- Discretization

For a sheet region *U*, we consider a cellular paving K with nodes N , edges E, and faces F. We write the relations among nodes, edges, and faces as

$$
\partial \mathcal{C}\_i = \sum\_{j=1}^{|\mathcal{N}|} n\_j \Delta^j\_{\ i} \quad (i = 1, 2, \cdots, \ |\mathcal{E}| ), \tag{87}
$$

$$\partial f\_i = \sum\_{j=1}^{|\mathcal{E}|} e\_j \Pi^{\bar{j}}\_{\bar{i}} \quad (i = 1, 2, \cdots, |\mathcal{F}|) \tag{88}$$

for *ei* ∈ E and *fi* ∈ F. The totality of *p*-forms on *U* is denoted by *Cp*(*U*). We can define *φp* : *Cp*(*U*) → *Cp*(K) as

$$\left[\phi^p(\omega)\right](\mathfrak{c}) = \int\_{\mathfrak{c}} \omega \tag{89}$$

for *ω* ∈ *Cp*(*U*) and all *c* ∈ *Cp*(K). Here, *φp* makes a continuous field *ω* discretized. Now, we obtain a commutative diagram:

$$\begin{array}{c} \mathcal{C}^{0}(\mathcal{U}) \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{C}^{1}(\mathcal{U}) \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{C}^{2}(\mathcal{U})\\ \bigcup^{\mathfrak{g}^{0}} \qquad \bigcup^{\mathfrak{g}^{1}} \qquad \bigcup^{\mathfrak{g}^{2}}\\ \mathcal{C}^{0}(\mathcal{K}) \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{C}^{1}(\mathcal{K}) \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{C}^{2}(\mathcal{K}) \end{array} \tag{90}$$

Commutativity can be checked as follows. We can calculate for all *f* ∈ *<sup>C</sup>*<sup>0</sup>(*U*), *c* ∈ *<sup>C</sup>*1(K) as

$$\left[\phi^1(\mathrm{d}f)\right](\mathrm{c}) = \int\_{\mathcal{c}} \mathrm{d}f = \int\_{\partial \mathcal{c}} f = \left[\phi^0(f)\right](\mathrm{\partial}\mathbf{c}) = \left[\mathrm{d}\phi^0(f)\right](\mathrm{c}),\tag{91}$$

where we used Stokes' theorem. Similarly, we obtain *φ*<sup>2</sup>(d*α*) = <sup>d</sup>[*φ*<sup>1</sup>(*α*)], for all *α* ∈ *<sup>C</sup>*<sup>1</sup>(*U*). The commutative diagram of Equation (90) indicates that the discretization by *φ* keeps the algebraic structure of d.

*Symmetry* **2019**, *11*, 1336

Using *φ*, we can discretize an electric field (untwisted 1-form) *E* as *V* = ∑|E| *i*=1 *Vie<sup>i</sup>* with *Vi* = *ei E*. Moreover, KVL (d*E* = 0) is discretized as

$$dV = 0,\tag{92}$$

which is explicitly expressed as ∑|E| *j*=1 *Vj*Π*ji* = 0. Therefore, Π*ji* represents the discretized rotation of the field.

For a current density *K* (twisted 1-form), we should consider the integration on the dual lattice. The set of all twisted *p*-forms on *U* is represented as *C* ˇ *<sup>p</sup>*(*U*). We have

$$\left[\dot{\Phi}^p(\omega)\right](\mathfrak{k}) = \int\_{\mathcal{L}} \check{\omega}$$

for *ω*<sup>ˇ</sup> ∈ *C* ˇ *p*(*U*) and all *c*ˇ ∈ *Cp*(K). Then, another diagram similar to Equation (90) is obtained:

$$\begin{array}{c} \dot{\mathsf{C}}^{0}(\mathsf{U}) \xleftarrow{\mathrm{d}} \stackrel{\mathrm{d}}{\longleftarrow} \dot{\mathsf{C}}^{1}(\mathsf{U}) \xrightarrow{\mathrm{d}} \mathsf{d}^{2}(\mathsf{U})\\ \downarrow \dot{\mathsf{\varphi}}^{0} & \downarrow \dot{\mathsf{\varphi}}^{1} & \downarrow \dot{\mathsf{\varphi}}^{2}\\ \mathsf{C}^{0}(\mathsf{K}^{\star}) \xrightarrow{\mathrm{d}} \mathrm{C}^{1}(\mathsf{K}^{\star}) \xrightarrow{\mathrm{d}} \mathrm{C}^{2}(\mathsf{K}^{\star}) \end{array} \tag{94}$$

