Finite Element Method for minimizing geometric error in the bending of large sheets

Finite Element Method for minimizing geometric error in the bending of large sheets | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Finite Element Method for minimizing geometric error in the bending of large sheets Alain Gil Del Val, Mariluz Penalva, Fernando Veiga, Bilal El Moussaoui This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4551326/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 31 Oct, 2024 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted 5 You are reading this latest preprint version Abstract Minimizing geometric error in the bending of large sheets remains a challenging endeavor in the industrial environment. This specific industrial operation is characterized by protracted cycles and limited batch sizes. Coupled with extended cycle times, the process involves a diverse range of dimensions and materials. Given these operational complexities, conducting practical experimentation for data extraction and control of industrial process parameters proves to be unfeasible. To gain insights into the process, finite element models serve as invaluable tools for simulating industrial processes for reducing experimental cost. Consequently, the primary objective of this research endeavor is to develop an intelligent finite element model capable of providing operators with pertinent information regarding the optimal range of key parameters to mitigate geometric error in the bending of large sheets. The average geometric error in curvature is recorded at 0.97%, thereby meeting the stringent industrial requirement for achieving such bending with minimal equivalent plastic deformation. As such, these findings present promising prospects for the automation of the industrial process. bending rolling finite element method response surface method Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Introduction The sectors of oil/gas, tunnel construction, and large industrial vessels have experienced a surge in demand for large industrial components, driven by the ongoing global energy diversification efforts. Within this landscape, the forming operation has emerged as a critical industrial process to produce large ducts. This industrial operation is characterized by unique challenges. On one hand, the design complexity of large pipes, often featuring single or double bends, coupled with the intricate and time-consuming nature of tube manufacturing, adds layers of complexity. Tube fabrication can be carried out sequentially or in phases, typically involving preforming or initial bending with rollers followed by final geometry generation. On the other hand, this manufacturing process entails prolonged production times, small batch sizes, and a broad spectrum of dimensions and materials. Moreover, efficient folding is paramount for achieving accurate formation of bending surfaces [1]. Traditionally, this operation has been manual, relying heavily on the skill, knowledge, and experience of operators, which significantly influences the final quality of metal sheet bending. Additionally, the setup phase is critical but inherently manual, with process parameter knowledge typically acquired through on-the-job experience within the work cell. Consequently, conducting real experimentation to extract and analyze the main parameters controlling the sheet forming process becomes challenging. Considering these challenges, the development of models for metal forming processes in the production of pipe-shaped components is crucial in production engineering [2]. Recent years have witnessed considerable technological advancements in the development of such models, resulting in a diverse array that continues to expand [3]. While numerous publications have addressed finite element simulations of rolling bending for small-sized pieces (dimensions less than 100 mm) [4–9], there is a scarcity of literature focusing on the forming of large-sized sheets (exceeding 1000 mm) for generating pipes, particularly for the gas and oil industry, due to prohibitively high computational costs. Consequently, there is a growing trend in manufacturing engineering towards advancing the development of models in industrial processes and exploring their boundaries to reduce manufacturing costs. Parameterization and robust process design are essential to effective modeling [9], with the last goal of meeting increasingly stringent tolerance demands in the final product [10]. The integration of these models and the virtualization of the workspace/piece facilitate the digitization of manufacturing in the era of new smart factories [11]. In this context, the development of models in the manufacturing process plays a crucial role in defining critical parameters for the production of large pieces through forming, as emphasized by [12]. Furthermore, the generation of models aligns with the digitization strategy of these processes [11], giving rise to various approaches and typologies of models [13]. The significance of these solutions is underscored by the growing demand for stricter tolerances in the final product [10]. Hence, the design and parameterization of robust processes can be effectively assessed with appropriate models [9]. To achieve the dual objectives of automation and cost reduction in manufacturing, machine learning techniques are indispensable. Various machine learning techniques are commonly employed in the literature to develop a supervised regression model, which forms the basis of metamodeling strategies to predict continuous variables [14]. Examples include Smith's neural networks [15], kriging or Williams and Rasmussen's Gaussian process [16], ensemble methods like random forest or gradient boosting [2], among others. However, implementing these strategies in industrial settings can be complex. Therefore, Orzechowski et al. propose the response surface model (RSM) due to its competitive accuracy in regression problems [17]. Two key reasons for selecting this technique are its ability to avoid overfitting, particularly beneficial in small data problems, and its relatively lower computational time compared to kriging or neural networks. The aim of this research is to develop an intelligent finite element model capable of guiding the operator on the optimal range of key parameters to minimize geometric error in the bending of large sheets. Such a model would facilitate the automation of the industrial process, mitigating the need for extensive experimentation and minimizing industrial costs. Materials and methods Industrial case The industrial pipes for the Oil/Gas sector involve creating a series of bent sheets throughout the forming process. The operation has four steps. Firstly, the base material plates are received in the rolling machine. Then, they are compressed in the forming machine and, finally, the rolled plates are composed to manufacture the pipe. Figure 1 shows an industrial pipe of the Oil/Gas sector after the manufacturing process in the last stage of commercial cycle. Table 1 outlines the main commercial attributes of the large-scale sheets. The two cases constitute the extremes of the typical reference pipes manufactured for the oil and gas sector. As can be seen in Table 1 , the first and second pipes are the smallest and the biggest ones, respectively. Leveraging this genuine process data enables a focused study to execute a minimal simulation while maximizing information acquisition to minimize geometric error. Table 1 Characteristics of the two common manufactured large dimension sheets Case 1 2 Diameter (mm) 3,500 7,000 Width (mm) 4,000 Thickness (mm) 30 100 Weight (Tn) 10.5 70 Material SA-516 Gr. 70N This manufacturing range will be important to stablish the range of the manufacturing parameters to develop the design of experiments the following sections. The configuration of the forming process essentially comprises a metal sheet throughout the use of a top roller and two lower ones (front and rear) as is illustrated in Fig. 2 . Each roller has distinct diameters and the lengths of the rollers is the same. This value is 4,000 mm, respectively. The maximum distance between the upper roller and the lower rollers is 100 mm, adjustable based on the metal sheet thickness. The bending process unfolds in two stages: the first step is the bending. The rollers feed on the metal sheet and bend it to a curvature by repositioning the rear roller to 60º relative to the centers of the front and rear rollers. The second stage is the rolling. The metal sheet is bended throughout the entire metal piece with the front roller rotating continuously to adjust the sheet and achieve the programmed curvature and shape. Notably, the front and rear rollers support solely the sheet and rotate to facilitate sheet passage without exerting pressure for bending. The efficacy of the three-roller bending process heavily relies on the operator's expertise and dexterity, being a predominantly manual process. Plate bending typically follows a multi-pass approach, contingent upon manual adjustments guided by the operator's experience to optimize bending capacity on the rollers. However, this method