# Optimization and Mapping of the Deep Drawing Force Considering Friction Combination
## 1 Introduction  
In terms of production capacity, energy consumption, and process improvement potential, the deep drawing process is one of the most significant technologies in current manufacturing [1, 2]. During the deep drawing process, a variety of factors might contribute to product failure [3, 4]. This failure typically manifests as wrinkling, tearing, earning, and rupture. These studies contained descriptions of material properties, die design, and operating factors such as blank holder load, friction coefficients, deep drawing ratio, and maximum drawing force; careful control of these parameters helps lower the likelihood of failure for drawing parts [5]. [PERSON] et al. [6] discovered the effectiveness of deep drawing variables, including die shoulder radii, punch nose radii, friction coefficients, and deep drawing ratios for cylinder-shaped parts. Using a genetic algorithm, this research created an evolutionary neural system, i.e., error back transmission. The evolutionary algorithm uses the results of the system to find factors that are preferable to maintaining a consistent thickness in the final cup. A complex blank stamping procedure may be examined under particular assumptions, allowing the use of the multi-objective optimization approach to be used [7]. After multi-objective optimization methods are created for a sheet metal forming problem, special considerations are given to conflicting objectives [8]. The maximal punch load (Fd) to perform the process is a main factor in the deep drawing process. This load is utilized in the choice of the instrument press and equipment, and it controls the creation of wrinkling on the lip of the drawn product. There exists no special mathematical calculation to compute the necessary deep drawing force in the drawing process [9]. Generally, thedrawing load for the first draw of circular components can be achieved using two methods, one from theoretical analysis taking into account the plasticity principle, and the other by utilizing well-established practical equations. The generalized function or expression takes the following form:  
\[Fd=f(\sigma_{1},\,FBH,\,\mu,\,h,\,t,\,n,\,K,\,r), \tag{1}\]  
where \(\sigma_{1}\) is the principal stress value, \(FBH\) is the blank holder load, \(\mu\) is the friction coefficient, \(h\) is the drawing cup height, \(t\) is the initial thickness of the sheet metal, \(n\) is the exponent of the strain hardening, \(K\) is the coefficient of the strain hardening, and \(r\) is the coefficient of normal anisotropy. Suitable records of \(\sigma 1\) and \(t\) can be achieved by differentiating \(Fd\) with regard to effective stress, such that  
\[dF/d\sigma=0. \tag{2}\]  
In the optimization of the operation of deep drawing, [PERSON] et al. [10] developed an optimization system to be used in determining the optimum blank holder force (BHF) scheme that minimizes punch force without running into any of the two operation restrictions: wrinkling and tearing. The design factors of the deep drawing machine included the geometrical factors for the punch and the die, in addition to the blank holder load. [PERSON] et al. [11] described the forecast of the drawing load and forming rise in perspective of optimizing the operating parameters included in the deep drawing operation. A model was employed to discover that friction has substantial effects on the punch force, and the low friction between the die and blank holder can significantly decrease the failure probability and increase the quality of the deep drawing process. The lubrication system and the type of lubricant used can significantly affect the process output and the amount of energy consumed [12]. [PERSON] [13] studied the minimization of drawing power used in the deep drawing operation by reducing the effect of friction and proposed the utilization of a solid lubricating polymer film barrier rather than oil lubrication. This lubricating barrier significantly decreases the drawing energy of precoated sheet metal at small tolerances.