Remembering (1)−<sup>1</sup> : *C*<sup>1</sup>(K) → *<sup>C</sup>*1(K), we can define *I* = (1)−<sup>1</sup>*φ*ˇ1(*K*) = ∑|E| *i*=1 *Iiei* with *Ii* = *e*ˇ*i K* = (<sup>1</sup>)−1*e<sup>i</sup> K*. Now, the discretized KCL is obtained as

$$
\partial I = 0,\tag{95}
$$

for which component representation is ∑|E| *j*=1 Δ*ijIj* = 0. Therefore, Δ*ij* indicates the discretized minus divergence.

For the discretization of Ohm's law, we interpolate *E* from *Ei* as

$$E \approx \sum\_{i=1}^{|\mathcal{E}|} E\_i w\_{c\_i \nu} \tag{96}$$

where {*wei*} are called interpolation forms. As the interpolation forms {*wei*}, so-called Whitney forms can be used [62,79–82]. Now, Ohm's law is discretized as

$$I^i = \sum\_{j=1}^{|\mathcal{E}|} \left( \int\_{\mathcal{E}\_i} G \star w\_{\mathcal{E}\_j} \right) E\_j. \tag{97}$$


Now, we establish correspondence between Keller–Dykhne duality and circuit duality. These two dualities are related through the diagram shown in Figure 35. In this section, we prove the commutativity of the two paths (1) and (2) in the figure.

**Figure 35.** Relation between Keller–Dykhne duality and circuit duality.

(1) For *I* and *V* discretized from *K* and *E*, we can consider the dual circuit with current *I* := *<sup>G</sup>*ref(<sup>1</sup>)−<sup>1</sup>(*V*) and voltage *V* := *R*ref 1 (*I*) distributions for a circuit on K:

$$I^\star = G\_{\text{rod}} \sum\_{i=1}^{|\mathcal{E}|} V\_i(\star^1)^{-1}(\mathcal{e}^i) = G\_{\text{rod}} \sum\_{i=1}^{|\mathcal{E}|} V\_i \mathbb{X}\_i \tag{98}$$

$$V^\star = R\_{\text{ref}} \sum\_{i=1}^{|\mathcal{E}|} I^i \star\_1 \left( \mathfrak{e}\_i \right) = R\_{\text{ref}} \sum\_{i=1}^{|\mathcal{E}|} I^i \bar{\mathfrak{e}}^i. \tag{99}$$

(2) On the other hand, we discretize *E* = *R*ref*K* and *K* = −*G*refΩ*E* in a dual mesh K. We need to choose a specific orientation of the plane, and (*e*<sup>ˇ</sup>*i*) is regarded as an inner-oriented edge in K. Here, we introduce the -conjugate operation to give an outer-oriented dual edge as *ei* = *e*ˇ*i*. The dual edge (*e*<sup>ˇ</sup>*i*) is outer-oriented, and represented as (*e*<sup>ˇ</sup>*i*) = −*ei* (Figure 36), which reflects the complex algebraic structure of the plane. Then, discretized *E* and *K* are given as

$$\left(\int\_{\left(\delta^i\_i\right)\_{\mathcal{O}}} E^\star \right) (\delta^i)\_{\mathcal{O}} = R\_{\text{ref}} \left(\int\_{\delta^i\_i} K \right) (\delta^i)\_{\mathcal{O}} = (V^\star)\_{\mathcal{O}\prime} \tag{100}$$

$$\left(\int\_{\left(\ell\_i^{\prime}\right)\_{\mathcal{Q}}\ast} K^{\star}\right) \left(\mathfrak{k}\_i^{\prime}\right)\_{\mathcal{Q}} = \left(\int\_{-\varepsilon\_i^{\prime}} (K^{\star})\_{\mathcal{Q}}\right) \left(\mathfrak{k}\_i^{\prime}\right)\_{\mathcal{Q}} = \mathfrak{G}\_{\mathsf{ref}} \left(\int\_{\varepsilon\_i} E\right) \left(\mathfrak{k}\_i^{\prime}\right)\_{\mathcal{Q}} = (I^{\star})\_{\mathcal{Q}}.\tag{101}$$

These equations indicate the commutativity of the diagram shown in Figure 35. Thus, Keller–Dykhne duality corresponds to circuit duality through discretization.

**Figure 36.** Interchange between a primary mesh and dual mesh.