incurs significant industrial costs due to material wastage and production time. Consequently, this process requires the development of an optimized production method to enhance repeatability, precision and productivity. Furthermore, the operator would acquire comprehensive knowledge of the process to achieve the desired diameter of the pipe component. Material The chosen mathematical method for modeling sheet bending is the finite element method, implemented through the LS-Dyna finite element program in this study. The finite element model of the forming process comprises a sheet using three rollers. Assumptions include isotropic material properties for the metal sheet and an elastoplastic material model. Implicit/explicit equations are used for computation with gravitational effects factored into the sheet behavior. However, during the simulation, LS-Dyna's algorithm integrates temporally and simulates the model accordingly. The rollers, constructed from carbon steel, are treated as rigid bodies. The dimensions of the sheet vary from 10,990 to 23,000 mm (length) × 4,000 mm (width) × 30 to 100 mm (thickness). Additionally, deformability of the piece is assumed, with the material identified as SA-516 Gr.70N steel. Table 2 provides key properties of this material for the mathematical method. Table 2 Mechanical properties of material Strain limit (MPa) 805 Young module (GPa) 200 Density (kg/mm 3 ) 7,85E-06 Poisson module 0,29 Furthermore, a uniaxial tensile test is conducted to estimate the stress-strain curve, facilitating definition of the material plastic behavior, as is illustrated in Fig. 3 . Finite element model The challenge encountered in forming processes requires the use of advanced simulation software. When opting for a material model for the sheet, amidst a plethora of options available in LS-Dyna, several considerations must be taken into account: a) suitability for metals/non-metals, b) addressing plasticity, and c) capability to study fracture. Aligning with the aforementioned specifications, the piecewise linear plasticity material model is selected for this study. This model leverages the material's stress-strain curve as input, with the yield stress estimated using Eq. 1 . $$ɸ\left(\sigma \right)={\sigma }_{eq}-{\sigma }_{y}\left({\epsilon }^{p}\right)=0$$ 1 However, in this scenario, the process is independent of the deformation rate. Thus, within the numerical algorithm, the increment of plastic deformation can be readily calculated employing Eq. 2 . $$\varDelta {\epsilon }^{p}=\frac{{\sigma }_{eq}^{trial}-{\sigma }_{y}\left({\epsilon }_{n}^{p}\right)}{3G+H}$$ 2 Here, \({\sigma }_{eq}^{trial}\) represents the equivalent stress in incremental plastic deformation, \({\sigma }_{y}\) denotes the yield stress, G signifies the shear modulus of the material (76.8 GPa), and H denotes the hardening modulus, evaluable at the slope where the stress-strain curve alters at the yield point. All parameters are derived from the stress-strain curve obtained from uniaxial tensile tests, as depicted in Fig. 3 . The piecewise linear plasticity model incorporates a failure criterion, where the equivalent plastic strain at failure and the fracture stress are incorporated as values in the material options of LS-Dyna, determined from the curve obtained from the uniaxial tensile test in Fig. 3 . Nonetheless, this criterion proves less pertinent for the current application, as the primary objective is to examine the roundness of the curved pipe with minimal equivalent plastic deformation. Material model parameters are optimized based on the stress-strain curve obtained from tensile tests (refer to Fig. 3 ) and are further validated against FEM simulations of tension before being integrated into the actual simulation of the bending process. All loads and constraints are defined with respect to the global coordinate system, while the local coordinate is referenced for the rear lower roller, as illustrated in Fig. 4 . The modeling process entails two stages: firstly, the bending of the sheet, simply denoted as "bending," and secondly, the circumferential bending of the sheet, referred to as bending or rolling, as depicted in Fig. 4 . The prescribed movements of the rigid bodies of the rollers are represented by the central point of the roller, where constraints and loads are accordingly applied. Boundary conditions are outlined in Fig. 4 . During the first stage, the upper and lower front rollers are free to rotate about the global X-axis, whereas the rear roller is confined to movement solely along the Z' local coordinate axis at a 60⁰ angle relative to the horizontal. In the second stage, the lower front roller rotates, with a speed of 6.23 m/min (approximately ω = 2 rpm/min), while the upper roller maintains its previous configuration. Finally, following the adjustment of the rear lower roller, it is constrained to the Z-axis in the second stage. The model is developed utilizing automatic surface-to-surface contact and interactions between various parts of a roller through the LS-Dyna code. Contacts are established employing the master/slave approach, wherein the rollers are considered master surfaces, and the opposing surfaces to the rollers and the sheet contact are designated as slaves. Furthermore, Coulomb's friction law is employed between the sheet and the rollers, with dynamic and static friction coefficients set as 0.25 and 0.5, respectively. Moreover, 3D solid elements are adopted for mesh definition, ensuring the capability to generate new simulations of industrial processes in the future. Metamodel The metamodel will encompass the development of intelligent algorithms predicated on the outcomes derived from the proposed simulations aimed at model enhancement. The strategy hinges upon a supervised regression model, which constitutes the fundamental framework in a metamodeling approach for prognosticating continuous variables. One viable stratagem within response surface methods (RSM) is the Box-Behnken design, widely employed across various engineering domains [18]. The objective entails establishing a correlation between a target output variable (y) and a set of controllable or input variables {x 1 , x 2 , ..., x n }. If the nature of the association between input and output values is discerned, a model can be formulated in the format delineated by Eq. 3 , $$y=f({x}_{1},{x}_{1},....{x}_{n})+\epsilon$$ 3 where ε signifies the error inherent to the response (y). If the anticipated response is denoted as E(y) = f(x 1 , x 2 , …x n ) + η, the resultant surface expressed by Eq. 4 is termed a response surface. $$\eta =f({x}_{1},{x}_{1},....{x}_{n})$$ 4 Typically, a second-order model, as detailed in Eq. 5 , is adopted in response surface methodology, with the coefficients β computed via the least squares method [18]. $$y={\beta }_{0}+\sum _{i=1}^{k}{\beta }_{i}{x}_{i}+\sum _{i=1}^{k}{\beta }_{ii}{x}_{i}^{2}+\sum _{i}\sum _{j}{\beta }_{i}{x}_{i}{x}_{j}+\epsilon$$ 5 Employing response surface methodology facilitates the determination of optimal values for the controllable parameters, thereby enhancing the response optimization or elucidating which parameter values of x engender a product or process aligned with diverse requirements or specifications. The comparison between Finite Element Method (FEM) and Response Surface Methodology (RSM) models via the experimental datasets utilized by machine operators for the bending of large-sized metal sheets has been conducted. Results Finite element model Figure 5 illustrates the assembly of the Finite Element Model (FEM) and the sequential stages of the rolling process, as obtained through numerical simulation calculations. Notably, asymmetry in curvature distribution is evident at the point of contact between the rear bottom roller and the metal sheet. This asymmetry arises due to the non-uniform deformation occurring over the region of maximum bending moment. Given the critical nature of CPU time, the numerical model undergoes optimization through mesh sensitivity analysis. This entails assessing the impact of mesh type and element count (mesh size) on metal sheet bending, as well as their influence on CPU time and geometric bending error. The quadrilateral element is chosen for this application due to its common usage in FEM simulations of sheet forming, which results in reduced calculation time compared to triangular elements. Table 3 consolidates all pertinent information outlined earlier. Table 3 Sensitivity analysis of FEM Mesh Elements CPU time Geom error. (%) Coarse 3.560 300 1,85 Normal 8.898 420 1,71 Fine 17.796 830 1,72 Consequently, the sensitivity analysis recommends adopting a standard mesh comprising 8,898 quadrilateral elements, which incurs a CPU time of approximately 420 minutes while achieving minimal geometric bending error. Figure 6 shows the distribution of equivalent plastic deformation generated during the forming process, along with the distribution of equivalent plastic deformation across the thickness of the sheet. The simulation employs the GM unit (kg, mm, ms, KN, GPa, KN-mm) as the unit of measurement, with stress expressed in GPa. Initially, the equivalent plastic deformation during the metal