# Real-time forging process control: integrating billet-related surrogate and machine behavior models

David Uribe, Camille Durand, Cyrille Baudouin, Régis Bigot

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David Uribe, Camille Durand, Cyrille Baudouin, Régis Bigot. Real-time forging process control: integrating billet-related surrogate and machine behavior models. Journal of Intelligent Manufacturing, 2025, 14p. 10.1007/s10845-025-02603-7. hal-05049326

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<figure><img src="image_1.png" /></figure>
# Real-time forging process control: integrating billet-related surrogate and machine behavior models

David Uribe<fnref n="1" />☐ Camille Durand<fnref n="1" />☐ Cyrille Baudouin<fnref n="1" />☐ Régis Bigot<fnref n="1" />☐

Received: 11 April 2024 / Accepted: 28 March 2025

© The Author(s) 2025

## Abstract

This study introduces a predictive surrogate model for real-time control in cold upsetting processes, incorporating both material and machine behaviors. Traditional approaches often simplify machine behavior as rigid or with constant stiffness; however, the proposed method dynamically couples material and machine responses, accounting for efficiency changes across different upsetting operations. This is achieved through the integration of a data-driven billet-related surrogate model with a machine-related analytical blow efficiency prediction, accurately capturing elastic energy losses. For the construction of the surrogate model in this use case, a multilayer perceptron artificial neural network (MLP ANN) was employed, demonstrating high predictive accuracy with a dataset comprising 2000 entries generated using Latin Hypercube Sampling (LHS) and numerical simulations. The model provides precise predictions for key outputs like forging load and plastic energy. Experimental validation shows prediction errors below 5% for energy setpoints, reduced to under 1% with blow efficiency correction. The general methodology of surrogate model creation can be adapted for various metal-forming processes, providing a versatile framework for real-time simulation and control.

**Keywords** Surrogate model · Forging processes · Real-time control · Blow efficiency · Numerical simulation · Data-driven model · Machine behavior

## Introduction

Forging processes play a crucial role in the manufacturing industry, enabling the production of high-strength and complex-shaped metal components. However, piloting the forging process presents challenges due to the inherent variabilities associated with the process (Douglas & Kuhlmann, 2000; Jia et al., 2023). These variabilities can arise from factors such as material properties, tool dynamics, and process

parameters, leading to uncertainties in the final product's quality and energy consumption (Wang et al., 2023).

It is crucial to employ highly responsive approaches to address the challenges in real-time piloting forging processes and improving process control, especially considering the dynamics of the forging machine. One such approach is the Closed-Loop Control system, such as the Generalized Predictive Control (GPC) (Dindorf et al., 2021; Dindorf & Wos, 2020). GPC utilizes a mathematical model of the system and employs a predictive algorithm to determine the control actions. By predicting the system's behavior and optimizing control inputs, GPC offers the potential for precise control of parameters such as temperature, pressure, and deformation. However, GPC requires an accurate model of the system, including the interactions with the material, which can be challenging to obtain in practice. Another approach to pilot forging processes is through adaptive control techniques, such as fuzzy logic systems or expert systems (Garth Frazier, 1998; Osakada et al., 1990; Osakada & Yang, 1991). These methods aim to adjust the control strategy based on real-time feedback from the forging process.

<figure><img src="image_2.png" /></figure>

David Uribe
david_santiago.uribe_espitia@ensam.eu

Camille Durand
camille.durand@ensam.eu

Cyrille Baudouin
cyrille.baudouin@ensam.eu

Régis Bigot
regis.bigot@ensam.eu

<footnote n="1">¹ Arts et Métiers Institute of Technology, HESAM Université,
Université de Lorraine, LCFC, Metz F-57070, France</footnote>
However, their design is subjective and heavily relies on the expertise and experience of the control engineer.

