# Optimization and Mapping of the Deep Drawing Force Considering Friction Combination
## 1 Introduction
The polymer film contributed successfully in saving surface reliability of the sheet metal [14]. An optimized lubricant application method through a Taguchi and regression model during deep drawing was proposed [15]. The optimum combination of lubricant and sheet surface microstructure was investigated experimentally. The experimental results were analyzed statistically to select the optimum combination of the two factors. Minimum work done while avoiding tearing, wrinkling, and spring back was achieved [16; 17; 18]. [PERSON] et al. [19] mentioned that variability in the deep drawing operation may result in effects which the deterministic models are not able to expect. A common method was applied to determine the doubts and combine them through the response surface method (RSM) in order to direct probabilistic established optimization [20]. The probabilistic design model was used to discover the optimum arrangement of blank holder load and friction coefficient through the existence of a variety of material characteristics. The outcome demonstrated that, by utilizing the probabilistic model, the wrinkling and failure decreased by 42% compared with the conventional design [19]. The uncertainness and unreliability of the deterministic optimization models utilized in deep drawing encouraged further researchers to implement quality enhancement applications, e.g., six sigma [21]. [PERSON] and [PERSON] [22] examined the influence of diverse factors on the deep drawing operation of steel/brass blanks. Finite element and experimental investigations of the deep drawing process of steel/brass blanks were conducted to display the positions of tearing for the cup. In this research, a straight-line relationship was obtained between the maximum punch load and the initial blank diameter. [PERSON] and [PERSON] [23] developed a combined finite element (FE), artificial neural network (ANN), and simulated annealing (SA) algorithm as a method to model and optimize the deep drawing operation for stainless steel 304 (SUS304). In this method, the parameters (die radius, punch radius, blank holder load, and frictional coefficients) were considered as input factors. [PERSON] and [PERSON] [24] suggested a new procedure combining two methods for optimizing and decreasing sheet metal thickness for the cover of automotive sealing. Mathematical equations were generated to join input operation factors and thinning. The optimum process was expressed for thinning using a genetic algorithm. [PERSON] et al. [25] generated a new optimizing method for optimizing the deep drawing operation utilizing the finite element technique incorporated with the response surface method (RSM). This optimizing method was a procedure for obtaining the optimal values of blank holder loads and then the desirable final product. [PERSON] et al. [26] presented an innovative method for optimizing shell structures taking into consideration their middle surface areas, such as undercuts. The authors presented a production restriction to the 3D topology optimization depending on the stiffness technique to be able to get an optimum structure with no undercuts, along with consistent thickness. Then, these shell structures could be produced by deep drawing within one stage. Lastly, the thickness of the shell enhanced the efficiency of the shell structure using this technique. [PERSON] et al. [27] suggested a procedure to simultaneously identify the ideal blank shape to minimize the earing and the ideal value of the blank holder load in the deep drawing operation for the circular cup. The total production cost was decreased by utilizing this procedure. In the present work, the optimization MATLAB toolbox (Pattern Search) was utilized to minimize the maximum deep drawing force. The deep drawing force and the value of the dimensionless ratio of die shoulder radius to sheet thickness (\(r_{d}\)/t) were optimized to minimize the forming work. The other operating parameters and geometrical parameters were constrained according to logical and practical limits documented in the literature [28].