## 1 Introduction  
Electrical contacts are crucial circuit components present in most electric devices, such as relays, switches, breakers, connectors, breakers and integrated circuits. The theory and analysis of electrical contacts [1, 2, 3, 4, 5] is a multidisciplinary issue since it involves electrical, mechanical and thermal concepts, all aimed at understanding and improving the performance of these electrical connections used in various applications. A contact system always consists of a pair of electric contacts and a source of contact force [6], with each contact made of conductive material, usually metal or metallic alloys. When a pair of contacts touch, electrical current passes with a certain electrical contact resistance. There are several factors [7] that influence the current flow and thus this resistance: contact area [8], shape [9], roughness [10], oxidation and coatings [11, 12],mechanical loads [13], etc. The electrical contacts' resistance should be low and stable, since its increase leads to contact failure and very high Joule heat losses [14]. The typical failure modes for electrical contacts consist of: (a) temperature increase in the contact area producing melting or quick aging, (b) elevated electrical contact resistance caused by fretting wear or oxidation, and (c) excessively small or large closing/opening forces. Failures of electrical contacts and the causes leading to them are extensively studied in the scientific literature, based on several different analysis approaches. First, there are theoretical analyses based on model types such as the single-point (or Holm tube) contact model [15; 16], the multi-point contact model [17; 18], and the fractal geometry theory model [19; 20; 21]. Several other scholars have used analytical models to study electrical contact behavior, with the authors of [22] presenting a study built on the improved elastic rough surface model, [23] introducing a new analytical model for the analysis of switching devices' contact areas for different voltage and current magnitudes, and [24] proposing a semi-analytical model based on the single asperity concept to highlight the flow of current between two contacting surfaces. Next, electrical contacts have also been widely studied using experimental setups and measurements, employing a combined analytical and experimental investigation [25] to describe the electrical-thermal behavior of electrical contact systems, using a low-cost tabletop indenter setup [26] to measure the in-situ electrical contact resistance, determining experimentally the adhesion force and energy at contact interfaces for conductive, isotropic, and rough surfaces [27]. High-current automotive connectors were tested [28] to assess their thermal performance, experimental observations on contact resistance were summed up [29] to be used as an electrical contact guide, two distinct techniques were used during investigations [30] for different pressures, temperatures, and contact types, and trials were conducted [31] on copper/brass plug-and-sockets types of contacts. Finally, there is the vast array of numerical methods. These started several decades ago with finite elements discretization [32; 33] and solving in codes like FEAP [34; 35; 36] or discrete elements simulations using the LMGC90 open platform or MULTICOR software [37; 38; 39; 40], but in recent years the continuous advancement of computer technology and the leaps in performance and processing power have allowed us to model, analyze, optimize and solve more and more complex problems of electrical contact systems. Nowadays, the numerical methods use dedicated engineering computer software, and are based on various techniques such as the boundary element method (BEM) [41; 42; 43], finite element method (FEM) [44; 45; 46] and coupled/multiphysics methods [47; 48; 49; 50] to model and simulate contact behavior. [PERSON] et al.