Furthermore, achieving efficient control of forging pro- cesses necessitates considering not only the material and process parameters but also the dynamics of the forging machine (Vajpayee & Sadek, 1978). The dynamics of the machine encompass its mechanical response and vibrations, which can substantially impact the quality of the forged components, altering the energy distribution throughout the process. Consequently, comprehending and optimizing the blow's efficiency becomes of utmost importance. The blow's efficiency directly affects the energy allocation in the pro- cess and can influence both the energy consumption and the quality of the forged components. Hence, precise prediction and optimization of the blow's efficiency emerge as criti- cal objectives for enhancing process efficiency and curtail- ing energy consumption (Mull et al., 2020). However, these dynamics have not yet been integrated into real-time appli- cations. Typically, blow efficiency is assessed post-opera- tion rather than predicted beforehand. This highlights the need to integrate real-time machine models with real-time product models. Furthermore, numerical simulations are often conducted under the assumption of rigid dies, which do not accurately account for the real behavior of the dies and the press during operation. This oversight can lead to suboptimal predictions and control strategies, emphasizing the importance of incorporating more realistic die behavior into the modeling framework. This approach aligns with the broader perspective outlined by (Zhang et al., 2019), which advocates for a comprehensive consideration of both infra- structure and material systems in process modeling.

Combining the inherent variabilities of forging processes with the energy losses resulting from the involved dynamics poses a dual and significant challenge. Effectively address- ing these issues within the control system is essential to ensure the production of high-quality components. In this context, a range of predictive control models are available, from analytical approaches like the slab method (Zhang et al., 2012), which offers reactivity but lower accuracy, to numerical simulations (Yin et al., 2021), which provide higher accuracy but lack in reactivity. To bridge the gap between accuracy and reactivity, an intermediate model type widely employed in the literature is the simulation- based surrogate model (Liu et al., 2022; Song et al., 2022).

To address the limitations of these individual models, combining analytical models with surrogate models has emerged as a promising approach. Analytical models can provide a solid foundation for understanding the underlying mechanical behavior of the forging process, while surrogate models can be used to speed up predictions and account for the complex, nonlinear interactions that might be difficult to model analytically. By coupling these two approaches,

we can achieve a model that balances the precision of FEM with the computational efficiency of surrogate models, lead- ing to better real-time control of forging processes. The integration of machine learning models, such as artificial neural networks (ANNs), with analytical models has also proven successful in improving prediction accuracy in other domains, such as offshore gas field modeling (Etesami et al., 2021).

Despite the use of various algorithms in the literature, no methodology has yet focused on integrating variable selection into surrogate model development (Alizadeh et al., 2020). This step is crucial, as the selection of influential process parameters varies with the application case. Exist- ing studies primarily concentrate on training algorithms (Hürkamp et al., 2020, 2020; Ryser & Bambach, 2021; Song et al., 2022), data preparation and database size (Liu et al., 2022; Slimani et al., 2023), or the use of model reduction techniques in their development (de Gooijer et al., 2021; Hamdaoui et al., 2014; Uribe et al., 2023).

In this context, this article presents a methodology for developing these product-related surrogate models and their integration with machine-related analytical models, with an application case in metal forming processes. This method- ology spans crucial stages, including sensitivity analysis, database creation, and surrogate model algorithm selection and development. The developed surrogate model is cou- pled with a machine analytic model, or domain knowledge model, which considers the overall stiffness of the machine and tooling to predict the blow's efficiency (Brecher et al., 2009; Vajpayee et al., 1979), thereby extending beyond numerical simulation predictions, typically made in nomi- nal tooling conditions. By integrating the blow's efficiency into the surrogate model, the accuracy of energy predictions is enhanced, improving overall process control. Both billet- related surrogate and machine-related analytical models enable the prediction of the energy setpoint, accounting for the elastic energy losses of the process.

## Materials

### Experimental setup

The mechanical screw press studied is a screw press LASCO® SPR400 of the VULCAIN platform in the LCFC laboratory at the Arts et Métiers in Metz (Durand et al., 2018). This press can provide a maximum forging energy of 28,9 kJ for a ram speed of 680 mm/s. A percent- age of this maximum energy must be given as a setpoint. This setpoint is adjustable from 1 to 100%.

During the forging process, the ram displacement and the load along the forging axis are acquired respectively with
three laser sensors and a strain gauge, all located on the lower die. For the lower and upper tools, dies with smooth flat surfaces were used, as it is typically used in open-die forging. In the tool holders, composite modular insulating plates were set up to ensure the tool's versatility and protect the sensors, allowing the tool to perform cold, warm, and hot-forging operations (see Fig. 1).

The operation studied is a cold single blow upsetting of a copper cylinder (Fig. 2).

