[7] G. Liu, A.T. Vu, O. Dambon, F. Klocke, Glass Material Modeling and its Molding Behavior, MRS Advances 2 (2017) 875-885. https://doi.org/10.1557/adv.2017.64

[8] F. Wang, Simulating the precision glass molding process. Zugl.: Aachen, Techn. Hochsch., Diss., 2013, 1st ed., Apprimus-Verl., Aachen, 2014.

[9] A.T. Vu, T. Grunwald, T. Bergs, Thermo-viscoelastic Modeling of Nonequilibrium Material Behavior of Glass in Nonisothermal Glass Molding, Procedia Manufacturing 47 (2020) 1561. https://doi.org/10.1016/j.promfg.2020.04.350

[10] T.D. Pallicity, A.T. Vu, K. Ramesh, P. Mahajan, G. Liu, O. Dambon, Birefringence measurement for validation of simulation of precision glass molding process, J Am Ceram Soc 100 (2017) 4680-4698. https://doi.org/10.1111/jace.15010

[11] F. Wang, Y. Chen, F. Klocke, G. Pongs, A.Y. Yi, Numerical Simulation Assisted Curve Compensation in Compression Molding of High Precision Aspherical Glass Lenses, Journal of Manufacturing Science and Engineering 131 (2009). https://doi.org/10.1115/1.3063652

[12] A.T. Vu, Modeling Relaxation Nature of Nonequilibrium Glass in Nonisothermal Glass Molding. Dissertation, 2023.

[13] A.T. Vu, A.N. Vu, G. Liu, T. Grunwald, O. Dambon, F. Klocke, T. Bergs, Experimental investigation of contact heat transfer coefficients in nonisothermal glass molding by infrared thermography, J Am Ceram Soc 102 (2019) 2116-2134. https://doi.org/10.1111/jace.16029

[14] A.T. Vu, T. Helmig, A.N. Vu, Y. Frekers, T. Grunwald, R. Kneer, T. Bergs, Numerical and experimental determinations of contact heat transfer coefficients in nonisothermal glass molding, J Am Ceram Soc 103 (2020) 1258-1269. https://doi.org/10.1111/jace.16756

[15] A.T. Vu, R.d.l.A. Avila Hernandez, T. Grunwald, T. Bergs, Modeling nonequilibrium thermoviscoelastic material behaviors of glass in nonisothermal glass molding, J Am Ceram Soc 105 (2022) 6799-6815. https://doi.org/10.1111/jace.18605

[16] A.T. Vu, T. Grunwald, T. Bergs, Friction in Glass Forming: Tribological Behaviors of Optical Glasses and uncoated Steel near Glass Transition Temperature, J Non-Cryst Solids (2024).

[17] B. Shahriari, K. Swersky, Z. Wang, R.P. Adams, N. de Freitas, Taking the Human Out of the Loop: A Review of Bayesian Optimization, Proc. IEEE 104 (2016) 148-175. https://doi.org/10.1109/JPROC.2015.2494218

[18] S. Greenhill, S. Rana, S. Gupta, P. Vellanki, S. Venkatesh, Bayesian Optimization for Adaptive Experimental Design: A Review, IEEE Access 8 (2020) 13937-13948. https://doi.org/10.1109/ACCESS.2020.2966228

[19] S. Ranftl, W. von der Linden, Bayesian Surrogate Analysis and Uncertainty Propagation, in: The 40th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MDPI, Basel Switzerland (2021) 6. https://doi.org/10.3390/psf2021003006

[20] F. Mekki-Berrada, Z. Ren, T. Huang, W.K. Wong, F. Zheng, J. Xie, I.P.S. Tian et al., Two-step machine learning enables optimized nanoparticle synthesis, npj Comput Mater 7 (2021). https://doi.org/10.1038/s41524-021-00520-w

[21] Y. Tian, M.K. Luković, T. Erps, M. Foshey, W. Matusik, AutoOED: Automated Optimal Experiment Design Platform, arXiv, 2021.

[22] M.K. Luković, Y. Tian, W. Matusik, Diversity-Guided Multi-Objective Bayesian Optimization with Batch Evaluations, in: Proceedings of the 34th International Conference on Neural Information Processing Systems, Curran Associates Inc, Red Hook, NY, USA, 2020.

[23] P. Saves et al., SMT 2.0: A Surrogate Modeling Toolbox with a focus on hierarchical and mixed variables Gaussian processes, Advances in Engineering Software 188 (2024) 103571. https://doi.org/10.1016/j.advengsoft.2023.103571
# VForm-xSteels: virtual materials database

ANDRADE-CAMPOS António<sup>1,a*</sup>, CAMPOS Afonso<sup>1,b</sup>, HENRIQUES João<sup>1,c</sup>, FILHO Lucius<sup>1,d</sup>, TÚLIO Marcos<sup>1,e</sup>, CONDE Mariana<sup>1,f</sup>, GONÇALVES Mafalda<sup>1,g</sup>, PRATES Pedro<sup>1,h</sup> and LOURENÇO Rúben<sup>1,i</sup>

<sup>1</sup>Centre for Mechanical Technology and Automation (TEMA), LASI Associate Laboratory, Mechanical Engineering Department, University of Aveiro, Portugal

agilac@ua.pt, bafonso.campos@ua.pt, cjoaodiogofh@ua.pt, dliciusviniciusf@ua.pt, emarcostulio@ua.pt, fmarianaconde@ua.pt, mafalda.goncalves@ua.pt, hprates@ua.pt, irubenl@ua.pt

**Keywords:** Material Database, Numerical Characterization, Model Identification, Full-Field Data

**Abstract.** Nowadays, most of the product designs rely on the aid of simulation software, particularly Finite Element Analysis (FEA) programs. However, an accurate simulation requires a proper virtual/numerical material behavior reproduction, meaning a precise material characterization through constitutive models and their parameters. To numerically characterize a material, particularly a metal, (i) experimental tests, (ii) model selection and (iii) inverse procedures are required. All these three tasks can be expensive and time-consuming. Therefore, product development engineers resort to materials databases to obtain the virtual materials, i.e. the constitutive models and their parameters adequate for the desired material. However, the information provided by the materials databases does not include experimental data nor provide information on the testing procedures. Due to this absence, users cannot verify the information nor its accuracy on the material database. Moreover, data related to material constitutive models, required for accurate simulations seems to be absent [1]. This work presents the development of a new material database that revises the previous problem. This database has the focus on virtual materials and their importance in product simulation and design. The presented VForm-xSteels material database includes (a) mechanical models and their implementation in FEA software, (b) experimental data and (c) the parameters identified for each material, and (d) indications concerning the quality of the material behavior reproduction associated with each model/parameters set. This database can be enlarged by the contributions of all users and present the following benefits for the engineering community: (i) increasing the precision and reliability of numerical FEA simulations by providing accurate input data, filling then a gap of the FEA market and answering to the request of the FEA users; (ii) reducing the development lead-time of metallic parts and the development of robust technological solutions with highly improved quality, consequently decreasing cost and time in the overall development process.

**Introduction**

Nowadays, the use of numerical simulation and particularly Finite Element Analysis (FEA) has become a mandatory step of material processing optimization [2]. According to the TechNavio "Global Simulation and Analysis Software Market" report (2023) [3], FEA software dominates around 50% of the simulation market and for the automotive industry alone, FEA is likely to exceed $968 million by 2024 (see Fig. 1). Despite the size of the FEA market and the regular use of FEA in the engineering design industry, the problem of obtaining reliable FEA input data, especially the description of non-linear material behavior, is a constant and has not been answered yet by the FEA software market. A solution for this problem is required for both the FEA users and providers.

<figure><img src="image_1706.png" /></figure>
# VForm-xSteels: virtual materials database

ANDRADE-CAMPOS António<sup>1,a*</sup>, CAMPOS Afonso<sup>1,b</sup>, HENRIQUES João<sup>1,c</sup>, FILHO Lucius<sup>1,d</sup>, TÚLIO Marcos<sup>1,e</sup>, CONDE Mariana<sup>1,f</sup>, GONÇALVES Mafalda<sup>1,g</sup>, PRATES Pedro<sup>1,h</sup> and LOURENÇO Rúben<sup>1,i</sup>

<sup>1</sup>Centre for Mechanical Technology and Automation (TEMA), LASI Associate Laboratory, Mechanical Engineering Department, University of Aveiro, Portugal

agilac@ua.pt, bafonso.campos@ua.pt, cjoaodiogofh@ua.pt, dliciusviniciusf@ua.pt, emarcostulio@ua.pt, fmarianaconde@ua.pt, mafalda.goncalves@ua.pt, hprates@ua.pt, irubenl@ua.pt

**Keywords:** Material Database, Numerical Characterization, Model Identification, Full-Field Data

**Abstract.** Nowadays, most of the product designs rely on the aid of simulation software, particularly Finite Element Analysis (FEA) programs. However, an accurate simulation requires a proper virtual/numerical material behavior reproduction, meaning a precise material characterization through constitutive models and their parameters. To numerically characterize a material, particularly a metal, (i) experimental tests, (ii) model selection and (iii) inverse procedures are required. All these three tasks can be expensive and time-consuming. Therefore, product development engineers resort to materials databases to obtain the virtual materials, i.e. the constitutive models and their parameters adequate for the desired material. However, the information provided by the materials databases does not include experimental data nor provide information on the testing procedures. Due to this absence, users cannot verify the information nor its accuracy on the material database. Moreover, data related to material constitutive models, required for accurate simulations seems to be absent [1]. This work presents the development of a new material database that revises the previous problem. This database has the focus on virtual materials and their importance in product simulation and design. The presented VForm-xSteels material database includes (a) mechanical models and their implementation in FEA software, (b) experimental data and (c) the parameters identified for each material, and (d) indications concerning the quality of the material behavior reproduction associated with each model/parameters set. This database can be enlarged by the contributions of all users and present the following benefits for the engineering community: (i) increasing the precision and reliability of numerical FEA simulations by providing accurate input data, filling then a gap of the FEA market and answering to the request of the FEA users; (ii) reducing the development lead-time of metallic parts and the development of robust technological solutions with highly improved quality, consequently decreasing cost and time in the overall development process.

**Introduction**

Nowadays, the use of numerical simulation and particularly Finite Element Analysis (FEA) has become a mandatory step of material processing optimization [2]. According to the TechNavio "Global Simulation and Analysis Software Market" report (2023) [3], FEA software dominates around 50% of the simulation market and for the automotive industry alone, FEA is likely to exceed $968 million by 2024 (see Fig. 1). Despite the size of the FEA market and the regular use of FEA in the engineering design industry, the problem of obtaining reliable FEA input data, especially the description of non-linear material behavior, is a constant and has not been answered yet by the FEA software market. A solution for this problem is required for both the FEA users and providers.

<figure><img src="image_1705.png" /></figure>
<figure><img src="image_1707.png" /><figcaption>Fig. 1. The market for simulation and analysis software is expected to grow at a CAGR of 14.22% between 2022 and 2027, with a forecasted increase of USD 9,680.14 million. This growth is driven by factors such as the rising demand for simulation and analysis software, the integration of advanced safety technologies in luxury cars to achieve higher safety ratings, and the increasing developments related to autonomous vehicles by OEMs (adapted from [3]).</figcaption></figure>

Reliable virtual forming would lead to stiffer, stronger, safer and lighter industrial parts through using advanced models, capable of accurately predicting the thermo-mechanical response and ductile failure behavior of materials when subjected to complex loading conditions. Simple constitutive models do not present any difficulties to be calibrated to known metals. However, these models do not provide accurate and robust results, such as the ones demanded by manufacturing, automotive, aerospace, etc. industries. Additionally, the continuous development of new materials, such as Advanced High Strength Steels (AHSS) that allow higher performance at lower weight, urgently requires (well calibrated) complex models to reproduce their behavior in FEA programs. An additional complexity is that the stiffness and yield stress of these materials are so high that their forming processing is frequently done in warm conditions.

Therefore, the characterization of materials has received increasing attention due to the need for precise input data to computational analysis software at a lower cost. Simulation software uses complex material constitutive models, and the successful prediction of the real thermo-mechanical and ductile failure behavior is inherently dependent on the quality of the model and the related material parameters. In general, these parameters are determined using multiple and different standard tests. However, the homogeneous thermal, stress and strain fields generated in these relatively simple tests do not fully represent the complex heterogeneous thermal, stress and strain fields occurring in warm metal forming operations. Additionally, inverse methodologies commonly used, e.g. minimization of experimental vs FE model, are not reliable enough, due to the non-uniqueness of the solution. Furthermore, for complex constitutive models with many parameters, a high number of classical standard tests must be included in the experimental
<figure><img src="image_1706.png" /><figcaption>Fig. 1. The market for simulation and analysis software is expected to grow at a CAGR of 14.22% between 2022 and 2027, with a forecasted increase of USD 9,680.14 million. This growth is driven by factors such as the rising demand for simulation and analysis software, the integration of advanced safety technologies in luxury cars to achieve higher safety ratings, and the increasing developments related to autonomous vehicles by OEMs (adapted from [3]).</figcaption></figure>

Reliable virtual forming would lead to stiffer, stronger, safer and lighter industrial parts through using advanced models, capable of accurately predicting the thermo-mechanical response and ductile failure behavior of materials when subjected to complex loading conditions. Simple constitutive models do not present any difficulties to be calibrated to known metals. However, these models do not provide accurate and robust results, such as the ones demanded by manufacturing, automotive, aerospace, etc. industries. Additionally, the continuous development of new materials, such as Advanced High Strength Steels (AHSS) that allow higher performance at lower weight, urgently requires (well calibrated) complex models to reproduce their behavior in FEA programs. An additional complexity is that the stiffness and yield stress of these materials are so high that their forming processing is frequently done in warm conditions.

Therefore, the characterization of materials has received increasing attention due to the need for precise input data to computational analysis software at a lower cost. Simulation software uses complex material constitutive models, and the successful prediction of the real thermo-mechanical and ductile failure behavior is inherently dependent on the quality of the model and the related material parameters. In general, these parameters are determined using multiple and different standard tests. However, the homogeneous thermal, stress and strain fields generated in these relatively simple tests do not fully represent the complex heterogeneous thermal, stress and strain fields occurring in warm metal forming operations. Additionally, inverse methodologies commonly used, e.g. minimization of experimental vs FE model, are not reliable enough, due to the non-uniqueness of the solution. Furthermore, for complex constitutive models with many parameters, a high number of classical standard tests must be included in the experimental
database, leading to an expensive and time-consuming experimental characterization and identification processes.

The main goal of the VForm-xSteels project [2] is to develop an efficient and accurate methodology for determining the material parameters of advanced mechanical and ductile damage models from a dedicated single test that involves non-homogeneous strain fields. Indeed, this non-homogeneity leads to richer information than more traditional approaches with quasi-homogeneous tests, thus leading to a decrease in the number of experiments and therefore to a cost-effective solution. However, this methodology requires performing a suitable mechanical test using Digital Image Correlation (DIC) technique and using post-processing numerical tools for the precise calibration of the constitutive models. Several numerical tools and strategies have emerged to solve the inverse calibration problem, such as the Finite Element Model Updating (FEMU) [4], the Virtual Field Method (VFM) [5], Constitutive Equation Gap Method (CEGM) [6], the Equilibrium Gap Method (EGM) [7], and very recently, with the explosion of data-driven methods, Machine Learning (ML) and Artificial Intelligence (AI) techniques [8]. A commercial numerical tool available to straightforwardly calibrate constitutive numerical models, which uses some of the previously listed methodologies, is MatchID [9]. Nevertheless, expertise on both experimental mechanical tests and inverse calibration is required. Therefore, a more straightforward solution should be available for most FEA users.

FEA software's libraries of material properties and material databases could be the solution for the need of material models and their parameters. However, the included materials are generic or the most common and do not represent the entire materials market, and these databases do not include experimental data nor provide information on the testing procedures [1]. Due to this absence, users cannot verify the information or its accuracy on the material. Moreover, data related to material constitutive models, required for accurate simulations seems to be absent.

Therefore, the only solution is the creation of a new database of material constitutive models calibrated to a large number of materials, which include (a) thermo-mechanical and ductile damage models and their implementation in FEA software (as user subroutines), (b) experimental data and (c) the parameters identified for each material, and (d) indications concerning the quality for reproducing the material behavior of each model/parameters set. This database should be enlarged by the contributions of all users even after the conclusion of the project. This solution is also an outcome of the vForm-xSteels project, as illustrated in Fig. 2.

Recently, this need was also highlighted by ANSYS Granta, launching their product called Granta Materials Data for Simulation (Granta MDS) [10], using sources like MaterialsUniverse™ database and the JAHM [11] simulation data set.

## Development methodology

In the design stage, it was decided to build a web platform, including the online open database, instead of a desktop application. This task is critical because this platform must be ready for easy upscaling, storage and management of a very large amount of data (the experimental full-field data of a single non-linear mechanical test can easily reach 5Gb without thermal data), be intuitive (in access, retrieving information, download and upload data) for an engineer, technician, or scientist and well documented. This platform must show a long longevity and must be developed by an expert engineer in the thematic of this project.
database, leading to an expensive and time-consuming experimental characterization and identification processes.

The main goal of the VForm-xSteels project [2] is to develop an efficient and accurate methodology for determining the material parameters of advanced mechanical and ductile damage models from a dedicated single test that involves non-homogeneous strain fields. Indeed, this non-homogeneity leads to richer information than more traditional approaches with quasi-homogeneous tests, thus leading to a decrease in the number of experiments and therefore to a cost-effective solution. However, this methodology requires performing a suitable mechanical test using Digital Image Correlation (DIC) technique and using post-processing numerical tools for the precise calibration of the constitutive models. Several numerical tools and strategies have emerged to solve the inverse calibration problem, such as the Finite Element Model Updating (FEMU) [4], the Virtual Field Method (VFM) [5], Constitutive Equation Gap Method (CEGM) [6], the Equilibrium Gap Method (EGM) [7], and very recently, with the explosion of data-driven methods, Machine Learning (ML) and Artificial Intelligence (AI) techniques [8]. A commercial numerical tool available to straightforwardly calibrate constitutive numerical models, which uses some of the previously listed methodologies, is MatchID [9]. Nevertheless, expertise on both experimental mechanical tests and inverse calibration is required. Therefore, a more straightforward solution should be available for most FEA users.

FEA software's libraries of material properties and material databases could be the solution for the need of material models and their parameters. However, the included materials are generic or the most common and do not represent the entire materials market, and these databases do not include experimental data nor provide information on the testing procedures [1]. Due to this absence, users cannot verify the information or its accuracy on the material. Moreover, data related to material constitutive models, required for accurate simulations seems to be absent.

Therefore, the only solution is the creation of a new database of material constitutive models calibrated to a large number of materials, which include (a) thermo-mechanical and ductile damage models and their implementation in FEA software (as user subroutines), (b) experimental data and (c) the parameters identified for each material, and (d) indications concerning the quality for reproducing the material behavior of each model/parameters set. This database should be enlarged by the contributions of all users even after the conclusion of the project. This solution is also an outcome of the vForm-xSteels project, as illustrated in Fig. 2.

Recently, this need was also highlighted by ANSYS Granta, launching their product called Granta Materials Data for Simulation (Granta MDS) [10], using sources like MaterialsUniverse™ database and the JAHM [11] simulation data set.

## Development methodology

In the design stage, it was decided to build a web platform, including the online open database, instead of a desktop application. This task is critical because this platform must be ready for easy upscaling, storage and management of a very large amount of data (the experimental full-field data of a single non-linear mechanical test can easily reach 5Gb without thermal data), be intuitive (in access, retrieving information, download and upload data) for an engineer, technician, or scientist and well documented. This platform must show a long longevity and must be developed by an expert engineer in the thematic of this project.
<figure><img src="image_1708.png" /><figcaption>Fig. 2. Graphical abstract of the project proposal VForm-xSteels. While full-field mechanical tests and calibration methodologies are scientifically sound, they may not be practical for industrial and FEA users. Instead, a material database can provide engineers with accurate and readily available data, enabling them to create more reliable simulations.</figcaption></figure>

Considering that the numerical characterization of a material requires experimental data and a selected constitutive model, and, consequently, the calibrated parameters of the model, the structure of the data is divided into these three groups, as illustrated in Fig. 3.

<figure><img src="image_1709.png" /><figcaption>Fig. 3. Metadata of the open online platform.</figcaption></figure>

However, the numerical material database must be prepared to effortlessly obtain the parameters using an inverse identification strategy. Nevertheless, the inclusion of these calibration software and codes in the platform is not practical in the sense that these require large computational effort to derive a precise result. Therefore, the structure of the database must be designed in a way of serving both the FEA users and the providers of calibration results. The solution found was the development of an API that could be used directly by parameter identification codes (for getting the experimental data and providing the results from/to the material database) and by the Front end (webpage) of the database. This solution is illustrated in
<figure><img src="image_1707.png" /><figcaption>Fig. 2. Graphical abstract of the project proposal VForm-xSteels. While full-field mechanical tests and calibration methodologies are scientifically sound, they may not be practical for industrial and FEA users. Instead, a material database can provide engineers with accurate and readily available data, enabling them to create more reliable simulations.</figcaption></figure>

Considering that the numerical characterization of a material requires experimental data and a selected constitutive model, and, consequently, the calibrated parameters of the model, the structure of the data is divided into these three groups, as illustrated in Fig. 3.

<figure><img src="image_1708.png" /><figcaption>Fig. 3. Metadata of the open online platform.</figcaption></figure>

However, the numerical material database must be prepared to effortlessly obtain the parameters using an inverse identification strategy. Nevertheless, the inclusion of these calibration software and codes in the platform is not practical in the sense that these require large computational effort to derive a precise result. Therefore, the structure of the database must be designed in a way of serving both the FEA users and the providers of calibration results. The solution found was the development of an API that could be used directly by parameter identification codes (for getting the experimental data and providing the results from/to the material database) and by the Front end (webpage) of the database. This solution is illustrated in
Fig. 4. The use of API as backend and processing engine offers several benefits, including increased interoperability, improved efficiency, better user experience, increased flexibility, and new revenue streams.