sheet's initial bending stage amounts to 1.55%. However, this deformation remains at a level of 0.78% throughout the entire bending process. Notably, the maximum plastic deformation occurs under the initial bending condition, indicating localized stress concentration. Furthermore, Fig. 6 illustrates the stress distribution in the bent part of the metal sheet. The minimum residual stress, observed at the neutral fiber, measures 0.040 GPa (40 MPa). However, as the distance from the neutral fiber to the outer surface increases, the residual stress progressively rises, reaching 0.1306 GPa (130.6 MPa) at the sheet's surface. This finding suggests that the model for metal sheet bending and curving aligns with industrial requirements, achieving minimal equivalent plastic deformation. Metamodel The CPU times required to obtain simulation results are quite long, as confirmed in Table 3 during mesh optimization. In the simulations conducted for this study, CPU times ranged between 4 hours and 8 hours for each simulation. The variations in CPU times were associated with different dimensions of the sheet metal plate. Sometimes, it was also necessary to repeat the simulation to obtain the correct diameter of the pipe. As explained, an effort was made to minimize the number of simulations due to the lengthy calculation times, so a reduced set of input variables was selected. The selected input variables are shown in the following Table 4 . Table 4 Design of experiments Inputs + - Lenght (mm) 10,990 23,000 Thickness (mm) 30 100 Roller displacement (mm) 0 200 Output (mm) Obtained diameter With this objective, the Response Surface Methodology (RSM) is employed to find values for controllable parameters that result in optimizing the response or to discover which ones are most important in the process to meet requirements or specifications with a minimum number of experiments. In this study, three input variables (sheet length, thickness, and displacement of the rear bottom roller) were included, each considering two levels (minimum and maximum values within their ranges), as shown in Table 4 . The output variable or response surface will be the obtained diameter. A design with four central points was implemented for each block. Thus, a Box-Behnken design consisting of 13 experiments was obtained, combining different values for the variables. Since this type of approach is generally used for designing experiments, some additional tests were conducted to verify the experiment's variability under the same conditions. Therefore, randomly, some extra tests are included within the defined range for each of them, resulting in new combinations of 15 experiments. The results obtained from the Box-Behnken RSM model are analyzed in terms of the correlation between different inputs and outputs. The response surface associated with the obtained diameter is modeled in the reduced centered space. The equation is derived using the DOE (Design-Expert 13) software. $$d\_o=269.903 + 0.973253 \text{*} l -0.115865 \text{*} e -0.798903 \text{*} d$$ 6 Where d_o is the diameter obtained in the simulation, l is the length of the sheet, e is the thickness, and d is the displacement distance of the rear bottom roller. It can be observed that there is a linear relationship between the output and the inputs. In the following lines, the effect of the input variables on different responses is analyzed in detail. Table 5 presents the study applying the response surface methodology. Table 5 Box-Behken RSM Model for intelligent rolling Source Sum of Square df Mean Square F-value p-value Model 5.490E + 07 3 1.830E + 07 6363.51 < 0.0001 significant Lenght 1.610E + 06 1 1.610E + 06 559.77 < 0.0001 significant Thickness 102.14 1 102.14 0.0355 0.8520 Roller displacement 223.76 1 223.76 0.0778 0.8826 Residuals 71892.25 25 2875.69 Lack of fit 71892.25 16 4493.27 Error pure 0.0000 9 0.0000 Cor Total 5.497E + 07 28 The model's F-value reaching 6,363.51 implies that the model is significant. There is only a 0.01% probability of obtaining such a large F-value due to noise. P-values below 0.0500 indicate that the model terms are significant. In this case, the sheet length (l) is a significant model term. P-values above 0.1000 indicate that model terms are not significant. If there are many insignificant model terms (excluding those necessary to support hierarchy), model reduction can enhance the model. Figure 7 represents the response surface of the diameter obtained as a function of the two most significant parameters (length and thickness of the sheets). The predicted R² value of 0.9987 indicates a reasonable fit for all sheet lengths and thicknesses. The obtained diameter shows little deviation; however, this parameter also depends on the displacement of the rear bottom roller, even though the RSM has classified it as the third least significant. To observe the simulated geometric error, Table 6 compiles the extreme points of the designed experiments that have been tested. It shows the values of the obtained diameter after simulating the conditions of that point, along with the geometric bending error. Table 6 Geometric bending errors and diameters obtained from the RSM Length (mm) Thickness (mm) Displacement (mm) Obtained diameter (mm) Error (%) 23,000 30 132 7.341 0,27 10,995 30 189 3560 1.71 23,000 100 118 7.298 -0.31 10,995 100 175 3.537 1.05 These geometric errors do not exceed 2%, and therefore, they meet the industrial specifications required in manufacturing. To clarify the curvature error calculation, Fig. 8 illustrates the estimated diameter obtained in cases with a sheet length of 10,995 mm and sheet thicknesses of 30 mm and 10 mm, respectively, as indicated in Table 6 . Finally, the average geometric bending error for the fifteen simulations based on the RSM metamodel is calculated. The values for this mean error and its standard deviation are 0.97% and 0.43, respectively. Therefore, these results are consistent with the manufacturing tolerances of the industrial user and enable them to have an intelligent optimizer for the process variables in the bending and curving of metal sheets for pipe manufacturing. Conclusions A sophisticated finite element model has been developed with the capability to guide operators on optimal parameter ranges to minimize geometric errors when bending large sheets for pipes within the oil and gas sector. Several conclusions derived from this study include: - The finite element model facilitates the prediction of tube dimensions utilizing input parameters from the intelligent model. - The FEM model for sheet bending and curving fulfills industrial requirements by achieving curvature with minimal equivalent plastic deformation. The Von Mises stress at the neutral fiber is 40 MPa, and the distribution of equivalent plastic deformation and Von Mises stress exhibits symmetry across the sheet thickness, with a maximum value of 130.6 MPa. - The intelligent model functions as a metamodel based on the Box-Behnken Response Surface Methodology (RSM), minimizing simulations while generating a highly dependent response surface, particularly on sheet length. The goodness of fit (R²) is 0.9987. - The average geometric bending error and its dispersion stand at 0.97% and 0.43, respectively, aligning with the manufacturing tolerances of industrial users. - A smart advisor for bending and curving process variables has been developed to enable autonomous manufacturing of pipes. Declarations Funding The research is partially supported the European Union's Horizon 2020 research and innovation program under grant agreement no. 958303. Competing interests The authors declare no competing interests. Author contribution Conceptualization, Alain Gil Del Val.; Data curation, F.V., Alain Gil Del Val.; Formal analysis, Fernando., Alain Gil Del Val.; Investigation, Alain Gil Del Val, Mariluz.; Methodology, Alain Gil Del Val and Mariluz.; Project administration, Alain Gil Del Val.; Supervision, Bilal.; Validation, Alain Gil Del Val. and Fernando; Writing—original draft, Fernando. and Alain Gil Del Val.; Writing–review and editing, Mariluz Penalva, Alain Gil Del Val., Bilal., and Fernando. All authors have read and agreed to the published version of the manuscript. References Batalov GS, Lunev AA, Radionova LV, Lezin VD (2021) Development of New Methods for the Production of Large-Diameter Double-Seam Pipes. In Solid State Phenomena; Trans Tech Publications Ltd Switzerland 316:538–548. Breiman L (2001) Random forests. Mach. Learn 45:5–32. Volk W, Groche P, Brosius, A.; Ghiotti, A.; Kinsey, B.L.; Liewald, M.; Madej, L.; Min, J.; Yanagimoto, J. 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Volk W, Groche P, Brosius A, Ghiotti A, Kinsey BL, Liewald M, Madej L, Min J, Yanagimoto J (2019) Models and modelling for process limits in metal forming. CIRP Annals 68(2):775-798. Smith M (1993) Neural Networks for Statistical Modeling. Thomson Learning: Boston, MA, USA. Williams CK, Rasmussen CE (2006) Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, Volume 2. Orzechowski P; La Cava W, Moore JH (2018) Where are we now?: A large benchmark study of recent symbolic regression methods. In Proceedings of the Genetic and Evolutionary Computation Conference, Kyoto, Japan, 15–19 July 2018; 1183–1190. Montgomery DC (2017) Design and analysis of experiments. 