For this operation, a cylindrical billet defined by its Initial Diameter (ID) and its Initial Height (IH) is forged along its axis of revolution until it reaches its Final Height (FH) (see Fig. 3).

## Numerical setup

In addition to the experimental setup, a 2D axisymmetric Finite Element Method (FEM) simulation was performed to investigate the sensitivity of the forging process to various input parameters and to generate a database for the development of a surrogate model (see Fig. 4). The simulations were conducted using the commercial software package

FORGE® by Transvalor. The FEM model employed tetrahedral meshing, with the number of finite elements ranging between 4000 and 8000, varying with the dimensions of the billets. The length of a mesh edge averaged 0.6 mm in the core of the billet and was refined to 0.3 mm in the external parts. Default FORGE® solver remeshing was activated during the simulation. The numerical setup was conducted with a rigid machine, and it is important to note that coupling with machine behavior is performed a posteriori. The average computation time is 15 min.

The numerical model was developed to represent the forging process, considering the capabilities of the screw press. This includes the forging tool, the workpiece, process parameters, and the appropriate boundary conditions (see Table 1). To capture the material behavior during forging, a reduced elastoviscoplastic Hansel-Spittel material model was employed in the simulation:

$$ \sigma_s = A \cdot e^{m1 \cdot T} \cdot \epsilon^{m2} \cdot \dot{\epsilon}^{m3} \cdot e^{\frac{m4}{\epsilon}} \quad (1) $$

Here, $\sigma_s$ denotes the flow stress, $\epsilon$ and $\dot{\epsilon}$ represent strain and strain rate, respectively; and $T$ stands for temperature.

<figure><img src="image_3.png" /><figcaption>Fig. 1 Diagrammatic representation of the experimental forging setup</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2 The screw press and the tool holder for the cold copper upsetting</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3 Diagrammatic representation of copper billet upsetting process: a) Before Upsetting, b) After Upsetting</figcaption></figure>

The material constants, namely $A, m_1, m_2, m_3, m_4$, are 411.19 MPa, -0.00121, 0.13, 0.01472, and 0.002 respectively. The numerical model underwent thorough calibration against experimental data to ensure its precision, adjusting

parameters such as the friction conditions, material rheology, Material physical properties are listed in Table 2.

Friction conditions follow a Coulomb-limited Tresca Model (D.-W. Zhang et al., 2021):
<figure><img src="image_6.png" /><figcaption>Fig. 4 Numerical setup: a) before upsetting. b) after upsetting</figcaption></figure>

<table><caption>**Table 1** Numerical model description</caption><thead><tr><th>Parameter's type</th><th>Parameters</th><th>Value/Range</th></tr></thead><tbody><tr><td>Thermal</td><td>Initial temperature of the Billet<br>Initial temperature of the Tool<br>Room temperature</td><td>20 °C</td></tr><tr><td>Billet geometry</td><td>Initial Diameter<br>Upsetting ratio (Height/Diameter)</td><td>[15–35] mm<br>[1.5–2.5]</td></tr><tr><td>Piloting parameters</td><td>Energy Setpoint</td><td>(0–11.56] kJ<br>– (0–40] %</td></tr><tr><td rowspan="3">Boundary conditions</td><td>Heat Transfer coefficient Billet-tool</td><td>2000<br>W/m<sup>2</sup>K</td></tr><tr><td>Heat transfer coefficient Billet-room</td><td>10 W/m<sup>2</sup>K</td></tr><tr><td>Coulomb-Tresca friction model</td><td>m&#773;= 0.2, μ<br>= 0.1</td></tr></tbody></table>

<table><caption>**Table 2** Physical properties of copper billet at 20 °C</caption><thead><tr><th>Material</th><th>Young's Modulus (MPa)</th><th>Poisson's ratio</th><th>Density (kg/m<sup>3</sup>)</th><th>Thermal Conductivity (W/m-K)</th><th>Specific Heat (J/kg-K)</th></tr></thead><tbody><tr><td>Pure Copper</td><td>110,000</td><td>0.3</td><td>8100</td><td>401</td><td>435</td></tr></tbody></table>

$$ \tau = \min \left( \mu \cdot \sigma_n; \bar{m} \cdot \frac{\sigma_s}{\sqrt{3}} \right) \qquad (2) $$

where $\sigma_n$ is the contact pressure; $\mu$ and $\bar{m}$ the Coulomb and Tresca friction coefficients, respectively.

