Development of an optimization algorithm in deep drawing of the double bowl cups with segmented blank holder

The deep drawing of square and rectangular cups is more difficult than that of cylindrical cups. Material flows at the straight and the corner sides of the blank in deep drawing of rectangular cups are completely different from each other. At the straight sides tensile stresses develop along the direction of material flow, whereas at the corner sides compressive stresses in the circumferential direction and tensile stresses in the radial direction develop during forming. Complex stress state constraint the material flow at the corner sides during the process. This restraint in the material flow induces the concentration of deformation on the corners and fracture of the cup from these regions. Application of different blank holder pressures (BHP) in the different regions of the blank is required to avoid the difference between material flows of the straight sides and corners. Thus, material flow in each region of the blank could be controlled separately and could be prevented tearing with the ease of material flow in corner regions. Determination of the BHP in each piston and programming of hydraulic press are required to apply variable BHP in a multipoint control hydraulic press. BHP are determined by trial and error in today's industry. However, determination of BHP by trial and error is difficult and costly because of the large number of variables involved in the multipoint control system. Before the manufacturing, performing of multiple trials until a defect-free part is obtained lead to increase in the scrap rate and expenditure. Therefore, development of a method for the estimation of BHP in each piston in multipoint control hydraulic press is required. In this study, a method is based on optimization techniques in conjunction with finite element (FE) analysis was developed for the prediction of variable blank holder pressure in multipoint control system. In the method, the estimation of variable BHP was formulated as a multiobjective optimization problem and two objective functions were defined in the problem. One of the functions was created according to minimization of thinning, while the other was created according to minimization of wrinkling and BHP were defined as design variable as well. Objective functions for thinning and wrinkling were expressed by the minimum thickness in the part and distance from the major strains in the formed part to wrinkling limit curve (WLC) respectively. Method developed within the scope of thesis was applied to forming of double bowl cups and double bowl sink was selected as the part. Application of the method was carried out in two stages: In the first stage space variant BHP were estimated, while in the second stage space and stroke variant BHP were estimated. In the prediction of space variant BHP, sequential response surface method (SRSM) was applied and the relationship between inputs (BHP in each piston) and output (multiobjective function) was defined with a second order function. After obtaining the function, search was performed with a hybrid algorithm composed of simulated annealing and leap frog algorithms and then minimum point of the function and optimum values of the design variables were estimated. Interaction between FE analyses and optimization algorithms was realized by using LS-DYNA and LS-OPT software. In the prediction of space and stroke variant BHP, BHP – stroke curve in each piston was represented by piece-wise lines and control points on the line were defined as design variables. Then FE analyses were carried out according to Taguchi L54 orthogonal array and mathematical model was created between inputs and multiobjective function by stepwise regression method. Optimum points were predicted by simulated annealing algorithm, Taguchi optimization and optimum pressure-stroke curves in each piston were obtained. In the last section of the study, experiments were carried out with the predicted optimum BHP and numerical and experimental thickness distributions of the part were compared to validate the method. It was determined that the differences between simulation and experiment in the points, which minimum thickness were observed, were %4,53 in diagonal direction, %3,57 in longitudinal direction for space variant BHP; they were %6,3 in diagonal direction, %4,83 in longitudinal direction for space and stroke variant BHP. The results obtained with optimum BHP in multipoint control system were compared with the results obtained from the deep drawing of the same part in traditional system and method was evaluated in the study. %26 reduction in thinning in the corner points of the part was observed for the optimum BHP in multipoint control hydraulic press compared to traditional system. Results obtained from experiments carried out with both BHP type were compared and it was seen that the difference between the thickness distributions in diagonal direction was insignificant, but reduction in thinning of %11 in longitudinal direction was obtained with space and stroke variant BHP.