9th Edition. John Wiley & Sons, Inc. Cite Share Download PDF Status: Published Journal Publication published 31 Oct, 2024 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted Editorial decision: Major Revisions Needed 22 Aug, 2024 Reviewers agreed at journal 26 Jun, 2024 Reviewers invited by journal 18 Jun, 2024 Editor assigned by journal 13 Jun, 2024 First submitted to journal 11 Jun, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4551326","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":316031956,"identity":"61da1fe6-8e85-4bd1-ab65-a80b086828fe","order_by":0,"name":"Alain Gil Del 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Innovation","correspondingAuthor":false,"prefix":"","firstName":"Mariluz","middleName":"","lastName":"Penalva","suffix":""},{"id":316031958,"identity":"f07bad07-93aa-4f66-9347-cec4d93830cc","order_by":2,"name":"Fernando Veiga","email":"","orcid":"","institution":"Universidad Publica de Navarra - Campus de Arrosadia: Universidad Publica de Navarra","correspondingAuthor":false,"prefix":"","firstName":"Fernando","middleName":"","lastName":"Veiga","suffix":""},{"id":316031959,"identity":"b4d97f5d-f450-41f1-8410-1eddecef883a","order_by":3,"name":"Bilal El Moussaoui","email":"","orcid":"","institution":"Tecnalia Research \u0026 Innovation Foundation: Fundacion Tecnalia Research \u0026 Innovation","correspondingAuthor":false,"prefix":"","firstName":"Bilal","middleName":"El","lastName":"Moussaoui","suffix":""}],"badges":[],"createdAt":"2024-06-08 16:38:46","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4551326/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4551326/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00170-024-14685-3","type":"published","date":"2024-10-31T16:20:31+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":59649024,"identity":"d199898f-3c5e-444f-b8eb-9a9cd7c4875f","added_by":"auto","created_at":"2024-07-04 09:21:26","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":193842,"visible":true,"origin":"","legend":"\u003cp\u003eIndustrial pipe for the Oil/Gas Sector\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/da62d3bc6174a7fa5b8c72da.png"},{"id":59649010,"identity":"49a8a709-7a4d-4779-9794-7189645fbd56","added_by":"auto","created_at":"2024-07-04 09:21:26","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":92894,"visible":true,"origin":"","legend":"\u003cp\u003eBending process configuration\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/daa8a74ca670318dbde8fb8a.png"},{"id":59649013,"identity":"e26a1445-05a5-439e-8db0-33e13b5aa5fc","added_by":"auto","created_at":"2024-07-04 09:21:26","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":19429,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain profile of SA-516 Gr. 70N.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/78e5ddc8e8983d2d912d6061.png"},{"id":59649049,"identity":"9bcf4494-5a38-4608-abfa-1c1ee7eeb93f","added_by":"auto","created_at":"2024-07-04 09:21:29","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":176107,"visible":true,"origin":"","legend":"\u003cp\u003eGlobal and local coordinates, boundary conditions, and constraints of the model.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/efbf67420a4d66d3b54636d1.png"},{"id":59649040,"identity":"c742e6aa-0043-40bb-805e-cc1422290337","added_by":"auto","created_at":"2024-07-04 09:21:28","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":257516,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element method and initial, bending and the last rolling stages, respectively.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/e2fb36a1a833c203f376804e.png"},{"id":59649007,"identity":"0aa1c90d-49a0-4485-a90f-a09009ca240e","added_by":"auto","created_at":"2024-07-04 09:21:26","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":553898,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of equivalent plastic strain along the entire sheet and distribution of residual stress in the initial bending stage.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/1ce1f05b3ce1b2dd12bd60a2.png"},{"id":59649046,"identity":"37e2e4f6-a26f-4072-9aa0-349cd7d1b8dd","added_by":"auto","created_at":"2024-07-04 09:21:29","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":130589,"visible":true,"origin":"","legend":"\u003cp\u003e3D response surface of the diameter obtained as a function of the required diameter and the thickness of the sheet.\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/726643e9a51592a11baa5dd6.png"},{"id":59649050,"identity":"a7f44a6b-36f8-4965-8408-b034cf05a99d","added_by":"auto","created_at":"2024-07-04 09:21:29","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":159567,"visible":true,"origin":"","legend":"\u003cp\u003eObtained diameters for sheet length of 10.995 mm and sheet thicknesses of 30 mm and 100 mm, respectively.\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/178259a2c9dacf56dbd3cbe8.png"},{"id":68207261,"identity":"e5401cca-5da4-4219-a50d-4e0c61d2a330","added_by":"auto","created_at":"2024-11-04 16:36:17","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2122388,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4551326/v1/038742bf-fde6-4c1a-8567-5ea998120a3f.pdf"}],"financialInterests":"","formattedTitle":"Finite Element Method for minimizing geometric error in the bending of large sheets","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe sectors of oil/gas, tunnel construction, and large industrial vessels have experienced a surge in demand for large industrial components, driven by the ongoing global energy diversification efforts. Within this landscape, the forming operation has emerged as a critical industrial process to produce large ducts. This industrial operation is characterized by unique challenges. On one hand, the design complexity of large pipes, often featuring single or double bends, coupled with the intricate and time-consuming nature of tube manufacturing, adds layers of complexity. Tube fabrication can be carried out sequentially or in phases, typically involving preforming or initial bending with rollers followed by final geometry generation.\u003c/p\u003e \u003cp\u003eOn the other hand, this manufacturing process entails prolonged production times, small batch sizes, and a broad spectrum of dimensions and materials. Moreover, efficient folding is paramount for achieving accurate formation of bending surfaces [1].\u003c/p\u003e \u003cp\u003eTraditionally, this operation has been manual, relying heavily on the skill, knowledge, and experience of operators, which significantly influences the final quality of metal sheet bending. Additionally, the setup phase is critical but inherently manual, with process parameter knowledge typically acquired through on-the-job experience within the work cell. Consequently, conducting real experimentation to extract and analyze the main parameters controlling the sheet forming process becomes challenging.\u003c/p\u003e \u003cp\u003eConsidering these challenges, the development of models for metal forming processes in the production of pipe-shaped components is crucial in production engineering [2]. Recent years have witnessed considerable technological advancements in the development of such models, resulting in a diverse array that continues to expand [3]. While numerous publications have addressed finite element simulations of rolling bending for small-sized pieces (dimensions less than 100 mm) [4\u0026ndash;9], there is a scarcity of literature focusing on the forming of large-sized sheets (exceeding 1000 mm) for generating pipes, particularly for the gas and oil industry, due to prohibitively high computational costs.\u003c/p\u003e \u003cp\u003eConsequently, there is a growing trend in manufacturing engineering towards advancing the development of models in industrial processes and exploring their boundaries to reduce manufacturing costs. Parameterization and robust process design are essential to effective modeling [9], with the last goal of meeting increasingly stringent tolerance demands in the final product [10]. The integration of these models and the virtualization of the workspace/piece facilitate the digitization of manufacturing in the era of new smart factories [11].\u003c/p\u003e \u003cp\u003eIn this context, the development of models in the manufacturing process plays a crucial role in defining critical parameters for the production of large pieces through forming, as emphasized by [12]. Furthermore, the generation of models aligns with the digitization strategy of these processes [11], giving rise to various approaches and typologies of models [13]. The significance of these solutions is underscored by the growing demand for stricter tolerances in the final product [10]. Hence, the design and parameterization of robust processes can be effectively assessed with appropriate models [9].