5.2.1 Domain Discretization

The three-dimensional simulation domain is discretized in the same way as for the scalar finite element method on tetrahedrons. Unfortunately this is not a simple problem, especially for arbitrary shaped regions and complicated field distributions [86,87]. The mesh properties have considerable impact on the finite element analysis and strongly affect the solution quality and the calculation duration time. Generally these properties include geometry conformity, mesh density, and element quality. The geometry conformity requires that the area defined by the mesh elements should sufficiently good approximate the domain of the problem. As already mentioned in Subsection 4.1.1 the mesh density must be sufficiently high and the mesh size sufficiently small, respectively, to minimize the discretization error and to achieve accurate solutions. A good idea is to use denser mesh, which means smaller elements only in those regions, where a high spacial variation of the investigated fields is anticipated. Of course this can be realized only with unstructured meshes used in this work. Automatic mesh adaptation and improvement based on refinement in the regions presenting the highest error in the approximation procedure [88,89,90,91] or on user expertise on the specific problem [92] are often used. Parallel mesh refinement algorithms can be applied to speed up this process [93,94]. The finite element method leads to a linear equation system solved iteratively. As discussed in [95,96] large element dihedral angles increase the discretization error in the finite element solution [97] and for a good condition number of the system matrix the discretization should avoid the generation of elements with a small inner solid angle or narrow elements [98,99]. Unfortunately, if there are small solid angles predefined in the domain, the mesh generation software will likely produce bad elements at this place.

As an example a conducting inductor (Fig. <5.1>) placed in an insulating rectangular brick domain (Fig. <5.2>) is discretized and visualized. Similar structures will be used in the application part of this work, where the simulation results of the vector finite element algorithm are demonstrated. The inductor is colored bright and the insulator dark. Normally the inductance and the resistance of the inductor at different frequencies must be calculated, which requires the extraction of the magnetic field distribution and the current density distribution in the simulation domain. If the operating frequency is sufficiently high for the dimensions of the corresponding domain parts, skin effect is observed. The current is forced to flow only on or near the surface of the inductor wire. There is no current flow inside of the inductor wire. In such cases it is convenient to produce the finest mesh on the surface of the wire and in the area close to the boundary between the inductor and the insulating environment. Deep inside and far outside the wire coarser mesh elements can be used as shown in Fig. <5.3> and Fig. <5.4>.

Figure 5.1: The inductor inside the simulation domain in Fig. <5.2>.

Figure 5.2: The entire simulation domain.

Figure 5.3: Inside view of the simulation domain in Fig. <5.2>.

Figure 5.4: The conductor entry.