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Research Papers

# Hierarchical Metamodeling of the Air Bending Process

[+] Author and Article Information
Matteo Strano

Dipartimento di Meccanica,
Politecnico di Milano,
Via La Masa 1,
Milan 20156, Italy
e-mail: matteo.strano@polimi.it

Quirico Semeraro

Dipartimento di Meccanica,
Politecnico di Milano,
Via La Masa 1,
Milan 20156, Italy
e-mail: quirico.semeraro@polimi.it

Lorenzo Iorio

MUSP Lab,
Str. della Torre della Razza,
Piacenza 29122, Italy
e-mail: lorenzo.iorio@polimi.it

Roberto Sofia

Via Giovanni Agnelli, 15,
Santena 10026, Turin, Italy

1Corresponding author.

Manuscript received June 14, 2017; final manuscript received April 11, 2018; published online May 21, 2018. Assoc. Editor: Yannis Korkolis.

J. Manuf. Sci. Eng 140(7), 071018 (May 21, 2018) (10 pages) Paper No: MANU-17-1373; doi: 10.1115/1.4040025 History: Received June 14, 2017; Revised April 11, 2018

## Abstract

Despite the tremendous effort of researchers and manufacturing engineers in improving the predictability of the air bending process, there is still a strong need for comprehensive and dependable prediction models. Currently, available modeling approaches all present some relevant limitations in practical applications. In this paper, we propose a new method, which represents an improvement over all existing modeling and prediction techniques. The proposed method can be used for accurate prediction of the main response variables of the air bending process: the angle α after springback and the bend deduction BD. The metamodeling method is based on the hierarchical fusion of different kinds of data: the deterministic low-fidelity response of numerical finite element method (FEM) simulations and the stochastic high fidelity response of experimental tests. The metamodel has been built over a very large database, unprecedented in the scientific literature on air bending, made of more than 500 numerical simulations and nearly 300 experimental tests. The fusion is achieved first by interpolating the FEM simulations with a kriging predictor; then, the hierarchical metamodel is built as a linear regression model of the experimental data, using the kriging predictor among the regressors. The accuracy of the method has been proved using a variant of the leave-one-out cross validation technique. The quality of the prediction yielded by the proposed method significantly over-performs the current prediction of the press brake on-line numerical control.

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## References

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## Figures

Fig. 3

Scheme of the simulation setup (a) and boundary conditions of the springback stage (b)

Fig. 2

Logical flowchart of the proposed method for a generic response variable y

Fig. 1

Geometrical process parameters of air bending; the bend deduction BD is here defined under the assumption of a symmetric process and for a final angle α ≤ 90 deg

Fig. 4

Definition of the flange length from the standard DIN-6935 [12]; the initial sheet length is L, as defined in Fig. 1

Fig. 5

Two couples of variables tested in the FEM plan of simulations

Fig. 6

Plot of simulated BDFEM versus αFEM; data are grouped by levels of sheet thickness t0

Fig. 7

(a) Experimental gauge block system and (b) scheme of the flange length measurement

Fig. 8

(a) Photograph of specimen 51 and (b) results of the threshold Huang algorithm

Fig. 9

Absolute prediction errors εαNC versus εBDNC of the press brake; data are grouped by levels of target angle αNC

Fig. 10

Absolute prediction errors εαh versus εBDh of the hierarchical metamodels; data are grouped by levels of target angle αNC

Fig. 11

Boxplots of the absolute error differences for the angle (left) and the bend deduction (right); the hierarchical metamodels calculated over reduced data sets still overperform the NC prediction

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