<figure><img src="image_1710.png" /><figcaption>Fig. 4. Information and data processing structure. The use of an API allows to download and upload data for both the Front end and other inverse calibration codes, as well as some processing tasks.</figcaption></figure>

Other difficulty on the development of the database is the structure for saving the data and an intuitive way to present to the user or even to access the database through the API. The chosen structure can be seen in the scheme of Fig. 5, where the hierarchy of the information was based on the file structure of a DIC test:

* material > • physical properties
> • experimental data > • mechanical tests > • DIC data
> • behavior models > • parameters

The result of the design phase has been developed in the last months. The actual stage of the VForm-xSteels Front end can be seen in Fig. 6. The database and its Front end include (i) documentation, scientific papers and description, (ii) FEA user routines (e.g. UMAT) with constitutive models, (iii) experimental data of mechanical tests for material characterization (see Fig. 6d), (iv) parameters identified for the available constitutive models and experimental data, and (v) Manual with instruction to use the database directly through the API. In this database, no files are saved, just data. However, the API engine can read all DIC files as well as write them to the user, which can reuse them for other purposes. The parameters resulting from the calibration operation are also presented to the user in a graphical manner, as shown in Fig. 7.

## Results: the example of the VForm-xSteels benchmark

In the VForm-xSteels project, the inverse identification of the constitutive behavior of a dual-phase steel was benchmarked. It was assumed that the DP600 steel is well reproduced by a known hardening law and yield criterion (Swift + Yld2000-2D, respectively). Both the experimental and numerical stages required for the behavior characterization were analyzed and the current practices in the framework of MT2.0 were assessed. University-designed tools and equipment were used. In this example, only the results obtained by the University of Aveiro are exhibited as an example of the potential of the database, however, several partners have participated and all project consortium have contributed.
Fig. 4. The use of API as backend and processing engine offers several benefits, including increased interoperability, improved efficiency, better user experience, increased flexibility, and new revenue streams.

<figure><img src="image_1709.png" /><figcaption>Fig. 4. Information and data processing structure. The use of an API allows to download and upload data for both the Front end and other inverse calibration codes, as well as some processing tasks.</figcaption></figure>

Other difficulty on the development of the database is the structure for saving the data and an intuitive way to present to the user or even to access the database through the API. The chosen structure can be seen in the scheme of Fig. 5, where the hierarchy of the information was based on the file structure of a DIC test:

* material > • physical properties
> • experimental data > • mechanical tests > • DIC data
> • behavior models > • parameters

The result of the design phase has been developed in the last months. The actual stage of the VForm-xSteels Front end can be seen in Fig. 6. The database and its Front end include (i) documentation, scientific papers and description, (ii) FEA user routines (e.g. UMAT) with constitutive models, (iii) experimental data of mechanical tests for material characterization (see Fig. 6d), (iv) parameters identified for the available constitutive models and experimental data, and (v) Manual with instruction to use the database directly through the API. In this database, no files are saved, just data. However, the API engine can read all DIC files as well as write them to the user, which can reuse them for other purposes. The parameters resulting from the calibration operation are also presented to the user in a graphical manner, as shown in Fig. 7.

## Results: the example of the VForm-xSteels benchmark

In the VForm-xSteels project, the inverse identification of the constitutive behavior of a dual-phase steel was benchmarked. It was assumed that the DP600 steel is well reproduced by a known hardening law and yield criterion (Swift + Yld2000-2D, respectively). Both the experimental and numerical stages required for the behavior characterization were analyzed and the current practices in the framework of MT2.0 were assessed. University-designed tools and equipment were used. In this example, only the results obtained by the University of Aveiro are exhibited as an example of the potential of the database, however, several partners have participated and all project consortium have contributed.
<figure><img src="image_1711.png" /><figcaption>Fig. 5. Storage general view.</figcaption></figure>

<figure><img src="image_1712.png" /><figcaption>Fig. 6. VForm-xSteels database Front end. (a) Home page, (b) available material list, (c) instruction for the direct use of the API by external codes and softwares, and (d) DIC data exhibits as full-field results.</figcaption></figure>
<figure><img src="image_1710.png" /><figcaption>Fig. 5. Storage general view.</figcaption></figure>

<figure><img src="image_1711.png" /><figcaption>Fig. 6. VForm-xSteels database Front end. (a) Home page, (b) available material list, (c) instruction for the direct use of the API by external codes and softwares, and (d) DIC data exhibits as full-field results.</figcaption></figure>
<figure><img src="image_1713.png" /><figcaption>Fig. 7. Example of behavior model representation in the VForm-xSteels Front end. The model is composed of an elastic law, yield criterion and hardening law.</figcaption></figure>

Using a known and simple specimen geometry (the heterogeneous dog-bone specimen, depicted in Figure 8) in a uniaxial tensile test at different material orientations (0, 45 and 90 degrees from the rolling direction), the hardening and anisotropy behavior of the DP600 steel must be characterized considering that these behaviors can be reproduced by the Swift's hardening model and the Barlat's yld2000-2d yield criterion, respectively. Therefore, the final goal of this exercise is to retrieve the parameters k, $\sigma_0$ and n of the hardening law and $\alpha_1...\alpha_8$ for the yield criterion (considering the yield's exponent a=6).

<figure><img src="image_1714.png" /><figcaption>Fig. 8. Heterogeneous dog-bone specimen in a uniaxial tensile test at 0, 45 and 90° from the RD. (a) Geometry of the specimen and measurement points and (b) samples of the specimen before and after the specimen manufacture.</figcaption></figure>

This benchmark has several steps, however, here only the stages of experimental data analysis; parameter identification and parameter analysis are included to show the potential of the database, as the process illustrated in Fig. 9.

The use of the API by an external code can be made in an effortless manner using Python code. After performing the mechanical test using DIC techniques, the user can send all the data to the VForm-xSteels as exemplified in table 1. Then, the inverse calibration program can read the data in a similar proceeding and code, performing the parameter identification without requiring the multiple DIC files. In our example, a FEMU code named ParamID [12,13] was used and their
<figure><img src="image_1712.png" /><figcaption>Fig. 7. Example of behavior model representation in the VForm-xSteels Front end. The model is composed of an elastic law, yield criterion and hardening law.</figcaption></figure>

Using a known and simple specimen geometry (the heterogeneous dog-bone specimen, depicted in Figure 8) in a uniaxial tensile test at different material orientations (0, 45 and 90 degrees from the rolling direction), the hardening and anisotropy behavior of the DP600 steel must be characterized considering that these behaviors can be reproduced by the Swift's hardening model and the Barlat's yld2000-2d yield criterion, respectively. Therefore, the final goal of this exercise is to retrieve the parameters k, $\sigma_0$ and n of the hardening law and $\alpha_1...\alpha_8$ for the yield criterion (considering the yield's exponent a=6).

<figure><img src="image_1713.png" /><figcaption>Fig. 8. Heterogeneous dog-bone specimen in a uniaxial tensile test at 0, 45 and 90° from the RD. (a) Geometry of the specimen and measurement points and (b) samples of the specimen before and after the specimen manufacture.</figcaption></figure>

This benchmark has several steps, however, here only the stages of experimental data analysis; parameter identification and parameter analysis are included to show the potential of the database, as the process illustrated in Fig. 9.

The use of the API by an external code can be made in an effortless manner using Python code. After performing the mechanical test using DIC techniques, the user can send all the data to the VForm-xSteels as exemplified in table 1. Then, the inverse calibration program can read the data in a similar proceeding and code, performing the parameter identification without requiring the multiple DIC files. In our example, a FEMU code named ParamID [12,13] was used and their
results can be seen in Table 2. This inverse calibration code used a multistarting strategy to avoid the initial set dependence [13]. Then, the best parameter set, i.e. the ones that the difference between the strain full-field and force evolution is the lowest, can be directly written in the VForm-xSteels database using again the API and its Python scripts.

<figure><img src="image_1715.png" /><figcaption>Fig. 9. Interaction stages of the inverse calibration process with the VForm-xSteels database. The advantages of the VForm-xSteels database are the inverse calibration process, the direct access to experimental data and writing of the achieved result with subsequent quality assessment.</figcaption></figure>

<caption>Table 1. Code example for the VForm-xSteels data intake. Upload of DIC data. The file mappings function reads the DIC files, writing in the most common format.</caption>

```python
1 from API import *
2 import os
3
4 token = authenticate("tester", "secretPass1234")
5
6 # Directory where the DIC files and load data are stored, consult the user manual for formatting and naming of the files
7 dir = "test\\"
8
9 # ID of the test we wish to populate
10 test_id = 3
11
12 f = []
13 for (dirpath, dirnames, filenames) in os Walk(d1r):
14 f. extend(filenames)
15 break
16
17 file mappings = {name : open(d1r + name, "rb") for name in f}
18
19 test = get_test(test_id)
20
21 # Upload the data
22 test. upload_test_data(token, file mappings, file_format=UploadFileFormat.MatchId, _3d=False)
23
24 # Download the data as a ZIP file
25 test. download_test_data()
26
27 # Download the data as a pandas dataframe
28 stages df = test. load_test_data()
29
30 print(stages df. keys())
31 print(stages df[2])
```
results can be seen in Table 2. This inverse calibration code used a multistarting strategy to avoid the initial set dependence [13]. Then, the best parameter set, i.e. the ones that the difference between the strain full-field and force evolution is the lowest, can be directly written in the VForm-xSteels database using again the API and its Python scripts.

<figure><img src="image_1714.png" /><figcaption>Fig. 9. Interaction stages of the inverse calibration process with the VForm-xSteels database. The advantages of the VForm-xSteels database are the inverse calibration process, the direct access to experimental data and writing of the achieved result with subsequent quality assessment.</figcaption></figure>

<caption>Table 1. Code example for the VForm-xSteels data intake. Upload of DIC data. The file mappings function reads the DIC files, writing in the most common format.</caption>

```python
1 from API import *
2 import os
3
4 token = authenticate("tester", "secretPass1234")
5
6 # Directory where the DIC files and load data are stored, consult the user manual for formatting and naming of the files
7 dir = "test\\"
8
9 # ID of the test we wish to populate
10 test_id = 3
11
12 f = []
13 for (dirpath, dirnames, filenames) in osWalk(d1):
14 f.extend(filenames)
15 break
16
17 fileCOmappings = {name : open(d1+name, "rb") for name in f}
18
test = get_test(test_id)
20
21 # Upload the data
test.upload_test_data(token, fileCOmappings, file_format=UploadFileFormat.MatchId, _3d=False)
23
24 # Download the data as a ZIP file
test.download_test_data()
26
27 # Download the data as a pandas dataframe
stages_df = test.load_test_data()
29
30 print(stages_df.keys())
31 print(stages_df[2])
```
<table><caption>Table 2. Results of the calibration of the elastoplasticity constitutive model (Yld2000-2D+Swift hardening) using a multistarting FEMU strategy and the heterogeneous dog-bone specimen.</caption><thead><tr><th></th><th>M fixed to 6</th><th>alpha 1</th><th>alpha 2</th><th>alpha 3</th><th>alpha 4</th><th>alpha 5</th><th>alpha 6</th><th>alpha 7</th><th>alpha 8</th><th>M</th><th>K</th><th>epsilon_0</th><th>n</th><th>Eval</th><th>CF</th></tr></thead><tbody><tr><td rowspan="2">Run 0 (iso)</td><td>Initial set</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>6.000</td><td>1200.000</td><td>0.00550</td><td>0.200</td><td>1</td><td>1.20E-02</td></tr><tr><td>Final set</td><td>1.029</td><td>0.855</td><td>0.664</td><td>0.914</td><td>0.983</td><td>1.497</td><td>0.868</td><td>0.677</td><td>6.000</td><td>1048.377</td><td>0.00471</td><td>0.236</td><td>276</td><td>3.60E-04</td></tr><tr><td rowspan="2">Run 1</td><td>Initial set</td><td>1.321</td><td>1.238</td><td>1.075</td><td>1.237</td><td>0.529</td><td>0.908</td><td>1.250</td><td>1.298</td><td>6.000</td><td>1610.791</td><td>0.00541</td><td>0.179</td><td>1</td><td>6.46E-02</td></tr><tr><td>Final set</td><td>1.183</td><td>0.968</td><td>0.696</td><td>1.055</td><td>1.109</td><td>1.499</td><td>1.002</td><td>0.889</td><td>6.000</td><td>1261.792</td><td>0.00393</td><td>0.235</td><td>255</td><td>3.63E-04</td></tr><tr><td rowspan="2">Run 2</td><td>Initial set</td><td>0.963</td><td>1.038</td><td>0.876</td><td>0.560</td><td>0.917</td><td>0.783</td><td>0.812</td><td>1.076</td><td>6.000</td><td>1499.560</td><td>0.00334</td><td>0.280</td><td>1</td><td>6.89E-02</td></tr><tr><td>Final set</td><td>0.648</td><td>1.434</td><td>1.231</td><td>1.081</td><td>0.975</td><td>0.500</td><td>0.963</td><td>1.040</td><td>6.000</td><td>1249.834</td><td>0.00374</td><td>0.233</td><td>230</td><td>3.81E-04</td></tr><tr><td rowspan="2">Run 3</td><td>Initial set</td><td>0.583</td><td>0.741</td><td>1.482</td><td>1.079</td><td>0.834</td><td>1.326</td><td>0.537</td><td>0.569</td><td>6.000</td><td>887.544</td><td>0.00636</td><td>0.252</td><td>1</td><td>1.45E-02</td></tr><tr><td>Final set</td><td>0.809</td><td>0.702</td><td>0.502</td><td>0.744</td><td>0.768</td><td>0.965</td><td>0.702</td><td>0.658</td><td>6.000</td><td>817.292</td><td>0.00548</td><td>0.234</td><td>284</td><td>3.67E-04</td></tr><tr><td rowspan="2">Run 4</td><td>Initial set</td><td>1.207</td><td>1.390</td><td>0.500</td><td>0.705</td><td>1.267</td><td>1.104</td><td>0.934</td><td>0.728</td><td>6.000</td><td>695.224</td><td>0.00473</td><td>0.117</td><td>1</td><td>1.32E-01</td></tr><tr><td>Final set</td><td>0.723</td><td>0.966</td><td>0.500</td><td>0.500</td><td>0.735</td><td>1.091</td><td>0.760</td><td>0.500</td><td>6.000</td><td>732.463</td><td>0.00275</td><td>0.189</td><td>276</td><td>2.04E-03</td></tr><tr><td rowspan="2">Run 5</td><td>Initial set</td><td>0.837</td><td>0.644</td><td>1.163</td><td>1.359</td><td>1.358</td><td>0.518</td><td>1.304</td><td>1.436</td><td>6.000</td><td>1103.603</td><td>0.007</td><td>0.21832</td><td>1</td><td>8.21E-02</td></tr><tr><td>Final set</td><td>0.500</td><td>1.500</td><td>0.619</td><td>1.038</td><td>1.037</td><td>0.500</td><td>0.924</td><td>1.356</td><td>6.000</td><td>1331.195</td><td>0.004</td><td>0.23247</td><td>286</td><td>3.94E-04</td></tr><tr><td>Best run</td><td>Final set</td><td>1.029</td><td>0.855</td><td>0.664</td><td>0.914</td><td>0.983</td><td>1.497</td><td>0.868</td><td>0.677</td><td>6.000</td><td>1048.377</td><td>0.004713</td><td>0.236</td><td>276</td><td>3.60E-04</td></tr></tbody></table>

The identified parameters once kept in the database can be accessed by all users. Classical curves (yield surface, hardening curve, tensile test curve, shear test curve, etc.) are presented in the Front end to the user infer its general behavior (see Fig. 10). Other Key Performance Indicators (KPI), which evaluate the quality of both the model and their parameters, can be presented.

<figure><img src="image_1716.png" /><figcaption>Fig. 10. Yield surface and hardening in the rolling direction of the constitutive model calibrated.</figcaption></figure>

## Conclusions

This paper presents the development of a numerical material database useful for FEA users. This database presents the following benefits for the engineering community: (i) increasing the precision and reliability of numerical FEA simulations by providing accurate input data, filling then a gap in the FEA market, and answering the request of the FEA users; (ii) reducing the development lead-time of metallic parts and the development of robust technological solutions with highly improved quality, consequently decreasing cost and time in the overall development process.

## Disclaimer

The results reflect only the authors' view, and the European Commission is not responsible for any use that may be made of the information it contains.

## Acknowledgments

This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. The authors also gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) under the projects UIDB/00481/2020 (https://doi.org/10.54499/UIDB/00481/2020) and UIDP/00481/2020-FCT (https://doi.org/10.54499/UIDP/00481/2020) under CENTRO-01-0145-FEDER-022083. This
<table><caption>Table 2. Results of the calibration of the elastoplasticity constitutive model (Yld2000-2D+Swift hardening) using a multistarting FEMU strategy and the heterogeneous dog-bone specimen.</caption><thead><tr><th></th><th>M fixed to 6</th><th>alpha 1</th><th>alpha 2</th><th>alpha 3</th><th>alpha 4</th><th>alpha 5</th><th>alpha 6</th><th>alpha 7</th><th>alpha 8</th><th>M</th><th>K</th><th>epsilon_0</th><th>n</th><th>Eval</th><th>CF</th></tr></thead><tbody><tr><td rowspan="2">Run 0 (iso)</td><td>Initial set</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>6.000</td><td>1200.000</td><td>0.00550</td><td>0.200</td><td>1</td><td>1.20E-02</td></tr><tr><td>Final set</td><td>1.029</td><td>0.855</td><td>0.664</td><td>0.914</td><td>0.983</td><td>1.497</td><td>0.868</td><td>0.677</td><td>6.000</td><td>1048.377</td><td>0.00471</td><td>0.236</td><td>276</td><td>3.60E-04</td></tr><tr><td rowspan="2">Run 1</td><td>Initial set</td><td>1.321</td><td>1.238</td><td>1.075</td><td>1.237</td><td>0.529</td><td>0.908</td><td>1.250</td><td>1.298</td><td>6.000</td><td>1610.791</td><td>0.00541</td><td>0.179</td><td>1</td><td>6.46E-02</td></tr><tr><td>Final set</td><td>1.183</td><td>0.968</td><td>0.696</td><td>1.055</td><td>1.109</td><td>1.499</td><td>1.002</td><td>0.889</td><td>6.000</td><td>1261.792</td><td>0.00393</td><td>0.235</td><td>255</td><td>3.63E-04</td></tr><tr><td rowspan="2">Run 2</td><td>Initial set</td><td>0.963</td><td>1.038</td><td>0.876</td><td>0.560</td><td>0.917</td><td>0.783</td><td>0.812</td><td>1.076</td><td>6.000</td><td>1499.560</td><td>0.00334</td><td>0.280</td><td>1</td><td>6.89E-02</td></tr><tr><td>Final set</td><td>0.648</td><td>1.434</td><td>1.231</td><td>1.081</td><td>0.975</td><td>0.500</td><td>0.963</td><td>1.040</td><td>6.000</td><td>1249.834</td><td>0.00374</td><td>0.233</td><td>230</td><td>3.81E-04</td></tr><tr><td rowspan="2">Run 3</td><td>Initial set</td><td>0.583</td><td>0.741</td><td>1.482</td><td>1.079</td><td>0.834</td><td>1.326</td><td>0.537</td><td>0.569</td><td>6.000</td><td>887.544</td><td>0.00636</td><td>0.252</td><td>1</td><td>1.45E-02</td></tr><tr><td>Final set</td><td>0.809</td><td>0.702</td><td>0.502</td><td>0.744</td><td>0.768</td><td>0.965</td><td>0.702</td><td>0.658</td><td>6.000</td><td>817.292</td><td>0.00548</td><td>0.234</td><td>284</td><td>3.67E-04</td></tr><tr><td rowspan="2">Run 4</td><td>Initial set</td><td>1.207</td><td>1.390</td><td>0.500</td><td>0.705</td><td>1.267</td><td>1.104</td><td>0.934</td><td>0.728</td><td>6.000</td><td>695.224</td><td>0.00473</td><td>0.117</td><td>1</td><td>1.32E-01</td></tr><tr><td>Final set</td><td>0.723</td><td>0.966</td><td>0.500</td><td>0.500</td><td>0.735</td><td>1.091</td><td>0.760</td><td>0.500</td><td>6.000</td><td>732.463</td><td>0.00275</td><td>0.189</td><td>276</td><td>2.04E-03</td></tr><tr><td rowspan="2">Run 5</td><td>Initial set</td><td>0.837</td><td>0.644</td><td>1.163</td><td>1.359</td><td>1.358</td><td>0.518</td><td>1.304</td><td>1.436</td><td>6.000</td><td>1103.603</td><td>0.007</td><td>0.21832</td><td>1</td><td>8.21E-02</td></tr><tr><td>Final set</td><td>0.500</td><td>1.500</td><td>0.619</td><td>1.038</td><td>1.037</td><td>0.500</td><td>0.924</td><td>1.356</td><td>6.000</td><td>1331.195</td><td>0.004</td><td>0.23247</td><td>286</td><td>3.94E-04</td></tr><tr><td>Best run</td><td>Final set</td><td>1.029</td><td>0.855</td><td>0.664</td><td>0.914</td><td>0.983</td><td>1.497</td><td>0.868</td><td>0.677</td><td>6.000</td><td>1048.377</td><td>0.004713</td><td>0.236</td><td>276</td><td>3.60E-04</td></tr></tbody></table>

The identified parameters once kept in the database can be accessed by all users. Classical curves (yield surface, hardening curve, tensile test curve, shear test curve, etc.) are presented in the Front end to the user infer its general behavior (see Fig. 10). Other Key Performance Indicators (KPI), which evaluate the quality of both the model and their parameters, can be presented.

