UDC 538.566

MATHEMATICAL PHYSICS

P. E. KRASNUShKIN

THE USE OF EVOLUTION GRAPHS OF FUNCTIONAL NETWORKS IN SOLVING BOUNDARY-VALUE PROBLEMS OF ELECTRODYNAMICS BY THE METHOD OF MATCHING

(Presented by Academician I. M. Vinogradov, June 17, 1965)

1. Let a volume \(V\), in which Maxwell’s equations for the complex amplitudes \(E\) and \(H\) hold, be divided into partial regions \(i=1, 2, 3,\ldots, P\). The solution of the first boundary-value problem of electrodynamics \((^{1,3})\) for a region \(i\) adjacent to other regions \(i'\) along surfaces \(S_j^{(i)}\), \(j=1, 2, 3,\ldots, M^{(i)}\), establishes a unique functional relation between the tangential components of the fields \(E_k^{(i)}\) and \(H_j^{(i)}\) on \(S_j^{(i)}\):

\[ \{H_j^{(i)}\}_{j=1}^{M}=\left(Y_{jk}^{(i)}\right)\{E_k^{(i)}\}_{k=1}^{M}, \tag{1} \]

where \(\left(Y_{jk}^{(i)}\right)\) is an \(M \times M\) matrix composed of the linear conductivity operators \((^1)\). The continuity equations for \(E_k\) and \(H_j\) on \(S_j^{(i)}\) will be

\[ E_{k_m}^{(i'')}=\mathscr{E}H_{j_m}^{(i')}, \qquad m=1,2,\ldots,R, \tag{2} \]

where \(\mathscr{E}\) is the identity-transformation operator. The set of partial regions is a functional network of \(M \times M\) cells (\(M\) input and \(M\) output channels), connected by ideal links described by the system of operator equations (1), (2). The solution of the first boundary-value problem for the volume \(V\) is now reduced to the solution of the network equations. Having represented (1), (2) by a two-color evolution graph according to the rules given in \((^2)\), we carry out the investigation and solution of (1), (2) graphically.

1. Consider the case of a partition (Fig. 1a) in which the network is a chain of adjacently interacting \(2 \times 2\) cells, described by \(2 \times 2\) matrices composed of conductivity operators \(Y_{11}^{(i)}\), \(Y_{12}^{(i)}\), \(Y_{21}^{(i)}\), and \(Y_{22}^{(i)}\), \(i=1,2,3,\ldots,R-1\). At the ends of the network two-channel cells \(i=0\) and \(i=R\) are connected, with input-conductivity operators \(Y^{(0)}=[Z^0]^{-1}\) and \(Y^{(R)}=[Z^{(R)}]^{-1}\). A time-harmonic \((e^{-i\omega t})\) external action is localized in the section \(S_{i+1}\) and is specified by the amplitude functions \(E_0\) and \(H_0\). The evolution graph of such a network is shown in Fig. 1b. For its construction an inversion of the arcs \(Y_{11}^{(i)}\), \(PR-(P=2, R=2)\) of the graphs of all cells, \(i=1, 2,\ldots,R-1\), has been carried out according to the rules \((^2)\). The right half of the graph in Fig. 1b has one path connecting the vertices \(4^{(i+1)}\) with \(1^{(i+1)}\) \((42142\ldots1421-431431431\ldots43141)\), transmitting states from left to right and then back. The left half of the graph in Fig. 1b also has one path connecting the vertices \(3^{(i)}\) with \(2^{(i)}\) \((314314\ldots431-421421\ldots4\ldots42)\), transmitting states from right to left and back. Both paths are loaded with feedback loops, and for computing the operators of the loaded paths \((^2)\) it is convenient to apply the recurrence formula (3), obtained directly from the graph of Fig. 1b, where \(\tilde{Y}_i\) is the path operator between the vertices \(1^{(i)}\) and \(4^{(i)}\), and \(\tilde{Y}_{i+1}\) is that between \(1^{(i+1)}\) and \(4^{(i+1)}\),

\[ \tilde{Y}_i=Y_{11}^{(i)}+Y_{21}^{(i)}\left[\tilde{Y}_{i+1}-Y_{22}^{(i)}\right]^{-1}Y_{21}^{(i)}. \tag{3} \]

