## 3 Modeling of ESD Textiles in General-Purpose Multiphysics Software  
Several software environments are available for the modeling of complex physical problems, such as Ansys [49], CST Studio Suite(r) [50], and COMSOL Multiphysics(r) [42]. In this paper, COMSOL software is used. The software offers fully coupled capabilities for multiphysics modeling. It allows for control of the flow of the numerical process by selecting methods of numerical integration and solving linear and nonlinear algebraic equations. The simulation's data management, export ability, and graphics are also very advanced. Developing reliable models to design protective textiles that meet existing standards for the evaluation of electrostatic discharge is necessary. The problem is very complex because the diameters of the conductive threads are small relative to the dimensions of the specimen and test stand, and the electrical conductivities of the base and conductive materials differ by up to several orders of magnitude. [PERSON] and [PERSON] [27] focused on modeling fabrics to determine surface and vertical resistance. They used sophisticated modeling at the thread level, with an average electrical conductivity value for the conductive yarn. The analysis used the cooperation of electrical current and electrical circuit modules. The developed model is very complex, so the analysis time is long. However, due to the measuring electrodes' dimensions and the static analysis, the model proved to be a good starting point for the development of simplified circuit models. This article focuses on modeling fabrics and nonwoven fabrics to determine charge decay times by induction. Measuring the charge decay time in a textile is one of the basic electrostatic tests and can be used to characterize all types of fabrics exposed to ESD. The models proposed in this paper can be applied to the design of textiles according to the EN 1149-3 standard. The electrodes and the specimen are much larger for the test than for tests in which surface and vertical resistances are determined. Moreover, the test itself leads outputs characteristics versus time. Thus, the model must be significantly simplified to obtain results in an acceptable time. Three-dimensional electromagnetic models of the electrode (6) and rings (3, 4, and 5) (see Figure 3) were developed on a 1:1 scale in the electric currents module. The geometry of the electrodes and the distances on the laboratory bench comply with the technical conditions presented in the EN 1149-3 standard. A probe (point probe) was placed in the center of the ring (3) instead of the measuring electrode. The probe collects the electric field norm over time. The specimen (8) is modeled using a cylinder with a radius of 128 mm  
Figure 4: Exemplary plots of charge decay measurement records (sketches). (**a**) Dissipative material without shielding effect; (**b**) dissipative material with shielding effect. and a thickness corresponding to the actual specimen. Relative permeability and electrical conductivity parameters of the cylinder correspond to the base textile material (polyester, polyamide, or cotton), taking into account any treatment (e.g., singeing/firing (a process whereby the textile is passed over an open flame or heated plates to burn off protruding fibers on the surface; this results in a smoother fabric with an improved appearance and reduced pilling ) or impregnation). A grid of conductive fibers is placed in the middle of the thickness of the cylinder. Preliminary studies have shown that such grid placement avoids many problems in numerical meshing. In addition, as verified in many other models with higher accuracy but smaller geometric dimensions, direct contact of the conductive thread with the rings does not significantly affect the results obtained with textiles less than 3 mm thick. The cross-section of the grid elements is rectangular, with dimensions as close to the actual measurements as possible. Such a grid is easily generated using the difference operation in COMSOL.