\u003c/p\u003e \u003cp\u003eTo achieve the dual objectives of automation and cost reduction in manufacturing, machine learning techniques are indispensable. Various machine learning techniques are commonly employed in the literature to develop a supervised regression model, which forms the basis of metamodeling strategies to predict continuous variables [14]. Examples include Smith's neural networks [15], kriging or Williams and Rasmussen's Gaussian process [16], ensemble methods like random forest or gradient boosting [2], among others. However, implementing these strategies in industrial settings can be complex. Therefore, Orzechowski et al. propose the response surface model (RSM) due to its competitive accuracy in regression problems [17]. Two key reasons for selecting this technique are its ability to avoid overfitting, particularly beneficial in small data problems, and its relatively lower computational time compared to kriging or neural networks.\u003c/p\u003e \u003cp\u003eThe aim of this research is to develop an intelligent finite element model capable of guiding the operator on the optimal range of key parameters to minimize geometric error in the bending of large sheets. Such a model would facilitate the automation of the industrial process, mitigating the need for extensive experimentation and minimizing industrial costs.\u003c/p\u003e"},{"header":"Materials and methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eIndustrial case\u003c/h2\u003e \u003cp\u003eThe industrial pipes for the Oil/Gas sector involve creating a series of bent sheets throughout the forming process. The operation has four steps. Firstly, the base material plates are received in the rolling machine. Then, they are compressed in the forming machine and, finally, the rolled plates are composed to manufacture the pipe. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows an industrial pipe of the Oil/Gas sector after the manufacturing process in the last stage of commercial cycle.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e outlines the main commercial attributes of the large-scale sheets. The two cases constitute the extremes of the typical reference pipes manufactured for the oil and gas sector. As can be seen in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the first and second pipes are the smallest and the biggest ones, respectively. Leveraging this genuine process data enables a focused study to execute a minimal simulation while maximizing information acquisition to minimize geometric error.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCharacteristics of the two common manufactured large dimension sheets\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCase\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDiameter (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3,500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eWidth (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003e4,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eThickness (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eWeight (Tn)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eMaterial\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eSA-516 Gr. 70N\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThis manufacturing range will be important to stablish the range of the manufacturing parameters to develop the design of experiments the following sections.\u003c/p\u003e \u003cp\u003eThe configuration of the forming process essentially comprises a metal sheet throughout the use of a top roller and two lower ones (front and rear) as is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eEach roller has distinct diameters and the lengths of the rollers is the same. This value is 4,000 mm, respectively. The maximum distance between the upper roller and the lower rollers is 100 mm, adjustable based on the metal sheet thickness. The bending process unfolds in two stages: the first step is the bending. The rollers feed on the metal sheet and bend it to a curvature by repositioning the rear roller to 60\u0026ordm; relative to the centers of the front and rear rollers. The second stage is the rolling. The metal sheet is bended throughout the entire metal piece with the front roller rotating continuously to adjust the sheet and achieve the programmed curvature and shape. Notably, the front and rear rollers support solely the sheet and rotate to facilitate sheet passage without exerting pressure for bending. The efficacy of the three-roller bending process heavily relies on the operator's expertise and dexterity, being a predominantly manual process. Plate bending typically follows a multi-pass approach, contingent upon manual adjustments guided by the operator's experience to optimize bending capacity on the rollers. However, this method incurs significant industrial costs due to material wastage and production time. Consequently, this process requires the development of an optimized production method to enhance repeatability, precision and productivity. Furthermore, the operator would acquire comprehensive knowledge of the process to achieve the desired diameter of the pipe component.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eMaterial\u003c/h2\u003e \u003cp\u003eThe chosen mathematical method for modeling sheet bending is the finite element method, implemented through the LS-Dyna finite element program in this study. The finite element model of the forming process comprises a sheet using three rollers. Assumptions include isotropic material properties for the metal sheet and an elastoplastic material model. Implicit/explicit equations are used for computation with gravitational effects factored into the sheet behavior. However, during the simulation, LS-Dyna's algorithm integrates temporally and simulates the model accordingly.\u003c/p\u003e \u003cp\u003eThe rollers, constructed from carbon steel, are treated as rigid bodies. The dimensions of the sheet vary from 10,990 to 23,000 mm (length) \u0026times; 4,000 mm (width) \u0026times; 30 to 100 mm (thickness). Additionally, deformability of the piece is assumed, with the material identified as SA-516 Gr.70N steel. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e provides key properties of this material for the mathematical method.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMechanical properties of material\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrain limit (MPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e805\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eYoung module (GPa)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDensity (kg/mm\u003c/b\u003e\u003csup\u003e\u003cb\u003e3\u003c/b\u003e\u003c/sup\u003e\u003cb\u003e)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7,85E-06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePoisson module\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0,29\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFurthermore, a uniaxial tensile test is conducted to estimate the stress-strain curve, facilitating definition of the material plastic behavior, as is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eFinite element model\u003c/h2\u003e \u003cp\u003eThe challenge encountered in forming processes requires the use of advanced simulation software. When opting for a material model for the sheet, amidst a plethora of options available in LS-Dyna, several considerations must be taken into account: a) suitability for metals/non-metals, b) addressing plasticity, and c) capability to study fracture. Aligning with the aforementioned specifications, the piecewise linear plasticity material model is selected for this study. This model leverages the material's stress-strain curve as input, with the yield stress estimated using Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$ɸ\\left(\\sigma \\right)={\\sigma }_{eq}-{\\sigma }_{y}\\left({\\epsilon }^{p}\\right)=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHowever, in this scenario, the process is independent of the deformation rate. Thus, within the numerical algorithm, the increment of plastic deformation can be readily calculated employing Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\varDelta {\\epsilon }^{p}=\\frac{{\\sigma }_{eq}^{trial}-{\\sigma }_{y}\\left({\\epsilon }_{n}^{p}\\right)}{3G+H}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{eq}^{trial}\\)\u003c/span\u003e\u003c/span\u003e represents the equivalent stress in incremental plastic deformation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{y}\\)\u003c/span\u003e\u003c/span\u003e denotes the yield stress, G signifies the shear modulus of the material (76.8 GPa), and H denotes the hardening modulus, evaluable at the slope where the stress-strain curve alters at the yield point. All parameters are derived from the stress-strain curve obtained from uniaxial tensile tests, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThe piecewise linear plasticity model incorporates a failure criterion, where the equivalent plastic strain at failure and the fracture stress are incorporated as values in the material options of LS-Dyna, determined from the curve obtained from the uniaxial tensile test in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Nonetheless, this criterion proves less pertinent for the current application, as the primary objective is to examine the roundness of the curved pipe with minimal equivalent plastic deformation. Material model parameters are optimized based on the stress-strain curve obtained from tensile tests (refer to Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) and are further validated against FEM simulations of tension before being integrated into the actual simulation of the bending process.