<figure><img src="image_1715.png" /><figcaption>Fig. 10. Yield surface and hardening in the rolling direction of the constitutive model calibrated.</figcaption></figure>

## Conclusions

This paper presents the development of a numerical material database useful for FEA users. This database presents the following benefits for the engineering community: (i) increasing the precision and reliability of numerical FEA simulations by providing accurate input data, filling then a gap in the FEA market, and answering the request of the FEA users; (ii) reducing the development lead-time of metallic parts and the development of robust technological solutions with highly improved quality, consequently decreasing cost and time in the overall development process.

## Disclaimer

The results reflect only the authors' view, and the European Commission is not responsible for any use that may be made of the information it contains.

## Acknowledgments

This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. The authors also gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) under the projects UIDB/00481/2020 (https://doi.org/10.54499/UIDB/00481/2020) and UIDP/00481/2020-FCT (https://doi.org/10.54499/UIDP/00481/2020) under CENTRO-01-0145-FEDER-022083. This
work has contributions from all members of the VForm-xSteels consortium, including the
profitable discussion in the multiple meetings of the project.

References

[1] L. Rumor, A. Andrade-Campos, On the need for material model databases – A state-of-the-art review, Advances in Mechanical Engineering, 14(10) (2022) 1-12
https://doi.org/10.1177/16878132221130575

[2] Information on http://www.vform-xsteels.eu/

[3] Technavio, Simulation and Analysis Software Market by End-user, Development and Geography - Forecast and Analysis 2023-2027, https://www.technavio.com/report/simulation-and-analysis-software-market-industry-analysis

[4] K.T. Kavanagh and R.W. Clough, Finite element applications in the characterization of elastic solids, International Journal of Solids and Structures, 7(1) (1971) 11-23

[5] F. Pierron and M. Grédiac, The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements, Springer Science & Business Media (2012)

[6] P. Ladeveze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM Journal on Numerical Analysis, 20(3) (1983) 485-509

[7] D. Claire, F. Hild, and S. Roux, A finite element formulation to identify damage fields: the equilibrium gap method, International Journal for Numerical Methods in Engineering, 61(2) (2004) 189-208

[8] A Andrade-Campos et al., IOP Conf. Ser.: Mater. Sci. Eng. 1238 012059 (2022)

[9] Information on https://www.matchid.eu/

[10] Information on https://www.ansys.com/products/materials/materials-data-for-simulation

[11] Information on https://www.jahm.com/

[12] J. Henriques, M. Conde, A. Andrade-Campos, J. Xavier, Identification of Swift law parameters using FEMU by a synthetic image DIC-based approach, Key Engineering Materials 926 (2022) 2211-2221 https://doi.org/10.4028/p-33un7m

[13] M. Conde, J. Henriques, S. Coppieters, A. Andrade-Campos, Parameter identification of Swift law using a FEMU-based approach and an innovative heterogeneous mechanical test, Key Engineering Materials 926 (2022) 2238-2246 https://doi.org/10.4028/p-1n7iop
work has contributions from all members of the VForm-xSteels consortium, including the
profitable discussion in the multiple meetings of the project.

References

[1] L. Rumor, A. Andrade-Campos, On the need for material model databases – A state-of-the-art review, Advances in Mechanical Engineering, 14(10) (2022) 1-12
https://doi.org/10.1177/16878132221130575

[2] Information on http://www.vform-xsteels.eu/

[3] Technavio, Simulation and Analysis Software Market by End-user, Development and Geography - Forecast and Analysis 2023-2027, https://www.technavio.com/report/simulation-and-analysis-software-market-industry-analysis

[4] K.T. Kavanagh and R.W. Clough, Finite element applications in the characterization of elastic solids, International Journal of Solids and Structures, 7(1) (1971) 11-23

[5] F. Pierron and M. Grédiac, The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements, Springer Science & Business Media (2012)

[6] P. Ladeveze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM Journal on Numerical Analysis, 20(3) (1983) 485-509

[7] D. Claire, F. Hild, and S. Roux, A finite element formulation to identify damage fields: the equilibrium gap method, International Journal for Numerical Methods in Engineering, 61(2) (2004) 189-208

[8] A Andrade-Campos et al., IOP Conf. Ser.: Mater. Sci. Eng. 1238 012059 (2022)

[9] Information on https://www.matchid.eu/

[10] Information on https://www.ansys.com/products/materials/materials-data-for-simulation

[11] Information on https://www.jahm.com/

[12] J. Henriques, M. Conde, A. Andrade-Campos, J. Xavier, Identification of Swift law parameters using FEMU by a synthetic image DIC-based approach, Key Engineering Materials 926 (2022) 2211-2221 https://doi.org/10.4028/p-33un7m

[13] M. Conde, J. Henriques, S. Coppieters, A. Andrade-Campos, Parameter identification of Swift law using a FEMU-based approach and an innovative heterogeneous mechanical test, Key Engineering Materials 926 (2022) 2238-2246 https://doi.org/10.4028/p-1n7iop
# Determining the residual formability of shear-cut sheet metal edges by utilizing an ML based prediction model

GÖRZ Marcel¹,ᵃ*, SCHENEK Adrian¹, VO Trong Quan¹, RIEDMÜLLER Kim Rouven¹ and LIEWALD Mathias¹

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

ªmarcel.goerz@ifu.uni-stuttgart.de

**Keywords:** Machine Learning, Edge Crack, Punching Force

**Abstract.** When forming high-strength steel sheet material, premature failure can occur at shear-cut component edges because formability of the base material is reduced due to work hardening caused by the previous punching process. Here, the digitization of production processes provides new possibilities for quality monitoring of such forming and stamping processes. In this context, the present paper deals with a novel machine learning (ML) based method for determining the residual formability of sheet metal materials from measured punching force curves. The specific objective of the study carried out was to develop an efficient and accurate method for predicting the residual formability of shear-cut edges. The methodology proposed for this purpose involves collecting a comprehensive dataset comprising experimental measurements of material properties, cutting conditions and punching-force curves measured during blanking. To determine the residual formability of the sheet metal materials investigated, hole tensile tests were performed and the maximum major and minor principal strain at initiation of cracking were measured. This dataset was then used to train and validate different AI prediction models, which employ machine learning algorithms to establish complex relationships between input parameters and residual formability.

**Introduction and State of the Art**

Shearing is generally one of the most economically important manufacturing processes in sheet metal working industry [1]. During its production process, almost every sheet metal component is cut as a blank from a semi-finished product, trimmed or perforated prior to forming [2]. In addition to constantly increasing demands on the quality of formed sheet metal components, the component edges produced during shear cutting therefore also have to meet ever higher quality requirements. These quality requirements for cut edges and surfaces are characterised by a small edge radius, a high proportion of smooth cuts, freedom from burrs, low fracture surface heights and tight manufacturing tolerances [3]. Furthermore, the quality criterion of the residual formability of shear-cut edges of sheet metal components is becoming increasingly important, especially with regard to modern lightweight materials such as high-strength aluminium grades [4] or dual-phase steel sheets [5]. The residual formability of sheared sheet metal component edges is reduced due to the high plastic deformations and strain hardening within the shear affected zone (edge effect). Low residual formability can lead to edge cracks during the subsequent forming of cut and punched sheet metal components, emanating from the edge of the component or from the cut-outs. The formation of edge cracks on shear cut component edges crucially depends on the selected cutting parameters and the wear condition of the cutting elements used [6].

The only standardised method for determining the susceptibility to edge cracking is the hole expansion test described in DIN 16630 [7]. This two-stage process consists of a hole punching operation followed by a forming operation to expand the punched hole. The hole expansion ratio achieved up to crack formation provides a value for assessing the edge crack sensitivity of a sheet metal material. The crack initiation during these tests is usually detected using a camera system.

<figure><img src="image_1717.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
# Determining the residual formability of shear-cut sheet metal edges by utilizing an ML based prediction model

GÖRZ Marcel¹,ᵃ*, SCHENEK Adrian¹, VO Trong Quan¹, RIEDMÜLLER Kim Rouven¹ and LIEWALD Mathias¹

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

ªmarcel.goerz@ifu.uni-stuttgart.de

**Keywords:** Machine Learning, Edge Crack, Punching Force

**Abstract.** When forming high-strength steel sheet material, premature failure can occur at shear-cut component edges because formability of the base material is reduced due to work hardening caused by the previous punching process. Here, the digitization of production processes provides new possibilities for quality monitoring of such forming and stamping processes. In this context, the present paper deals with a novel machine learning (ML) based method for determining the residual formability of sheet metal materials from measured punching force curves. The specific objective of the study carried out was to develop an efficient and accurate method for predicting the residual formability of shear-cut edges. The methodology proposed for this purpose involves collecting a comprehensive dataset comprising experimental measurements of material properties, cutting conditions and punching-force curves measured during blanking. To determine the residual formability of the sheet metal materials investigated, hole tensile tests were performed and the maximum major and minor principal strain at initiation of cracking were measured. This dataset was then used to train and validate different AI prediction models, which employ machine learning algorithms to establish complex relationships between input parameters and residual formability.

**Introduction and State of the Art**

Shearing is generally one of the most economically important manufacturing processes in sheet metal working industry [1]. During its production process, almost every sheet metal component is cut as a blank from a semi-finished product, trimmed or perforated prior to forming [2]. In addition to constantly increasing demands on the quality of formed sheet metal components, the component edges produced during shear cutting therefore also have to meet ever higher quality requirements. These quality requirements for cut edges and surfaces are characterised by a small edge radius, a high proportion of smooth cuts, freedom from burrs, low fracture surface heights and tight manufacturing tolerances [3]. Furthermore, the quality criterion of the residual formability of shear-cut edges of sheet metal components is becoming increasingly important, especially with regard to modern lightweight materials such as high-strength aluminium grades [4] or dual-phase steel sheets [5]. The residual formability of sheared sheet metal component edges is reduced due to the high plastic deformations and strain hardening within the shear affected zone (edge effect). Low residual formability can lead to edge cracks during the subsequent forming of cut and punched sheet metal components, emanating from the edge of the component or from the cut-outs. The formation of edge cracks on shear cut component edges crucially depends on the selected cutting parameters and the wear condition of the cutting elements used [6].

The only standardised method for determining the susceptibility to edge cracking is the hole expansion test described in DIN 16630 [7]. This two-stage process consists of a hole punching operation followed by a forming operation to expand the punched hole. The hole expansion ratio achieved up to crack formation provides a value for assessing the edge crack sensitivity of a sheet metal material. The crack initiation during these tests is usually detected using a camera system.

<figure><img src="image_1716.png" /></figure>
However, due to the wide range of parameters (punch diameter and radius on the drawing die) allowed by the standard, it is often difficult to compare different tests with each other. Other methods for determining edge crack sensitivity include the Diabolo Test developed at the IFU Stuttgart, which can only be used for open cut edges, and the Edge-Fracture-Tensile-Test (EFTT) method, which allows for analysis of both open and closed cut edges [8]. Watanabe developed the Open Hole Tensile Test (OHTT) method [9], which was adapted from the field of composite materials to metallic materials. In this method, a hole is punched in a rectangular specimen with a closed cutting line and subsequently subjected to uniaxial tensile loading. The tests thereby are analysed using a DIC capable camera system. However, a major disadvantage of the testing methods mentioned in this section is that none of them can be directly integrated into production processes for inline monitoring.

Based on the guidelines of Industry 4.0, meanwhile also the value chain in the stamping sector has been digitized, e.g. by integrating sensors into the stamping tools to collect large amounts of process data. Combined with analysis using machine learning (ML) methods, this enabled the development of new systems for monitoring stamping and forming processes. As a result, parameters that previously had to be determined in time-consuming procedures can now be measured inline during the process using ML methods. For example, the wear condition of a punch can be characterized by evaluating cutting force-displacement curves [10, 11]. Molitor was also able to predict the wear state of the punch from images of the stamped parts using convolutional neural networks [12]. Schenek et al. showed that the regression capability of artificial neural networks (ANN) enables material properties to be determined from cutting force-displacement curves [13]. In another study, the same authors showed that ANN can also be used to determine cutting surface quality parameters (edge draw-in height, clean cut height, fracture surface height, and burr height) from punching force curves measured during the stamping process [14].

The current state of the art shows that a number of properties that could previously only be measured offline can now be determined inline by using measuring systems based on ML models. In this context, a novel ML based method for determining edge crack sensitivity will be presented in this paper. Investigations will be described, which were performed to examine whether an ML model can be used to predict the maximum achievable major and minor principal strain at the onset of fracture based on cutting force displacement curves measured inline during punching. In these investigations, training data was determined experimentally in a first step. Here, cutting tests were carried out to determine the punching force and OHTT were performed to determine the major and minor principal strains. This data set subsequently was used to train and compare ML models with increasing complexity. The models considered here were linear regression (LR), support vector machine regression (SVM), random forest (RF) and artificial neural network (ANN).

## Experimental Setup

A data set consisting of input and output data is required to create an ML model. The following section describes how this data set was determined experimentally. Punching force curves recorded during a stamping process were used as input data for training the ML models. The experimental investigations for recording the punching force curves were carried out using a modular test tool. Figure 1 shows a 3D rendering and cross-section of the tool used. The tool was equipped with a load cell for direct force measurement (Kistler 9104a) and an additional displacement measurement system (TR 8710-100). The measurement was performed with measuring frequency of 10 kHz, ensuring a high resolution of the measured punching force curves. During the experimental investigations, the punch was firmly clamped in the shank area of the punch over a length of 22 mm. The shank diameter of the punch was 13 mm in all tests performed in this study, in accordance with the ISO8020 standard. The initial cutting edge radius of the punch was approx. 50 µm and was determined using an optical microscope. The parameters used in the tests are shown in Table 1. The 90 punching force curves measured during the tests are shown in Figure 1 (a). The
However, due to the wide range of parameters (punch diameter and radius on the drawing die) allowed by the standard, it is often difficult to compare different tests with each other. Other methods for determining edge crack sensitivity include the Diabolo Test developed at the IFU Stuttgart, which can only be used for open cut edges, and the Edge-Fracture-Tensile-Test (EFTT) method, which allows for analysis of both open and closed cut edges [8]. Watanabe developed the Open Hole Tensile Test (OHTT) method [9], which was adapted from the field of composite materials to metallic materials. In this method, a hole is punched in a rectangular specimen with a closed cutting line and subsequently subjected to uniaxial tensile loading. The tests thereby are analysed using a DIC capable camera system. However, a major disadvantage of the testing methods mentioned in this section is that none of them can be directly integrated into production processes for inline monitoring.

Based on the guidelines of Industry 4.0, meanwhile also the value chain in the stamping sector has been digitized, e.g. by integrating sensors into the stamping tools to collect large amounts of process data. Combined with analysis using machine learning (ML) methods, this enabled the development of new systems for monitoring stamping and forming processes. As a result, parameters that previously had to be determined in time-consuming procedures can now be measured inline during the process using ML methods. For example, the wear condition of a punch can be characterized by evaluating cutting force-displacement curves [10, 11]. Molitor was also able to predict the wear state of the punch from images of the stamped parts using convolutional neural networks [12]. Schenek et al. showed that the regression capability of artificial neural networks (ANN) enables material properties to be determined from cutting force-displacement curves [13]. In another study, the same authors showed that ANN can also be used to determine cutting surface quality parameters (edge draw-in height, clean cut height, fracture surface height, and burr height) from punching force curves measured during the stamping process [14].

The current state of the art shows that a number of properties that could previously only be measured offline can now be determined inline by using measuring systems based on ML models. In this context, a novel ML based method for determining edge crack sensitivity will be presented in this paper. Investigations will be described, which were performed to examine whether an ML model can be used to predict the maximum achievable major and minor principal strain at the onset of fracture based on cutting force displacement curves measured inline during punching. In these investigations, training data was determined experimentally in a first step. Here, cutting tests were carried out to determine the punching force and OHTT were performed to determine the major and minor principal strains. This data set subsequently was used to train and compare ML models with increasing complexity. The models considered here were linear regression (LR), support vector machine regression (SVM), random forest (RF) and artificial neural network (ANN).

## Experimental Setup

A data set consisting of input and output data is required to create an ML model. The following section describes how this data set was determined experimentally. Punching force curves recorded during a stamping process were used as input data for training the ML models. The experimental investigations for recording the punching force curves were carried out using a modular test tool. Figure 1 shows a 3D rendering and cross-section of the tool used. The tool was equipped with a load cell for direct force measurement (Kistler 9104a) and an additional displacement measurement system (TR 8710-100). The measurement was performed with measuring frequency of 10 kHz, ensuring a high resolution of the measured punching force curves. During the experimental investigations, the punch was firmly clamped in the shank area of the punch over a length of 22 mm. The shank diameter of the punch was 13 mm in all tests performed in this study, in accordance with the ISO8020 standard. The initial cutting edge radius of the punch was approx. 50 µm and was determined using an optical microscope. The parameters used in the tests are shown in Table 1. The 90 punching force curves measured during the tests are shown in Figure 1 (a). The
output data for training the ML models were recorded through open-hole tensile tests. The tests carried out according to the method proposed by Watanabe [9]. Described tests were performed on a Roell + Korthaus RKM 100 material testing machine. The ARAMIS 12M digital image correlation system was used to measure the specimen strains. Figure 1 (b) shows two exemplary images of the Aramis 12M system taken out of the test series. Here, the left image shows the distribution of the major principal strain at the start of the test and the right image shows the major principal strain at the initiation of the crack. The maximum tolerable major and minor strain up to failure is determined at a distance of 1 mm from the edge of the introduced hole. Figure 1 (c) shows an overview of the results of all OHTTs performed.

<figure><img src="image_1718.png" /><figcaption>Fig. 1. 3D rendering of the modular test tool (a); sectional view of the modular test tool.</figcaption></figure>

<table><caption>Table 1. Punching parameters investigated in the present study.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Length of punch</td><td>80 mm</td></tr><tr><td>Cutting clearance</td><td>10%, 15%</td></tr><tr><td>Sheet Thickness</td><td>1 mm</td></tr><tr><td>Punch diameter</td><td>10 mm</td></tr><tr><td>Material</td><td>DP600, DP1000, DC03</td></tr></tbody></table>

## Feature Engineering, Model Design and Training

The data-specific investigations described in this paper were carried out using the Python programming language. The deep learning library TensorFlow (TF) was used to correlate features with the major and minor principal strain at crack initiation. TF was developed by researchers at Google specialising in artificial intelligence. Other Python libraries used in the research presented were Numpy for data preparation, Matplotlib for plotting, Pandas for reading data from measurement logs and Scikit-learn for performing the hyperparameter tuning, standardization and the training of the RF, SVM and LR Model.

In the first step of the data processing, the punching force curves recorded in the experiments were processed with a proprietary script and the shearing process itself was extracted from the time series data (Figure 1a). In addition to reducing the amount of data to be processed, this extraction also improved the prediction quality of the models, as only the relevant physical regions of the recorded measurement signal were taken into account. Previous studies have shown that the prediction quality of an AI model is strongly dependent on the quality and preparation of the training data. The process of systematic preparation of the training data is also referred to as feature
output data for training the ML models were recorded through open-hole tensile tests. The tests carried out according to the method proposed by Watanabe [9]. Described tests were performed on a Roell + Korthaus RKM 100 material testing machine. The ARAMIS 12M digital image correlation system was used to measure the specimen strains. Figure 1 (b) shows two exemplary images of the Aramis 12M system taken out of the test series. Here, the left image shows the distribution of the major principal strain at the start of the test and the right image shows the major principal strain at the initiation of the crack. The maximum tolerable major and minor strain up to failure is determined at a distance of 1 mm from the edge of the introduced hole. Figure 1 (c) shows an overview of the results of all OHTTs performed.

<figure><img src="image_1717.png" /><figcaption>Fig. 1. 3D rendering of the modular test tool (a); sectional view of the modular test tool.</figcaption></figure>

<table><caption>Table 1. Punching parameters investigated in the present study.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Length of punch</td><td>80 mm</td></tr><tr><td>Cutting clearance</td><td>10%, 15%</td></tr><tr><td>Sheet Thickness</td><td>1 mm</td></tr><tr><td>Punch diameter</td><td>10 mm</td></tr><tr><td>Material</td><td>DP600, DP1000, DC03</td></tr></tbody></table>

## Feature Engineering, Model Design and Training

The data-specific investigations described in this paper were carried out using the Python programming language. The deep learning library TensorFlow (TF) was used to correlate features with the major and minor principal strain at crack initiation. TF was developed by researchers at Google specialising in artificial intelligence. Other Python libraries used in the research presented were Numpy for data preparation, Matplotlib for plotting, Pandas for reading data from measurement logs and Scikit-learn for performing the hyperparameter tuning, standardization and the training of the RF, SVM and LR Model.

In the first step of the data processing, the punching force curves recorded in the experiments were processed with a proprietary script and the shearing process itself was extracted from the time series data (Figure 1a). In addition to reducing the amount of data to be processed, this extraction also improved the prediction quality of the models, as only the relevant physical regions of the recorded measurement signal were taken into account. Previous studies have shown that the prediction quality of an AI model is strongly dependent on the quality and preparation of the training data. The process of systematic preparation of the training data is also referred to as feature
engineering od feature extraction. Feature engineering is based on domain-specific expert knowledge and feature extraction on algorithms for the statistical analysis of the data. For the research presented in this contribution, only feature extraction based on Principal component analysis (PCA) was used.

PCA was applied to the resulting data set, as it provides a method to summarise the training data and extract features that represent the individual differences in the process. PCA was thus used to identify a reduced data set based on components that still represent the original training data in a lower dimensional subspace, but with minimal loss of information [15]. Here, it was ensured that the calculated components described 95% of the variance of the original data set, which could be accomplished with ten components. This data set, reduced to 10 components, was used for the following model training. To train the ML model, the data set was divided as follows: 80% of the total punch force curves and the associated parameters from the hole tensile tests were used as the training data set. The remaining 20% of the total data set were further split in a validation (10%) and a test data set (10%).

