**4.2 Base functions of spaces S<sup>i</sup>**

Consider the function pi(**x**) of coordinates of point **x** and relative to node ni, which is equal to 1 at this node, varies continuously in geometric elements having this node in common, and becomes equal to 0 in other elements without discontinuity (**Figure 6**). This function is nothing else than the base function, relative to node ni, of the function space of nodal finite elements built on the considered geometric elements. The function subspaces associated with each of the finite elements have respective dimensions 4, 8 or 6, for tetrahedra, hexahedra and prisms [4, 6, 9].

With **node** ni = {i}, is associated the function

$$\mathbf{s}\_{\mathbf{n}\_i}(\mathbf{x}) = \mathbf{p}\_i(\mathbf{x}). \tag{34}$$

The finite dimensional space generated by all sni's is denoted by S<sup>0</sup> . With **edge** eij = {i, j}, is associated the vector field

$$\mathbf{s}\_{\mathbf{e}\_{\vec{\mathbb{I}}}} = \mathbf{p}\_{\vec{\mathbb{I}}} \mathbf{grad} \sum\_{\mathbf{r} \in \mathcal{N}\_{\mathbb{F}, \vec{\mathbb{I}}}} \mathbf{p}\_{\mathbf{r}} - \mathbf{p}\_{\mathbf{i}} \mathbf{grad} \sum\_{\mathbf{r} \in \mathcal{N}\_{\mathbb{F}, \vec{\mathbb{I}}}} \mathbf{p}\_{\mathbf{r}},\tag{35}$$

where NF,mn is the set of nodes which belong to the facet of the geometrical element including evaluation point **x**, and including node m but not node n; such a facet is uniquely defined for three-edge-per-node elements. Its determination is shown in **Figure 8**, where either a triangular or a quadrangular facet is involved, and where shown edges belong to the geometric element including point **x**. Directions of dotted edges can be modified in order to schematize either a tetrahedron, a hexahedron or a prism. The defined set of nodes comes into view as being either {{m}, {o}, {p}} or {{m}, {o}, {p}, {q}}, respectively. The set NF,mn depends on point **x**, thus on elements. Particularly, it is empty (no node) in elements which have not

**Figure 8.** *Determination of the facet associated with* NF,mn *[6].*

edge {m, n} in common. Consequently, field **s**eij is zero in all the elements non adjacent to edge eij.

The vector field space generated by all **s**<sup>e</sup> is denoted by S<sup>1</sup> .

With **facet** f=fijk = {i, j, k} = {q1, q2, q3} or f = fijkl = {i, j, k, l} = {q1, q2, q3, q4}, is associated the vector field

$$\mathbf{s}\_{\mathbf{f}} = \mathbf{a}\_{\mathbf{f}} \sum\_{\mathbf{c}=1}^{\mathsf{s} \cdot \mathsf{N}\_{\mathbf{f}}} \mathbf{p}\_{\mathbf{c}\_{\mathbf{c}}} \operatorname{grad} \left( \sum\_{\mathbf{r} \in \mathsf{N}\_{\mathbf{f},\mathbf{q}\_{\mathbf{c}}} \overline{\mathbf{q}\_{\mathbf{c}}} + 1} \right) \times \operatorname{grad} \left( \sum\_{\mathbf{r} \in \mathsf{N}\_{\mathbf{f},\mathbf{q}\_{\mathbf{c}}} \overline{\mathbf{q}\_{\mathbf{c}}} - 1} \mathbf{p}\_{\mathbf{r}} \right) \tag{36}$$

where #Nf is the number of nodes of facet f, af = 2 if #Nf = 3, af = 1 if #Nf = 4, and the list of qi's is made circular by setting q0 � q#Nf and q#Nf + 1 � q1. Field **s**<sup>f</sup> is zero in all the elements non adjacent to facet f.

Vector fields sfijk(l)'s generate the space S<sup>2</sup> .

With **volume** v, is associated the function sv, equal to 1/vol(v) on v and 0 elsewhere. The space S3 is generated by these functions.

Some developments give the following results: sni is equal to 1 at node ni, and to 0 at other nodes; the circulation of **s**eij is equal to 1 along edge eij, and to 0 along other edges; the flux of sfijk(l) is equal to 1 across facet sfijk(l), and to 0 across other facets; and the volume integration of sv is equal to 1 over volume v, and to 0 over other volumes; that is

$$\mathbf{s}\_{\mathbf{i}}(\mathbf{x}\_{\mathbf{j}}) = \delta\_{\mathbf{i}\mathbf{j}}, \quad \forall \mathbf{i}, \mathbf{j} \in \mathbf{N} \tag{37}$$

$$\int\_{\mathbf{j}} \mathbf{s}\_{\mathbf{i}} \cdot \mathbf{dl} = \delta\_{\mathbf{i}\mathbf{j}}, \quad \forall \mathbf{i}, \mathbf{j} \in \mathbf{E} \tag{38}$$

$$\int\_{\mathfrak{j}} \mathbf{s}\_{\mathfrak{i}} \cdot \mathbf{n} \, \mathbf{ds} = \mathfrak{delta}\_{\mathfrak{i}\mathfrak{j}}, \quad \forall \mathfrak{i}, \mathfrak{j} \in \mathcal{F} \tag{39}$$

$$\int\_{\mathbf{j}} \mathbf{s}\_{\mathbf{i}} \mathbf{d} \mathbf{v} = \mathbf{\delta}\_{\mathbf{i}\mathbf{j}}, \quad \forall \mathbf{i}, \mathbf{j} \in \mathbf{V} \tag{40}$$

where δij = 1 if i = j and δij = 0 if i 6¼ j.

