## 3 The Establishment and Analysis of BBD Experimental Design and Prediction Model
### BBD Experimental Design and Grey Relation Analysis
#### 3.1.1 Experimental Design  
Considering the high simulation time cost and the strict requirements for the safety zone of design variables, this paper adopts the BBD method for its experimental design. BBD is a commonly used design method for response surface optimization. In this paper, the friction factor (F), blank initial temperature (T), and velocity of downward pressure (V) are taken as the design parameters, and the forming load (Y), equivalent stress (S), and damage value (Z) are taken as response variables.  
The friction coefficient plays a critical role in uniform deformation inside forgings. Considering the guidelines from Deform and practical expertise within the company, a friction coefficient range between 0.3 and 0.7 is chosen. The initial temperature of the blank is also important for the quality of the forgings. Increasing the appropriate temperature helps with grain refinement and boosting forging quality. However, excessively high temperatures may cause the forgings to overheat or burn, while excessively low temperatures increase the deformation resistance of the metal, affecting deformation efficiency and reducing the quality of the forgings. After consulting the forging manual, a temperature range from 1150 \({}^{\circ}\)C to 1250 \({}^{\circ}\)C is selected. The forging speed is another key factor in the plastic deformation of metal. Increasing the forging speed can enhance production efficiency, but too fast a downward pressing speed will increase the friction between the mold and the blank, easily leading to surface cracks. Too slow a downward pressing speed will increase heat loss between the forgings and the mold, impeding metal flow. According to the actual values of the equipment, a speed range from 7.5 mm/s to 12.5 mm/s is selected.  
A three-factor and three-level experiment was designed based on the BBD principle. The experimental factor level table is shown in Table 2. Numerical simulations were conducted for each group of experiments using Deform 10.1. This experimental design includes 15 experimental points, with three sets of repeated experiments for error assessment. Table 3 presents the experimental design and the resultant findings.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Level** & **F** & **T (\({}^{\circ}\)C)** & **V (mm/s)** \\ \hline \(-1\) & 0.3 & 1150 & 7.5 \\ \(0\) & 0.5 & 1200 & 10 \\ \(1\) & 0.7 & 1250 & 12.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Factor level table.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Trial No.** & **F** & **T (\({}^{\circ}\)C)** & **V (mm/s)** & **Y (MN)** & **S (MPa)** & **Z** \\ \hline \(1\) & 0.3 & 1150 & 10 & 746 & 54.5 & 0.594 \\ \(2\) & 0.7 & 1150 & 10 & 1100 & 63 & 0.624 \\ \(3\) & 0.3 & 1250 & 10 & 509 & 39.3 & 0.592 \\ \(4\) & 0.7 & 1250 & 10 & 784 & 43.3 & 0.56 \\ \(5\) & 0.3 & 1200 & 7.5 & 447 & 32.4 & 0.748 \\ \(6\) & 0.7 & 1200 & 7.5 & 530 & 51.2 & 0.587 \\ \(7\) & 0.3 & 1200 & 12.5 & 720 & 37.7 & 0.526 \\ \(8\) & 0.7 & 1200 & 12.5 & 976 & 40.5 & 0.818 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experimental design and results.