Let us denote the obtained path operators between \(1^{(i+1)}\) and \(4^{(i+1)}\), and between \(3^{(i)}\) and \(2^{(i)}\), respectively, by \(\widetilde{Y}_i^{(R)}\) and \(\widetilde{Y}_i^{(0)}\) (these are \(Y^{(R)}\) and \(Y^{(0)}\), recalculated from the ends of the chain in Fig. 1a to the section \(i, i+1\)). Now the graph in Fig. 1b can be replaced by the graph in Fig. 1c, and with its aid one can determine the \(2\times2\) matrices of the operators \(Z_1\) and \(Z_2\), relating

\[ \left\|\begin{array}{c} E_1^{(i+1)}\\ H_1^{(i+1)} \end{array}\right\| \quad \text{and} \quad \left\|\begin{array}{c} E_2^{(i)}\\ H_2^{(i)} \end{array}\right\| \quad \text{to} \quad \left\|\begin{array}{c} E_0\\ H_0 \end{array}\right\|: \]

\[ Z_1= \begin{pmatrix} -\widetilde{Z}_i^{(R)}\Pi''\widetilde{Y}_i^{(0)} & \widetilde{Z}_i^{(R)}\Pi''\\ \Pi''\widetilde{Y}_i^0 & -\Pi'' \end{pmatrix}, \qquad Z_2= \begin{pmatrix} \Pi' & \Pi'\widetilde{Z}_i^{(R)}\\ \widetilde{Y}_i^{(0)}\Pi' & \widetilde{Y}_i^{(0)}\Pi'\widetilde{Z}_i^{(R)} \end{pmatrix}; \tag{4} \]

here \(\Pi''=(\mathscr{E}+\widetilde{Y}_i^0\widetilde{Z}_i^R)^{-1}\) and \(\Pi'=(\mathscr{E}+\widetilde{Z}_i^R\widetilde{Y}_i^0)^{-1}\) are the feedback-loop operators of the circuit \(6145326\) with contour operators \((-\widetilde{Y}_i^0 Z_i^R)\) and \((-\widetilde{Z}_i^R Y_i^0)\), counted from vertices 5 for \(\Pi'\) and 6 for \(\Pi''\). The graph in Fig. 1c gives the solution for forced oscillations in the volume \(V\). They exist if the operator of the circuit \(6145326\) differs from \(\mathscr{E}\), which follows from (2). Otherwise, natural oscillations exist. To find the fields \(E^{(i)}\) and \(H^{(i)}\) in any section \(i\), let us replace, by inverting the arcs, the matrices of the conductance operators \((Y_{jk}^{(i)})\) (the graph in Fig. 1b) by the Breisig matrices \((A_{jk}^{(i)})\) (1), explicitly relating \(E_2^{(i)}\) and \(H_2^{(i)}\) to \(E_1^{(i)}\) and \(H_1^{(i)}\).

Fig. 1

Combining the states \(E_1^{(i)}\) and \(H_1^{(i)}\) into one, determined by the vector-function

\[ \left\|\begin{array}{c} E_1\\ H_1 \end{array}\right\|, \]

we replace the graph in Fig. 1b by the simpler graph in Fig. 1d. With its help it is easy to construct expressions relating

\[ \left\|\begin{array}{c} E_1^{(i')}\\ H_1^{(i')} \end{array}\right\| \]

to \(E_0^{(i+1)}\) and \(H_0^{(i+1)}\) in any section \(i', i'+1\) (see (1)).

1. If the cells into which the volume \(V\) is divided are identical and are loaded at the ends by the cells 0 and \(R\) with \(Y^0\) and \(Y^R\), determined from equation (3), where

\(\widetilde{Y}_i=\widetilde{Y}_{i+1}\) (for the case \(Y_{11}=Y_{22}\) and \(Y_{12}=Y_{21}\), \(Y^{0,R}=Y_{11}\pm Y_{12}\)), then it makes sense to carry out further investigation of the chain by the method of normal waves [4]. The problem of finding the normal waves can be reduced to the problem of the natural oscillations of a cell closed by a conditional waveguide according to the periodicity condition

\[ E_2=-e^{-i\psi}E_1,\qquad H_2=-e^{-i\psi}H_1. \]

As a result we obtain the closed graph of Fig. 2a. Reading from it the system of equations for \(E_1\), we obtain

\[ (Y_{11}+Y_{22})E_1= \]
\[ =(e^{i\psi}Y_{21}+e^{-i\psi}Y_{12})E_1, \]

where \(Y_{ik}\) are the conductance operators of the cell.