These properties show up various kinds of functionals and involve that functions sn, **s**e, **s**f, sv form bases for the spaces they generate. They are then called nodal, edge, facet and volume **base functions**. The associated finite elements are called **nodal**, **edge**, **facet** and **volume finite elements**.

#### **4.3 Geometric interpretation of edge and facet functions**

A geometric interpretation of edge and facet functions may be helpful to verify some of their properties. The vector field [4, 6, 9]

$$\mathbf{gradP\_{F,m\overline{m}}} = \mathbf{grad} \sum\_{\mathbf{r} \in \mathbb{N}\_{\mathbb{F},m\overline{m}}} \mathbf{p}\_{\mathbf{r}},\tag{41}$$

involved in both expressions (31) and (32), should be analyzed at first. The continuous scalar field,

$$\mathbf{P\_{F,m\bar{m}}} = \sum\_{\mathbf{r} \in \mathcal{N}\_{\mathcal{F},m}} \mathbf{p\_r},\tag{42}$$

has the characteristic of being equal to 1 at every point on the facet associated with NF,mn. This is a property of the nodal base functions. Therefore, vector field (42) is orthogonal to this facet at every point on it (**Figure 9**).

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

**Figure 9.** *Geometric interpretation of the edge function s<sup>e</sup> (35) [6].*

The vector field which is the product of pm and (35),

$$\mathbf{p\_m} \text{ grad } \sum\_{\mathbf{r} \in \mathcal{N}\_{\mathbb{F}, \mathbf{mn}}} \mathbf{p\_{r}},\tag{43}$$

is considered now. This field is said to be associated with edge {m, n}. As far as the function pm is concerned, it is equal to 0 on all the edges of the geometric element including point **x**, except those which are incident to node {m}. Therefore, the circulation of (43) is equal to 0 along all the edges except emn; field (43) is either simply equal to zero on them, or orthogonal to them (**Figure 9**). The combination of two fields of form (43) associated with edges {j, i} and {i, j}, as in (35), leads to a vector field which has the same properties as (43) (**Figure 9**), and

**Figure 10.** *Vector field a b involved in s<sup>f</sup> (33) [6].*

has consequently the announced properties of **s**eij. The fact that its circulation along edge eij is equal to 1 needs some calculation to be proved.

The vector field

$$\mathbf{p}\_{\mathbf{q}\_c} \mathbf{grad} \mathbf{P}\_{\mathbf{F}, \mathbf{q}\_c} \mathbf{\bar{q}}\_{c+1} \times \mathbf{grad} \mathbf{P}\_{\mathbf{F}, \mathbf{q}\_c \bar{\mathbf{q}}\_{c-1}} \tag{44}$$

which appears in expression (35) of **s**f, is considered now. Both gradients in (44) are shown in **Figure 10**. Each one is orthogonal to its associated facet and, therefore, their cross product (i.e., **a** � **b** in **Figure 10**) is parallel to both these facets.

The flux of this cross product, and in consequence the one of (44), is then equal to 0 across these facets. The term pqc in (44) enables the flux of (44) to be equal to zero across all other facets except facet f. The summation in (44) keeps the same property. The flux of **s**<sup>f</sup> across facet f is then the only one to differ from zero (**Figure 11**).

#### **4.4 Degrees of freedom**

The expression of a field in the base of a space Si –S<sup>0</sup> or S<sup>3</sup> for a scalar field, S<sup>1</sup> or S<sup>2</sup> for a vector field– gives scalar coefficients, called **degrees of freedom**. Fields ϕ ∈ S0 , **h** ∈ S<sup>1</sup> , **j** ∈ S<sup>2</sup> and ρ ∈ S3 can be expressed as [4, 6, 9]

$$\Phi = \sum\_{\mathbf{n} \in \mathcal{N}} \phi\_{\mathbf{n}} \mathbf{s}\_{\mathbf{n}}, \phi \in \mathbb{S}^{0}, \phi\_{\mathbf{n}} = \phi(\mathbf{x}\_{\mathbf{n}}), \mathbf{n} \in \mathcal{N}, \tag{45}$$

$$\mathbf{h} = \sum\_{\mathbf{e} \in \mathcal{E}} \mathbf{h}\_{\mathbf{e}} \mathbf{s}\_{\mathbf{e}}, \mathbf{h} \in \mathbb{S}^1, \mathbf{h}\_{\mathbf{e}} = \int\_{\mathbf{e}} \mathbf{h} \cdot \mathbf{dl}, \mathbf{e} \in \mathcal{E} \tag{46}$$