In order to determine optimal hyperparameters for the models the validation data set was used. The ANN was tuned with the "Hyperband" tuner integrated into Keras. The aim of the optimization was to minimize the validation loss of the model. Table 2 shows an overview of the parameters varied and the range of parameters varied during the hyperparameter tuning of the ANN and the results of the tuning process. The SVM was tuned by varying the kernel functions. The model with the best R2 Score was choosen. Table 3 shows an overview of the varied kernel functions and the results of the tuning process. The RF model was optimized by using the "GridSearchCV" function of Scikit-learn. The aim of the optimization was to minimize the validation loss of the model during the tuning process. The parameters varied and the range of parameters varied during the hyperparameter tuning of the RF are shown in

<table><caption>Table 2. Hyperparameter tuning ANN.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Number of Hidden Layers</td><td>3 to 6</td><td>6</td></tr><tr><td>Neurons per Hidden Layer</td><td>32 to 512 (steps of 32)</td><td>320, 286, 192, 416, 32, 32</td></tr><tr><td>Activation function</td><td>Sigmoid, ReLU</td><td>ReLU</td></tr></tbody></table>

<table><caption>Table 3. Hyperparameter tuning SVM.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Kernel</td><td>sigmoid, rbf, poly, linear</td><td>sigmoid</td></tr></tbody></table>

<table><caption>Table 4. Hyperparameter tuning RF.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Number of Tress</td><td>10,50,100</td><td>50</td></tr><tr><td>Depth of the Tree</td><td>10,50,100</td><td>10</td></tr><tr><td>Number of samples at leaf node</td><td>1,2,4</td><td>2</td></tr><tr><td>Samples to split an internal node</td><td>2,5,10</td><td>2</td></tr></tbody></table>
engineering od feature extraction. Feature engineering is based on domain-specific expert knowledge and feature extraction on algorithms for the statistical analysis of the data. For the research presented in this contribution, only feature extraction based on Principal component analysis (PCA) was used.

PCA was applied to the resulting data set, as it provides a method to summarise the training data and extract features that represent the individual differences in the process. PCA was thus used to identify a reduced data set based on components that still represent the original training data in a lower dimensional subspace, but with minimal loss of information [15]. Here, it was ensured that the calculated components described 95% of the variance of the original data set, which could be accomplished with ten components. This data set, reduced to 10 components, was used for the following model training. To train the ML model, the data set was divided as follows: 80% of the total punch force curves and the associated parameters from the hole tensile tests were used as the training data set. The remaining 20% of the total data set were further split in a validation (10%) and a test data set (10%).

In order to determine optimal hyperparameters for the models the validation data set was used. The ANN was tuned with the "Hyperband" tuner integrated into Keras. The aim of the optimization was to minimize the validation loss of the model. Table 2 shows an overview of the parameters varied and the range of parameters varied during the hyperparameter tuning of the ANN and the results of the tuning process. The SVM was tuned by varying the kernel functions. The model with the best R2 Score was choosen. Table 3 shows an overview of the varied kernel functions and the results of the tuning process. The RF model was optimized by using the "GridSearchCV" function of Scikit-learn. The aim of the optimization was to minimize the validation loss of the model during the tuning process. The parameters varied and the range of parameters varied during the hyperparameter tuning of the RF are shown in

<table><caption>Table 2. Hyperparameter tuning ANN.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Number of Hidden Layers</td><td>3 to 6</td><td>6</td></tr><tr><td>Neurons per Hidden Layer</td><td>32 to 512 (steps of 32)</td><td>320, 286, 192, 416, 32, 32</td></tr><tr><td>Activation function</td><td>Sigmoid, ReLU</td><td>ReLU</td></tr></tbody></table>

<table><caption>Table 3. Hyperparameter tuning SVM.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Kernel</td><td>sigmoid, rbf, poly, linear</td><td>sigmoid</td></tr></tbody></table>

<table><caption>Table 4. Hyperparameter tuning RF.</caption><thead><tr><td>Parameter</td><td>Parameter range</td><td>Results of tuning</td></tr></thead><tbody><tr><td>Number of Tress</td><td>10,50,100</td><td>50</td></tr><tr><td>Depth of the Tree</td><td>10,50,100</td><td>10</td></tr><tr><td>Number of samples at leaf node</td><td>1,2,4</td><td>2</td></tr><tr><td>Samples to split an internal node</td><td>2,5,10</td><td>2</td></tr></tbody></table>
<figure><img src="image_1719.png" /><figcaption>Fig. 2. Experimentally determined punching force curves (a), Strain distribution at the beginning and during crack initiation in an OHTT (DC03, cutting clearance: 10%)(b), Results of the OHTT(c).</figcaption></figure>

## Results and Discussion

Figure 2 shows a comparison of the model training results for the different ML models examined. The bar plot shows the mean squared error (MSE), mean squared error (MAE) and the R2-Score for all models. These results were determined for the training data set Figure 2 (left) and the test data set not used for training or hyperparameter tuning Figure 2 (right). Due to the small difference in model error between the training and test datasets, overfitting and underfitting of the generated models is not observed. Because Support Vector Machine models can only predict one parameter due to their particular characteristics, two separate models were trained for each model type for major and minor principal strain at crack initiation. In order to achieve better comparability, the results for the ANN were also presented separately for major and minor principal strain. The comparison of the results shows that all trained models achieve a high Pearson correlation coefficient (>0.8), demonstrating the models' fundamental industrial applicability [16]. The comparison of prediction quality shows that the RF model has the highest regression quality and the lowest MAE for both predicted variables. The LR model has the second best prediction quality. The SVM model and the ANN are comparable in their performance. Both models show a significant difference in regression quality when comparing the predictions for major and minor principal strain. This shows that increasing model complexity does not improve the prediction results. At first this result seems surprising as the sheet metal forming process is characterized by strong non-linearities. It is expected that the ANN performs best and LR worst due to the higher model complexity. We assume that this might be due to two factors. The use of PCA leads to a type of linearization that simplifies the problem and may have a positive effect on the less complex
<figure><img src="image_1718.png" /><figcaption>Fig. 2. Experimentally determined punching force curves (a), Strain distribution at the beginning and during crack initiation in an OHTT (DC03, cutting clearance: 10%)(b), Results of the OHTT(c).</figcaption></figure>

## Results and Discussion

Figure 2 shows a comparison of the model training results for the different ML models examined. The bar plot shows the mean squared error (MSE), mean squared error (MAE) and the R2-Score for all models. These results were determined for the training data set Figure 2 (left) and the test data set not used for training or hyperparameter tuning Figure 2 (right). Due to the small difference in model error between the training and test datasets, overfitting and underfitting of the generated models is not observed. Because Support Vector Machine models can only predict one parameter due to their particular characteristics, two separate models were trained for each model type for major and minor principal strain at crack initiation. In order to achieve better comparability, the results for the ANN were also presented separately for major and minor principal strain. The comparison of the results shows that all trained models achieve a high Pearson correlation coefficient (>0.8), demonstrating the models' fundamental industrial applicability [16]. The comparison of prediction quality shows that the RF model has the highest regression quality and the lowest MAE for both predicted variables. The LR model has the second best prediction quality. The SVM model and the ANN are comparable in their performance. Both models show a significant difference in regression quality when comparing the predictions for major and minor principal strain. This shows that increasing model complexity does not improve the prediction results. At first this result seems surprising as the sheet metal forming process is characterized by strong non-linearities. It is expected that the ANN performs best and LR worst due to the higher model complexity. We assume that this might be due to two factors. The use of PCA leads to a type of linearization that simplifies the problem and may have a positive effect on the less complex
model. Furthermore, the data set used is relatively small and contains just 90 force curves. This means that the optimum result may not be achieved during hyperparameter tuning or subsequent training. In conclusion, the objective of the study was achieved. By using an RF model, it is possible to determine the maximum principal and minor strains at crack initiation from cutting force curves measured during punching operations.

<figure><img src="image_1720.png" /><figcaption>Fig. 3. Comparison of the prediction quality of the different models for the major and minor principal strain at crack initiation for the training data set (left) and the test data set (right).</figcaption></figure>

## Summary and Outlook

During the processing of high strength steel and aluminium sheet metal materials, premature failure can occur at the shear cut edges of components. This is due to the reduced formability induced by the punching process, particularly when compared to the inherent formability of the base material. Conventional testing methods are unable to measure the resulting residual formability inline during the blanking or punching process. In this contribution a novel method for characterising the maximum tolerable major and minor strain in combined shearing and forming processes based on ML was presented. As part of the investigations, it was shown that the
model. Furthermore, the data set used is relatively small and contains just 90 force curves. This means that the optimum result may not be achieved during hyperparameter tuning or subsequent training. In conclusion, the objective of the study was achieved. By using an RF model, it is possible to determine the maximum principal and minor strains at crack initiation from cutting force curves measured during punching operations.

<figure><img src="image_1719.png" /><figcaption>Fig. 3. Comparison of the prediction quality of the different models for the major and minor principal strain at crack initiation for the training data set (left) and the test data set (right).</figcaption></figure>

## Summary and Outlook

During the processing of high strength steel and aluminium sheet metal materials, premature failure can occur at the shear cut edges of components. This is due to the reduced formability induced by the punching process, particularly when compared to the inherent formability of the base material. Conventional testing methods are unable to measure the resulting residual formability inline during the blanking or punching process. In this contribution a novel method for characterising the maximum tolerable major and minor strain in combined shearing and forming processes based on ML was presented. As part of the investigations, it was shown that the
regression capability of an Random Forrest model is able to predict the maximum residual formability after punching on the basis of inline measured cutting force-displacement curves.

To advance the research presented in this publication, further investigations will be undertaken at IFU by transferring the results to other material classes such as copper or aluminum. Additionally, the influence of batch-related fluctuations in the semi-finished product properties will also be investigated. Future research will also investigate transfer learning methods to reduce the amount of data required for model training. Furthermore, the influence of wear of the tool parts on the prediction quality of the models will be investigated through endurance tests.

## References

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[2] K. Siegert, Ed. *Blechumformung*. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015.

[3] P. Sachnik, "Methodik für gratfreie Schnittflächen beim Scherschneiden," Dissertation, Technische Universität München, 2017.

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[5] N. Habibi, T. Beier, H. Richter, M. Könemann, and S. Münstermann, "The effects of shear affected zone on edge crack sensitivity in dual-phase steels," *IOP Conf. Ser.: Mater. Sci. Eng.*, vol. 651, no. 1, p. 12073, 2019. https://doi.org/10.1088/1757-899X/651/1/012073

[6] M. Liewald and M. Gall, "Experimental investigation of the influence of shear cutting parameters on the edge crack sensitivity of dual phase steels," in *Towards zero failure production methods by advanced modeling techniques and a process integrated virtual control: IDDRG, International Deep Drawing Research Group conference, Zurich, Switzerland, June 2 - 5, 2013*, P. Hora, Ed., Zürich: Institute of Virtual Manufacturing ETH Zurich, 2013, pp. 219–224.

[7] *ISO 16630:2017 Metallic materials — Sheet and strip — Hole expanding test*, Internationale Organisation für Normung, Berlin, 2017.

[8] M. Feistle, R. Golle, and W. Volk, "Determining the Influence of Shear Cutting Parameters on the Edge Cracking Susceptibility of High-strength-steels Using the Edge-fracture-tensile-test," *Procedia CIRP*, vol. 41, pp. 1078–1083, 2016. https://doi.org/10.1016/j.procir.2016.01.007

[9] K. Watanabe and M. Tachibana, "Simple prediction method for the edge fracture of steel sheet during vehicle collision (1st report)," in *5. LS-DYNA Anwenderforum, Ulm 2006*, DYNAmore GmbH, Ed., 2006.

[10] C. Kubik, D. A. Molitor, M. Rojahn, and P. Groche, "Towards a real-time tool state detection in sheet metal forming processes validated by wear classification during blanking," *IOP Conf. Ser.: Mater. Sci. Eng.*, vol. 1238, no. 1, p. 12067, 2022. doi: 10.1088/1757-899X/1238/1/012067. [Online]. Available: https://iopscience.iop.org/article/10.1088/1757-899X/1238/1/012067

[11] S. Niessner and M. Liewald, "Identification of methods for the in-situ measurement of cutting forces in a tool-bound punching machine," *IOP Conf. Ser.: Mater. Sci. Eng.*, vol. 967, no. 1, p. 12025, 2020. doi: 10.1088/1757-899X/967/1/012025. [Online]. Available: https://iopscience.iop.org/article/10.1088/1757-899X/967/1/012025

[12] C. Kubik, D. A. Molitor, S. Varchmin, D. S. Leininger, J. Ohrenberg, and P. Groche, "Image-based feature extraction for inline quality assurance and wear classification in high-speed blanking processes," *Int J Adv Manuf Technol*, 2023. https://doi.org/10.1007/s00170-023-12653-x
regression capability of an Random Forrest model is able to predict the maximum residual formability after punching on the basis of inline measured cutting force-displacement curves.

To advance the research presented in this publication, further investigations will be undertaken at IFU by transferring the results to other material classes such as copper or aluminum. Additionally, the influence of batch-related fluctuations in the semi-finished product properties will also be investigated. Future research will also investigate transfer learning methods to reduce the amount of data required for model training. Furthermore, the influence of wear of the tool parts on the prediction quality of the models will be investigated through endurance tests.

References

[1] H. Hoffmann, R. Neugebauer, and G. Spur, Eds. *Handbuch Umformen*. München: Carl Hanser Verlag GmbH & Co. KG, 2012. https://doi.org/10.1007/978-3-446-43004-4

[2] K. Siegert, Ed. *Blechumformung*. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015.

[3] P. Sachnik, "Methodik für gratfreie Schnittflächen beim Scherschneiden," Dissertation, Technische Universität München, 2017.

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# Influence of data filtering and noise on the calibration of constitutive models using machine learning techniques

PRATES Pedro¹,²,³,a*, PINTO José⁴, MARQUES João¹, HENRIQUES João¹,², PEREIRA André³ and ANDRADE-CAMPOS António¹,²

¹Centre for Mechanical Technology and Automation (TEMA), Mechanical Engineering Department, University of Aveiro, Portugal

²Intelligent Systems Associate Laboratory (LASI), Portugal

³Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), ARISE, Department of Mechanical Engineering, University of Coimbra, Coimbra, Portugal

⁴Physics Department, University of Aveiro, Portugal

a prates@ua.pt

**Keywords:** Parameter Identification, Machine Learning, Feature Analysis, Noise, Sheet Metal Forming

**Abstract.** This work focuses on predicting material parameters that describe the plastic behaviour of metallic sheets using the XGBoost machine learning algorithm, with a dual focus on the influence of data filtering and data noise. A dataset was populated with finite element simulation results of cruciform tensile tests, including strain field data during the test. Different noise levels were added to the strain-related features of the dataset; additionally, a feature importance study was carried out to identify and select the most relevant features of the dataset. A systematic analysis shows how feature noise and selection individually and simultaneously influence the predictive performance of machine learning models. The results show that feature selection will greatly accelerate model training, without losing its predictive performance. Also, adding noise to the features does not have significant impact on model performance, highlighting the robustness of the models.

## Introduction

The essential role of numerical simulations in sheet metal forming processes for automotive and aerospace industries requires precise constitutive modeling. While existing strategies like Finite Element Model Updating (FEMU) and Virtual Fields Method (VFM) have been proposed, they present computational challenges [1, 2]. This study builds upon previous work by using machine learning, specifically the XGBoost algorithm, for material parameter identification. It extends prior work by employing machine learning, specifically the XGBoost algorithm, for material parameter identification. Initially, a dataset was generated using finite element simulation results from cruciform tensile tests. After preprocessing the dataset, the XGBoost algorithm was trained, and its performance in predicting material parameters was assessed. Subsequently, noise was introduced to the strain-related features to emulate scenarios like measurement-affecting errors in Digital Image Correlation (DIC), impacting computed strains via DIC. The performance of the model was re-evaluated under these conditions. Finally, a feature analysis was conducted to assess the robustness of machine learning models, considering only the most important features; the influence of added noise to the strain-related important features was again assessed.

## Numerical simulation model

This study employs the biaxial tensile test on a cruciform sample as the chosen mechanical test. The sample's geometry, previously designed in a previous work [3], allows for the generation of

<figure><img src="image_1721.png" /></figure>
# Influence of data filtering and noise on the calibration of constitutive models using machine learning techniques

PRATES Pedro<sup>1,2,3,a*</sup>, PINTO José<sup>4</sup>, MARQUES João<sup>1</sup>, HENRIQUES João<sup>1,2</sup>, PEREIRA André<sup>3</sup> and ANDRADE-CAMPOS António<sup>1,2</sup>

<sup>1</sup>Centre for Mechanical Technology and Automation (TEMA), Mechanical Engineering Department, University of Aveiro, Portugal

<sup>2</sup>Intelligent Systems Associate Laboratory (LASI), Portugal

<sup>3</sup>Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), ARISE, Department of Mechanical Engineering, University of Coimbra, Coimbra, Portugal

<sup>4</sup>Physics Department, University of Aveiro, Portugal

<sup>a</sup>prates@ua.pt

**Keywords:** Parameter Identification, Machine Learning, Feature Analysis, Noise, Sheet Metal Forming

**Abstract.** This work focuses on predicting material parameters that describe the plastic behaviour of metallic sheets using the XGBoost machine learning algorithm, with a dual focus on the influence of data filtering and data noise. A dataset was populated with finite element simulation results of cruciform tensile tests, including strain field data during the test. Different noise levels were added to the strain-related features of the dataset; additionally, a feature importance study was carried out to identify and select the most relevant features of the dataset. A systematic analysis shows how feature noise and selection individually and simultaneously influence the predictive performance of machine learning models. The results show that feature selection will greatly accelerate model training, without losing its predictive performance. Also, adding noise to the features does not have significant impact on model performance, highlighting the robustness of the models.

## Introduction

The essential role of numerical simulations in sheet metal forming processes for automotive and aerospace industries requires precise constitutive modeling. While existing strategies like Finite Element Model Updating (FEMU) and Virtual Fields Method (VFM) have been proposed, they present computational challenges [1, 2]. This study builds upon previous work by using machine learning, specifically the XGBoost algorithm, for material parameter identification. It extends prior work by employing machine learning, specifically the XGBoost algorithm, for material parameter identification. Initially, a dataset was generated using finite element simulation results from cruciform tensile tests. After preprocessing the dataset, the XGBoost algorithm was trained, and its performance in predicting material parameters was assessed. Subsequently, noise was introduced to the strain-related features to emulate scenarios like measurement-affecting errors in Digital Image Correlation (DIC), impacting computed strains via DIC. The performance of the model was re-evaluated under these conditions. Finally, a feature analysis was conducted to assess the robustness of machine learning models, considering only the most important features; the influence of added noise to the strain-related important features was again assessed.

## Numerical simulation model

This study employs the biaxial tensile test on a cruciform sample as the chosen mechanical test. The sample's geometry, previously designed in a previous work [3], allows for the generation of

<figure><img src="image_1720.png" /></figure>
heterogeneous stress and strain fields, encompassing a diverse range of stress and strain paths commonly encountered in sheet metal forming processes. The cruciform sample's geometry and dimensions in the sheet plane are presented in Fig. 1(a). The numerical simulation model considers only one quarter of the sample, as indicated by the grey region in Fig. 1(a), due to symmetries in boundary conditions, sample geometry, and material behaviour. The simulation assumes plane stress conditions and maintains a constant thickness of the sheet. Fig. 1(b) depicts the boundary conditions and finite element mesh employed in the numerical model. Symmetry boundary conditions were applied on the 0x and 0y axes ($u_x = u_y = 0$ mm). Additionally, displacement boundary conditions are applied to nodes at the ends of both arms of the sample to ensure equal displacements along both 0x and 0y axes ($u_x = u_y = 2$ mm). The numerical model uses a regular mesh consisting of 405 CPS4R elements with bilinear shape functions and reduced integration. All finite element analysis (FEA) simulations are carried out using the ABAQUS CAE software [4]. Each simulation consists of twenty equally spaced time-steps, during which the forces along the 0x and 0y directions and the strain field ($\epsilon_x, \epsilon_y$, and $\epsilon_{xy}$) are obtained at each time-step.

<figure><img src="image_1722.png" /><figcaption>Fig. 1. Biaxial tensile test on a cruciform sample: (a) geometry and dimensions; (b) boundary conditions and finite element mesh.</figcaption></figure>

<figure><img src="image_1723.png" /></figure>

The constitutive model of the material assumes an isotropic elastic behavior, described by the Hooke's law (with Young's modulus $E = 210$ GPa and Poisson's ratio $\nu = 0.3$), and an orthotropic plastic behavior, described by Hill's yield criterion with isotropic hardening described by Swift law under an associated flow rule. For plane stress conditions, the Hill's yield criterion can be written as follows:

$$ (F+H)\sigma_{yy}^2 + (G+H)\sigma_{xx}^2 - 2H\sigma_{xx}\sigma_{yy} + 2N\tau_{xy}^2 = Y^2, \quad (1) $$

where $F, G, H$ and $N$ are anisotropy coefficients, $\sigma_{xx}, \sigma_{yy}$ and $\tau_{xy}$ are the components of the Cauchy stress tensor in the material axes system of the metal sheet, and $Y$ is the yield stress.

