## 1 Introduction
Hence, the UBT can be expressed as follows:  
\[\int_{S_{V}}T_{i}\ \times\ v_{i}\ \times\ dS_{v}\leq\int_{S_{D}}k\ \times\ v^{*}dS_{D}^{*}+\int_{S_{F}}T_{i}v_{i}^{*}dS_{F} \tag{1}\]  
where:  
\(T_{i}\): Workpiece external strengths;  
\(v_{i}\): Real velocities field;  
\(S_{v}\): Surface where external load are applied;  
\(k\): Shear yield stress;  
\(S_{D}^{*}\): Discontinuity surface;  
\(p_{i}^{*}\): Kinematically Admissible Velocity Field (KAVF);  
and \(S_{F}\): External surface exposed to virtual velocity field. Additionally, under plane strain hypothesis, the material deformed offers its maximum resistance when \(\tau=k\), where \(\tau\) represents the shear stress. Thus, the dissipative power value due to the internal energy will not exceed the outcome value of \(k\times s\times v^{*}\), with \(s\) being the length of the tangential velocity discontinuity line. Working with UBT, the Triangular Rigid Zone (TRZ) alternative is the kinematic-geometrical option that allows a more accurate solution with high capacity of analysis [18]. The study of the material flow field is possible applying the TRZ choice. Each TRZ slides one over the other along the discontinuity lines of the KAVF represented by the corresponding hodograph (Figure 1). Having the KAVF defined, it is possible to calculate the forming forces and represent the velocities, strain rates and strains of the process considered. Figure 1 shows a quarter of an indentation, where the TRZ division identifies each triangle with a number, being \(V\) the velocity of the punch over the workpiece. This velocity causes a reaction over the whole part, being the blue arrow the movement of the part. The hodograph (Figure 1B) is built with the absolute (\(V_{i}\)) and relative (\(V_{ij}\)) velocity of each TRZ. In previous published works, we present and validate the study on indentation process by means of the new Modular Upper Bound technique (MUB) [19,20]. MUB was developed for its implementation with a flat and inclined punch (Figure 2). Particularly in this paper, the study of the punch geometry is extended to another four geometries--combinations of the flat and inclined punch--including a curve punch, which gives rise to future research where more complex shapes can be treated. Thereby, the purpose of this research is to validate MUB for new punch geometries, extending its application. MUB consideration is designed with three modules of only two TRZs each. The analysis of the single modules simplifies the problem evaluation. It is known that, when working with UBT, the more TRZ the workpiece divided into, the more complex the mathematical equation gets, which highly hinders it application. An example of the mathematical complexity compared with the versatility of MUB is clarified in Equations (2)-(7) from [PERSON]'s research [21]. The traditional UBT method is expressed in Equation (2), being values of \(x_{i}\) illustrated in Equations (3)-(5). With MUB, this complication is avoided and the problem resolution requires less calculation effort, as seen in Equation (6) for module A and Equation (7) for module B. For this reason, the approach's interesting feature lies in the fact that the complexity of the UBT with MUB does not apply. In this case, the complexity of the analysis is not considerably increased by incrementing the number of modules:  
\[\left(\frac{p}{2k}\right)_{T}=\frac{1}{2b}\ \times\ \left[4h^{2}+x_{1}^{2}+x_{2}^ {2}+x_{3}^{2}+(b-x_{1}-x_{2}-x_{3})^{2}+m\ \times\ \left(x_{1}x_{2}+x_{1}x_{3}+x_{2}^{2}+x_{2}x_{3}\right)+\right. \tag{2}\]  
\[mb\ \times\ (b-x_{1}-x_{2}-x_{3})\big{]}\]  
where:  
\[x_{1}=\frac{m\ \times\ (b-x_{2}-x_{3})-2x_{2}-2x_{3}+2b}{4} \tag{3}\]  
\[x_{2}=\frac{b\ \times\ \left(4-m^{2}\right)+x_{3}\ \times\ \left(m^{2}+4m-12 \right)}{\left(4-m^{2}\right)} \tag{4}\]  
Figure 1: Upper Bound Theorem (UBT) Indentation Analysis. Triangular Rigid Zone (TRZ) distribution (**A**) and Kinematically Admissible Velocity Field (KAVF) analysis (**B**). Figure 2: Flat (**A**) and inclined (**B**) punch Modular Upper Bound (MUB) application.