**5. Practical information about finite elements**

#### **5.1 Isoparametric elements**

An **isoparametric element** is a finite element whose nodal base functions, which enable the interpolation of scalar fields, are also used to parametrize the associated geometric element. The base functions are usually piecewise defined, in each of the geometric elements which cover the studied domain, and some continuity conditions have to be satisfied at the interfaces between elements. Then, there will be no discontinuity of the interpolated scalar fields, nor of the coordinates after transformation from the reference elements towards the real ones. Such base functions are said to be **conformal**.

Consider a nodal finite element (K, PK, ΣK). If NK is the set of nodes of K, whose coordinates are **x**i, i ∈ N, and if the pi(**u**), i ∈ NK, are its base functions expressed in the coordinates **u** of the reference element Kr associated with K, then the parametrization of K (i.e., **x** = **x**(**u**)) is given by [6]

$$\mathbf{x} = \sum\_{\mathbf{i} \in \mathcal{N}\_{\mathbb{K}}} \mathbf{x}\_{\mathbf{i}} \mathbf{p}\_{\mathbf{i}}(\mathbf{u}) \tag{53}$$

where **x**∈K, **u**∈Kr; this element is isoparametric.

#### **5.2 Reference elements**

We define here the reference elements which are associated with the considered geometric elements, that is, with tetrahedra, hexahedra and prisms. Nodal, edge, facet and volume finite elements are defined in these geometric elements.

#### *5.2.1 Reference tetrahedron of type I*

The reference tetrahedron of type I is an element with 4 nodes whose coordinates are given in **Figure 13**. The associated geometric entities, as well as their notation, are shown in **Figure 13**. The nodal and edge base functions of this element are given in **Tables 1** and **2**. **Table 3** shows the notation of facets. The incidence matrices are given by (53), (54) and (55) (**Figure 14**).

**Figure 13.** *Reference tetrahedron of type I [6].*

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*


**Table 1.**

*Nodal base functions of the tetrahedron of type I.*


#### **Table 2.**

*Notation of the edges of the tetrahedron of type I and associated edge base functions (se).*


#### **Table 3.**

*Notation of the facets of the tetrahedron of type I.*

Edge-node incidence matrix

$$\mathbf{G}\_{\text{AN}} = \begin{bmatrix} \cdot \cdot \cdot \cdot & 1 & 2 & 3 & 4\\ & 1 & 1 & \cdot & \cdot\\ & 2 & \begin{bmatrix} -1 & 1 & \cdot & \cdot\\ -1 & \cdot & 1 & \cdot\\ -1 & \cdot & \cdot & 1\\ \cdot & -1 & 1 & \cdot\\ \cdot & -1 & \cdot & 1\\ \cdot & -1 & \cdot & 1\\ \cdot & \cdot & -1 & 1 \end{bmatrix} \tag{54}$$


#### **Figure 14.**

*Geometric entities defined on a tetrahedron of type I [6].*

Facet-edge incidence matrix

$$\mathbf{R}\_{\rm FA} = \begin{bmatrix} \cdot \cdot \cdot & \cdot & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & \cdot & -1 & \cdot & 1 & \cdot & \cdot \\ 1 & 1 & \cdot & -1 & \cdot & 1 & \cdot \\ -1 & 1 & \cdot & -1 & \cdot & \cdot & \cdot \\ \cdot & -1 & 1 & \cdot & \cdot & -1 \\ \cdot & \cdot & \cdot & \cdot & 1 & -1 & 1 \end{bmatrix},\tag{55}$$

Volume-facet incidence matrix

$$\mathbf{D}\_{\text{vlr}} = \frac{\mathbf{v}^{\prime} \cdot \mathbf{v}^{\prime}}{1} + \frac{1}{1} \frac{2}{1} \frac{3}{1} \frac{4}{1} \frac{4}{1} \frac{3}{1} \dots \frac{4}{1} \dots \tag{56}$$

**Figure 15.** *Reference hexahedron of type I [6].*

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*

#### *5.2.2 Reference hexahedron of type I*

The reference hexahedron of type I is an element with 8 nodes whose coordinates are given in **Figure 15**. The associated geometric entities, as well as their notation, are shown in **Figure 16**. The nodal and edge base functions of this element are given in **Tables 4** and **5**. **Table 6** shows the notation of facets. The incidence matrices are given by (56), (57) and (58).


**Figure 16.**

*Geometric entities defined on a hexahedron of type I [4].*


**Table 4.** *Nodal base functions of the hexahedron of type I.*


**Table 5.**

*Notation of the edges of the hexahedron of type I and associated edge base functions (se).*


#### **Table 6.**

*Notation of the facets of the hexahedron of type I.*

Edge-node incidence matrix


ð57Þ

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*

Facet-edge incidence matrix

$$\mathbf{R}\_{\mu\mu} = \begin{bmatrix} \cdot, \cdot & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 1 & \cdot & -1 & \cdot & 1 & \cdot & \cdot & -1 & \cdot & \cdot & \cdot \\ -1 & 1 & \cdot & -1 & \cdot & -1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & -1 & 1 & \cdot & \cdot & \cdot & \cdot & -1 & \cdot & 1 & \cdot & \cdot \\ \cdot & -1 & 1 & \cdot & \cdot & \cdot & -1 & \cdot & 1 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 1 & -1 & \cdot & 1 & \cdot & \cdot & -1 & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & 1 & -1 & 1 & \cdot & \cdot & \cdot & -1 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 & -1 & 1 \end{bmatrix} \tag{58}$$

