# Robust Optimization and Kriging Metamodeling of Deep-Drawing Process to Obtain a Regulation Curve of Blank Holder Force

^{*}

## Abstract

**:**

## 1. Introduction

- Wrinkles are generally caused by insufficient force on blank holder.
- Cracks occur when the applied load exceeds the maximum resistance of the material.
- Spring-back is due to the deformations of the component in the elastic–plastic field. Spring-back is a critical aspect of drawing processes especially in the automotive industry, where high-dimensional accuracy is often required. Many parameters influence this phenomenon such as Young modulus, yield stress, punch radius and sheet thickness.

## 2. Materials and Methods

_{0}), tensile strength (Rm) and strain hardening exponent (n) are highlighted.

_{m}is the average of plastic strain ratio at 0°, 45° and 90° of rolling direction; r

_{b}is plastic strain ratio at biaxial stress, which is defined as the ratio of strains ε

_{2}and ε

_{1}; σ

_{b}/σ

_{0}is the ratio between onset of yielding at equi-biaxial stress and yield stress; σ

_{ps0}/σ

_{0}is the ratio between plane strain stress at 0° of rolling direction and yield stress; σ

_{ps90}/σ

_{0}is the ratio between plane strain stress at 90° of rolling direction and yield stress; and σ

_{shear}/σ

_{0}is the ratio between shear stress and yield stress.

_{1}and ε

_{2}measured at the onset of material failure.

^{2}covered by the design sites. After kriging meta-modeling phase, there was the multi-objective optimization phase. The approach used for optimization is that of desirability. With the optimization phase, the combination of input parameters (force on the blank holder, friction coefficient and yield stress) which guarantees a stamped component without defects (wrinkles, thinning, thickening and breakage) was identified. (4) Considering combination of input parameters that give high desirability, force regulation curves on the blank holder were obtained as a function of the yield stress of the material for three different values of friction coefficient. Finally, by comparing an optimized solution with a non-optimized one, the draw-in of metal sheet was evaluated, and it was observed that the sheet has different sliding in the two conditions.

## 3. Results

#### 3.1. Design of Stamping Process Using Finite Element Model (FEM)

#### 3.2. Robust Analysis

_{k}), which indicates controllability of the process around the defined specification limits. This index is calculated as:

- $c{p}_{k}<0.67$: The process is not acceptable, as more than 2.25% of the results do not meet the specification limits.
- $c{p}_{k}\in \left[0.67:1\right)$: The process is unreliable, as 0.14–2.25% of results do not meet the specification limits.
- $c{p}_{k}\in \left[1:1.33\right):$ The process may be acceptable because the results fall within limit imposed. However, a check is required.
- $c{p}_{k}\ge 1.33$: The process is reliable, as less than 0.004% of the results exceed the limits.

#### 3.3. Metamodeling with Kriging Methodology

#### 3.4. Multi-Objective Optimization and Regulation Curve

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Hardening curve; (

**b**) yield surface with BBC criterion; and (

**c**) Formability Limit Curve (FLC).

**Figure 8.**(

**a**) Upper $c{p}_{k}$ for potential wrinkles after drawing phase; and (

**b**) upper $c{p}_{k}$ for potential wrinkles at the end of the process.

**Figure 9.**Metamodel of the percentage of thickened zone as friction coefficient and yield stress vary.

**Figure 10.**Metamodel of the percentage of area with insufficient stretching as friction coefficient and yield stress vary.

**Figure 12.**Metamodel of the percentage of area with potential splits as friction coefficient and yield stress vary.

**Figure 13.**(

**a**) Metamodel of the percentage of thinning at Critical Point A as friction coefficient and yield stress vary; and (

**b**) metamodel of the percentage of thinning at Critical Point B as friction coefficient and yield stress vary.

**Figure 15.**Draw-in as a function of the punch stroke at Points D, F and I of the sheet in the optimized condition (safe) and in the non-optimal condition (cracks).

C Max | Si Max | Mn Max | P Max | S Max | Al | Ti+Nb Max | Cr+Mo Max | B Max | Cu Max |
---|---|---|---|---|---|---|---|---|---|

0.18 | 0.50 | 2 | 0.05 | 0.010 | 0.015–2 | 0.15 | 1 | 0.01 | 0.2 |

Symbol | Definition |
---|---|

${Y}_{i}\left(x\right)$ | ith response as a function of x parameter |

${d}_{i}\left({Y}_{i}\right)$ | ith desirability function correlated to the response parameter $\left({Y}_{i}\right)$ |

${L}_{i}$ | Minimum ith value |

${U}_{i}$ | Maximum ith value |

${T}_{i}$ | Target ith value |

s | Exponent that defines the shape of the function (s = 0.1 convex function) |

D | Total desirability |

k | Number of responses |

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**MDPI and ACS Style**

Palmieri, M.E.; Lorusso, V.D.; Tricarico, L.
Robust Optimization and Kriging Metamodeling of Deep-Drawing Process to Obtain a Regulation Curve of Blank Holder Force. *Metals* **2021**, *11*, 319.
https://doi.org/10.3390/met11020319

**AMA Style**

Palmieri ME, Lorusso VD, Tricarico L.
Robust Optimization and Kriging Metamodeling of Deep-Drawing Process to Obtain a Regulation Curve of Blank Holder Force. *Metals*. 2021; 11(2):319.
https://doi.org/10.3390/met11020319

**Chicago/Turabian Style**

Palmieri, Maria Emanuela, Vincenzo Domenico Lorusso, and Luigi Tricarico.
2021. "Robust Optimization and Kriging Metamodeling of Deep-Drawing Process to Obtain a Regulation Curve of Blank Holder Force" *Metals* 11, no. 2: 319.
https://doi.org/10.3390/met11020319