\u003c/p\u003e \u003cp\u003eAll loads and constraints are defined with respect to the global coordinate system, while the local coordinate is referenced for the rear lower roller, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe modeling process entails two stages: firstly, the bending of the sheet, simply denoted as \"bending,\" and secondly, the circumferential bending of the sheet, referred to as bending or rolling, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The prescribed movements of the rigid bodies of the rollers are represented by the central point of the roller, where constraints and loads are accordingly applied. Boundary conditions are outlined in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. During the first stage, the upper and lower front rollers are free to rotate about the global X-axis, whereas the rear roller is confined to movement solely along the Z' local coordinate axis at a 60⁰ angle relative to the horizontal. In the second stage, the lower front roller rotates, with a speed of 6.23 m/min (approximately ω\u0026thinsp;=\u0026thinsp;2 rpm/min), while the upper roller maintains its previous configuration. Finally, following the adjustment of the rear lower roller, it is constrained to the Z-axis in the second stage.\u003c/p\u003e \u003cp\u003eThe model is developed utilizing automatic surface-to-surface contact and interactions between various parts of a roller through the LS-Dyna code. Contacts are established employing the master/slave approach, wherein the rollers are considered master surfaces, and the opposing surfaces to the rollers and the sheet contact are designated as slaves. Furthermore, Coulomb's friction law is employed between the sheet and the rollers, with dynamic and static friction coefficients set as 0.25 and 0.5, respectively. Moreover, 3D solid elements are adopted for mesh definition, ensuring the capability to generate new simulations of industrial processes in the future.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003eMetamodel\u003c/h2\u003e \u003cp\u003eThe metamodel will encompass the development of intelligent algorithms predicated on the outcomes derived from the proposed simulations aimed at model enhancement. The strategy hinges upon a supervised regression model, which constitutes the fundamental framework in a metamodeling approach for prognosticating continuous variables. One viable stratagem within response surface methods (RSM) is the Box-Behnken design, widely employed across various engineering domains [18]. The objective entails establishing a correlation between a target output variable (y) and a set of controllable or input variables {x\u003csub\u003e1\u003c/sub\u003e, x\u003csub\u003e2\u003c/sub\u003e, ..., x\u003csub\u003en\u003c/sub\u003e}. If the nature of the association between input and output values is discerned, a model can be formulated in the format delineated by Eq.\u0026nbsp;\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e,\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$y=f({x}_{1},{x}_{1},....{x}_{n})+\\epsilon$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere ε signifies the error inherent to the response (y). If the anticipated response is denoted as E(y)\u0026thinsp;=\u0026thinsp;f(x\u003csub\u003e1\u003c/sub\u003e, x\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;x\u003csub\u003en\u003c/sub\u003e) + η, the resultant surface expressed by Eq.\u0026nbsp;\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e is termed a response surface.\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\eta =f({x}_{1},{x}_{1},....{x}_{n})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTypically, a second-order model, as detailed in Eq.\u0026nbsp;\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, is adopted in response surface methodology, with the coefficients β computed via the least squares method [18].\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$y={\\beta }_{0}+\\sum _{i=1}^{k}{\\beta }_{i}{x}_{i}+\\sum _{i=1}^{k}{\\beta }_{ii}{x}_{i}^{2}+\\sum _{i}\\sum _{j}{\\beta }_{i}{x}_{i}{x}_{j}+\\epsilon$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEmploying response surface methodology facilitates the determination of optimal values for the controllable parameters, thereby enhancing the response optimization or elucidating which parameter values of x engender a product or process aligned with diverse requirements or specifications. The comparison between Finite Element Method (FEM) and Response Surface Methodology (RSM) models via the experimental datasets utilized by machine operators for the bending of large-sized metal sheets has been conducted.\u003c/p\u003e \u003c/div\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eFinite element model\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e illustrates the assembly of the Finite Element Model (FEM) and the sequential stages of the rolling process, as obtained through numerical simulation calculations. Notably, asymmetry in curvature distribution is evident at the point of contact between the rear bottom roller and the metal sheet. This asymmetry arises due to the non-uniform deformation occurring over the region of maximum bending moment.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eGiven the critical nature of CPU time, the numerical model undergoes optimization through mesh sensitivity analysis. This entails assessing the impact of mesh type and element count (mesh size) on metal sheet bending, as well as their influence on CPU time and geometric bending error. The quadrilateral element is chosen for this application due to its common usage in FEM simulations of sheet forming, which results in reduced calculation time compared to triangular elements. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e consolidates all pertinent information outlined earlier.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSensitivity analysis of FEM\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMesh\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eElements\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCPU time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeom error. (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCoarse\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.560\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1,85\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNormal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8.898\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e420\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1,71\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFine\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e17.796\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e830\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1,72\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eConsequently, the sensitivity analysis recommends adopting a standard mesh comprising 8,898 quadrilateral elements, which incurs a CPU time of approximately 420 minutes while achieving minimal geometric bending error.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the distribution of equivalent plastic deformation generated during the forming process, along with the distribution of equivalent plastic deformation across the thickness of the sheet. The simulation employs the GM unit (kg, mm, ms, KN, GPa, KN-mm) as the unit of measurement, with stress expressed in GPa.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eInitially, the equivalent plastic deformation during the metal sheet's initial bending stage amounts to 1.55%. However, this deformation remains at a level of 0.78% throughout the entire bending process. Notably, the maximum plastic deformation occurs under the initial bending condition, indicating localized stress concentration.