacements of nodes in y and z-direction respectively.</figcaption></figure>

Several neural network architectures are suitable for this type of problem. However, a simple feed-forward artificial neural network was used for this study. This type of network has a fixed number of inputs and outputs and is well suited to approximate non-linearity in the functions. Other commonly used image-based alternatives limit the inputs to a fixed square grid of pixels. But in the case of this problem, restricting workpieces of different lengths to a fixed image size will make it resolution dependent.

In this study, MATLAB is used to program the neural networks. The input, output and the structure of the neural network is determined by defining reference configuration and deformed configuration at $t=t_0$ and $t$ respectively. This is similar to the formulation of non-linear problems in continuum mechanics. In this case, deformed configuration is the workpiece after the force is applied and the material is deformed for the given stroke, whereas the reference configuration is the workpiece before the application of the force. Let P be a position vector to an arbitrary point in the workpiece such that in point P($x_0$, $y_0$) becomes $P_1(x_1, y_1)$ in the deformed configuration. The displacement of any point between the two configurations can be calculated as,

$$u = P_1 - P_0 \qquad (4)$$

Let f be any function that maps any point in the deformed configuration to its displacement. Additionally, this function depends on the process parameters that determine the deformation and therefore the displacement of any point in the given configuration. Thus, this gives the position of the point in the deformed configuration:

$$P_1 = P - f(P, \alpha_0, x_b, \epsilon_h) \qquad (5)$$

The function f is replaced by a neural network (final adaption consists of 16 layers with a total of 4800 neurons), as shown in Fig. 3. It can predict the coordinates in the deformed configuration from the process parameters and a given coordinate in the reference configuration. The output vector is the two components of the displacement which is then used to calculate the corresponding nodes in the deformed configuration. This network can run multiple times for each node in the single geometry to thus create a result for the complete body, since the input vector is only defined