## 2 Materials and Methods
### Numerical Analysis: Finite Element Method (FEM)  
The problem consists of a one-stroke operation where a punch performs a single indentation in the workpiece, applying a 30% reduction. Different punch shapes are considered at the surface in contact with the workpiece. The displacement of the workpiece is partially restricted horizontally due to the vertical symmetry imposed. The dimensions and geometry of the workpiece are presented in Figure 3.  
The workpiece material is A92024 (Unified Numbering System (UNS) designation, Scientific Forming Technologies Corporation, Columbus, OH, USA), cold worked without hardening effect. The simulation is contemplated under plain strain conditions taking the adhesion friction into account. Friction in bulk forming processes--like indentation or forging--has been deeply studied [22; 23]. In forming processes where high pressure takes place, the adhesion friction offers a more precise approximation than the Coulomb model.  
Due to the plain strain condition, software DEFORM 2D (Scientific Forming Technologies Corporation, version 8.1, Columbus, OH, USA, 2004) is chosen for the FEM analysis. Furthermore, working with 2D, the computational resolution times are lower than with 3D simulations.  
Figure 3: Dimensions and geometries (red frame) of the indentation process simulated.  
The punch is considered a rigid object, modelled as a non-deformable material, and the workpiece a plastic object, modelled as a rigid-plastic material. Only half of the workpiece is implemented to reduce workload (Figure 3). Focusing resolution on simulation times and results accuracy, the mesh is implemented, taking advantage of mesh windows. This tool allows a division of the part studied, having higher density meshes--more elements per unit area/volume--where needed. Therefore, a higher density mesh window is placed at the workpiece part in contact with the punch and a second mesh window with a coarse density is placed for the remaining workpiece. Working with very high density mesh increases computational times. Consequently, in this case, the refined mesh is applied only to the deformation area, attempting to acquire the exact indentation shape. Although coarse FE meshes influences solutions' accuracy, results are obtained in a reasonable time. Using the mesh window tool, we can reduce simulation times without sacrificing accuracy.  
The relative element size relation between mesh windows is 1/20. That is to say, the elements of the second mesh are twenty times larger than the elements in the first mesh, having a total number of five thousand elements. The optimal mesh configuration is obtained from several simulations carried out until results repeatability is achieved (Figure 4). Along with the element size, velocity can be assigned to the window, making it possible to follow the punch throughout the entire penetration (Figure 5).  
Figure 4: Mesh optimization study. Initial stage (**A**) and optimized mesh (**B**).  
The material assigned from the database is UNS A92024, selecting the tabular data format as the flow stress law (Equation (8)). This method is recommended when the true behaviour of the material needs to be implemented. Figure 6 shows the material behaviour in this case study:  
\[\overline{\sigma}=\overline{\sigma}\ \times\ \left(\overline{\epsilon},\tilde{ \overline{\epsilon}},T\right) \tag{8}\]  
where:  
\(\overline{\sigma}\) = Flow stress;  
\(\overline{\epsilon}\) = Effective plastic strain;  
\(\tilde{\overline{\epsilon}}\) = Effective strain rate;  
\(T\) = Temperature.  
Figure 5: Mesh window display.  
Figure 6: Material flow law.  
Finally, the general simulation settings are shown in Table 1.  
These conditions are maintained hereinafter for the different geometries considered (Figure 7).