Volume-facet incidence matrix

$$\mathbf{D}\_{\text{VF}} = \frac{\mathbf{v}^{\prime} \cdot \mathbf{v}^{\prime}}{\mathbf{1}} + \frac{1}{\mathbf{1}} \frac{2}{\mathbf{1}} \frac{3}{\mathbf{1}} \frac{4}{\mathbf{1}} \frac{5}{\mathbf{1}} \frac{6}{\mathbf{1}} \frac{6}{\mathbf{1}} \tag{59}$$

#### *5.2.3 Reference prism of type I*

The reference prism of type I is an element with 6 nodes whose coordinates are given in **Figure 17**. The associated geometric entities, as well as their notation, are shown in **Figure 18**. The nodal and edge base functions of this element are given in **Tables 7** and **8**. **Table 9** shows the notation of facets. The incidence matrices are given by (59), (60) and (61).

**Figure 17.** *Reference prism of type I [6].*


#### **Figure 18.**

*Geometric entities defined on a prism of type I [6].*


#### **Table 7.**

*Nodal base functions of the prism of type I.*

Edge-node incidence matrix

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*


#### **Table 8.**

*Notation of the edges of the prism of type I and associated edge base functions (se).*


#### **Table 9.**

*Notation of the facets of the prism of type I.*

Facet-edge incidence matrix


Volume-facet incidence matrix

$$\mathbf{D}\_{\rm vF} = -\frac{\mathbf{v}^{\cdot} \cdot \mathbf{v}^{\prime}}{1} + \frac{1}{1} \frac{2}{1} \dots \frac{2}{1} \dots \frac{3}{1} \dots \frac{4}{1} \dots \frac{5}{1} \dots \tag{62}$$

#### **6. Applications**

The practical test problem is a 3-D model based on the benchmark problem 19 of the TEAM workshop including a stranded inductor (coil) and an aluminum plate (**Figure 19**) [10].

The coil is excited by a sinusoidal current which generates the distribution of time varying magnetic fields around the coil (**Figure 20**). The relative permeability and electric conductivity of the plate are *μr*,*plate* ¼ 1, *σr*,*plate* ¼ 35*:*26 MS*=*m, respectively. The source of the magnetic field is a sinusoidal current with the maximum ampere turn being 2742AT. The problem is tested with two cases of frequencies of the 50 Hz and 200 Hz.

The 3-D dimensional mesh with edge elements is depicted in **Figure 21** (*left*). The distribution of magnetic flux density generated by the excited electric current in the coil is pointed out in **Figure 21** (*right*). The computed results on the of the *z*-component of the magnetic flux density along the lines A1-B1 and A2-B2 (**Figure 19**) is checked to be close to the measured results for different frequencies of exciting currents (already proposed by authors in [10]) are shown in **Figure 21**. The mean errors between calculated and measured methods [10] on the magnetic flux density are lower than 10%.

**Figure 19.** *Modeling of TEAM problem 7: Coil and conducting plate [10].*

#### **Figure 20***.*

*The 3-D mesh model with edge elements of the coil and conducting plate, and the limited boundary [4] (*left*), and distribution of magnetic flux density generated by the excited sinusoidal current in the coil, with μ<sup>r</sup>*,*plate* ¼ 1, *<sup>σ</sup><sup>r</sup>*,*plate* <sup>¼</sup> <sup>35</sup>*:*<sup>26</sup> *MS <sup>m</sup> and f = 50 Hz.*

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*

**Figure 21**

*.*

*The comparison of the calculated results with the measured results on magnetic flux densities at y = 72 mm, with <sup>μ</sup><sup>r</sup>*,*plate* <sup>¼</sup> 1, *<sup>σ</sup><sup>r</sup>*,*plate* <sup>¼</sup> <sup>35</sup>*:*<sup>26</sup> *MS <sup>m</sup> and different frequencies [4].*

The *y* component of the varying of the eddy current losses with different frequencies (50 Hz and 200 Hz) along the lines A3-B3 and A4-B4 (**Figure 19**) is shown in **Figure 22**. The computed results are also compared with the measured results as well [7]. The obtained results from the theory modeling are quite similar as what measured from the measurements. The maximum error near the end of the conductor plate on the eddy currents between two methods are below 20% for both cases (50 Hz and 200 Hz).

#### **7. Conclusions**

In the 3D computation of the magnetic flux density and eddy current, thanks to the set of Maxwell's equations, it has been successfully developed for two weak formulations, where the discretization of the fields is performed by Whitney edge elements [2, 3, 8]: magnetostatic formulation and magnetodynamic formulation. The developments of the method is validated on the actual problem (TEAM problem 7) [10]. The numerical error between simlated and measured results on the magnetic flux densities and eddy current is lower than 10%. This is also proved that there is a very good validation between two methods. The results have been achieved by a detailed study of the magnetodynamic formulation.

#### **Figure 22**

*The comparison of the calculated results with the measured results at z = 19 mm, with μ<sup>r</sup>*,*plate* ¼ 1, *σ<sup>r</sup>*,*plate* ¼ 35*:*26 *MS <sup>m</sup> and different frequencies [4].*