1. Let us apply the stitching method to such a cell and divide it into partial regions (for example, according to Fig. 2b) with the prescribed matrices of the conductance operators (1). Joining the cells \(i=I\) and \(i=II\) according to the periodicity and continuity conditions, we obtain the graph of Fig. 2c. It consists of a single isolated contour. Computing the operator of this contour and equating it to \(\mathcal{E}\), we obtain a system of homogeneous equations

\[ \begin{pmatrix} Y_{11}^{(I)}-Y_{22}^{(II)} & Y_{12}^{(I)}+e^{i\psi}Y_{21}^{(II)}\\ Y_{21}^{(I)}+Y_{12}^{(II)}e^{-i\psi} & Y_{22}^{(I)}-Y_{11}^{(II)} \end{pmatrix} \times \left\| \begin{matrix} E_1\\ E_2 \end{matrix} \right\|=0 \tag{5} \]

\(E_1\) and \(E_2\) are functions of the fields \(E_\tau\) in sections \(\langle 1\rangle\) and \(\langle 2\rangle\). In Fig. 2c another division of the cell into partial regions \(I\) and \(II\) is given. Region \(I\) corresponds to a 6-channel, and \(II\) to a 2-channel, cell of the network described by the matrices (1). The graph of such a division is presented in Fig. 2d. It also consists of a single isolated contour. Equating the operator of this contour to \(\mathcal{E}\), we obtain an equation for \(E_3\) in section \(\langle 1\rangle\):

\[ \bigl[(Y_{11}-Y_{22}-e^{-i\psi}Y_{12}+e^{i\psi}Y_{21})(Y_{31}-e^{-i\psi}Y_{32})^{-1}(Y_{00}-Y_{33})+ \]
\[ +(Y_{13}+e^{i\psi}Y_{33})\bigr]E_3=0. \tag{6} \]

(5) and (6) are valid for regions \(I\) and \(II\) of arbitrary shape. The conditions of their compatibility are the dispersion equations relating the wave numbers of the normal waves \(\psi\) with the frequency \(\omega\): \(\psi=f[\omega]\). For numerical calculations they must be algebraized. If \(I\) and \(II\) are cylindrical waveguide sections with arbitrary contours \(\Gamma_I\) and \(\Gamma_{II}\) of the cross sections (Fig. 2e), then, introducing the coordinates \(r,\varphi,z\), we represent \((Y_{j,k}^{1,2})\) by diagonal matrices in the basis of normal waves traveling along the \(z\)-axis. In this case \(E\) and \(H\) are expanded in terms of the forms of the normal waves \(e_{m,n}^{(1)}(r,\varphi)\) and \(e_{m,n}^{(2)}(r,\varphi)\), traveling along \(z\) in \(I\) and \(II\) with wave numbers \(\alpha_{m,n}\) and \(\gamma_{m,n}\), respectively. If the fields \(E_1\) and \(E_2\) are approximated by Fourier series in the basis \(\{\mathcal{E}_s\}_s\), different from \(\{e_{m,n}\}_{m,n}\), i.e.

\[ E_1=\sum_{s=1}^{N}K_s^{(1)}\mathcal{E}_s \quad\text{and}\quad E_2=\sum_{s=1}^{N}K_s^{(2)}\mathcal{E}_s(r,\varphi), \]

then the scalar pro-

admittances \((Y_{j,k}\mathscr E_s,\mathscr E_{s'})\), which are elements of the matrices \(\|Y_{jk}\|\), will be convolutions over the indices \(m\) and \(n\) (see formula (8)). The dispersion equation of the waveguide composed of the cells of Fig. 2e, in matrix form, will be

\[ \det\begin{bmatrix} a-\alpha-\beta\cos\psi & -\beta\sin\psi\\ -\beta\sin\psi & a+\alpha+\beta\cos\psi \end{bmatrix}=0, \tag{7} \]

where \(a\), \(\alpha\), and \(\beta\) are square \(N\times N\) matrices with elements

\[ a_{s,s}=\sum_{m,n}^{\infty}\frac{\operatorname{cth}\gamma_{m,n}d}{\gamma_{m,n}}\, \beta_{m,n,s}\beta_{m,n,s'} + \sum_{m,n}^{\infty}\frac{\operatorname{cth}\alpha_{m,n}\tau}{\alpha_{m,n}}\, \alpha_{m,n,s}\alpha_{m,n,s'}, \tag{8} \]

