## 2 Materials and Methods
The values of the function at the boundary nodes, which represent the solution to the problem, are not known beforehand. Typically, the coefficients of the approximating functions are determined by enforcing the equality of neighboring function values at the nodes that define the boundaries between elements. These coefficients are then expressed in terms of the function values at the nodes within each element. A set of linear algebraic equations is constructed with the number of equations directly proportional to the number of unknown values at nodes where the solution is sought. This number is limited only by the computer's capabilities and is determined by the number of elements. As each element is only related to a limited number of neighboring elements, the resulting system of equations is sparse, making it much easier to solve. In matrix terms, stiffness and mass matrices (also known as Dirichlet matrices) are assembled. Boundary conditions are then applied to these matrices, such as Neumann conditions that leave the matrices unchanged or Dirichlet conditions that remove rows and columns corresponding to boundary nodes where the solution components are already known. The system of linear equations is then constructed and solved using some of the known methods [30]. The application of the method in electrodynamics is based on [PERSON]'s equations, which are formulated as follows. [PERSON]'s law:  
\[\oint\mathbf{D}\mathbf{s}=\mathbf{Q}, \tag{1}\]  
where \(\mathbf{D}\) is the flux of electric field strength through a closed surface, \(s\) is the surface area, and \(\mathbf{Q}\) is the charge enclosed within that surface. Figure 6: SolidWorks. 3D model of an induction motor AIR 63V2. [PERSON]'s law for the magnetic field:  
\[\oint\mathbf{B}ds=0, \tag{2}\]  
where \(\mathbf{B}\) is the flux of the magnetic induction vector through any closed surface. [PERSON]'s law of induction:  
\[\oint\mathbf{E}dl=-\frac{d}{dl}\int\mathbf{B}ds, \tag{3}\]  
where \(\mathbf{E}\) is the electric vector field, and \(dl\) is an infinitesimal vector element of the contour. Biot-Savart-Laplace law:  
Figure 1:0  
where \(\mathbf{H}\) is the magnetising force or magnetising, \(I\) is electric current, and \(\mathbf{D}\) is electric induction. Additional relations:  
Figure 1:1  
where \(\mu\) is the property of the magnetic material called its permeability, and \(\mu_{0}\) is the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum. Figure 1:2  
where \(\varepsilon\) is relative permittivity, and \(\epsilon_{0}\) is dielectric constant. Figure 1:3  
where \(\mathbf{j}\) is free charge current density, and \(\sigma\) is specific conductivity. Equations (1)-(7) are utilized as a basic principle for the calculations in the Maxwell software, which utilizes the finite element method (FEM) to accurately compute static, harmonic electromagnetic and electric fields, as well as transients in electric motors. These equations are not utilized for optimization purposes but rather to calculate the electrodynamics of an induction motor using the finite element method. The computation starts with the creation of a new project, wherein the category of electrical machine to be evaluated is explicitly stated (see Figure 7). Within the project manager, four classifications materialize in the established project: machine, which pertains to the configuration of model parameters; analysis, which relates to the establishment of model calculation; optimetrics, which involves the determination of model optimization parameters; and results, which deals with the output of analysis findings. The preliminary data for the IM AIR 63V2 motor is then introduced into the project. By utilizing a previously generated 3D model via the SolidWorks software, the dimensions of the stator and rotor slots are entered into the graphs. Unlike the 3D model, the simplification of the stator winding is deemed inadmissible in this computation.