Optimization of multilayer perceptron neural network structure for simulating the effect of input variables on the spring-back phenomenon in the ultrasonic vibration assisted single point incremental forming

Applying the ultrasonic vibration to the forming tool in single-point incremental forming reduces the forming force, increases the sheet formability, and reduces the spring-back. In the present research, with the aim of minimizing the sheet spring-back, the optimal structure of the multilayer perceptron neural network was extracted using three algorithms which are genetic algorithm, imperialistic competition algorithm, and equilibrium optimizer. Analyzing the optimal network with an R-value of 0.99973 and a root mean squared error of 0.0084 shows that the optimized network performs excellently in simulating the considered system. Then, the best network was used to optimize the variables affecting the objective functions. These objective functions include the average of measured depth (Have) and the spring-back coefficient (K). The input variables are: vertical step size, sheet thickness, tool diameter, wall inclination angle, and tool feed rate. The results showed that the optimized multilayer perceptron network can simulate the process with very good precision. Also, the extraction of optimal values shows that the maximum of Have and the minimum of K can be achieved with very good accuracy. Finally, the comparison of the three algorithms showed that the performance of the equilibrium optimizer was better in optimizing the neural network structure. On the other hand, in the optimizing process of the input variables, the imperialistic competition algorithm has been more efficient.