The condition $G+H=1$ (i.e. $\sigma_{xx} = Y$) was assumed, which corresponds to the following relationships:

$$ F = \frac{r_0}{r_{90}(r_0 + 1)}; \quad G = \frac{1}{r_0 + 1}; \quad H = \frac{r_0}{r_0 + 1}; \quad N = \frac{1}{2} \frac{(r_0 + r_{90})(2r_{45} + 1)}{r_{90}(r_0 + 1)}, \quad (2) $$

where $r_0, r_{45}$ and $r_{90}$ are the Lankford coefficients obtained at 0°, 45° and 90° w.r.t. the rolling direction of the sheet, respectively. The Swift law describes the yield stress evolution during plastic deformation as follows:
heterogeneous stress and strain fields, encompassing a diverse range of stress and strain paths commonly encountered in sheet metal forming processes. The cruciform sample's geometry and dimensions in the sheet plane are presented in Fig. 1(a). The numerical simulation model considers only one quarter of the sample, as indicated by the grey region in Fig. 1(a), due to symmetries in boundary conditions, sample geometry, and material behaviour. The simulation assumes plane stress conditions and maintains a constant thickness of the sheet. Fig. 1(b) depicts the boundary conditions and finite element mesh employed in the numerical model. Symmetry boundary conditions were applied on the 0x and 0y axes ($u_x = u_y = 0$ mm). Additionally, displacement boundary conditions are applied to nodes at the ends of both arms of the sample to ensure equal displacements along both 0x and 0y axes ($u_x = u_y = 2$ mm). The numerical model uses a regular mesh consisting of 405 CPS4R elements with bilinear shape functions and reduced integration. All finite element analysis (FEA) simulations are carried out using the ABAQUS CAE software [4]. Each simulation consists of twenty equally spaced time-steps, during which the forces along the 0x and 0y directions and the strain field ($\varepsilon_x, \varepsilon_y$, and $\varepsilon_{xy}$) are obtained at each time-step.

<figure><img src="image_1721.png" /><figcaption>Fig. 1. Biaxial tensile test on a cruciform sample: (a) geometry and dimensions; (b) boundary conditions and finite element mesh.</figcaption></figure>

The constitutive model of the material assumes an isotropic elastic behavior, described by the Hooke's law (with Young's modulus $E = 210$ GPa and Poisson's ratio $\nu = 0.3$), and an orthotropic plastic behavior, described by Hill's yield criterion with isotropic hardening described by Swift law under an associated flow rule. For plane stress conditions, the Hill's yield criterion can be written as follows:

$$ (F + H)\sigma_{yy}^2 + (G + H)\sigma_{xx}^2 - 2H\sigma_{xx}\sigma_{yy} + 2N\tau_{xy}^2 = Y^2, \quad (1) $$

where $F, G, H$ and $N$ are anisotropy coefficients, $\sigma_{xx}, \sigma_{yy}$ and $\tau_{xy}$ are the components of the Cauchy stress tensor in the material axes system of the metal sheet, and $Y$ is the yield stress.

The condition $G+H=1$ (i.e. $\sigma_{xx} = Y$) was assumed, which corresponds to the following relationships:

$$ F = \frac{r_0}{r_{90}(r_0 + 1)}; \quad G = \frac{1}{r_0 + 1}; \quad H = \frac{r_0}{r_0 + 1}; \quad N = \frac{1}{2} \frac{(r_0 + r_{90})(2r_{45} + 1)}{r_{90}(r_0 + 1)}, \quad (2) $$

where $r_0, r_{45}$ and $r_{90}$ are the Lankford coefficients obtained at 0°, 45° and 90° w.r.t. the rolling direction of the sheet, respectively. The Swift law describes the yield stress evolution during plastic deformation as follows:
$$Y = K \left[ \left( \frac{\sigma_0}{K} \right)^{1/n} + \bar{\varepsilon}^p \right]^n, \quad (3)$$

where $\bar{\varepsilon}^p$ is the equivalent plastic strain and $\sigma_0$, $K$ and $n$ are material parameters.

## Dataset generation, model training and performance evaluation

A total of 5733 sets of material parameters $\sigma_0$, $K$, $n$, $r_0$, $r_{45}$ and $r_{90}$ (i.e. samples) were generated using the Latin Hypercube Sampling (LHS) method. The input space the material parameters is: $\sigma_0 \in [120, 300]$ MPa, $K \in [280, 700]$ MPa, $n \in [0.10, 0.30]$; $r_0$, $r_{45}$, $r_{90} \in [0.600, 2.500]$. The step size for $\sigma_0$ and $K$ is 1 MPa, for $n$ is 0.01 and for $r_0$, $r_{45}$ and $r_{90}$ is 0.001. Numerical simulations of the cruciform tensile test were performed for each sample while maintaining the same geometry, boundary conditions and elastic properties. Fig. 2 presents an example of numerical results of the cruciform test that were used to build the dataset [2].

<figure><img src="image_1724.png" /><figcaption>Fig. 2. FEA results of the cruciform test ($Y_0=172$MPa, $n=0.16$; $K=486$MPa; $r_0=2.38$; $r_{45}=1.8$; $r_{90}=1.06$) [2]: (a) load vs. displacement along the $0x$ and $0y$ axes; strain fields (b) $\varepsilon_{xx}$, (c) $\varepsilon_{yy}$, and (d) $\varepsilon_{xy}$. The strain fields ($\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon_{xy}$) were obtained for $u_x=u_y=2$mm.</figcaption></figure>

Among the 5000 samples, 4671 exhibited no decrease in load during the simulation and were used for populating the dataset with synthetic data. Then, the dataset was randomly split into two subsets: a training set consisting of 4000 samples and a test set comprising 671 simulations. Each of the training and test sets consist of two matrices: a feature matrix and a target matrix. The feature matrix has shape $n_{samples} \times n_{features}$, where $n_{samples}$ is the total number of samples (i.e. numerical simulations) of the set and $n_{features}$ is the total number of features (20 timesteps × 2 forces + 3 strain components × 405 elements) = 24340). The target matrix has shape $n_{samples} \times n_{parameters}$, where $n_{outputs}$ is the total number of model outputs (i.e. material parameters). The feature and target matrices of the training set are shaped 4000×24340 and 4000×6, respectively, while the feature and target matrices of the test set have shape 671×24340 and 671×6, respectively. Fig. 3 shows the
$$Y = K \left[ \left( \frac{\sigma_0}{K} \right)^{1/n} + \bar{\varepsilon}^p \right]^n, \quad (3)$$

where $\bar{\varepsilon}^p$ is the equivalent plastic strain and $\sigma_0$, $K$ and $n$ are material parameters.

## Dataset generation, model training and performance evaluation

A total of 5733 sets of material parameters $\sigma_0$, $K$, $n$, $r_0$, $r_{45}$ and $r_{90}$ (i.e. samples) were generated using the Latin Hypercube Sampling (LHS) method. The input space the material parameters is: $\sigma_0 \in [120, 300]$ MPa, $K \in [280, 700]$ MPa, $n \in [0.10, 0.30]$; $r_0$, $r_{45}$, $r_{90} \in [0.600, 2.500]$. The step size for $\sigma_0$ and $K$ is 1 MPa, for $n$ is 0.01 and for $r_0$, $r_{45}$ and $r_{90}$ is 0.001. Numerical simulations of the cruciform tensile test were performed for each sample while maintaining the same geometry, boundary conditions and elastic properties. Fig. 2 presents an example of numerical results of the cruciform test that were used to build the dataset [2].

<figure><img src="image_1722.png" /><figcaption>Fig. 2. FEA results of the cruciform test ($Y_0=172MPa$, $n=0.16$; $K=486MPa$; $r_0=2.38$; $r_{45}=1.8$; $r_{90}=1.06$) [2]: (a) load vs. displacement along the 0x and 0y axes; strain fields (b) $\varepsilon_{xx}$, (c) $\varepsilon_{yy}$, and (d) $\varepsilon_{xy}$. The strain fields ($\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon_{xy}$) were obtained for $u_x=u_y=2mm$.</figcaption></figure>

Among the 5000 samples, 4671 exhibited no decrease in load during the simulation and were used for populating the dataset with synthetic data. Then, the dataset was randomly split into two subsets: a training set consisting of 4000 samples and a test set comprising 671 simulations. Each of the training and test sets consist of two matrices: a feature matrix and a target matrix. The feature matrix has shape $n_{samples} \times n_{features}$, where $n_{samples}$ is the total number of samples (i.e. numerical simulations) of the set and $n_{features}$ is the total number of features (20 timesteps × 2 forces + 3 strain components × 405 elements) = 24340). The target matrix has shape $n_{samples} \times n_{parameters}$, where $n_{outputs}$ is the total number of model outputs (i.e. material parameters). The feature and target matrices of the training set are shaped 4000×24340 and 4000×6, respectively, while the feature and target matrices of the test set have shape 671×24340 and 671×6, respectively. Fig. 3 shows the
distribution patterns of the material parameters for the 4671 samples. Distributions associated with parameter *n* exhibit constrained coverage in the input space, a result of a too large step size (0.01). Also, samples featuring combinations of high values of *σ*₀ and low values of *K* were excluded from the initial 5000 sets due to an observed decrease in load during the simulation; the same occurred for samples having *σ*₀ values in between 188 and 190 MPa (only 1 sample having 190MPa was not excluded).

<figure><img src="image_1725.png" /><figcaption>Fig. 3. Distribution patterns for material parameters in the training set (black markers) and test set (red markers).</figcaption></figure>

The training set was used for training the Extreme Gradient Boosting (XGBoost) regression algorithm; more details regarding the XGBoost algorithm can be found in [1, 5]. The hyperparameters of the XGBoost regression algorithm were kept as default, except for the learning rate (=0.02), max_depth (=15) and n_estimators (=1000) [5]. The test set was used for evaluating the performance of the ML model in predicting the material parameters. Fig. 4 presents the performance evaluation of the ML model. It compares the σ₀, *n* and *K*, *r*₀, *r*₄₅ and *r*₉₀ values
distribution patterns of the material parameters for the 4671 samples. Distributions associated with parameter *n* exhibit constrained coverage in the input space, a result of a too large step size (0.01). Also, samples featuring combinations of high values of *σ*₀ and low values of *K* were excluded from the initial 5000 sets due to a observed decrease in load during the simulation; the same occurred for samples having *σ*₀ values inbetween 188 and 190 MPa (only 1 sample having 190MPa was not excluded).

<figure><img src="image_1723.png" /><figcaption>Fig. 3. Distribution patterns for material parameters in the training set (black markers) and test set (red markers).</figcaption></figure>

The training set was used for training the Extreme Gradient Boosting (XGBoost) regression algorithm; more details regarding the XGBoost algorithm can be found in [1, 5]. The hyperparameters of the XGBoost regression algorithm were kept as default, except for the learning rate (=0.02), max_depth (=15) and n_estimators (=1000) [5]. The test set was used for evaluating the performance of the ML model in predicting the material parameters. Fig. 4 presents the performance evaluation of the ML model. It compares the σ₀, *n* and *K*, *r*₀, *r*₄₅ and *r*₉₀ values
predicted by the ML model with the real values considered in the FEA simulations of the test set. Generally, the machine learning model demonstrates superior predictive performance, as evidenced by the R² values presented in the figures.

<figure><img src="image_1726.png" /><figcaption>Fig. 4. Evaluation of constitutive parameter predictions for the test set: (a) $\sigma_0$; (b) $K$; (c) $n$; (d) $r_0$; (e) $r_{45}$ and (f) $r_{90}$.</figcaption></figure>

## Performance evaluation with training set noise

Now, random noise following a uniform distribution is added to all strain-related features of the training dataset. This aims to replicate scenarios such as measurement-affecting errors in Digital Image Correlation (DIC) that influence the strains computed via DIC. Four distinct noise levels were taken into account (± 0%, ± 5%, ± 10%, ± 20%), with a model trained and evaluated for each level. Table 1 presents the R² values for the constitutive parameters under different noise ranges added to the training set. As noise levels increase, the model's adaptability and predictive performance are evaluated. Increasing the level of noise does not significantly diminish the model's performance in predicting the constitutive parameters. In fact, incorporating noise during training can enhance the robustness of the model and can contribute to a reduction in generalization error.

<table><caption>Table 1. Influence of added noise to the training set on model performance in predicting the material parameters.</caption><thead><tr><th>Noise level</th><th>R<sup>2</sup> σ<sub>0</sub></th><th>R<sup>2</sup> K</th><th>R<sup>2</sup> n</th><th>R<sup>2</sup> r<sub>0</sub></th><th>R<sup>2</sup> r<sub>45</sub></th><th>R<sup>2</sup> r<sub>90</sub></th></tr></thead><tbody><tr><td>± 0%</td><td>0.9972</td><td>0.9925</td><td>0.9808</td><td>0.9991</td><td>0.9959</td><td>0.9989</td></tr><tr><td>± 5%</td><td>0.9983</td><td>0.9917</td><td>0.9777</td><td>0.9992</td><td>0.9952</td><td>0.9991</td></tr><tr><td>± 10%</td><td>0.9983</td><td>0.9913</td><td>0.9771</td><td>0.9991</td><td>0.9951</td><td>0.9988</td></tr><tr><td>± 20%</td><td>0.9982</td><td>0.9908</td><td>0.9739</td><td>0.9981</td><td>0.9944</td><td>0.9980</td></tr></tbody></table>
predicted by the ML model with the real values considered in the FEA simulations of the test set. Generally, the machine learning model demonstrates superior predictive performance, as evidenced by the R² values presented in the figures.

<figure><img src="image_1724.png" /><figcaption>Fig. 4. Evaluation of constitutive parameter predictions for the test set: (a) $\sigma_0$; (b) $K$; (c) $n$; (d) $r_0$; (e) $r_{45}$ and (f) $r_{90}$.</figcaption></figure>

## Performance evaluation with training set noise

Now, random noise following a uniform distribution is added to all strain-related features of the training dataset. This aims to replicate scenarios such as measurement-affecting errors in Digital Image Correlation (DIC) that influence the strains computed via DIC. Four distinct noise levels were taken into account (± 0%, ± 5%, ± 10%, ± 20%), with a model trained and evaluated for each level. Table 1 presents the R² values for the constitutive parameters under different noise ranges added to the training set. As noise levels increase, the model's adaptability and predictive performance are evaluated. Increasing the level of noise does not significantly diminish the model's performance in predicting the constitutive parameters. In fact, incorporating noise during training can enhance the robustness of the model and can contribute to a reduction in generalization error.

<table><caption>Table 1. Influence of added noise to the training set on model performance in predicting the material parameters.</caption><thead><tr><th>Noise level</th><th>R<sup>2</sup> σ<sub>0</sub></th><th>R<sup>2</sup> K</th><th>R<sup>2</sup> n</th><th>R<sup>2</sup> r<sub>0</sub></th><th>R<sup>2</sup> r<sub>45</sub></th><th>R<sup>2</sup> r<sub>90</sub></th></tr></thead><tbody><tr><td>± 0%</td><td>0.9972</td><td>0.9925</td><td>0.9808</td><td>0.9991</td><td>0.9959</td><td>0.9989</td></tr><tr><td>± 5%</td><td>0.9983</td><td>0.9917</td><td>0.9777</td><td>0.9992</td><td>0.9952</td><td>0.9991</td></tr><tr><td>± 10%</td><td>0.9983</td><td>0.9913</td><td>0.9771</td><td>0.9991</td><td>0.9951</td><td>0.9988</td></tr><tr><td>± 20%</td><td>0.9982</td><td>0.9908</td><td>0.9739</td><td>0.9981</td><td>0.9944</td><td>0.9980</td></tr></tbody></table>
An analysis of the overall relative error in predicting material parameters was carried out to assess the predictive performance of the models across the 671 simulations of the test set. This evaluation was performed using the Euclidean norm $||\delta_j||$, expressed as follows:

$$ ||\delta_j||(\%) = \sqrt{\frac{1}{m} \sum_{i=1}^{m} \left( \frac{x'_{i,j} - x_{i,j}}{x_{i,j}} \right)^2} \times 100\%, \quad i=1,...,m; j=1,...,671, \quad (4) $$

where $x'_{i,j}$ and $x_{i,j}$ represent the estimated and true values of the i-th constitutive parameter in the j-th simulation in the testing set, and m is the total number of constitutive parameters to be identified (m=6). Fig. 5 presents $||\delta_j||$ as a function of the 671 simulations in the test set and compares the influence of noise added to the training set (± 0% vs. ± 20% added noise). In the case of the model trained on a noise-free training set, only 6 samples among the 671 exceed the 5% error threshold. However, for the model trained with ± 20% noise, only 11 samples exceed the 5% error threshold. These instances are minimal in comparison to the overall size of the test set, again highlighting the model's robustness in estimating material parameters even with added noise.

<figure><img src="image_1727.png" /><figcaption>Fig. 5. Relative error $||\delta_j||$ in the prediction of material parameters for the 671 samples of the test set: (a) model trained with a noise-free training set; (b) model trained with ± 20% added noise.</figcaption></figure>

## Feature analysis and selection

As the previous ML models have demonstrated robustness to noise, a question arises: how does the sensitivity of the models to noise in the training set evolve when the dataset undergoes a substantial reduction in the number of features? To tackle this question, a features analysis using Shapley Additive Explanations (SHAP) is performed [6], to provide insights into the impact of employing a reduced dataset containing only the most influential features for the models to estimate the material parameters. Fig. 6 presents the 20 most influential features to predict each of the 6 material parameters. Each force feature is labelled as “Force_direction_timestep” (direction can be “x”, or “y”; timestep ranges between 1 and 20), and each strain feature is labelled as “Strain_component_elementnumber_timestep” (component can be “x”, “y”, or “xy”; elementnumber ranges between 1 and 405 - see Fig. 1, and timestep ranges between 1 and 20). For example, according to Fig. 6 (a) and (b) the most influential features for predicting $\sigma_0$ and K are the force along the y direction at the first and last timesteps, respectively (which is naturally expected). For the trained model, an increase in the magnitudes of these features corresponds to an increase in the predicted values of the parameters, highlighting a positive correlation between these input features and the predicted outcomes. The remaining features shown in Fig. 6 (a) and
An analysis of the overall relative error in predicting material parameters was carried out to assess the predictive performance of the models across the 671 simulations of the test set. This evaluation was performed using the Euclidean norm $||\delta_j||$, expressed as follows:

$$ ||\delta_j||(\%) = \sqrt{\frac{1}{m} \sum_{i=1}^{m} \left( \frac{x'_{i,j} - x_{i,j}}{x_{i,j}} \right)^2} \times 100\%, \quad i=1,...,m; j=1,...,671, \quad (4) $$

where $x'_{i,j}$ and $x_{i,j}$ represent the estimated and true values of the i-th constitutive parameter in the j-th simulation in the testing set, and m is the total number of constitutive parameters to be identified (m=6). Fig. 5 presents $||\delta_j||$ as a function of the 671 simulations in the test set and compares the influence of noise added to the training set (± 0% vs. ± 20% added noise). In the case of the model trained on a noise-free training set, only 6 samples among the 671 exceed the 5% error threshold. However, for the model trained with ± 20% noise, only 11 samples exceed the 5% error threshold. These instances are minimal in comparison to the overall size of the test set, again highlighting the model's robustness in estimating material parameters even with added noise.

<figure><img src="image_1725.png" /><figcaption>Fig. 5. Relative error $||\delta_j||$ in the prediction of material parameters for the 671 samples of the test set: (a) model trained with a noise-free training set; (b) model trained with ± 20% added noise.</figcaption></figure>

## Feature analysis and selection

As the previous ML models have demonstrated robustness to noise, a question arises: how does the sensitivity of the models to noise in the training set evolve when the dataset undergoes a substantial reduction in the number of features? To tackle this question, a features analysis using Shapley Additive Explanations (SHAP) is performed [6], to provide insights into the impact of employing a reduced dataset containing only the most influential features for the models to estimate the material parameters. Fig. 6 presents the 20 most influential features to predict each of the 6 material parameters. Each force feature is labelled as “Force_direction_timestep” (direction can be “x”, or “y”; timestep ranges between 1 and 20), and each strain feature is labelled as “Strain_component_elementnumber_timestep” (component can be “x”, “y”, or “xy”; elementnumber ranges between 1 and 405 - see Fig. 1, and timestep ranges between 1 and 20). For example, according to Fig. 6 (a) and (b) the most influential features for predicting $\sigma_0$ and K are the force along the y direction at the first and last timesteps, respectively (which is naturally expected). For the trained model, an increase in the magnitudes of these features corresponds to an increase in the predicted values of the parameters, highlighting a positive correlation between these input features and the predicted outcomes. The remaining features shown in Fig. 6 (a) and
(b) are mostly strain-related features, which present little influence for predicting $\sigma_0$ and K. A similar analysis conducted for the remaining parameters reveals subtle insights that may not be immediately apparent. SHAP analysis contributes to a more comprehensive understanding of the relationships between the input features and the predicted outcomes for these material parameters.

<figure><img src="image_1728.png" /><figcaption>Fig. 6. SHAP feature importance analysis: (a) $\sigma_0$; (b) K; (c) n; (d) $r_0$; (e) $r_{45}$ and (f) $r_{90}$.</figcaption></figure>

## Model noise sensitivity after feature selection

New training and test sets were generated by excluding features not displayed in Fig. 6 from the original training and testing sets. These refined new sets, each comprising 118 features (as opposed to the original 24340 features), served as the basis for further analysis. Additionally, a perturbed training set was derived from this refined training set, adding a noise level of ± 20% on the strain-related features. A model was trained with the refined training set, and another underwent training with the training set containing added noise. Both models achieved training completion in less than 1 minute, in contrast to the original training set, which required about 3 hours (13th Gen Intel(R) Core(TM) i9-13900 24-core @2.00 GHz, 32 Gb RAM). The validation of both models was conducted using the same training set. Table 2 presents the performance of the ML models trained with a refined training set, where the comparison focuses on models trained with a refined training set under two conditions: noise-free and with ± 20% added noise. The results show that the model only requires a small fraction of the original 243340 features to have excellent results in predicting
(b) are mostly strain-related features, which present little influence for predicting $σ_0$ and K. A similar analysis conducted for the remaining parameters reveals subtle insights that may not be immediately apparent. SHAP analysis contributes to a more comprehensive understanding of the relationships between the input features and the predicted outcomes for these material parameters.