$$\mathbf{j} = \sum\_{f \in F} j\_f \mathbf{s}\_{\mathbf{f}}, \mathbf{j} \in \mathbb{S}^2, \quad \mathbf{j}\_{\mathbf{f}} = \int\_{\mathbf{f}} \mathbf{j} \cdot \mathbf{n} \mathbf{ds}, \mathbf{f} \in \mathbf{F} \tag{47}$$

$$\sigma = \sum\_{\mathbf{v} \in \mathcal{V}} \sigma\_{\mathbf{v}} \mathbf{s}\_{\mathbf{v}}, \rho \in \mathbb{S}^3, \sigma\_{\mathbf{v}} = \int\_{\mathbf{v}} \sigma \mathbf{d} \mathbf{v}, \mathbf{v} \in \mathcal{V} \tag{48}$$

The degrees of freedom ϕn, he, jf and ρ<sup>v</sup> are thus, respectively, values at nodes, circulations along edges, fluxes across facets or volume integrals, of the associated fields. This is a consequence of the base functions. The associated linear functionals,

**Figure 11.** *Geometric interpretation of the facet function s<sup>f</sup> (36) [6].*

as mentioned in the definition of finite elements, are thus respectively pointwise evaluations, line, surface and volume integrals.

### **4.5 Continuity of base functions across facets**

It can be proved that the function sn is continuous across facets. The same holds true for the tangential component of **s**<sup>e</sup> and for the normal component of **s**f. As for function sv, it is discontinuous. This property, called **conformity**, allows to take exactly into account interface conditions for fields used in the modeling of physical problems. For example, in electromagnetic problems, vector fields of S<sup>1</sup> can represent vector fields like magnetic field **h** or electric field **e** whose tangential components are continuous across interfaces between materials, and those of S<sup>2</sup> can represent fields like induction field **b** or current density field **j** whose normal components are continuous across interfaces between these materials.

#### **4.6 Spaces Si form a sequence**

The notion of **incidence** is first defined [4, 6, 9]:

The incidence of node n in edge e, denoted by i (n, e), is equal to 1 if n is the extremity of e, �1 if n is the origin of e, and 0 if n does not belong to e.

Next, the incidence of edge e in facet f is denoted by i(e, f). If e belongs to f, and if the ordered set of nodes of e appears as a direct subset in the circular set of nodes of f, then it is equal to 1. It is equal to �1 in the case of an inverse subset. If e does not belong to f, it is equal to 0.

Finally, the incidence of facet f in volume v is denoted by i(f, v). If f belongs to v, and if the normal to f, whose direction is given by the ordered set of nodes of f (right-hand rule), is outer to v, then it is equal to 1. It is equal to �1 in the case of an inner normal. If f does not belong to v, it is equal to 0.

Thanks to this notion, the following equalities can be proved,

$$\sum\_{\mathbf{e}\in\mathcal{E}}\mathbf{i}(\mathbf{n},\mathbf{e})\mathbf{s}\_{\mathbf{e}}=\mathbf{grad}\mathbf{s}\_{\mathbf{n}}\tag{49}$$

$$\sum\_{\mathbf{f}\in\mathcal{F}}\mathbf{i}(\mathbf{e},\mathbf{f})\mathbf{s}\_{\mathbf{f}}=\mathbf{curl}\,\mathbf{s}\_{\mathbf{e}},\tag{50}$$

$$\sum\_{\mathbf{v}\in\mathcal{V}}\mathbf{i}(\mathbf{f},\mathbf{v})\mathbf{s}\_{\mathbf{v}}=\mathbf{div}\,\mathbf{s}\_{\mathbf{f}}.\tag{51}$$

The following inclusions are then verified,

$$\operatorname{grad}(\mathbb{S}^0) \subset \mathbb{S}^1, \operatorname{curl}(\mathbb{S}^1) \subset \mathbb{S}^2, \operatorname{div}(\mathbb{S}^2) \subset \mathbb{S}^3. \tag{52}$$

Therefore, the spaces Si, i = 0 to 3, form a **sequence**, that can be schematized by the diagram in **Figure 12**.

These spaces can then constitute approximation spaces for some continuous spaces Fi, i = 0 to 3, which contain scalar and vector fields associated with electromagnetic fields. The associated finite elements can then be called **mixed elements**.

$$\mathbf{s}^{0} \xrightarrow{\mathbf{grad}} \mathbf{s}^{1} \xrightarrow{\mathbf{curl}} \mathbf{s}^{2} \xrightarrow{\mathbf{div}} \mathbf{s}^{3}$$

**Figure 12.** *The sequence of spaces S<sup>i</sup> .*

All the established properties of base functions are **valid for any collection of considered geometric elements**, that is, for any mixing of tetrahedra, hexahedra and prisms.