## Methods

While several authors have developed FEM-based surrogate models for various manufacturing applications, to our knowledge, no methodology has yet incorporated a variable selection step or sensitivity analysis for creating these models, focusing instead on data processing, data reduction, and training algorithms. This study advances the field by introducing a methodology that incorporates sensitivity analysis for variable selection and integrates product-related surrogate models with machine-related analytical models. Additionally, the approach separates the simulation of the billet using a rigid machine and incorporates machine behavior, allowing for the dissociation of behaviors and the independent qualification of different components before facilitating their interaction.

Following the numerical simulation, the methodology encompasses the following stages (see Fig. 5):

1. *Sensitivity Analysis*, identifying the most influential process variables, guiding the selection of inputs for the surrogate model,
<figure><img src="image_7.png" /><figcaption>III) Database creation</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>II) Sensitivity analysis</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>IV) Model training and validation</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 5** Methodology for surrogate model creation in metal forming processes</figcaption></figure>

2. *Database Construction*, using a customized Design of Experiments,

3. *Surrogate Model Training*, using an interpolation or regression algorithm.

In our specific case, an additional step is undertaken, involv-
ing the coupling of the surrogate model with an analytical
counterpart to enhance prediction accuracy.

This multi-step methodology ensures the robustness and
the applicability of the surrogate model, thus enabling accu-
rate predictions and valuable insights in the area of metal-
forming processes.

Sensitivity analysis

The evaluation of the local dimensionless SA for both phys-
ical and model-related factors around specific output vari-
ables (such as the billet’s Final Height or Maximum Load) enables the identification of the most influential parameters in the process. This analysis helps in selecting the variables that will be integrated into the surrogate model (Liu et al., 2023).

For this, physical variables, such as billet’s temperatures
and billet dimensions, as well as model-related factors, such
as workpiece rheology and coefficients of friction, must be
exhaustively listed.

Once these parameters are established, it is crucial to
quantify their sensitivity on the final state of the part. In
our case, the influence on the Final Height of the billet is
studied.

A SA can be conducted to compare the influence of
different process variables. Various techniques can be
employed, including screening, global, and local analysis.
However, in metal forming cases, applying screening and
global approaches might pose challenges due to limited
knowledge of the range of variation for certain numerical
variables, such as rheological law coefficients, and the asso-
ciated uncertainty surrounding these values.

In this context, a local SA appears to be a suitable choice
for conducting such a comparison. This approach involves
perturbing the process variables around their nominal val-
ues, to allow a more focused examination of their impact.

In this regard, the following local adimensional SA is
employed:
$$S_{o,i} = \frac{\Delta Y_{o,i} \hat{X}}{\Delta X_i \hat{Y}_o} \qquad (3)$$

The sensitivity analysis S of the model output o against the input i is represented by $S_{o,i}$. The specific parameter variation $\Delta X_i = X_i - \hat{X}_i$. The output $Y_o$ variation of the model given the $X_i$ is represented by $\Delta Y_{o,i} = Y_{o,i} - \hat{Y}_o$. The nominal input i and output o values are $\hat{X}_i$ and $\hat{Y}_o$, respectively.

A Pareto diagram could illustrate the results of the SA (Mizgier, 2017). In this phase, the inputs and outputs of the model are chosen, which are all scalar variables studied in this SA. Additionally, given our core goal of process control, the machine's piloting variable becomes a necessary prediction. In the case of our screw press application, the energy setpoint stands out as a vital and obligatory output variable.

## Database creation

The numerical simulation described earlier is utilized to generate the database for constructing a surrogate model. During this phase, DoE techniques allow to systematically investigate the relationship between input variables and output responses. The full factorial design involves a comprehensive exploration of each variable's effects and interactions across the entire input space (Kroiß et al., 2013; Ryser et al., 2023), and the reduced factorial design, strategically selecting a subset of input variable combinations while capturing key factors and interactions, is commonly employed (Wiebenga et al., 2013, 2015). However, increasing variables lead to exponential simulations, posing computational complexity and time limitations. To address this, the reduced factorial design is employed, significantly reducing computational burden while maintaining crucial insights. Factors and their levels are chosen based on the surrogate model's type. The size and number of factors considered in the database creation phase are customized to ensure the effective capturing of underlying relationships, enabling accurate predictions and efficient analysis. At this initial stage, the determination of the number of points to be sampled is arbitrary. Nonetheless, certain authors have suggested starting with at least 10 points per active variable when employing a reduced factorial design (Schonlau, 1997). This recommendation aims to balance the acquisition of sufficient data for accurate modeling while considering computational time constraints. Additionally, if the surrogate model's performance is considered insufficient, additional points can be gradually added to enhance its accuracy and predictive capability.