<figure><img src="image_1726.png" /><figcaption>Fig. 6. SHAP feature importance analysis: (a) $σ_0$; (b) K; (c) n; (d) $r_0$; (e) $r_{45}$ and (f) $r_{90}$.</figcaption></figure>

## Model noise sensitivity after feature selection

New training and test sets were generated by excluding features not displayed in Fig. 6 from the original training and testing sets. These refined new sets, each comprising 118 features (as opposed to the original 24340 features), served as the basis for further analysis. Additionally, a perturbed training set was derived from this refined training set, adding a noise level of ± 20% on the strain-related features. A model was trained with the refined training set, and another underwent training with the training set containing added noise. Both models achieved training completion in less than 1 minute, in contrast to the original training set, which required about 3 hours (13th Gen Intel(R) Core(TM) i9-13900 24-core @2.00 GHz, 32 Gb RAM). The validation of both models was conducted using the same training set. Table 2 presents the performance of the ML models trained with a refined training set, where the comparison focuses on models trained with a refined training set under two conditions: noise-free and with ± 20% added noise. The results show that the model only requires a small fraction of the original 243340 features to have excellent results in predicting
the values of the six material parameters. Incorporating a noise level of ± 20% results does not significantly diminish the model's predictive performance, as shown in Table 2. Comparing Fig. 7 to Fig. 5, only 6 samples exceed the 5% error threshold for the model trained with a noise-free set, and 11 samples for the model trained with ± 20% added noise. Again, these instances are minimal in comparison to the overall size of the test set, again demonstrating the model's robustness in estimating material parameters in the presence of added noise and assuming a small fraction of the total number of features.

<table><caption>Table 2. Influence of added noise to the refined training set on model performance in predicting the material parameters.</caption><thead><tr><th>Noise range</th><th>R<sup>2</sup><sub>σ0</sub></th><th>R<sup>2</sup><sub>K</sub></th><th>R<sup>2</sup><sub>n</sub></th><th>R<sup>2</sup><sub>r0</sub></th><th>R<sup>2</sup><sub>r45</sub></th><th>R<sup>2</sup><sub>r90</sub></th></tr></thead><tbody><tr><td>± 0%</td><td>0.9982</td><td>0.9935</td><td>0.9826</td><td>0.9991</td><td>0.9947</td><td>0.9991</td></tr><tr><td>± 20%</td><td>0.9981</td><td>0.9923</td><td>0.9688</td><td>0.9983</td><td>0.9934</td><td>0.9981</td></tr></tbody></table>

<figure><img src="image_1729.png" /><figcaption>Fig. 7. Relative error $||\delta_j||$ in the prediction of material parameters for the 671 samples of the test set: (a) model trained with a noise-free training set; (b) model trained with ± 20% added noise. Models trained with only 118 features.</figcaption></figure>

In Fig. 8, a comparison is presented between the actual and predicted hardening curves, along with Lankford coefficients in the sheet plane. This comparison focuses on 3 samples surpassing the 5% error threshold (see Fig. 7 (a)), considering the model trained with a noise-free dataset. Generally, the prediction error for the hardening curve remains below 2.5%. However, the maximum error in predicting the Lankford coefficients tends to exceed 5%, particularly at $r_{45}$, except for the sample labeled "134". This discrepancy may be attributed to the limited sensitivity of the cruciform test results in estimating $r_{45}$.
the values of the six material parameters. Incorporating a noise level of ± 20% results does not significantly diminish the model's predictive performance, as shown in Table 2. Comparing Fig. 7 to Fig. 5, only 6 samples exceed the 5% error threshold for the model trained with a noise-free set, and 11 samples for the model trained with ± 20% added noise. Again, these instances are minimal in comparison to the overall size of the test set, again demonstrating the model's robustness in estimating material parameters in the presence of added noise and assuming a small fraction of the total number of features.

<table><caption>Table 2. Influence of added noise to the refined training set on model performance in predicting the material parameters.</caption><thead><tr><th>Noise range</th><th>R<sup>2</sup><sub>σ0</sub></th><th>R<sup>2</sup><sub>K</sub></th><th>R<sup>2</sup><sub>n</sub></th><th>R<sup>2</sup><sub>r0</sub></th><th>R<sup>2</sup><sub>r45</sub></th><th>R<sup>2</sup><sub>r90</sub></th></tr></thead><tbody><tr><td>± 0%</td><td>0.9982</td><td>0.9935</td><td>0.9826</td><td>0.9991</td><td>0.9947</td><td>0.9991</td></tr><tr><td>± 20%</td><td>0.9981</td><td>0.9923</td><td>0.9688</td><td>0.9983</td><td>0.9934</td><td>0.9981</td></tr></tbody></table>

<figure><img src="image_1727.png" /><figcaption>Fig. 7. Relative error $||\delta_j||$ in the prediction of material parameters for the 671 samples of the test set: (a) model trained with a noise-free training set; (b) model trained with ± 20% added noise. Models trained with only 118 features.</figcaption></figure>

In Fig. 8, a comparison is presented between the actual and predicted hardening curves, along with Lankford coefficients in the sheet plane. This comparison focuses on 3 samples surpassing the 5% error threshold (see Fig. 7 (a)), considering the model trained with a noise-free dataset. Generally, the prediction error for the hardening curve remains below 2.5%. However, the maximum error in predicting the Lankford coefficients tends to exceed 5%, particularly at $r_{45}$, except for the sample labeled "134". This discrepancy may be attributed to the limited sensitivity of the cruciform test results in estimating $r_{45}$.
<figure><img src="image_1730.png" /><figcaption>Fig. 8. Comparison between the actual and predicted hardening curves for samples labelled “26”, “474” and “134”, along with Lankford coefficients in the sheet plane.</figcaption></figure>

## Conclusions

This study demonstrates the effectiveness and competitiveness of machine learning-based techniques, namely using the XGBoost algorithm, in predicting material parameters for sheet metal forming simulations. Feature analysis highlights the significance of key features, accelerating model training without sacrificing predictive performance. The model's robustness is evident, with noise addition in key features minimally affecting predictive performance.
<figure><img src="image_1728.png" /><figcaption>Fig. 8. Comparison between the actual and predicted hardening curves for samples labelled “26”, “474” and “134”, along with Lankford coefficients in the sheet plane.</figcaption></figure>

## Conclusions

This study demonstrates the effectiveness and competitiveness of machine learning-based techniques, namely using the XGBoost algorithm, in predicting material parameters for sheet metal forming simulations. Feature analysis highlights the significance of key features, accelerating model training without sacrificing predictive performance. The model's robustness is evident, with noise addition in key features minimally affecting predictive performance.
## Acknowledgments

This work has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. The authors also gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) and by UE/FEDER through the programs CENTRO 2020 and COMPETE 2020, UIDB/00285/2020, UIDB/00481/2020 and UIDP/00481/2020-FCT, DOI 10.54499/UIDB/00481/2020 (https://doi.org/10.54499/UIDB/00481/2020) and DOI 10.54499/UIDP/00481/2020 (https://doi.org/10.54499/UIDP/00481/2020), CENTRO-01-0145-FEDER-022083, LA/P/0104/2020 and LA/P/0112/2020. It was also supported by the project RealForm (reference 2022.02370.PTDC), funded by Portuguese Foundation for Science and Technology. J. Henriques was supported by a grant for scientific research from the Portuguese Foundation for Science and Technology (ref. 2021.05692.BD).

## References

[1] A. Andrade-Campos, N. Bastos, M. Conde, M. Gonçalves, J. Henriques, R. Lourenço, J.M.P. Martins, M.G. Oliveira, P. Prates, L. Rumor, On the inverse identification methods for forming plasticity models using full-field measurements, IOP Conf. Ser. Mater. Sci. Eng. 1238 (2022) 012059. https://doi.org/10.1088/1757-899X/1238/1/012059

[2] P.A. Prates, J.D. Henriques, J. Pinto, N. Bastos, A. Andrade-Campos, Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Mater. Res. Proc. 28 (2023) 1193-1202. https://doi.org/10.21741/9781644902479-130

[3] J. Martins, A. Andrade-Campos, and S. Thuillier, Calibration of anisotropic plasticity models using a biaxial test and the virtual fields method, Int. J. Solids Struct. 172 (2019) 21-37. https://doi.org/10.1016/j.ijsolstr.2019.05.019

[4] Dassault Systèmes. Abaqus 2017 documentation, 2017.

[5] T. Chen and C. Guestrin, XGBoost: A scalable tree boosting system, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2016) 785-794. https://doi.org/10.1145/2939672.2939785

[6] S. Lundberg and S. Lee, "A Unified Approach to Interpreting Model Predictions", 2017.
## Acknowledgments

This work has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. The authors also gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) and by UE/FEDER through the programs CENTRO 2020 and COMPETE 2020, UIDB/00285/2020, UIDB/00481/2020 and UIDP/00481/2020-FCT, DOI 10.54499/UIDB/00481/2020 (https://doi.org/10.54499/UIDB/00481/2020) and DOI 10.54499/UIDP/00481/2020 (https://doi.org/10.54499/UIDP/00481/2020), CENTRO-01-0145-FEDER-022083, LA/P/0104/2020 and LA/P/0112/2020. It was also supported by the project RealForm (reference 2022.02370.PTDC), funded by Portuguese Foundation for Science and Technology. J. Henriques was supported by a grant for scientific research from the Portuguese Foundation for Science and Technology (ref. 2021.05692.BD).

## References

[1] A. Andrade-Campos, N. Bastos, M. Conde, M. Gonçalves, J. Henriques, R. Lourenço, J.M.P. Martins, M.G. Oliveira, P. Prates, L. Rumor, On the inverse identification methods for forming plasticity models using full-field measurements, IOP Conf. Ser. Mater. Sci. Eng. 1238 (2022) 012059. https://doi.org/10.1088/1757-899X/1238/1/012059

[2] P.A. Prates, J.D. Henriques, J. Pinto, N. Bastos, A. Andrade-Campos, Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Mater. Res. Proc. 28 (2023) 1193-1202. https://doi.org/10.21741/9781644902479-130

[3] J. Martins, A. Andrade-Campos, and S. Thuillier, Calibration of anisotropic plasticity models using a biaxial test and the virtual fields method, Int. J. Solids Struct. 172 (2019) 21-37. https://doi.org/10.1016/j.ijsolstr.2019.05.019

[4] Dassault Systèmes. Abaqus 2017 documentation, 2017.

[5] T. Chen and C. Guestrin, XGBoost: A scalable tree boosting system, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2016) 785-794. https://doi.org/10.1145/2939672.2939785

[6] S. Lundberg and S. Lee, "A Unified Approach to Interpreting Model Predictions", 2017.
# Intelligent control of ISBM process for recycled PET bottles

HAN William¹, a *, KERFRIDEN Pierre¹, b *, VIORA Laurianne², c, COMBEAUD Christelle², d, BOUVARD Jean-Luc², e, and CANTOURNET Sabine¹, f

¹Mines Paris, PSL University, Centre des Matériaux (CMAT), CNRS UMR 7633, BP 87, 91003 Evry, France

²Mines Paris, PSL University, Centre for Material Forming (CEMEF), UMR CNRS 7635, 1 rue Claude Daunesse, CS 10207, 06904 Sophia Antipolis, France

awilliam.han@minesparis.psl.eu, bpierre.kerfriden@minesparis.psl.eu,

claurianne.viora@minesparis.psl.eu, dchristelle.combeaud@minesparis.psl.eu,

ejean-luc.bouvard@minesparis.psl.eu, fsabine.cantournet@minesparis.psl.eu

**Keywords:** PET, Free Injection Stretch Blow Process, Machine Learning, Gaussian Process Regression

**Abstract.** To manufacture plastic bottles with an increased ratio of rPET (recycled Polyethylene terephthalate), the ISBM (Injection Stretch Blow Moulding) process must be controlled to account for the variable mechanical and thermal properties. Calibration and optimization of the process have been successfully realized in past works but cannot be used for real-time applications. To address this, a gaussian process regression model of the free blowing step is created. It can calibrate itself using the pressure curve from a previous blowing to obtain near instantaneous predictions of key properties of the bottle. To create the model, the process' characteristics are studied. Finite element simulations of the blowing where the properties follow a multivariate gaussian distribution are used to train the artificial intelligence. Then, an example is shown using the artificial intelligence predictions to optimize the thickness distribution of a bottle after blowing.

## Introduction

To manufacture plastic bottles, ISBM (Injection Stretch Blow Moulding) is the most employed process. PET (Polyethylene terephthalate) is commonly used for food packaging because of its advantageous mechanical, optical and barrier properties. To improve sustainability, usage of rPET for bottle forming is currently increasing.

Past studies have considered the simulation of the blowing step of the ISBM process [1,2] to help reducing costs in optimizing the process. However, the complex models used to produce accurate predictions make the simulations very expensive [3], and therefore, the time needed for calibration and optimization makes on-line control of the process unpractical.

It is especially true when considering rPET (recycled PET), since adding recycled PET will affect the mechanical and thermal properties of the preform [4,5] and disturb the high-paced production line (1.5 s per bottle from start to finish [6]). Therefore, to increase the ratio of recycled resin, it is necessary to be able to adapt the process to the varying amount of rPET.

In this study, we propose a method inspired by digital twinning methods devised recently [7,8] to drastically reduce the time needed for calibration and optimization. To do this, an AI (artificial intelligence) model is constructed to be able to predict outputs of the calibrated model given control inputs and historical process signature.

To illustrate this, we focus on the free blowing step to simplify the process. First, the settings of the process are enunciated following literature and real behaviour as much as possible. Then, a procedure to train an AI model to give quick prediction of a self-calibrated model is presented. Finally, the capability of the model is showcased by doing an optimization of the process.

<figure><img src="image_1731.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
# Intelligent control of ISBM process for recycled PET bottles

HAN William¹, a *, KERFRIDEN Pierre¹, b *, VIORA Laurianne², c, COMBEAUD Christelle², d, BOUVARD Jean-Luc², e, and CANTOURNET Sabine¹, f

¹Mines Paris, PSL University, Centre des Matériaux (CMAT), CNRS UMR 7633, BP 87, 91003 Evry, France

²Mines Paris, PSL University, Centre for Material Forming (CEMEF), UMR CNRS 7635, 1 rue Claude Daunesse, CS 10207, 06904 Sophia Antipolis, France

**Keywords:** PET, Free Injection Stretch Blow Process, Machine Learning, Gaussian Process Regression

**Abstract.** To manufacture plastic bottles with an increased ratio of rPET (recycled Polyethylene terephthalate), the ISBM (Injection Stretch Blow Moulding) process must be controlled to account for the variable mechanical and thermal properties. Calibration and optimization of the process have been successfully realized in past works but cannot be used for real-time applications. To address this, a gaussian process regression model of the free blowing step is created. It can calibrate itself using the pressure curve from a previous blowing to obtain near instantaneous predictions of key properties of the bottle. To create the model, the process' characteristics are studied. Finite element simulations of the blowing where the properties follow a multivariate gaussian distribution are used to train the artificial intelligence. Then, an example is shown using the artificial intelligence predictions to optimize the thickness distribution of a bottle after blowing.

## Introduction

To manufacture plastic bottles, ISBM (Injection Stretch Blow Moulding) is the most employed process. PET (Polyethylene terephthalate) is commonly used for food packaging because of its advantageous mechanical, optical and barrier properties. To improve sustainability, usage of rPET for bottle forming is currently increasing.

Past studies have considered the simulation of the blowing step of the ISBM process [1,2] to help reducing costs in optimizing the process. However, the complex models used to produce accurate predictions make the simulations very expensive [3], and therefore, the time needed for calibration and optimization makes on-line control of the process unpractical.

It is especially true when considering rPET (recycled PET), since adding recycled PET will affect the mechanical and thermal properties of the preform [4,5] and disturb the high-paced production line (1.5 s per bottle from start to finish [6]). Therefore, to increase the ratio of recycled resin, it is necessary to be able to adapt the process to the varying amount of rPET.

In this study, we propose a method inspired by digital twinning methods devised recently [7,8] to drastically reduce the time needed for calibration and optimization. To do this, an AI (artificial intelligence) model is constructed to be able to predict outputs of the calibrated model given control inputs and historical process signature.

To illustrate this, we focus on the free blowing step to simplify the process. First, the settings of the process are enunciated following literature and real behaviour as much as possible. Then, a procedure to train an AI model to give quick prediction of a self-calibrated model is presented. Finally, the capability of the model is showcased by doing an optimization of the process.

<figure><img src="image_1729.png" /></figure>

<footnote>awilliam.han@minesparis.psl.eu, bpierre.kerfriden@minesparis.psl.eu,</footnote>

<footnote>claurianne.viora@minesparis.psl.eu, dchristelle.combeaud@minesparis.psl.eu,</footnote>

<footnote>ejean-luc.bouvard@minesparis.psl.eu, fsabine.cantournet@minesparis.psl.eu</footnote>
# 1. Free blowing process

The study focuses on a simplified blowing step spanning 0.3 s and realized with no mould (free blowing).

Optimization of the bottle thickness distribution has been a focus of previous studies [9,10] because of its importance to have uniform properties throughout the bottle [6]. Since there is no mould in the study, the volume of the bottle formed is not controlled. Therefore, to optimize the thickness homogeneity, we aim to minimize the standard deviation of the bottle thickness distribution $\sigma_{Wd}$ for a given volume $V_f$.

The process is usually controlled by the injection pressure and how much heating energy is applied to the preform. Salomeia et al. [11] have shown that the injection pressure results in a choked flow, thus constant mass air flow $Q_{air}$ when the ratio between the internal pressure and the injection pressure is low enough. To simplify the problem, we consider the process controlled by a constant temperature $T$ throughout the preform and the blowing duration. The range of both control parameters is consistent with usual values in the literature [2,12].

<table><caption>Table 1. Control parameters and their range.</caption><thead><tr><td>Control Parameters (μ)</td><td>Range</td></tr></thead><tbody><tr><td>Mass air flow Q<sub>air</sub> [g s<sup>-1</sup>]</td><td>[10, 30]</td></tr><tr><td>Temperature T [°C]</td><td>[85, 120]</td></tr></tbody></table>

The preform is an 18.5 g preform following a geometry used by SIDEL, partner of the CARNOT 2020 AAP 2 BOUTEILLE project (Figure 1). The material is PET with varying amount of recycled PET.

<figure><img src="image_1732.png" /><figcaption>Fig. 1. Preform design.</figcaption></figure>

To monitor the process, the preform internal pressure is recorded during the blowing step. Indeed, past studies suggest that the behaviour pressure history is closely related to the course and characteristics of the process [1,13].

# 2. Uncertainties in the free blowing problem

In the process, there are several parameters that are not known beforehand, but will have a noticeable impact on the process control. In Table 2, such parameters are compiled with their expected standard deviations. They include the lack of accuracy of the sensors or the controls of the process, the environmental conditions for the bottle internal pressure $P$, mass air flow $Q_{air}$ and
# 1. Free blowing process

The study focuses on a simplified blowing step spanning 0.3 s and realized with no mould (free blowing).

Optimization of the bottle thickness distribution has been a focus of previous studies [9,10] because of its importance to have uniform properties throughout the bottle [6]. Since there is no mould in the study, the volume of the bottle formed is not controlled. Therefore, to optimize the thickness homogeneity, we aim to minimize the standard deviation of the bottle thickness distribution $\sigma_{Wd}$ for a given volume $V_f$.

The process is usually controlled by the injection pressure and how much heating energy is applied to the preform. Salomeia et al. [11] have shown that the injection pressure results in a choked flow, thus constant mass air flow $Q_{air}$ when the ratio between the internal pressure and the injection pressure is low enough. To simplify the problem, we consider the process controlled by a constant temperature $T$ throughout the preform and the blowing duration. The range of both control parameters is consistent with usual values in the literature [2,12].

<table><caption>Table 1. Control parameters and their range.</caption><thead><tr><td>Control Parameters (μ)</td><td>Range</td></tr></thead><tbody><tr><td>Mass air flow Q<sub>air</sub> [g s<sup>-1</sup>]</td><td>[10, 30]</td></tr><tr><td>Temperature T [°C]</td><td>[85, 120]</td></tr></tbody></table>

The preform is an 18.5 g preform following a geometry used by SIDEL, partner of the CARNOT 2020 AAP 2 BOUTEILLE project (Figure 1). The material is PET with varying amount of recycled PET.

<figure><img src="image_1730.png" /><figcaption>Fig. 1. Preform design.</figcaption></figure>

To monitor the process, the preform internal pressure is recorded during the blowing step. Indeed, past studies suggest that the behaviour pressure history is closely related to the course and characteristics of the process [1,13].

# 2. Uncertainties in the free blowing problem

In the process, there are several parameters that are not known beforehand, but will have a noticeable impact on the process control. In Table 2, such parameters are compiled with their expected standard deviations. They include the lack of accuracy of the sensors or the controls of the process, the environmental conditions for the bottle internal pressure $P$, mass air flow $Q_{air}$ and
temperature *T* measurements, and the thickness deviation from the specified thickness distribution *W*<sub>d</sub>.

