## 2 Methodology  
The limit analysis (lower and upper) provides analytical solutions that delimit the range where the solution must exist. The UBT is one analytical method that can provide quite accurate results when it is properly applied. Among the two approaches of limit analysis, the UBT is chosen because it is not so difficult to apply and also because of its theoretical fundamentals, where the deformation is guaranteed for the obtained solution. The UBT not only provides the minimum value of load that assesses the desired deformation but also allows to define the involved parameters so optimal process conditions can be determined. In the traditional application of the method, the profile of the part was configured by means of a series of triangular rigid zones which increased in number and shape with the result of equations more and more complex. The geometrical parameters of each TRZ configuration and their hodograph are represented in Figures 1 and 2 for an odd and an even number of TRZ to study the influence of the TRZ number in the _P_/2\(k\) relation, where \(p\) is the mean forging pressure and \(k\) is the shear yield stress of this material. The application of the UBT in each of these four cases should obtain optimized values of _x_\({}_{i}\) parameters in different boundary conditions. Figure 1: Distribution and hodograph with odd number of triangular rigid zones (TRZ). Figure 2: Distribution and hodograph with even number of TRZ. The consideration of a semi-adherence in die-workpiece contact has been solved including an \(m\) friction shear factor in affected terms of Equation (16). The \(p/2k\) relation is evaluated according to shape factor over a variable number of TRZ, obtaining a generalization for \(n\) TRZ expression. Equations (2) and (3) show the deformation power expressions for an odd number (Equation (2)) and an even number (Equation (3)) of TRZ:  
\[\frac{dW}{dt}=\dot{W}\leq kw\left[\sum_{i=1}^{N-1}v_{i,i+1}x_{i,i+1}+m\sum_{i= 1}^{(N-1)/2}v_{1,2i+1}(x_{2i}+x_{2i+1})\right];\;\;x_{N}=0 \tag{2}\]  
\[\frac{dW}{dt}=\dot{W}\leq kw\left[\sum_{i=1}^{N-1}v_{i,i+1}x_{i,i+1}+m\sum_{i= 1}^{N/2-1}v_{1,2i+1}(x_{2i}+x_{2i+1})\right] \tag{3}\]  
Applying the trigonometric ratios for the positions and velocities of each zone is possible to simplify these equations, obtaining Equation (4) for an odd number of TRZ  
\[\frac{dW}{dt}=\dot{W}\leq\frac{kwv}{h}\left[h^{2}(N-1)+\sum_{i=1}^{N-1}x_{i}^ {2}+m\sum_{i=1}^{(N-1)/2}\left(\sum_{j=1}^{2i}x_{j}\right)(x_{2i}+x_{2i+1}) \right];\;x_{N}=0 \tag{4}\]  
and Equation (5) for an even number of TRZ. \[\frac{dW}{dt}=\dot{W}\leq\frac{kwv}{h}\left[h^{2}(N-1)+\sum_{i=1}^{N-1}x_{i}^ {2}+m\sum_{i=1}^{(N-2)/2}\left(\sum_{i=1}^{2i}x_{j}\right)(x_{2i}+x_{2i+1})\right] \tag{5}\]  
As shown, this approach only considered plane-parallel surfaces, increasing the terms of the equation according to the number of TRZ defined. Following previous works developed by the authors of this paper, this study establishes a new approach in which the basic equation develops from the geometrical configuration of a generic module of three TRZ [25; 26; 27]. This module can connect to subsequent ones through the yield velocities of the material, being the output velocity from the first module, \(V_{s}\), the input velocity to the next module, \(V_{e}\). The nondimensional relation \(P/2k\) (\(p=\) pressure on the profile; \(k=\) shear yield strength of the material) will be always the same, whatever the module. This new approach, called modular approach by the authors, provides an extension of the upper bound application to geometric configurations in which the tool profile of the flat die combines parallel and inclined flat surfaces. The most important limitation of the UBT by means of TRZ model is the fact that a double symmetry in geometries subjected to a plane strain state is imposed. In this study, one of these boundary conditions is released, allowing to expand the possibilities of application of this method.