## Surrogate model training

Given that the model's inputs and outputs are scalar values, a fitting surrogate algorithm tailored to scalar prediction is implemented. The multilayer perceptron artificial neural networks (MLP-ANN) appear well-matched for various metal-forming process applications, especially when numerical simulations enable efficient database acquisition, as demonstrated by (Slimani et al., 2023). This substantiates our choice to employ this algorithm for scalar prediction in our particular case. Nevertheless, alternative regression or interpolation techniques could be explored, contingent on factors such as problem characteristics, data availability, and input/output parameter count (Hou & Behdinan, 2022).

The model is trained using the dataset generated from numerical simulations, which includes input variables and corresponding output responses. To enhance training efficiency, prevent gradient-related issues, and improve overall model stability and performance, the dataset is normalized (Shanker et al., 1996). Subsequently, it is partitioned into training and validation subsets.

There is no specific rule for the number of hidden layers and neurons in each layer of a Multi-Layer Perceptron Artificial Neural Network (MLP-ANN); these parameters are typically determined empirically. However, to fine-tune the network architecture, various hyperparameter optimization algorithms can be employed, such as Grid Search, Random Search, Hyperband, and Bayesian Optimization (Agrawal, 2021). These algorithms facilitate the systematic evaluation of different configurations, including varying numbers of hidden layers, neuron counts per layer, activation functions, and training optimizers. This approach aids in identifying the most effective network setup.

The learning process for the MLP-ANN utilizes the backpropagation algorithm. In this algorithm, the neurons' weights are updated in the gradient descent of the cumulative network error.

For regression tasks, the ReLU activation function is commonly recommended for hidden layers, while the linear activation function is suitable for the output layer (Goodfellow et al., 2016). To enhance generalization and prevent overfitting, dropout can be applied.

## Application: development of a surrogate model for a cold upsetting process

### Sensitivity analysis

Table 3 presents the list of physical and model-related factors employed in the SA. Each parameter was perturbed by 15% around its nominal value to assess its influence. The
<table><caption>**Table 3** List of parameters and outputs for the sensitivity analysis</caption><thead><tr><th>Parameter's Type</th><th>Parameters</th><th>Outputs</th></tr></thead><tbody><tr><td rowspan="2">Thermal</td><td>Initial temperature of the Billet (T<sub>0billet</sub>)</td><td>Billet's Final Height (FH)</td></tr><tr><td>Initial temperature of the Tool (T<sub>0tool</sub>)</td><td>Maximum forging load (F<sub>max</sub>)</td></tr><tr><td rowspan="3">Billet rheology</td><td>Room temperature (T<sub>room</sub>)</td><td></td></tr><tr><td>Hansel-Spittel's Law Parameters (A<sub>1</sub>, m<sub>1</sub>, m<sub>3</sub>, m<sub>4</sub>)</td><td></td></tr><tr><td>Young Modulus, Poisson's ratio (E<sub>m</sub>, ν)</td><td></td></tr><tr><td rowspan="2">Billet geometry</td><td>Initial Height (IH)</td><td></td></tr><tr><td>Initial Diameter (ID)</td><td></td></tr><tr><td>Piloting parameters</td><td>Energy Setpoint (E)</td><td></td></tr><tr><td rowspan="3">Boundary conditions</td><td>Heat Transfer coefficient Billet-tool (φ<sub>B-T</sub>)</td><td></td></tr><tr><td>Heat transfer coefficient Billet-room (φ<sub>B-R</sub>)</td><td></td></tr><tr><td>Coulomb-Tresca friction model (μ, μ̄)</td><td></td></tr></tbody></table>

results of the SA were presented using a Pareto Chart, highlighting in red the parameters chosen for inclusion in the surrogate model (see Fig. 6).

