A dynamic programming approach for minimizing the number of drawing stages and heat treatments in cylindrical shell multistage deep drawing

Deep drawing is an important sheet metal forming process that appears in many industrial fields. It involves pressing a blank sheet against a hollow cavity that takes the form of the desired product. Due to limitations related to the properties of the blank sheet material, several drawing stages may be needed before the required shape and dimensions of the final product can be obtained. Heat treatment may also be needed during the process in order to restore the formability of the material so that failure is avoided. In this paper, the problem of minimizing the number of drawing stages and heat treatments needed for the multistage deep drawing of cylindrical shells is addressed. This problem is directly related to minimizing manufacturing costs and lead time. It is required to determine the post-drawing shell diameters along with whether heat treatment is to be conducted after each drawing stage such that the aforementioned objectives are achieved and failure is avoided. Conventional computer-aided process planning (CAPP) rules are used to define the search space for a dynamic programming (DP) approach in which both the post-drawing shell diameter and material condition are used to define the states in the problem. By discretizing the range of feasible shell diameters starting from the initial blank diameter down to the final shell diameter, the feasible transitions from state to another is represented by a directed graph, based upon which the DP functional equation is easily defined. The DP generates a set of feasible optimized process plans that are then verified by carrying out finite element analysis in which the deformation severity and the resulting strains and thickness variations are investigated. Two case studies are presented to demonstrate the effectiveness of the developed approach. The results suggest that the proposed approach is a valuable, reliable and quick computer aided process planning approach to this complicated problem.