1.7 Discrete Electromagnetics

Several issues related to the differential formulation of the Maxwell equations were presented in Section 1.2. Finally, this section introduces a concise way of formulating physical problems regarding the fiber bundle and algebraic topology concepts. Starting with the integral formulation and partially reinserting the geometrical objects expressed in vector calculus notation, but omitting the orientation reads:

$$\int_{\partial V} \ensuremath{\mathbf{B}} \cdot d\ensuremath{\mathbf{S}}$$ (1.51)

$$\int_V \varrho  dV$$ (1.52)

A better-suited representation, which directly references the oriented geometric object a quantity is assigned to, is given with the formalism of ordinary and twisted differential forms which can be seen as the continuous counterpart of cochains, as introduced in Section 1.4. For a brief introduction of this topic, a $p$ -dimensional differential form, or short $p$ -form, can be seen as the subject to integration on $p$ -dimensional domains [60,24]. If the domain is internally oriented, then the $p$ -form is called ordinary $p$ -form which is denoted by $\omega^p$ and the corresponding externally oriented $p$ -form is called twisted, denoted by $\tilde \omega^p$ . By the concept of a multivector (see Section 1.4), a $p$ -form is given as a linear function on the space of multivectors with values in an algebraic field. Then it follows that the pairing of a multivector, or $p$ -vector $\ensuremath{\mathbf{v}}_p$ , and a $p$ -form $\omega^p$ gives a value like the pairing of a chain and cochain [61,35], as given in Section 1.4.4. This analogy suggests the following representation of the pairing of a $p$ -vector and a $p$ -form:

$$\langle \ensuremath{\mathbf{v}}_p, \omega^p \rangle$$ (1.53)

The duality property, stated by $\langle c_p, c^p \rangle$ , between chains and cochains transfers directly to the continuous multivectors and $p$ -forms, introduced in Section 1.5. This is an important step towards a formal, consistent, and computationally manageable concept. A $p$ -form $\omega^p$ on a continuous domain ${\ensuremath{\Omega}}$ can then be correlated to its discrete counterpart on a cell complex, a cochain $c^p$ , since it associates a value $c^i$ with each cell $\ensuremath{\tau_{p}^{i}} \in {\ensuremath{\mathfrak{K}}}$

$$c^i = \int_{\ensuremath{\tau_{p}^{i}}} \omega^p$$ (1.54)

Another correspondence between cochains and $p$ -forms is given by the concept of the coboundary operator. As introduced in Section 1.4.2, the coboundary operator is defined to allow the transition from a topological equation of the form

$$\langle \partial c_{p+1}, a^p \rangle = \langle c_{p+1}, b^{p+1} \rangle$$ (1.55)

to the following relation between cochains:

$$\delta a^p = b^{p+1}$$ (1.56)

The continuous differential forms can then be related

$$\int_{D_{p+1}} d \omega^p := \int_{\partial D_{p+1}} \omega^p \quad \forall D_{p+1} \in D \subseteq \Omega$$ (1.57)

where $d$ is an operator transforming $p$ -forms into $p+1$ -forms. This operator is called the exterior differential and mimics the property of the coboundary operator by transforming a topological equation given in integral form

$$\int_{\partial D_{p+1}} \alpha^p = \int_{D_{p+1}} \beta^{p+1} \quad \forall D_{p+1} \in D$$ (1.58)

into

$$d \alpha^p = \beta^{p+1}$$ (1.59)

Given the properties of the coboundary operator $\delta$ , the exterior differential $d$ can be seen as the continuous counterpart [53] of $\delta$ . The following table depicts the correspondence between discrete and continuous concepts [35].

discrete setting		continuous setting
$p$ -cell	$\ensuremath{\tau_{p}^{}}$	${\ensuremath{\Omega}}_p$	$p$ -dimensional domain
boundary of a $p$ -cell	$\partial \ensuremath{\tau_{p}^{}}$	$\partial {\ensuremath{\Omega}}_p$	boundary of a $p$ -dimensional domain
$p$ -chain	$c_p$	$\ensuremath{\mathbf{v}}_p$	weighted $p$ -domain
$p$ -cochain	$c^p$	$\omega_p$	$p$ -differential form
pairing of $p$ -chain and $p$ -cochain	$\langle c_p, c^p \rangle$	$\int_{\ensuremath{\mathbf{v}}_p} \omega^p$	weighted $p$ -integral of a $p$ -form
coboundary operator	$\delta$	$d$	exterior differential operator

Based on these concepts, the local vector field representation $\ensuremath{\mathbf{B}}$ becomes an ordinary 2-form $b^2$ and the scalar field $\varrho$ a twisted 3-form $\tilde \varrho^3$ :

$$\ensuremath{\mathbf{B}} \rightarrow b^2$$ (1.60)

$$\varrho \rightarrow \tilde \varrho^3$$ (1.61)

The numbers and orientation give the dimension on which these quantities are to be integrated. The adjoint of the exterior differential as the boundary of a weighted domain is the generalized Stoke's theorem:

$$\int_{\ensuremath{\mathbf{v}}_{p+1}} d \omega^p = \int_{\partial \ensuremath{\mathbf{v}}_{p+1}} \omega^p$$ (1.62)

It has to be noted, that the given differential form expression is more general than the vector calculus notation due to the fact that the expression is valid for $\mathrm{div} \left( \right)$ , $\mathrm{grad} \left( \right), \mathrm{curl} \left( \right)$ and the discrete chain and cochain representations automatically express the type of the dimension with the general notion of:

$$\langle c_{(p+1)} , \delta c^{(p)} \rangle = \langle \partial c_{(p+1)} , c^{(p)} \rangle$$ (1.63)

Examples of $p$ -form complexes for differential operators encountered in different works [62,24] for vector analysis in three dimensions are denoted by:

$$0 \rightarrow \left \{ \textnormal{scalar functions} \right \} \x... ... \textnormal{field vector} \right \} \xrightarrow{\mathrm{curl} \left( \right)}$$ (1.64)

$$\left \{ \textnormal{flux vectors} \right \} \xrightarrow{\mathrm... ... \left( \right)} \left \{ \textnormal{volume densities} \right \} \rightarrow 0$$

The concept of constitutive links closes the gap between ordinary and twisted cochains with discrete links between them. Two different types can be obtained:

$$L_1: C^p({\ensuremath{\mathfrak{K}}}) \rightarrow C^q({\ensuremath{\mathfrak{\tilde K}}})$$ (1.65)

$$L_2: C^r({\ensuremath{\mathfrak{\tilde K}}}) \rightarrow C^s({\ensuremath{\mathfrak{K}}})$$ (1.66)

The inherently discrete computer implementation can now be equipped with all the necessary information and structure regarding the physical entities.