The SA supports the choice of the input and output variables of the surrogate model. It includes three input parameters: Initial Diameter, Initial Height, and Final Height. The output parameters of interest were the Plastic Energy and the Maximum Forging Load.

## Database creation

For the DoE, a Latin Hypercube Sampling (LHS) approach was employed, consisting of 50 simulations. The variable

bounds were defined according to the physical constraints and design specifications of the process.

The Initial Diameter was varied within the range of 15 to 35 mm. The Initial Height was defined based on a specific slenderness ratio (Initial Height/Initial Diameter), ranging from 1.5 to 2.5 times the Initial Diameter to mitigate the risk of buckling during the upsetting process. The Energy parameter was constrained to a range of 1 to 40% of the machine's capabilities, depending on the size of the billets.

By implementing the Hypercube Latin approach with these predefined bounds, the DoE enabled comprehensive exploration of the parameter space while considering the physical and design constraints of the process. The resulting simulations generated a diverse dataset encompassing various combinations of five input and output parameter values (Initial Height, Initial Diameter, Energy, Final Height, and Maximum Forging Load). In total, 2000 simulations were executed, resulting in a comprehensive database of 2000 input-output combinations.

## Surrogate model training

The input and output data underwent a standardization process before training for consistent and normalized representation. The network was built using the Keras API within the TensorFlow framework in Python. GridSearchCV algorithm (Ma et al., 2023) has been chosen for the hyperparameter optimization process, various configurations were tested, including networks with 1, 2, and 3 hidden layers, as well as varying numbers of neurons in each hidden layer (48, 64, 128 and 254). Different activation functions were evaluated, including ReLU, Tanh, and Sigmoid. Additionally, both the ADAM and SGD optimizers were assessed.

The results from GridSearchCV revealed that the optimal configuration consisted of a neural network with two hidden layers, each containing 128 neurons and utilizing the ReLU activation function. The dropout rate was set to 0.2, and the ADAM optimizer was used. This configuration yielded the

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 6** Pareto chart for the sensitivity analysis around a) Billet's Final Height, b) Maximum Forging Load</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 7** Surrogate Model's (MLP-ANN) I/O architecture</figcaption></figure>

best performance based on the mean squared error (MSE) metric (see Fig. 7).

The computation time of the trained model is below 50 ms. The training metrics are presented in Fig. 8a. The training results are compared to the FEM results in Fig. 8b.

## Surrogate model results

To validate the accuracy of the surrogate model, an experimental campaign was conducted. Eight cylindrical billets with two different Initial Diameters, namely 32 mm and 18 mm, and different Initial Heights were selected for forging. This experimental campaign aimed to compare the actual results from the physical forging process with the predictions provided by the surrogate model. The energy setpoint for the experimental tests was carefully selected, ensuring a compression ratio below 50%. This energy setpoint selected, along with the maximum forging load measured were then compared to the energy setpoint and the maximum forging load predicted by the surrogate model. The prediction inputs were the initial conditions of the billet (Initial Height and Diameter), along with the measured experimental Final Height. Results are presented in Fig. 9.

Besides, comparing the surrogate model's maximum load prediction and the FE model prediction, the errors are below 2% corresponding to 4kN.

<figure><img src="image_14.png" /><figcaption>**Fig. 8** Surrogate Model Training: **a)** Evolution of loss function, **b)** Surrogate Model vs. FEM</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 9 Surrogate model prediction errors: a) Errors, b) Percentage Errors</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 10 Coupled model architecture</figcaption></figure>

## Coupling of surrogate and analytical models

The current model's prediction of the energy does not account for the elastic energy losses within the metal-forming process, as the simulation model does not consider the stiffness of the machine and tool. These losses can significantly affect overall energy consumption, process efficiency, and particularly, the accuracy of the energy setpoint prediction by the standalone surrogate model (Mull et al., 2020). Therefore, to further enhance the accuracy and comprehensiveness of the model, it is crucial to incorporate a blow efficiency prediction through an analytical model that considers the overall stiffness of the system. Furthermore, this late incorporation of machine stiffness will facilitate easier adaptation in cases involving modular tooling, eliminating the need to overhaul the entire surrogate model for modifications.

