## 3 Modeling and Simulation of the Electrical Contacts Using Numerical Methods
### The Step-by-Step Simulation Setup for Comsol  
The simulation was performed for a 3D model in two distinct steps within stationary studies of Structural Mechanics > Solid Mechanics (solid) and Heat Transfer > Electromagnetic Heating > Joule Heating type. The two contact elements presented above have a cylindrical body with the touching ends shaped similar to a hook. Here the coupling of apparent thermal and electrical resistances by means of mechanical contact pressure takes place. It was considered that the initial temperature of the contact is equal to the ambient temperature, and the difference of electric potential between the contact elements produces heating by the Joule effect. Figure 10 shows the geometry of the model imported into COMSOL. In the Definitions menu, the Contact Pair type was chosen for  
Figure 8: The model of the electrical contact. Figure 9: Geometry and dimensions of one contact element. the contact elements and their Source Boundaries and Destination Boundaries were defined as shown in Figure 11.  
Next, for the entire model, the material chosen was copper, with predefined characteristics set in the material library as described in Table 1.  
The fixed surfaces have been established as shown in Figure 12. These surfaces represent the inner part of the semi-cylindrical areas of the contacts. Since the imported geometric pattern is a cross-section through the two electrical contacts, we used the Boundary > Symmetry \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Variable** & **Value** & **Unit** \\ \hline Electrical conductivity & sigma\_iso & \(5.988\times 10^{7}\) & S/m \\ Heat capacity at constant pressure & Cp & 385 & J/(kg\(\cdot\)K) \\ Relative permittivity & epsilon\_iso & 1 & \\ Density & rho & 8960 & kg/m\({}^{3}\) \\ Thermal conductivity & k\_iso & 400 & W/(m\(\cdot\)K) \\ Young’s modulus & E & \(110\times 10^{9}\) & Pa \\ Poisson’s ratio & nu & 0.35 & 1 \\ Coefficient of thermal expansion & alpha\_iso & \(17\times 10^{-6}\) & 1/K \\ Reference resistivity & rho & \(1.72\times 10^{-8}\) & O-m \\ Resistivity temperature coefficient & alpha & 0.0039 & 1/K \\ Reference temperature & Tref & 298 & K \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material characteristics of the model. Figure 11: The contact elements, sources and destination boundaries as defined in COMSOL. Figure 10: The model geometry as imported into COMSOL. option from the Physics menu, and thus the plane of symmetry of the model has been defined (Figure 13). This approach simplifies computations for half of the model. The contact method was chosen from the COMSOL solver as the Augmented Lagrangian type with the initial value of the contact pressure being \(T_{n}\) = 1 \(\times\)\(10^{7}\) N/m\({}^{2}\). The Augmented Lagrangian method is widely used in simulations involving friction contacts [71], as it is the optimal method of treating constrained minimization problems [72]. Regarding the heat transfer, it was set as Convective heat flux type with a heat transfer coefficient \(h\) = 2. The difference in electric potential representing the voltage drop on the simulated contact was imposed by assigning zero voltage (ground) to the base surface of the cylinder to one of the contact elements, and a voltage of 1 mV (electric potential) to the other contact element, as shown in Figure 14a,b. Figure 12: Fixed surfaces definition. Figure 13: The plane of symmetry for the model. When modeling using FEA in general (COMSOL in this particular case), it is crucial to select appropriate convergence criteria in order to obtain accurate results. Mesh convergence refers to how small the elements need to be to ensure that the results of the finite element analysis are not affected by changing the size of the mesh. The choice of convergence criteria should be justified based on the specific phenomenon being studied, as well as the chosen model, simulation conditions, and engineer experience.