To evaluate the variance caused by the presence of recycled PET, measurements for the storage modulus *E*′ were realized by Viora et al. [5] by DMA using samples with three different ratios of recycled PET (0%, 50% or 100%). Temperature scans were realized between 25°C and 140°C, with a heating rate of 2 °C/min, at a frequency of 1 Hz, a preload of 1 N and a dynamic deformation of 0.04%. Five scans of the storage modulus are shown in Figure 2 and show the variations between different types, with a tendency for PET with lesser amounts of recycled resin to be more rigid in the rubbery plateau (around 90°C-115°C). It also shows that material variations seem to have a logarithmic distribution. Therefore, for the material variations, a geometric standard deviation factor σ<sub>m</sub>(*T*) was calculated (eq. 1) using *E*′(*T*) the geometric mean of the measurements *E*'<sub>i</sub>(*T*) at temperature *T*.

$$ \sigma_m(T) = \exp\left[\frac{1}{n}\sum_{i=1}^{n}\left(\ln\frac{E_i'(T)}{E'(T)}\right)\right] \quad (1) $$

To simplify, the arithmetic average σ̄<sub>m</sub> of σ<sub>m</sub>(*T*) at temperatures between 85°C and 120°C was chosen.

<table><caption>Table 2. Error between the expected value and reality.</caption><thead><tr><td>Uncertain parameters</td><td>Standard deviation</td></tr></thead><tbody><tr><td>Bottle internal pressure</td><td>σ<sub>P</sub> = 0.25%</td></tr><tr><td>Mass air flow</td><td>σ<sub>Qair</sub> = 2.5%</td></tr><tr><td>Temperature</td><td>σ<sub>T</sub> = 1°C</td></tr><tr><td>Geometry (thickness)</td><td>σ<sub>w</sub> = 0.01 mm</td></tr><tr><td>Material</td><td>σ̄<sub>m</sub> = 0.116</td></tr></tbody></table>

<figure><img src="image_1733.png" /><figcaption>Fig. 2. Storage modulus (*E*′) as a function of temperature for different samples, in tensile mode, at a frequency of 1 Hz and a heating rate of 2°C/min.</figcaption></figure>
temperature *T* measurements, and the thickness deviation from the specified thickness distribution *W*<sub>d</sub>.

To evaluate the variance caused by the presence of recycled PET, measurements for the storage modulus *E*′ were realized by Viora et al. [5] by DMA using samples with three different ratios of recycled PET (0%, 50% or 100%). Temperature scans were realized between 25°C and 140°C, with a heating rate of 2 °C/min, at a frequency of 1 Hz, a preload of 1 N and a dynamic deformation of 0.04%. Five scans of the storage modulus are shown in Figure 2 and show the variations between different types, with a tendency for PET with lesser amounts of recycled resin to be more rigid in the rubbery plateau (around 90°C-115°C). It also shows that material variations seem to have a logarithmic distribution. Therefore, for the material variations, a geometric standard deviation factor σ<sub>m</sub>(*T*) was calculated (eq. 1) using *E*′(*T*) the geometric mean of the measurements *E*<sub>i</sub>'(*T*) at temperature *T*.

$$ \sigma_m(T) = \exp\left[\frac{1}{n}\sum_{i=1}^{n}\left(\ln\frac{E_i'(T)}{E'(T)}\right)\right] \quad (1) $$

To simplify, the arithmetic average σ̄<sub>m</sub> of σ<sub>m</sub>(*T*) at temperatures between 85°C and 120°C was chosen.

<table><caption>Table 2. Error between the expected value and reality.</caption><thead><tr><td>Uncertain parameters</td><td>Standard deviation</td></tr></thead><tbody><tr><td>Bottle internal pressure</td><td>σ<sub>P</sub> = 0.25%</td></tr><tr><td>Mass air flow</td><td>σ<sub>Qair</sub> = 2.5%</td></tr><tr><td>Temperature</td><td>σ<sub>T</sub> = 1°C</td></tr><tr><td>Geometry (thickness)</td><td>σ<sub>w</sub> = 0.01 mm</td></tr><tr><td>Material</td><td>σ̄<sub>m</sub> = 0.116</td></tr></tbody></table>

<figure><img src="image_1731.png" /><figcaption>Fig. 2. Storage modulus (E') as a function of temperature for different samples, in tensile mode, at a frequency of 1 Hz and a heating rate of 2°C/min.</figcaption></figure>
### 3. Methodology

The gaussian process regression machine learning model is presented here, alongside the data used to train it.

**Gaussian Process Regression (GPR).** The blowing process has uncertainties which have been initially quantified. Assuming they follow a multivariate gaussian distribution, their knowledge can be carried in predictions by employing gaussian process regression machine learning [14]. Indeed, considering a point $x$ in the input matrix $X$, a prediction $f(x) = \tilde{f}(x) + \epsilon$ with a noise $\epsilon$ that quantify its uncertainty and $\tilde{f}(x)$ a gaussian process. To set up the prior distribution, $\tilde{f}(x)$ follows a covariance function $K$ (eq.2) which was chosen to be the product of its amplitude $A$ with a radial basis function kernel. It depends on the characteristic length-scale $l$ and the distance between the input points $x_i$ and $x_j$.

$$K(x_i, x_j) = A \exp\left(-\frac{d(x_i, x_j)^2}{2l_{ij}^2}\right) \qquad (2)$$

The covariance function follows eq.3, where $\delta_{ij}$ is a Kronecker delta which is one if $i = j$ and zero otherwise and $\sigma_n^2$ the variance of the noise.

$$\operatorname{cov}(f(\mathbf{x}_i), f(\mathbf{x}_j)) = K(\mathbf{x}_i, \mathbf{x}_j) + \sigma_n^2 \delta_{ij} \qquad (3)$$

With $\mathbf{y} = f(\mathbf{x})$ the output vector, predictions from test points $\mathbf{X}^*$ will consequently have the following gaussian distribution:

$$\begin{align}
& f_* | \{f(x_i) = y_i\}_{i \le n} \sim \mathcal{N}(\bar{f}_*, \operatorname{cov}(f_*)) \nonumber \\
& \bar{f}_* = \mathbf{K}(\mathbf{X}_*, \mathbf{X})[\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}]^{-1} \mathbf{y} \tag{4} \\
& \operatorname{cov}(f_*) = \mathbf{K}(\mathbf{X}_*, \mathbf{X}_*) - \mathbf{K}(\mathbf{X}_*, \mathbf{X})[\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}]^{-1} \mathbf{K}(\mathbf{X}, \mathbf{X}_*) \nonumber
\end{align}$$

The parameters $l$, $A$, and $\sigma_n$ are determined from data by maximizing the log-likelihood:

$$\ln P(y) = -\frac{1}{2}\ln \det(\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}) - \frac{1}{2}y^T(\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I})^{-1}y - \frac{N}{2}\ln(2\pi) \qquad (5)$$

**Simulations.** The training data was constructed using free blowing simulations using Abaqus/Explicit. Similarly to previous studies [1,2,9], they are conducted with the fluid cavity option, where a basic fluid-structure interaction is calculated between the air modelled as an ideal gas and the preform. Simulations were performed with 84 axisymmetric shell elements (SAX1) following the inner surface of the preform. Rigid elements (RAX2) were used at the base of the preform to close the cavity.

The Arruda-Boyce incompressible hyperelastic material law was chosen for its simplicity, fast computation (~30 s per simulation) and its ability to reproduce the bottle shape during the blowing step (Figure 3).
### 3. Methodology

The gaussian process regression machine learning model is presented here, alongside the data used to train it.

**Gaussian Process Regression (GPR).** The blowing process has uncertainties which have been initially quantified. Assuming they follow a multivariate gaussian distribution, their knowledge can be carried in predictions by employing gaussian process regression machine learning [14]. Indeed, considering a point $x$ in the input matrix $X$, a prediction $f(x) = \tilde{f}(x) + \epsilon$ with a noise $\epsilon$ that quantify its uncertainty and $\tilde{f}(x)$ a gaussian process. To set up the prior distribution, $\tilde{f}(x)$ follows a covariance function $K$ (eq.2) which was chosen to be the product of its amplitude $A$ with a radial basis function kernel. It depends on the characteristic length-scale $l$ and the distance between the input points $x_i$ and $x_j$.

$$K(x_i, x_j) = A \exp\left(-\frac{d(x_i, x_j)^2}{2l_{ij}^2}\right) \qquad (2)$$

The covariance function follows eq.3, where $\delta_{ij}$ is a Kronecker delta which is one if $i = j$ and zero otherwise and $\sigma_n^2$ the variance of the noise.

$$\operatorname{cov}(f(\mathbf{x}_i), f(\mathbf{x}_j)) = K(\mathbf{x}_i, \mathbf{x}_j) + \sigma_n^2 \delta_{ij} \qquad (3)$$

With $\mathbf{y} = f(\mathbf{x})$ the output vector, predictions from test points $\mathbf{X}^*$ will consequently have the following gaussian distribution:

$$\begin{align}
\mathbf{f}_* | \{f(\mathbf{x}_i) = y_i\}_{i \le n} &\sim \mathcal{N}(\bar{\mathbf{f}}_*, \operatorname{cov}(\mathbf{f}_*)) \nonumber \\
\bar{\mathbf{f}}_* &= \mathbf{K}(\mathbf{X}_*, \mathbf{X})[\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}]^{-1} \mathbf{y} \tag{4} \\
\operatorname{cov}(\mathbf{f}_*) &= \mathbf{K}(\mathbf{X}_*, \mathbf{X}_*) - \mathbf{K}(\mathbf{X}_*, \mathbf{X})[\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}]^{-1} \mathbf{K}(\mathbf{X}, \mathbf{X}_*) \nonumber
\end{align}$$

The parameters $l$, $A$, and $\sigma_n$ are determined from data by maximizing the log-likelihood:

$$\ln P(y) = -\frac{1}{2}\ln \det(\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I}) - \frac{1}{2}y^T(\mathbf{K}(\mathbf{X}, \mathbf{X}) + \sigma_n^2 \mathbf{I})^{-1}y - \frac{N}{2}\ln(2\pi) \qquad (5)$$

**Simulations.** The training data was constructed using free blowing simulations using Abaqus/Explicit. Similarly to previous studies [1,2,9], they are conducted with the fluid cavity option, where a basic fluid-structure interaction is calculated between the air modelled as an ideal gas and the preform. Simulations were performed with 84 axisymmetric shell elements (SAX1) following the inner surface of the preform. Rigid elements (RAX2) were used at the base of the preform to close the cavity.

The Arruda-Boyce incompressible hyperelastic material law was chosen for its simplicity, fast computation (~30 s per simulation) and its ability to reproduce the bottle shape during the blowing step (Figure 3).
<figure><img src="image_1734.png" /><figcaption>Fig. 3. Free blow simulation with an air mass flow of 14 g s⁻¹, (μr = 5.6 MPa, λm = 3).</figcaption></figure>

Its strain energy density is given by eq. 6 where $λ_{chain} = \sqrt{I_1/3}$ with $I_1$ the first invariant of the left Cauchy-Green deformation tensor, and $β = L^{-1}(λ_{chain}/λ_m)$ with $L(β) = \coth(β) - 1/β$ the Langevin function. It depends on two parameters, the shear modulus $μ_r$ and $λ_m$ which is the stretch at which the polymer chain network becomes locked.

$$W = G(λ_mλ_{chain}β + λ_m^2 \ln \frac{\sinh β}{β}) \qquad (6)$$

A rudimentary temperature dependency is inputted in the simulation through the material law. Since we consider the PET incompressible, $G = E'/(2(1 + ν)) = E'/3$. Thus, to obtain the average shear modulus $\overline{G}(T)$, an empirical model for $\overline{E'}(T)$ (eq.7) was fitted using the average curve of the DMA data described in section 2.

$$\overline{E'}(T) = A_0 \exp(-KT + B_1 \tanh(\frac{T_1-T}{T_1^*}) + B_2 \tanh(\frac{T-T_2}{T_2^*})) \qquad (7)$$

Since no data was readily available to identify $λ_m$, the limit of the viscous strain values measured by Chevalier et al. [15] were used to fit a model (eq. 8). A sigmoid was used to avoid values getting too high as the blowing behaviour already tends to neo-Hookean when $λ_m > 5$.

$$\overline{\lambda_m}(T) = (A_\lambda T + B_\lambda) \left(1 + \frac{2}{1 + \exp(T - T_\lambda)}\right) \qquad (8)$$

<table><caption>Table 3. Identified parameters of eq.7 and 8.</caption><tr><th>Parameters<br>of eq.7</th><th>A<sub>0</sub><br>1476 MPa</th><th>K<br>0.01704 °C<sup>-1</sup></th><th>B<sub>1</sub><br>2.242</th><th>T<sub>1</sub><br>79.64 °C</th><th>T<sub>1</sub><sup>*</sup><br>5.461 °C</th><th>B<sub>2</sub><br>0.7846</th><th>T<sub>2</sub><br>119.7 °C</th><th>T<sub>2</sub><sup>*</sup><br>2.87°C</th></tr><tr><th>Parameters<br>of eq. 8</th><th>A<sub>λ</sub><br>0.0184</th><th>B<sub>λ</sub><br>0.306</th><th>T<sub>λ</sub><br>111°C</th><td></td><td></td><td></td><td></td><td></td></tr></table>
<figure><img src="image_1732.png" /><figcaption>Fig. 3. Free blow simulation with an air mass flow of 14 g s⁻¹, (μr = 5.6 MPa, λm = 3).</figcaption></figure>

Its strain energy density is given by eq. 6 where $λ_{chain} = \sqrt{I_1/3}$ with $I_1$ the first invariant of the left Cauchy-Green deformation tensor, and $β = L^{-1}(λ_{chain}/λ_m)$ with $L(β) = \coth(β) - 1/β$ the Langevin function. It depends on two parameters, the shear modulus $μ_r$ and $λ_m$ which is the stretch at which the polymer chain network becomes locked.

$$W = G (λ_m λ_{chain} β + λ_m^2 \ln \frac{\sinh β}{β}) \qquad (6)$$

A rudimentary temperature dependency is inputted in the simulation through the material law. Since we consider the PET incompressible, $G = E'/(2(1 + ν)) = E'/3$. Thus, to obtain the average shear modulus $\overline{G}(T)$, an empirical model for $\overline{E'}(T)$ (eq.7) was fitted using the average curve of the DMA data described in section 2.

$$\overline{E'}(T) = A_0 \exp(-KT + B_1 \tanh(\frac{T_1-T}{T_1^*}) + B_2 \tanh(\frac{T-T_2}{T_2^*})) \qquad (7)$$

Since no data was readily available to identify $λ_m$, the limit of the viscous strain values measured by Chevalier et al. [15] were used to fit a model (eq. 8). A sigmoid was used to avoid values getting too high as the blowing behaviour already tends to neo-Hookean when $λ_m > 5$.

$$\overline{λ_m}(T) = (A_λ T + B_λ) \left(1 + \frac{2}{1 + \exp(T - T_λ)}\right) \qquad (8)$$

<table><caption>Table 3. Identified parameters of eq.7 and 8.</caption><tr><td>Parameters of eq.7</td><td>A<sub>0</sub></td><td>K</td><td>B<sub>1</sub></td><td>T<sub>1</sub></td><td>T<sub>1</sub><sup>*</sup></td><td>B<sub>2</sub></td><td>T<sub>2</sub></td><td>T<sub>2</sub><sup>*</sup></td></tr><tr><td></td><td>1476 MPa</td><td>0.01704 °C<sup>-1</sup></td><td>2.242</td><td>79.64 °C</td><td>5.461 °C</td><td>0.7846</td><td>119.7 °C</td><td>2.87°C</td></tr><tr><td>Parameters of eq. 8</td><td>A<sub>λ</sub></td><td>B<sub>λ</sub></td><td>T<sub>λ</sub></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>0.0184</td><td>0.306</td><td>111°C</td><td></td><td></td><td></td><td></td><td></td></tr></table>
<figure><img src="image_1735.png" /><figcaption>Fig. 4. Comparison of experimental data and eq. 7 (left) and eq. 8 (right).</figcaption></figure>

**Construction of the training data.** To create an input point **x** in the training data, a simulation is done to record the internal pressure history. However, since the pressure history **P** will change depending on the controls **μ** used to obtain it, it cannot serve alone as the process signature in a problem where the optimization is realized on those same controls. Hence, the internal pressure history **P**₁ is recorded alongside the control **μ**₁ = (Q<sub>air1</sub>, T<sub>1</sub>) to have the process signature (P<sub>1</sub>, μ<sub>1</sub>) and let the model learn the relationship P<sub>1</sub>(μ<sub>1</sub>) to calibrate the predictions. Then, a second simulation is needed to link the control μ<sub>2</sub> = (Q<sub>air2</sub>, T<sub>2</sub>) to the quantities y = (V<sub>f</sub>, σ<sub>w</sub>) we wish to predict. Two simulations were needed as the pressure curve **P** depends on the control parameters **μ** and therefore is not enough alone to constitute the process signature allowing for model calibration. To keep a reasonable dimensionality for the problem, we reduced the pressure history to a vector with three characteristic values (P<sub>1</sub> = (p<sub>1</sub><sup>I</sup>, p<sub>1</sub><sup>II</sup>, p<sub>1</sub><sup>III</sup>)) using principal component analysis. Therefore, the input vector x = (μ<sub>2</sub>, P<sub>1</sub>, μ<sub>1</sub>) is used to predict the output y.

As such, in the range of Table 1, the **μ**₁ and **μ**₂ matrices were sampled using a Sobol sequence [16] to create a low-discrepancy distribution in the parameter space. Then, to include the knowledge of the process described in section 2, a matrix **Θ** was constructed so that a distribution **θ**<sub>k</sub> ∈ **Θ** (with k ∈ {P, Q<sub>air</sub>, T, w, G, λ<sub>m</sub>}) represents the true values that are unknown either because of measurement inaccuracy, or because they are not measured. The distributions for each unknown parameter are described in Table 4, where **Z**<sub>k</sub>~N(0,1) are random distributions following the unitary normal law. As the two simulations used for an input point are part of the same process, we considered that **Z**<sub>k</sub> = **Z**<sub>1,k</sub> = **Z**<sub>2,k</sub>.

<table><caption>Table 4. Error between the expected value and reality.</caption><thead><tr><th>Unknown parameters</th><th>Distribution</th></tr></thead><tbody><tr><td>True bottle internal pressure</td><td>θ<sub>P</sub> = P(1 + σ<sub>P</sub>Z<sub>P</sub>)</td></tr><tr><td>True mass air flow</td><td>θ<sub>Qair</sub> = Q<sub>air</sub>(1 + σ<sub>Qair</sub>Z<sub>Qair</sub>)</td></tr><tr><td>True temperature</td><td>θ<sub>T</sub> = T + σ<sub>T</sub>Z<sub>T</sub></td></tr><tr><td>Geometry (thickness)</td><td>θ<sub>w</sub> = W<sub>d</sub> + σ<sub>w</sub>Z<sub>w</sub></td></tr><tr><td>Material (G)</td><td>θ<sub>G</sub> = <sup>1</sup>&frasl;<sub>3</sub> exp(log(E'(T)) + σ<sub>m</sub>Z<sub>E'</sub>)</td></tr><tr><td>Material (λ<sub>m</sub>)</td><td>θ<sub>λm</sub> = exp(log(λ<sub>m</sub>(T)) + σ<sub>m</sub>Z<sub>λm</sub>)</td></tr></tbody></table>

Thus, to create the training data, the simulations were controlled by the matrices **Θ**₁ and **Θ**₂ for all parameters aside from the bottle internal pressure (since θ<sub>P</sub> is the simulation result, **P** is calculated afterward).
<figure><img src="image_1733.png" /><figcaption>Fig. 4. Comparison of experimental data and eq. 7 (left) and eq. 8 (right).</figcaption></figure>

**Construction of the training data.** To create an input point **x** in the training data, a simulation is done to record the internal pressure history. However, since the pressure history **P** will change depending on the controls **μ** used to obtain it, it cannot serve alone as the process signature in a problem where the optimization is realized on those same controls. Hence, the internal pressure history **P**₁ is recorded alongside the control **μ**₁ = (Q<sub>air1</sub>, T₁) to have the process signature (P<sub>1</sub>, μ<sub>1</sub>) and let the model learn the relationship P<sub>1</sub>(μ<sub>1</sub>) to calibrate the predictions. Then, a second simulation is needed to link the control μ<sub>2</sub> = (Q<sub>air2</sub>, T<sub>2</sub>) to the quantities y = (V<sub>f</sub>, σ<sub>w</sub>) we wish to predict. Two simulations were needed as the pressure curve **P** depends on the control parameters **μ** and therefore is not enough alone to constitute the process signature allowing for model calibration. To keep a reasonable dimensionality for the problem, we reduced the pressure history to a vector with three characteristic values (P<sub>1</sub> = (p<sub>1</sub><sup>I</sup>, p<sub>1</sub><sup>II</sup>, p<sub>1</sub><sup>III</sup>)) using principal component analysis. Therefore, the input vector x = (μ<sub>2</sub>, P<sub>1</sub>, μ<sub>1</sub>) is used to predict the output y.

As such, in the range of Table 1, the **μ**₁ and **μ**₂ matrices were sampled using a Sobol sequence [16] to create a low-discrepancy distribution in the parameter space. Then, to include the knowledge of the process described in section 2, a matrix **Θ** was constructed so that a distribution **θ**<sub>k</sub> ∈ **Θ** (with k ∈ {P, Q<sub>air</sub>, T, w, G, λ<sub>m</sub>}) represents the true values that are unknown either because of measurement inaccuracy, or because they are not measured. The distributions for each unknown parameter are described in Table 4, where **Z**<sub>k</sub>~N(0,1) are random distributions following the unitary normal law. As the two simulations used for an input point are part of the same process, we considered that **Z**<sub>k</sub> = **Z**<sub>1,k</sub> = **Z**<sub>2,k</sub>.