\u003c/p\u003e \u003cp\u003eFurthermore, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the stress distribution in the bent part of the metal sheet. The minimum residual stress, observed at the neutral fiber, measures 0.040 GPa (40 MPa). However, as the distance from the neutral fiber to the outer surface increases, the residual stress progressively rises, reaching 0.1306 GPa (130.6 MPa) at the sheet's surface. This finding suggests that the model for metal sheet bending and curving aligns with industrial requirements, achieving minimal equivalent plastic deformation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003eMetamodel\u003c/h2\u003e \u003cp\u003eThe CPU times required to obtain simulation results are quite long, as confirmed in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e during mesh optimization. In the simulations conducted for this study, CPU times ranged between 4 hours and 8 hours for each simulation. The variations in CPU times were associated with different dimensions of the sheet metal plate. Sometimes, it was also necessary to repeat the simulation to obtain the correct diameter of the pipe. As explained, an effort was made to minimize the number of simulations due to the lengthy calculation times, so a reduced set of input variables was selected. The selected input variables are shown in the following Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDesign of experiments\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInputs\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e+\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eLenght (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10,990\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e23,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eThickness (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eRoller displacement (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e200\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eOutput (mm)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eObtained diameter\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWith this objective, the Response Surface Methodology (RSM) is employed to find values for controllable parameters that result in optimizing the response or to discover which ones are most important in the process to meet requirements or specifications with a minimum number of experiments.\u003c/p\u003e \u003cp\u003eIn this study, three input variables (sheet length, thickness, and displacement of the rear bottom roller) were included, each considering two levels (minimum and maximum values within their ranges), as shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The output variable or response surface will be the obtained diameter.\u003c/p\u003e \u003cp\u003eA design with four central points was implemented for each block. Thus, a Box-Behnken design consisting of 13 experiments was obtained, combining different values for the variables. Since this type of approach is generally used for designing experiments, some additional tests were conducted to verify the experiment's variability under the same conditions. Therefore, randomly, some extra tests are included within the defined range for each of them, resulting in new combinations of 15 experiments.\u003c/p\u003e \u003cp\u003eThe results obtained from the Box-Behnken RSM model are analyzed in terms of the correlation between different inputs and outputs. The response surface associated with the obtained diameter is modeled in the reduced centered space. The equation is derived using the DOE (Design-Expert 13) software.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$d\\_o=269.903 + 0.973253 \\text{*} l -0.115865 \\text{*} e -0.798903 \\text{*} d$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere d_o is the diameter obtained in the simulation, l is the length of the sheet, e is the thickness, and d is the displacement distance of the rear bottom roller. It can be observed that there is a linear relationship between the output and the inputs. In the following lines, the effect of the input variables on different responses is analyzed in detail. Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e presents the study applying the response surface methodology.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBox-Behken RSM Model for intelligent rolling\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSource\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSum of Square\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003edf\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMean Square\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eF-value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eModel\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.490E\u0026thinsp;+\u0026thinsp;07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.830E\u0026thinsp;+\u0026thinsp;07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6363.51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.0001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003esignificant\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLenght\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.610E\u0026thinsp;+\u0026thinsp;06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.610E\u0026thinsp;+\u0026thinsp;06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e559.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.0001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003esignificant\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThickness\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e102.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e102.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0355\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.8520\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRoller displacement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e223.76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e223.76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0778\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.8826\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eResiduals\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e71892.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2875.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLack of fit\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e71892.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4493.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eError pure\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCor Total\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.497E\u0026thinsp;+\u0026thinsp;07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe model's F-value reaching 6,363.51 implies that the model is significant. There is only a 0.01% probability of obtaining such a large F-value due to noise.\u003c/p\u003e \u003cp\u003eP-values below 0.0500 indicate that the model terms are significant. In this case, the sheet length (l) is a significant model term. P-values above 0.1000 indicate that model terms are not significant. If there are many insignificant model terms (excluding those necessary to support hierarchy), model reduction can enhance the model.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e represents the response surface of the diameter obtained as a function of the two most significant parameters (length and thickness of the sheets).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe predicted R\u0026sup2; value of 0.9987 indicates a reasonable fit for all sheet lengths and thicknesses. The obtained diameter shows little deviation; however, this parameter also depends on the displacement of the rear bottom roller, even though the RSM has classified it as the third least significant.\u003c/p\u003e \u003cp\u003eTo observe the simulated geometric error, Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e compiles the extreme points of the designed experiments that have been tested. It shows the values of the obtained diameter after simulating the conditions of that point, along with the geometric bending error.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGeometric bending errors and diameters obtained from the RSM\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLength (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eThickness (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDisplacement (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eObtained diameter (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eError (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e23,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e132\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.341\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0,27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10,995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3560\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.71\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e23,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e118\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.298\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.31\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10,995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e175\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.537\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.05\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThese geometric errors do not exceed 2%, and therefore, they meet the industrial specifications required in manufacturing. To clarify the curvature error calculation, Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e illustrates the estimated diameter obtained in cases with a sheet length of 10,995 mm and sheet thicknesses of 30 mm and 10 mm, respectively, as indicated in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFinally, the average geometric bending error for the fifteen simulations based on the RSM metamodel is calculated. The values for this mean error and its standard deviation are 0.97% and 0.43, respectively. Therefore, these results are consistent with the manufacturing tolerances of the industrial user and enable them to have an intelligent optimizer for the process variables in the bending and curving of metal sheets for pipe manufacturing.