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A sustainable hybrid FEA–AI optimization framework for multistage deep drawing of unidirectionally rolled copper micro-cups
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  • Published: 26 March 2026

A sustainable hybrid FEA–AI optimization framework for multistage deep drawing of unidirectionally rolled copper micro-cups

  • S. P. Sundar Singh Sivam1,
  • Stalin Kesavan2 &
  • A. Johnson Santhosh3 

Scientific Reports , Article number:  (2026) Cite this article

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We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply.

Subjects

  • Engineering
  • Materials science
  • Mathematics and computing

Abstract

Precision micro-cups for electronic and medical applications require stringent control of resultant tool force (RTF) and spring-back (SB) to prevent defects, excessive tool loading, and material waste. However, conventional micro deep-drawing processes remain reliant on costly trial-and-error experimentation and exhibit limited predictive reliability under severe plastic deformation and rolling-induced anisotropy. To address this challenge, a robust and computationally reproducible optimization framework is developed for sustainable, high-volume manufacturing. Multistage deep drawing of unidirectionally rolled copper strips subjected to 200% thickness reduction (true strain − 2.78) is investigated using validated finite element modeling (FEM), with clearance (C), punch nose radius (PNR), and coefficient of friction (µ) systematically varied through a Central Composite Design. The resulting FEM-generated dataset is used to train predictive models based on Response Surface Methodology (RSM), Levenberg–Marquardt artificial neural networks (LM-ANN), and Bayesian Regularized artificial neural networks (BR-ANN). Experimental trials are conducted independently for model validation and performance assessment, ensuring that the ANN models are FEM-trained and experimentally validated rather than experimentally trained. These predictive models are integrated with a Genetic Algorithm (GA) for global optimization of RTF and SB. Model performance is evaluated using R², RMSE, MAPE, MSRE, and Nash–Sutcliffe Efficiency (NSE) for both seen (simulation-based) and unseen (experimental) datasets. The BR-ANN–GA framework demonstrates superior predictive accuracy and robustness, identifying optimal parameters of C = 0.57, PNR = 2, and µ = 0.10, minimizing RTF and SB. Prediction errors are limited to 0.35% (RTF) and 0.41% (SB) for FEM-trained data, while experimental validation trials maintain low errors of 1.7% and 2.68%, respectively, outperforming FEM, RSM, and LM-ANN approaches. Although the study is constrained to axisymmetric geometries and unidirectionally rolled copper, the proposed FEM-trained and experimentally validated BR-ANN–GA framework significantly enhances predictive reliability while reducing energy consumption, material waste, and experimental cost, offering a scalable optimization methodology for precision micro deep drawing.

Data availability

All relevant data generated and analyzed during this study are included in the manuscript.

Abbreviations

PNR or R:

Punch nose radius

COF or µ:

Coefficient of friction

LDR:

Limiting drawing ratio

ID:

Inner diameter

OD:

Outer diameter

FEA:

Finite element analysis

FEM:

Finite element method

ANN:

Artificial neural network

GA:

Genetic Algorithm

RSM:

Response surface methodology

CCD:

Central composite design

BR-ANN:

Bayesian Regularized Artificial Neural Network

LM-ANN:

Levenberg–Marquardt Artificial Neural Network

CAE:

Computer-aided engineering

ANOVA:

Analysis of variance

R2 :

Coefficient of determination

RMSE:

Root mean square error

MAPE:

Mean absolute percentage error

MSRE:

Mean squared relative error

NSE:

Nash–Sutcliffe efficiency

MSE:

Mean squared error

ETP:

Electrolytic Tough Pitch (Copper)

UDR:

Unidirectionally rolled

UTS:

Ultimate tensile strength

EBSD:

Electron backscatter diffraction

KAM:

Kernel average misorientation

SEM:

Scanning electron microscopy

DSC:

Differential scanning calorimetry

RD:

Rolling direction

ND:

Normal direction

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Funding

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, 603203, Tamil Nadu, India

    S. P. Sundar Singh Sivam

  2. Department of Marine Engineering, Amet University, East Coast Road, Kanathur, Chennai, 603 112, India

    Stalin Kesavan

  3. Faculty of Mechanical Engineering, Jimma Institute of Technology, Jimma University, Jimma, Ethiopia

    A. Johnson Santhosh

Authors
  1. S. P. Sundar Singh Sivam
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  2. Stalin Kesavan
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  3. A. Johnson Santhosh
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Contributions

**S. P. Sundar Singh Sivam: ** Conceptualisation, Methodology, Resources, Software, Supervision, and Writing ‐ original draft.**Stalin Kesavan: ** Data curation, Formal analysis, Investigation, Visualisation, and Writing – review & editing.**Johnson Santhosh: ** Project administration, Validation, and Writing – review & editing.

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Correspondence to A. Johnson Santhosh.

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Sivam, S.P.S.S., Kesavan, S. & Santhosh, A.J. A sustainable hybrid FEA–AI optimization framework for multistage deep drawing of unidirectionally rolled copper micro-cups. Sci Rep (2026). https://doi.org/10.1038/s41598-026-45011-4

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  • Received: 30 December 2025

  • Accepted: 16 March 2026

  • Published: 26 March 2026

  • DOI: https://doi.org/10.1038/s41598-026-45011-4

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Keywords

  • Micro deep drawing
  • Resultant tool force
  • Spring-back
  • Bayesian regularized ANN
  • Genetic algorithm
  • Process optimization
  • Sustainable manufacturing
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