Numerical multi-objective optimization of segmented and variable blank holder force trajectories in deep drawing based on DNN-GA-MCS strategy

Abstract

In sheet metal forming, persistent challenges such as failure, wrinkling, and springback necessitate innovative solutions. This study elucidates a novel methodology tailored to mitigate these prevalent defects. Our primary aim was to bolster formability while concurrently diminishing defects throughout the forming cycle. To realize this objective, we deployed a segmented and variable blank holder force (S-VBHF) trajectory, facilitating precise adjustments to the blank holder force (BHF). In our quest to optimize process parameters, encompassing the S-VBHF, friction coefficient, and drawbead restraining force (DBRF), we amalgamated deep neural network, genetic algorithm, and Monte Carlo simulation techniques, collectively denoted as DNN-GA-MCS. The forming limit diagram (FLD) served as our rigorous evaluative framework, enabling a comprehensive assessment of sheet failure dynamics during the forming phase. Our proposed methodology underwent stringent validation via numerical simulations, with the cylindrical cup from NUMISHEET 2011 (BM1) providing the benchmark. Empirical outcomes underscored substantial enhancements in formed sheet quality: marked reductions of 8.33% in failure, 10.81% in wrinkling, and 5.88% in springback. In summation, our advanced methodology manifests potential in refining sheet metal forming processes, emphasizing its efficacy in substantially curtailing defects.

Appendices

Appendix 1. Deep neural network (DNN) modeling

Figure 16 illustrates the DNN model design that we used in this study. The input layer has three neurons including VBHF, trajectory, DBRF and the output layer has three neurons with failure, wrinkling and springback. We chose a DNN architecture with five hidden layers containing 128, 256, 512, 256, and 128 neurons, respectively, based on previous research that suggested deeper and wider networks could improve model performance by increasing the capacity for learning complex patterns and relationships [60, 61, 62]. This architecture has also been shown to be effective for non-linear regression tasks, as it allows for the modeling of non-linear relationships between the input and output variables [63, 64].

The rectified linear unit (ReLU) activation functions were employed in the hidden layers of our DNN model to introduce nonlinearity and enable better feature representation and extraction [65, 66]. The DNN was implemented using keras on top of TensorFlow.

Fig. 16

Schematic illustration of the proposed DNN

To assess the accuracy of the models, two metrics have been adopted: mean absolute error (MAE) metric and mean squared error (MSE) metric were used. MAE metric was employed to assess the performance quality of an approximative model by represented the average magnitude of the residuals in datasets. MSE metric evaluated the variance of the residuals and was used to assess the overall quality of the approximate model. While MSE metric evaluated the average of the squared discrepancies among the original values and their corresponding approximations in the datasets. The values of the terms mentioned were computed using the following mathematical formulas.

$$MSE=\frac{1}{n}\sum_{i=1}^{n}{\left({Y}_{i}-{\widehat{Y}}_{i}\right)}^{2}$$

(9)

$$MAE=\frac{1}{n}\sum_{i=1}^{n}\left|{Y}_{i}-{\widehat{Y}}_{i}\right|$$

(10)

where, Ŷ_i is approximated value of Y, Y_i is actual value of Y, n denote total number of samples.

Appendix 2. Design of experiment

This part describes the processing parameter design. To determine the effect of S-VBHF, punch stroke trajectory, DBRF and μ_s, mixed level and four factors simulation sets design experiments were implemented on deep drawing process. Four factors of the processing parameter determination are shown in the Fig. 7, the level of S-VBHF, punch stroke trajectory, DBRF and μ_s is selected as 3, 2, 6, and 3. The whole stroke was partitioned into three sub-stroke sections, thus the trajectory case could be separated as arranged as 6 cases where L, M, and H means low level, middle level, and high level in the three sub-stroke sections in Fig. 17a. Considering the configuration of the stroke trajectory that which level should be longer in whole stroke, reference the literature mentioned in the cylindrical cup cases, only two sub-strokes of 10 mm and one 20 mm sub-stroke were used. S-VBHF amplitude configuration was partitioned from 1 to 5kN as shown in Fig. 17c, because of space limitations, three magnitude of the BHF only considering trajectory case L-M-H is shown (Fig. 18).

Fig. 17

Design variables of the VBHF with (a) variable trajectory distribution of three level as low, middle and high, (b) variable trajectory stage of proportion 1:1:1,2:1:1 and 1:1:2, (c) variable force magnitude of 1kN-2kN-3kN level,2kN-3kN-4kN level and 3kN-4kN-5kN level

Fig. 18

Plots of level average values of S/N ratio with full factorial design

Appendix 3. Nondominated sorting genetic algorithm-II (NSGA-II)

NSGA-II algorithm was implemented to tackle the challenge of determining optimal process parameters in metal forming processes in this study. The NSGA-II algorithm adheres to a general framework by a genetic algorithm, incorporating a survival selection process that has been modified to suit the optimization problem. The process involves the classification of individuals based on their non-domination status, which reflects the trade-off relationship among the individuals with respect to the objectives. Subsequently, individuals are selected by their crowding distances, which quantify the diversity of solutions in the population. Each individual in the population consists of design variables, including the blank holder force (VBHF), drawbead restraining force (DBRF), friction coefficient (μ_s) and VBHF trajectory. The evaluation of the individuals was performed in terms of three defects, namely failure, wrinkling and springback.The NSGA-II was implemented using keras on top of TensorFlow (Fig. 19).

Fig. 19

Workflow of NSGA-II algorithm

Appendix 4. Monte Carlo simulation (MCS) method

Figure 20 present the probability density curves described by statistically normal distributions of process parameters (a) VBHF, (b) trajectory, (c) DBRF, and (d) μ_s. The MCS method was implemented using a model in MATLAB.

Fig. 20

Normal distribution of process parameters with (a) VBHF, b trajectory, c friction coefficient μ_s, and (d) DBRF