\u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusions","content":"\u003cp\u003eA sophisticated finite element model has been developed with the capability to guide operators on optimal parameter ranges to minimize geometric errors when bending large sheets for pipes within the oil and gas sector. Several conclusions derived from this study include:\u003c/p\u003e\n\u003cp\u003e- The finite element model facilitates the prediction of tube dimensions utilizing input parameters from the intelligent model.\u003c/p\u003e\n\u003cp\u003e- The FEM model for sheet bending and curving fulfills industrial requirements by achieving curvature with minimal equivalent plastic deformation. The Von Mises stress at the neutral fiber is 40 MPa, and the distribution of equivalent plastic deformation and Von Mises stress exhibits symmetry across the sheet thickness, with a maximum value of 130.6 MPa.\u003c/p\u003e\n\u003cp\u003e- The intelligent model functions as a metamodel based on the Box-Behnken Response Surface Methodology (RSM), minimizing simulations while generating a highly dependent response surface, particularly on sheet length. The goodness of fit (R\u0026sup2;) is 0.9987.\u003c/p\u003e\n\u003cp\u003e- The average geometric bending error and its dispersion stand at 0.97% and 0.43, respectively, aligning with the manufacturing tolerances of industrial users.\u003c/p\u003e\n\u003cp\u003e- A smart advisor for bending and curving process variables has been developed to enable autonomous manufacturing of pipes.\u003c/p\u003e\n"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe research is partially supported\u0026nbsp;the European Union\u0026apos;s Horizon 2020 research and innovation program under grant agreement no. 958303.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConceptualization, Alain Gil Del Val.; Data curation, F.V., Alain Gil Del Val.; Formal analysis, Fernando., Alain Gil Del Val.; Investigation, Alain Gil Del Val, Mariluz.; Methodology, Alain Gil Del Val and Mariluz.; Project administration, Alain Gil Del Val.; Supervision, Bilal.; Validation, Alain Gil Del Val. and Fernando; Writing\u0026mdash;original draft, Fernando. and Alain Gil Del Val.; Writing\u0026ndash;review and editing, Mariluz Penalva, Alain Gil Del Val., Bilal., and Fernando. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eBatalov GS, Lunev AA, Radionova LV, Lezin VD (2021) Development of New Methods for the Production of Large-Diameter Double-Seam Pipes. In Solid State Phenomena; Trans Tech Publications Ltd Switzerland 316:538\u0026ndash;548.\u003c/li\u003e\n\u003cli\u003eBreiman L (2001) Random forests. Mach. Learn 45:5\u0026ndash;32.\u003c/li\u003e\n\u003cli\u003eVolk W, Groche P, Brosius, A.; Ghiotti, A.; Kinsey, B.L.; Liewald, M.; Madej, L.; Min, J.; Yanagimoto, J. (2019) Models and modelling for process limits in metal forming. CIRP Ann. 68:775\u0026ndash;798.\u003c/li\u003e\n\u003cli\u003eParalikas J, Salonitis K, Chryssolouris G (2011) Investigation of the effect of roll forming pass design on main redundant deformations on profiles from AHSS. Int. J. Adv. Manuf. Technol. 56:475\u0026ndash;491.\u003c/li\u003e\n\u003cli\u003eYang M, Shima S (1988) Simulation of pyramid type three-roll bending process. Int. J. Mech. Sci. 30:877\u0026ndash;886.\u003c/li\u003e\n\u003cli\u003eFan ST, Meng SF (2007) Simulation of the three-roll pyramid type plate bending machine: Multi-bending process. Mach. Des. Manuf. 6:82\u0026ndash;84.\u003c/li\u003e\n\u003cli\u003eFeng Z, Champliaud H (2011) Modeling and simulation of asymmetrical three-roll bending process. Simul. Model. Pract. Theory 19:1913\u0026ndash;1917.\u003c/li\u003e\n\u003cli\u003eShin JG, Park TJ, Yim H (2001) Kinematics based determination of the rolling region in roll bending for smoothly curved plates. J. Manuf. Sci. Eng. 123:284\u0026ndash;290.\u003c/li\u003e\n\u003cli\u003eGandhi AH, Raval HK (2008) Analytical and Empirical Modeling of Top Roller Position for Three-Roller Cylindrical Bending of Plates and Its Experimental Verification. J. Mater. Process. Technol. 197:268\u0026ndash;278.\u003c/li\u003e\n\u003cli\u003eWang B, Tao F, Fang X, Liu C, Liu Y, Freiheit T (2021) Smart manufacturing and intelligent manufacturing: A comparative review. Engineering 7:738\u0026ndash;757.\u003c/li\u003e\n\u003cli\u003eStarman B, Cafuta G, Mole N (2021) A Method for Simultaneous Optimization of Blank Shape and Forming Tool Geometry in Sheet Metal Forming Simulations. Metals. 11:544.\u003c/li\u003e\n\u003cli\u003eRalph B, Stockinger M (2020) Digitalization and digital transformation in metal forming: Key technologies, challenges and current developments of industry 4.0 applications. In XXXIX Colloquium on Metal Forming Montanuniversit\u0026auml;t Leoben, Lehrstuhl f\u0026uuml;r Umformtechnik: Leoben, Austria, 13\u0026ndash;23.\u003c/li\u003e\n\u003cli\u003eTekkaya AE, Allwood JM, Bariani PF, Bruschi S, Cao J, Gramlich S, Groche P, Hirt G, Ishikawa T, Lobbe C, Lueg-Althoff J, Merklein M, Misiolek WZ, Pietrzyk M, Shivpuri R, Yanagimoto (2015) J. Metal forming beyond shaping: Predicting and setting product properties. CIRP Annals 64(2):629-653.\u003c/li\u003e\n\u003cli\u003eVolk W, Groche P, Brosius A, Ghiotti A, Kinsey BL, Liewald M, Madej L, Min J, Yanagimoto J (2019) Models and modelling for process limits in metal forming. CIRP Annals 68(2):775-798.\u003c/li\u003e\n\u003cli\u003eSmith M (1993) Neural Networks for Statistical Modeling. Thomson Learning: Boston, MA, USA.\u003c/li\u003e\n\u003cli\u003eWilliams CK, Rasmussen CE (2006) Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, Volume 2.\u003c/li\u003e\n\u003cli\u003eOrzechowski P; La Cava W, Moore JH (2018) Where are we now?: A large benchmark study of recent symbolic regression methods. In Proceedings of the Genetic and Evolutionary Computation Conference, Kyoto, Japan, 15\u0026ndash;19 July 2018; 1183\u0026ndash;1190.\u003c/li\u003e\n\u003cli\u003eMontgomery DC (2017) Design and analysis of experiments. 9th Edition. John Wiley \u0026amp; Sons, Inc.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"bending, rolling, finite element method, response surface method","lastPublishedDoi":"10.21203/rs.3.rs-4551326/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4551326/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMinimizing geometric error in the bending of large sheets remains a challenging endeavor in the industrial environment. This specific industrial operation is characterized by protracted cycles and limited batch sizes. Coupled with extended cycle times, the process involves a diverse range of dimensions and materials. Given these operational complexities, conducting practical experimentation for data extraction and control of industrial process parameters proves to be unfeasible.\u003c/p\u003e \u003cp\u003eTo gain insights into the process, finite element models serve as invaluable tools for simulating industrial processes for reducing experimental cost. Consequently, the primary objective of this research endeavor is to develop an intelligent finite element model capable of providing operators with pertinent information regarding the optimal range of key parameters to mitigate geometric error in the bending of large sheets. The average geometric error in curvature is recorded at 0.97%, thereby meeting the stringent industrial requirement for achieving such bending with minimal equivalent plastic deformation. As such, these findings present promising prospects for the automation of the industrial process.\u003c/p\u003e","manuscriptTitle":"Finite Element Method for minimizing geometric error in the bending of large sheets","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-07-04 09:21:12","doi":"10.21203/rs.3.rs-4551326/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2024-08-22T10:52:57+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2024-06-26T12:45:06+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-06-18T16:48:21+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-06-13T11:30:40+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2024-06-11T09:28:55+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"b3dde752-fec3-4338-9970-9675c4e9a53c","owner":[],"postedDate":"July 4th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-11-04T16:28:14+00:00","versionOfRecord":{"articleIdentity":"rs-4551326","link":"https://doi.org/10.1007/s00170-024-14685-3","journal":{"identity":"the-international-journal-of-advanced-manufacturing-technology","isVorOnly":false,"title":"The International Journal of Advanced Manufacturing Technology"},"publishedOn":"2024-10-31 16:20:31","publishedOnDateReadable":"October 31st, 2024"},"versionCreatedAt":"2024-07-04 09:21:12","video":"","vorDoi":"10.1007/s00170-024-14685-3","vorDoiUrl":"https://doi.org/10.1007/s00170-024-14685-3","workflowStages":[]},"version":"v1","identity":"rs-4551326","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4551326","identity":"rs-4551326","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}