\[ \alpha_{s,s'}=\sum_{m,n}^{\infty} \frac{\alpha_{m,n,s}\alpha_{m,n,s'}}{\alpha_{m,n}\operatorname{sh}\alpha_{m,n}\tau}, \qquad \beta_{s,s'}=\sum_{m,n}^{\infty} \frac{\beta_{m,n,s}\beta_{m,n,s'}}{\gamma_{m,n}\operatorname{sh}\gamma_{m,n}d}, \]

where \(\alpha_{m,n,s}\) and \(\beta_{m,n,s}\) are the Fourier coefficients of the expansion of \(\mathscr E_s\) in \(e_{m,n}^{(1)}\) and \(e_{m,n}^{(2)}\), respectively. Below we consider cases in which branches \(I\) and \(II\) have circular cross sections (Fig. 2ж). Restricting ourselves to the case of axially symmetric waves of type TH, we obtain, for \(\alpha_{m,n,s}=\alpha_{m,s}\) and \(\beta_{m,n,s}=\beta_{m,s}\), the following results.

If \(\{\mathscr E_s\}\) coincides with one of the bases of normal waves, for example with \(\{e_m(r/a)\}\), then

\[ \alpha_{m,s}=\delta_{m,s}= \begin{cases} 1, & m=s,\\ 0, & m\ne s; \end{cases} \qquad \beta_{m,s}= \frac{2ab\rho_{0,m}J_0(\rho_{0,m}a/b)J_1(\rho_{0,s})} {J_1^2(\rho_{0,m})(\rho_{0,s}^2b^2-\rho_{0,m}^2a^2)}. \tag{9} \]

To take into account the electrostatic singularity of \(E\) at the sharp edges of the cell, we choose the basis

\[ \mathscr E_s=(r/a)/[1-(r/a)^2]^{1/p}; \]

then

\[ \beta_{m,s}= \frac{2\sqrt{\pi a^2}}{b\,|J_1(\rho_{0,m})|} \sum_{r=0}^{s-1} (-1)_{s-1}^{r} C_r^r\,2^{r-1/p} \Gamma\!\left(r-\frac{1}{p}+1\right) I_{r-1/p+2}\!\left(\rho_{0,m}\frac{a}{b}\right). \tag{10} \]

\[ \alpha_{m,s}=\beta_{m,s}\quad (b=a). \]

For the case of subdividing the cell according to Fig. 2ж, when \(Y_{12}=Y_{21}\), \(Y_{23}=Y_{32}\), \(Y_{13}=Y_{31}\), \(Y_{11}=Y_{22}\), \(Y_{13}=Y_{23}\), and equation (6) takes the form \([Y_{13}Y_{12}^{-1}Y_{13}-Y_{33}+Y_0]E_3=0\), it is convenient in region \(I\) to use as a basis the forms of normal waves traveling along \(r\), with wave forms \(\exp\{-i\beta_m z\}\), \(\beta_m=(\psi+2\pi m)/D\) (wave numbers \(\mu_m=\sqrt{k_0^2-\beta_m^2}\)), and for \(II\) the basis of the forms of normal waves \(\cos \pi pz/d\), likewise traveling along \(r\), with wave numbers \(\chi_p=\sqrt{k_0^2-(p\pi/d)^2}\). As a result of algebraization, we obtain the dispersion equation in the form

\[ \det\left[ \sum_{m=-\infty}^{\infty} \frac{J_1(\mu_m a)}{\mu_m J_0(\mu_m a)} \beta_{m,s}\beta_{m,s'} - \sum_{p=0}^{\infty} \frac{D(a',b)}{\chi_p D(a,b)} \alpha_{p,s}\alpha_{p,s'} \right]=0, \tag{11} \]

where \(D(a,b)\) and \(D(a',b)\) are given in (5).

V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
24 V 1965

REFERENCES

1. P. E. Krasnushkin, Radiotechnics and Electronics, 10, 6 (1965).
2. P. E. Krasnushkin, DAN, 165, No. 6 (1965).
3. Ya. N. Fel’d, Foundations of the Theory of Slotted Antennas, 1948.
4. P. E. Krasnushkin, The Method of Normal Waves as Applied to Waveguides and Their Algebraic Transformations, Moscow, 1945.
5. P. E. Krasnushkin, DAN, 138, No. 5, 1055 (1961).