<table><caption>Table 4. Error between the expected value and reality.</caption><thead><tr><th>Unknown parameters</th><th>Distribution</th></tr></thead><tbody><tr><td>True bottle internal pressure</td><td>θ<sub>P</sub> = P(1 + σ<sub>P</sub>Z<sub>P</sub>)</td></tr><tr><td>True mass air flow</td><td>θ<sub>Qair</sub> = Q<sub>air</sub>(1 + σ<sub>Qair</sub>Z<sub>Qair</sub>)</td></tr><tr><td>True temperature</td><td>θ<sub>T</sub> = T + σ<sub>T</sub>Z<sub>T</sub></td></tr><tr><td>Geometry (thickness)</td><td>θ<sub>w</sub> = W<sub>d</sub> + σ<sub>w</sub>Z<sub>w</sub></td></tr><tr><td>Material (G)</td><td>θ<sub>G</sub> = <sup>1</sup>&frasl;<sub>3</sub> exp(log(E'(T)) + σ<sub>m</sub>Z<sub>E'</sub>)</td></tr><tr><td>Material (λ<sub>m</sub>)</td><td>θ<sub>λm</sub> = exp(log(λ<sub>m</sub>(T)) + σ<sub>m</sub>Z<sub>λm</sub>)</td></tr></tbody></table>

Thus, to create the training data, the simulations were controlled by the matrices **Θ**₁ and **Θ**₂ for all parameters aside from the bottle internal pressure (since **θ**<sub>P</sub> is the simulation result, **P** is calculated afterward).
**Training results.** The training input and output $X_{train}$ and $Y_{train}$ was constructed with 10000 points from 20000 simulations. Two gaussian processes to respectively predict $V_f$ and $\sigma_{W_d}$ were trained with the scikit-learn [17] implementation of gaussian process regression. To evaluate the generalization capability of the model, test data ($X_{test}$, $Y_{test}$) was also constructed with 1000 different points obtained in a similar manner as for the training data. The performance was rated by how many points are inside the 95% confidence interval of the prediction's distribution. Table 5 shows that the training goes well since around 95% of the predictions are indeed in the 95% confidence interval and that the model generalizes satisfactorily since the percentage of unknown data in the prediction's confidence interval is only slightly lower.

<table><caption>Table 5. Ratio of output data Y inside the 95% confidence interval.</caption><thead><tr><td>Quantity of interest</td><td>Training results</td><td>Test results</td></tr></thead><tbody><tr><td>Volume $V_f$</td><td>95.33%</td><td>93.7%</td></tr><tr><td>Thickness SD $\sigma_w$</td><td>94.69%</td><td>93.7%</td></tr></tbody></table>

**4. Test of the model: robust optimization of the control parameters**

In Figure 5, the result of a blowing taken from the validation set is shown, alongside its thickness distribution. The control parameters used were $Q_{air} = 14.8 \text{ g s}^{-1}$ and $T = 111^{\circ}\text{C}$. The bubble only formed at the base of the preform, thus the membrane is still very thick (~2.5 mm) at the top. As the base of the preform is clamped, the material stays thick in that area. The final volume obtained was $V_f = 1.74 \text{ L}$ with a thickness standard deviation $\sigma_{W_d} = 1.05 \text{ mm}$. To demonstrate the capability of the model, in this section, the aim is to optimize the controls to have a two-litre bottle with a more uniform thickness distribution.

<figure><img src="image_1736.png" /><figcaption>Fig. 5. Final shape of the bottle blown with $Q_{air} = 14.8 \text{ g s}^{-1}$, $T = 111^{\circ}\text{C}$ (left), thickness distribution of the bottle (right).</figcaption></figure>

**Objective function.** To obtain a more uniform thickness distribution, the objective function aims to minimize the standard deviation of the thickness. As $P_1$ and $\mu_1$ are here to calibrate the model, the minimization is realized on $\mu_2$. Since there is no mould, a constraint was added to define the desired volume of the bottle. The gaussian process regression method gives predictions in the form of distribution, thus the uncertainty on the volume prediction can also be accounted for. In eq.9, $V_{f, obj}$ is the desired volume, $\varepsilon_v$ is the volume error tolerance, $P_v$ is the probability of having $V_f \in [V_{f, obj} - \varepsilon_v, V_{f, obj} + \varepsilon_v]$, and $P_m$ is the minimal probability tolerated on the predicted volume being in the accepted error margin.
**Training results.** The training input and output $X_{train}$ and $Y_{train}$ was constructed with 10000 points from 20000 simulations. Two gaussian processes to respectively predict $V_f$ and $\sigma_{W_d}$ were trained with the scikit-learn [17] implementation of gaussian process regression. To evaluate the generalization capability of the model, test data ($X_{test}$, $Y_{test}$) was also constructed with 1000 different points obtained in a similar manner as for the training data. The performance was rated by how many points are inside the 95% confidence interval of the prediction's distribution. Table 5 shows that the training goes well since around 95% of the predictions are indeed in the 95% confidence interval and that the model generalizes satisfactorily since the percentage of unknown data in the prediction's confidence interval is only slightly lower.

<table><caption>Table 5. Ratio of output data Y inside the 95% confidence interval.</caption><thead><tr><td>Quantity of interest</td><td>Training results</td><td>Test results</td></tr></thead><tbody><tr><td>Volume V<sub>f</sub></td><td>95.33%</td><td>93.7%</td></tr><tr><td>Thickness SD σ<sub>w</sub></td><td>94.69%</td><td>93.7%</td></tr></tbody></table>

## 4. Test of the model: robust optimization of the control parameters

In Figure 5, the result of a blowing taken from the validation set is shown, alongside its thickness distribution. The control parameters used were $Q_{air} = 14.8 \text{ g s}^{-1}$ and $T = 111^{\circ}\text{C}$. The bubble only formed at the base of the preform, thus the membrane is still very thick (~2.5 mm) at the top. As the base of the preform is clamped, the material stays thick in that area. The final volume obtained was $V_f = 1.74 \text{ L}$ with a thickness standard deviation $\sigma_{W_d} = 1.05 \text{ mm}$. To demonstrate the capability of the model, in this section, the aim is to optimize the controls to have a two-litre bottle with a more uniform thickness distribution.

<figure><img src="image_1734.png" /><figcaption>Fig. 5. Final shape of the bottle blown with $Q_{air} = 14.8 \text{ g s}^{-1}$, $T = 111^{\circ}\text{C}$ (left), thickness distribution of the bottle (right).</figcaption></figure>

**Objective function.** To obtain a more uniform thickness distribution, the objective function aims to minimize the standard deviation of the thickness. As $P_1$ and $\mu_1$ are here to calibrate the model, the minimization is realized on $\mu_2$. Since there is no mould, a constraint was added to define the desired volume of the bottle. The gaussian process regression method gives predictions in the form of distribution, thus the uncertainty on the volume prediction can also be accounted for. In eq.9, $V_{f, obj}$ is the desired volume, $\varepsilon_v$ is the volume error tolerance, $P_v$ is the probability of having $V_f \in [V_{f, obj} - \varepsilon_v, V_{f, obj} + \varepsilon_v]$, and $P_m$ is the minimal probability tolerated on the predicted volume being in the accepted error margin.
$$ \begin{gathered} \min_{\mu_2} \sigma_{Wd}(\mathbf{x}(\mu_2, \mathcal{P}_1, \mu_1)) \\ \mathbb{P}_v(\mu_2) = \mathbb{P}(|V_{f, obj} - V_f(\mathbf{x}(\mu_2, \mathcal{P}_1, \mu_1))| < \varepsilon_v) > \mathbb{P}_m \end{gathered} \tag{9} $$

**Optimization results.** The optimization was realized using the trained machine learning model to evaluate $\sigma_w$ and $V_f$, with the differential evolution method [18] integrated in SciPy. In eq. 9, the following parameters were set: $\varepsilon_v = 0.2$, $\mathbb{P}_m = 70\%$ and $V_f = 2$ L. $\mathbb{P}_m$ is taken low because preliminary calculations of probabilities to obtain the desired volume did not go above 73.5%. Optimized control parameters given by the algorithm were $Q_{air} = 29.5 \text{ g s}^{-1}$ and $T = 107.7^{\circ}\text{C}$. A volume of $V_f = 1.84 \text{ L}$ is obtained, which is inside the range of the volume constraint. The thickness standard deviation is $\sigma_{Wd} = 0.59 \text{ mm}$ which is much lower than the original standard deviation. The results with optimized controls are presented in Figure 6. This time, the bubble at the base of the preform has propagated to nearly the whole preform, thus a thickness higher than 2 mm can only be observed at the end of the preform.

<figure><img src="image_1737.png" /><figcaption>Fig. 6. Final shape of the bottle blown with optimized parameters $Q_{air} = 29.5 \text{ g m}^{-3}, T = 107.7^{\circ}\text{C}$ (left), thickness distribution of the bottle (right).</figcaption></figure>

The optimization took 213 s and evaluated the objective function 1717 times. This is way faster than optimization using finite element simulation, as calibration of the process is already done, and even with simulations lasting one minute, the optimization would have taken 30 hours.

## Conclusion

In this study, a methodology able to adapt its results depending on PETs with different ratio of rPET and depending on the conditions of the process was presented. Gaussian process regression models were trained using free blowing simulations following the Arruda-Boyce model, where the variability of the parameters was determined according to the variability of DMTA measurements obtained using resin with different ratios of rPET. It was able to provide a very fast optimization of the blowing process of a bottle.

For future works, the aim will be to improve the physical model of the simulations used for training. The viscoplasticity of the PET and the influence of the heating conditions of the bottle will notably have to be implemented to obtain more realistic predictions. To improve the precision of the predictions, the influence of the size of the training database, and the influence of the bottle blowing history of the process should be studied.

## Nomenclature

* $\mathbf{x}$ Scalar

* $\mathbf{x}$ Vector

* $\mathbf{x}$ Matrix
$$ \begin{gathered} \min_{\mu_2} \sigma_{Wd}(\mathbf{x}(\mu_2, \mathcal{P}_1, \mu_1)) \\ \mathbb{P}_v(\mu_2) = \mathbb{P}(|V_{f, obj} - V_f(\mathbf{x}(\mu_2, \mathcal{P}_1, \mu_1))| < \varepsilon_v) > \mathbb{P}_m \end{gathered} \tag{9} $$

**Optimization results.** The optimization was realized using the trained machine learning model to evaluate $\sigma_w$ and $V_f$, with the differential evolution method [18] integrated in SciPy. In eq. 9, the following parameters were set: $\varepsilon_v = 0.2$, $\mathbb{P}_m = 70\%$ and $V_f = 2$ L. $\mathbb{P}_m$ is taken low because preliminary calculations of probabilities to obtain the desired volume did not go above 73.5%. Optimized control parameters given by the algorithm were $Q_{air} = 29.5 \text{ g s}^{-1}$ and $T = 107.7^{\circ}\text{C}$. A volume of $V_f = 1.84 \text{ L}$ is obtained, which is inside the range of the volume constraint. The thickness standard deviation is $\sigma_{Wd} = 0.59 \text{ mm}$ which is much lower than the original standard deviation. The results with optimized controls are presented in Figure 6. This time, the bubble at the base of the preform has propagated to nearly the whole preform, thus a thickness higher than 2 mm can only be observed at the end of the preform.

<figure><img src="image_1735.png" /><figcaption>Fig. 6. Final shape of the bottle blown with optimized parameters $Q_{air} = 29.5 \text{ g m}^{-3}, T = 107.7^{\circ}\text{C}$ (left), thickness distribution of the bottle (right).</figcaption></figure>

The optimization took 213 s and evaluated the objective function 1717 times. This is way faster than optimization using finite element simulation, as calibration of the process is already done, and even with simulations lasting one minute, the optimization would have taken 30 hours.

## Conclusion

In this study, a methodology able to adapt its results depending on PETs with different ratio of rPET and depending on the conditions of the process was presented. Gaussian process regression models were trained using free blowing simulations following the Arruda-Boyce model, where the variability of the parameters was determined according to the variability of DMTA measurements obtained using resin with different ratios of rPET. It was able to provide a very fast optimization of the blowing process of a bottle.

For future works, the aim will be to improve the physical model of the simulations used for training. The viscoplasticity of the PET and the influence of the heating conditions of the bottle will notably have to be implemented to obtain more realistic predictions. To improve the precision of the predictions, the influence of the size of the training database, and the influence of the bottle blowing history of the process should be studied.

## Nomenclature

* $\mathbf{x}$ Scalar

* $\mathbf{x}$ Vector

* $\mathbf{x}$ Matrix
## Acknowledgments

The authors gratefully acknowledge the financial support from ARMINES organization for the CARNOT MINES 2020 on Polymer recyclability.

## References

[1] J. Nixon, G.H. Menary, S. Yan, Finite element simulations of stretch-blow moulding with experimental validation over a broad process window, Int. J. Mater. Form. 10 (2017) 793–809.
https://doi.org/10.1007/s12289-016-1320-9

[2] Y. Luo, L. Chevalier, E. Monteiro, S. Yan, G. Menary, Simulation of the Injection Stretch Blow Molding Process: An Anisotropic Visco-Hyperelastic Model for Polyethylene Terephthalate Behavior, Polym. Eng. Sci. 60 (2020) 823–831. https://doi.org/10.1002/pen.25341

[3] Y. Luo, G. Tantchou Yakam, R. Charlot, L. Chevalier, R. Savajano, In situ adjustment of a visco hyper elastic model for stretch blow molding process simulation of poly-ethylene terephthalate bottles, Polym. Eng. Sci. 63 (2023) 3066–3082. https://doi.org/10.1002/pen.26428

[4] A.-D. Le, R. Gilblas, V. Lucin, Y.L. Maoult, F. Schmidt, Infrared heating modeling of recycled PET preforms in injection stretch blow molding process, Int. J. Therm. Sci. 181 (2022) 107762. https://doi.org/10.1016/j.ijthermalsci.2022.107762

[5] L. Viora, M. Combeau, M.F. Pucci, D. Perrin, P.-J. Liotier, J.-L. Bouvard, C. Combeaud, A Comparative Study on Crystallisation for Virgin and Recycled Polyethylene Terephthalate (PET): Multiscale Effects on Physico-Mechanical Properties, Polymers 15 (2023) 4613. https://doi.org/10.3390/polym15234613

[6] O. Brandau, Stretch blow molding, Third edition, William Andrew, an imprint of Elsevier, Amsterdam Boston Heidelberg, 2017.

[7] P.-B. Rubio, L. Chamoin, F. Louf, Real-time data assimilation and control on mechanical systems under uncertainties, Adv. Model. Simul. Eng. Sci. 8 (2021) 4. https://doi.org/10.1186/s40323-021-00188-3

[8] P. Pereira Álvarez, P. Kerfriden, D. Ryckelynck, V. Robin, Real-Time Data Assimilation in Welding Operations Using Thermal Imaging and Accelerated High-Fidelity Digital Twinning, Mathematics 9 (2021) 2263. https://doi.org/10.3390/math9182263

[9] M. Bordival, F.M. Schmidt, Y.L. Maoult, V. Velay, Optimization of preform temperature distribution for the stretch-blow molding of PET bottles: Infrared heating and blowing modeling, Polym. Eng. Sci. 49 (2009) 783–793. https://doi.org/10.1002/pen.21296

[10] J. Biglione, Y. Béreaux, J.-Y. Charmeau, J. Balcaen, S. Chhay, Numerical simulation and optimization of the injection blow molding of polypropylene bottles - a single stage process, Int. J. Mater. Form. 9 (2016) 471–487. https://doi.org/10.1007/s12289-015-1234-y

[11] Y. Salomeia, G.H. Menary, C.G. Armstrong, J. Nixon, S. Yan, Measuring and modelling air mass flow rate in the injection stretch blow moulding process, Int. J. Mater. Form. 9 (2015) 531–545. https://doi.org/10.1007/s12289-015-1240-0

[12] E. Gorlier, J.F. Agassant, J.M. Haudin, N. Billon, Experimental and theoretical study of uniaxial deformation of amorphous poly(ethylene terephthalate) above glass transition temperature, Plast. Rubber Compos. 30 (2001) 48–55. https://doi.org/10.1179/146580101101541435

[13] G.H. Menary, C.W. Tan, C.G. Armstrong, Y. Salomeia, M. Picard, N. Billon, E.M.A. Harkin-Jones, Validating injection stretch-blow molding simulation through free blow trials, Polym. Eng. Sci. 50 (2010) 1047–1057. https://doi.org/10.1002/pen.21555
## Acknowledgments

The authors gratefully acknowledge the financial support from ARMINES organization for the CARNOT MINES 2020 on Polymer recyclability.

## References

[1] J. Nixon, G.H. Menary, S. Yan, Finite element simulations of stretch-blow moulding with experimental validation over a broad process window, Int. J. Mater. Form. 10 (2017) 793-809. https://doi.org/10.1007/s12289-016-1320-9

[2] Y. Luo, L. Chevalier, E. Monteiro, S. Yan, G. Menary, Simulation of the Injection Stretch Blow Molding Process: An Anisotropic Visco-Hyperelastic Model for Polyethylene Terephthalate Behavior, Polym. Eng. Sci. 60 (2020) 823-831. https://doi.org/10.1002/pen.25341

[3] Y. Luo, G. Tantchou Yakam, R. Charlot, L. Chevalier, R. Savajano, In situ adjustment of a visco hyper elastic model for stretch blow molding process simulation of poly-ethylene terephthalate bottles, Polym. Eng. Sci. 63 (2023) 3066-3082. https://doi.org/10.1002/pen.26428

[4] A.-D. Le, R. Gilblas, V. Lucin, Y.L. Maoult, F. Schmidt, Infrared heating modeling of recycled PET preforms in injection stretch blow molding process, Int. J. Therm. Sci. 181 (2022) 107762. https://doi.org/10.1016/j.ijthermalsci.2022.107762

[5] L. Viora, M. Combeau, M.F. Pucci, D. Perrin, P.-J. Liotier, J.-L. Bouvard, C. Combeaud, A Comparative Study on Crystallisation for Virgin and Recycled Polyethylene Terephthalate (PET): Multiscale Effects on Physico-Mechanical Properties, Polymers 15 (2023) 4613. https://doi.org/10.3390/polym15234613

[6] O. Brandau, Stretch blow molding, Third edition, William Andrew, an imprint of Elsevier, Amsterdam Boston Heidelberg, 2017.

[7] P.-B. Rubio, L. Chamoin, F. Louf, Real-time data assimilation and control on mechanical systems under uncertainties, Adv. Model. Simul. Eng. Sci. 8 (2021) 4. https://doi.org/10.1186/s40323-021-00188-3

[8] P. Pereira Álvarez, P. Kerfriden, D. Ryckelynck, V. Robin, Real-Time Data Assimilation in Welding Operations Using Thermal Imaging and Accelerated High-Fidelity Digital Twinning, Mathematics 9 (2021) 2263. https://doi.org/10.3390/math9182263

[9] M. Bordival, F.M. Schmidt, Y.L. Maoult, V. Velay, Optimization of preform temperature distribution for the stretch-blow molding of PET bottles: Infrared heating and blowing modeling, Polym. Eng. Sci. 49 (2009) 783-793. https://doi.org/10.1002/pen.21296

[10] J. Biglione, Y. Béreaux, J.-Y. Charmeau, J. Balcaen, S. Chhay, Numerical simulation and optimization of the injection blow molding of polypropylene bottles - a single stage process, Int. J. Mater. Form. 9 (2016) 471-487. https://doi.org/10.1007/s12289-015-1234-y

[11] Y. Salomeia, G.H. Menary, C.G. Armstrong, J. Nixon, S. Yan, Measuring and modelling air mass flow rate in the injection stretch blow moulding process, Int. J. Mater. Form. 9 (2015) 531-545. https://doi.org/10.1007/s12289-015-1240-0

[12] E. Gorlier, J.F. Agassant, J.M. Haudin, N. Billon, Experimental and theoretical study of uniaxial deformation of amorphous poly(ethylene terephthalate) above glass transition temperature, Plast. Rubber Compos. 30 (2001) 48-55. https://doi.org/10.1179/146580101101541435

[13] G.H. Menary, C.W. Tan, C.G. Armstrong, Y. Salomeia, M. Picard, N. Billon, E.M.A. Harkin-Jones, Validating injection stretch-blow molding simulation through free blow trials, Polym. Eng. Sci. 50 (2010) 1047-1057. https://doi.org/10.1002/pen.21555
[14] C.E. Rasmussen, C.K.I. Williams, Gaussian processes for machine learning, MIT Press, Cambridge, Mass, 2006.

[15] L. Chevalier, Y.M. Luo, E. Monteiro, G.H. Menary, On visco-elastic modelling of polyethylene terephthalate behaviour during multiaxial elongations slightly over the glass transition temperature, Mech. Mater. 52 (2012) 103-116. https://doi.org/10.1016/j.mechmat.2012.05.003

[16] I.M. Sobol', On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comput. Math. Math. Phys. 7 (1967) 86-112. https://doi.org/10.1016/0041-5553(67)90144-9

[17] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, Scikit-learn: Machine Learning in Python, J. Mach. Learn. Res. 12 (n.d.) 2825--2830.

[18] R. Storn, K. Price, Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, J. Glob. Optim. 11 (1997) 341-359. https://doi.org/10.1023/A:1008202821328
[14] C.E. Rasmussen, C.K.I. Williams, Gaussian processes for machine learning, MIT Press, Cambridge, Mass, 2006.

[15] L. Chevalier, Y.M. Luo, E. Monteiro, G.H. Menary, On visco-elastic modelling of polyethylene terephthalate behaviour during multiaxial elongations slightly over the glass transition temperature, Mech. Mater. 52 (2012) 103-116. https://doi.org/10.1016/j.mechmat.2012.05.003

[16] I.M. Sobol', On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comput. Math. Math. Phys. 7 (1967) 86-112. https://doi.org/10.1016/0041-5553(67)90144-9

[17] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, Scikit-learn: Machine Learning in Python, J. Mach. Learn. Res. 12 (n.d.) 2825--2830.

[18] R. Storn, K. Price, Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, J. Glob. Optim. 11 (1997) 341-359. https://doi.org/10.1023/A:1008202821328