The first-order approximation model proposed by (Vajpayee et al., 1979; Vajpayee & Sadek, 1978) is used for the blow efficiency prediction:

$$ Blow efficiency = 1 - \frac{F_{max}^2}{2 * K_{eq} * E} \qquad (4) $$

Where $F_{max}$ is the maximum forming load, $K_{eq}$ is the equivalent stiffness of the system, and the energy setpoint is denoted by $E$. The surrogate model provides a prediction

for the maximum forming load and the energy setpoint. The value of the overall stiffness was extracted from (Mull et al., 2021), for the same experimental setup. The corrected energy setpoint is finally calculated using the equation:

$$ E_{corrected} = \frac{E}{Blow Efficiency} \qquad (5) $$

where the blow efficiency is obtained from the analytical model and the energy setpoint is taken from the surrogate model. The coupled model architecture is illustrated in Fig. 10.

By combining the surrogate model's energy prediction with the calculated blow efficiency, the actual energy setpoint, accounting for elastic losses in the forging process, was determined. The results of the coupled model's prediction are compared to the same experimental campaign as before (see Figs. 11 and 12).

## Discussion

The findings from the SA support the rational reduction of parameters for inclusion in the surrogate model, as demonstrated by effective performance achieved using just three inputs for the upsetting process.
<figure><img src="image_17.png" /><figcaption>Fig. 11 Coupled model and Surrogate Model vs. Experimental Data: a) Errors, b) Percentage Errors</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 12 Coupled model and surrogate model vs. Experimental Data: a) Energy Prediction in kJ for ID=32 mm. b) Energy Prediction in kJ for ID=18 mm</figcaption></figure>

The use of LHS in the DoE demonstrated its effectiveness in generating a comprehensive and efficient training dataset for the surrogate model. The well-distributed sampling allowed for a thorough exploration of the parameter space, ensuring that the model could capture the relationships between the input variables and the output responses. Moreover, the reduced computational requirements made the approach computationally feasible and resource-saving.

The training of the surrogate model, employing an MLP ANN, yields precise outcomes. Nevertheless, similar accuracy might have been attained using alternative surrogate models, potentially with a simpler implementation. The selection of MLP ANN is substantiated by both the preferences of various authors and the substantial 2000-size database, which have been proven sufficient for the training phase. The number of neurons of the MLP ANN developed follows the recommendation made by (Rao & Prasad, 1995):

$$h = \frac{N^{\circ} \text{Samples}}{10 * (N^{\circ} \text{Inputs} + N^{\circ} \text{Outputs})} \quad (6)$$

where h is the minimum number of neurons recommended, resulting in a specific count of 40 neurons for our particular scenario.

In contrast to conventional numerical 2D simulations for these upsetting processes, the proposed model demonstrates exceptional responsiveness, achieving computation times roughly 900 times faster than rigid-machine numerical simulations. It completes calculations in less than 1 s while preserving high-quality results.

The surrogate's prediction for maximum load consistently yields accurate results, with errors remaining below 4% which corresponds to 8kN compared to experimental data, and errors below 2% or 4kN compared to FE results. Compared to traditional analytical methods such as the slab method, our model offers superior accuracy, notably overcoming the slab method's tendency to underestimate the maximum forging load and where a correction term is necessary for better results (Foster et al., 2009). Nevertheless, it's important to mention that the slab method allows for the inclusion of the friction coefficient as a parameter, a capability not employed in our case due to its absence in the DoE.

The integration of the blow efficiency prediction into the surrogate model further enhanced the precision of the results. By incorporating the maximum load prediction and accounting for the press and tools' equivalent stiffness for calculating the blow's efficiency, the coupled model achieved even greater accuracy, with errors reduced to below 1% or 0.05 kJ in the energy setpoint predictions. It's worth mentioning that our previous studies for the same forging
process have shown errors below 0.2 kJ for the energy set-
point prediction (Uribe et al., 2023), indicating that the cor-
rection model in this case is effectively employed, leading
to improved accuracy.

The proposed methodology allows for the integration
of scalar variables as both inputs and outputs in the pre-
diction model. However, it does not directly accommodate
multidimensional variables, which are common in forg-
ing processes (e.g., temperature maps, deformation fields,
stress fields). Integrating these multidimensional variables
into surrogate model development would require an addi-
tional model reduction step, as suggested by various authors
(Dang et al., 2017; de Gooijer et al., 2021; Hamdaoui et al.,
2013).

Although the methodology provides a framework for
surrogate model development in forging operations, several
key considerations warrant attention. The performance of
the surrogate model is closely tied to the quality and com-
prehensiveness of the training dataset. Despite the imple-
mentation of LHS, there remains a potential risk of not
capturing critical parameter interactions, especially in com-
plex, high-dimensional contexts. Furthermore, although the
MLP-ANN demonstrates strong performance in various
applications in literature, it may not be universally opti-
mal; alternative models might provide simpler implementa-
tion or superior performance for specific applications. The
integration of the surrogate model with the analytical blow
efficiency model, while improving prediction accuracy,
introduces additional complexity and necessitates precise
knowledge of the system's equivalent stiffness, which may
not always be available. Additionally, the methodology's
focus on a cold one-blow upsetting process may limit its
direct applicability to other metal-forming processes, par-
ticularly those involving more dynamic or multi-step opera-
tions. These considerations underscore the necessity for
careful adaptation and interpretation of the methodology
when applied to varied forging contexts.

Conclusions

This study presents a methodology for developing coupled
models that enhance the real-time control of forging opera-
tions. This approach involves the creation of a surrogate
model specifically focused on billet-related parameters,
which is then integrated with a machine-related analytical
model. By considering elastic energy losses during the forg-
ing process, the methodology significantly improves control
predictions, leading to more accurate and efficient outcomes
in forging applications.

Sensitivity analysis is crucial to the surrogate model's
adaptability, enabling the selection of process parameters

based on the desired output variable, which varies with the
specific application. The surrogate model's database is gen-
erated using a Latin Hypercube Sampling (LHS) approach
and is constructed using a multilayer perceptron artificial
neural network. The methods for database creation and sur-
rogate model training are grounded in various studies from
the literature. Traditionally, blow efficiency calculations in
forging operations were performed post hoc by integrating
data from numerical or analytical models into a machine
model. However, this methodology allows for efficient
blow efficiency calculations to be made before the forging
process begins by coupling a billet-related surrogate model
with a machine-related analytical model.

The methodology has been validated through a single-
blow upsetting process, demonstrating its real-time predic-
tivity and accuracy. Specifically, the model's performance
was evaluated, revealing errors below 5% for the energy
setpoint prediction. Furthermore, the precision of the model
was notably enhanced by incorporating elastic energy loss,
a critical factor in the overall energy distribution, resulting
in errors below 1% for the energy setpoint. Additionally,
the model's flexibility allows for seamless adaptation to
variations in equivalent stiffness (press and tool), enabling
its application across diverse tool configurations without
necessitating alterations to the surrogate model itself.

To further enhance its capabilities, future research should
focus on incorporating additional energy losses such as fric-
tion or damping, as well as temperature fields in hot forging
processes. Extending the model to predict microstructure
evolution and deformation fields beyond the Final Height
will offer a more holistic understanding of the forging pro-
cess. Additionally, developing a model for multiple blow
operations considering material work hardening during suc-
cessive blows will improve the accuracy of predictions in
complex forging scenarios.

**Acknowledgements** We would like to express our sincere thanks to the Technical Center for Mechanical Industry (CETIM) for their financial support in this research project. Specifically, we would like to thank Valérie SULIS, Stéphane MAGRON, and Pierre KRUMPIPE for their project follow-up and advice. Additionally, we wish to express our thanks to ISEETECH for the provision of the VULCAIN Platform facilities. Finally, we would like to thank Daniel BOEHM and Alexandre FENDLER for their technical support during the various tests conducted.

**Author contributions** DU: Investigation, Data curation, Software, Writing-original draft, Writing-review & editing; CD: Methodology, Validation, Writing-review & editing; CB: Conceptualization, Formal Analysis, Writing- review & editing; RB: Resources, Supervision, Writing-review & editing, Funding acquisition.

**Funding** Open access funding provided by Arts et Metiers Institute of Technology.
This study was funded by the Technical Center for Mechanical Industry (CETIM) and the Carnot Institut ARTS (Research Actions for
Technology and Society).

Declarations

**Competing interests** The authors have no competing interests to declare that are relevant to the content of this article.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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