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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[1] In this paper, we consider new solutions and algorithms for initial boundary-value problems in the electromagnetic theory of open waveguide resonators. The approaches are based on a description of the scattering properties of such resonators in terms of the transform operators for an evolutionary basis of the non-stationary signal, that have the same significance in the course of analyses in time domain as generalized scattering matrixes in frequency domain. All suggested approaches imply using mathematically correct computational procedures at the key stages of forming the solution, in particular, the analytical regularization method—the semi-inversion method.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[2] In this paper we describe the complicated and canonic problems and accurate solutions to them that are to create the “basis” for Time Domain (TD) system of mathematical modeling of transmission lines (resonant phenomena based units like converters, filters, hermetic matching windows, multiplexes, changers of waveguide dimensions etc) that is complementary to the developed by authors [Shestopalov et al.,1986a, 1986b; Sirenko et al., 2000; Yashina, 1999; Pochanina et al., 1991] and references to them) in Frequency Domain (FD). The corresponding two parts of algorithm sets (algorithms for analysis in Time and Frequency Domain) being based on rigorous mathematical models, are to create together the unique system, providing simulation of rather complicated, requiring specific accuracy, resonant electromagnetic phenomena that is only possible when carried out in parallel and interacting both in TD and FD.

[3] The partial analytical inversion of the operator by regularization of the problem is a powerful classical mathematical tool. Its gradual and well-validated application in course of developing efficient methods of the solution to homogeneous (spectral) and inhomogeneous problems of the wave diffraction theory has allowed to almost solve the problem of an adequate electromagnetic modeling for a number of canonic open resonance structures (gratings, waveguide discontinuities, open-ended screens; see, e.g. books [Shestopalov et al., 1986a, 1986b] and references to them). The mathematical models of the analytical regularization methods and other rigorous analytical-numerical methods of FD have some incontrovertible advantages had been shown being worth by extensive and intensive studies of the physic nature of the resonance effects and phenomena. Unfortunately they are suitable only for a limited number of idealized scatterers having sufficiently simple geometry. The scope of problems that can be solved with these methods can be considerably enhanced by using the approaches that are based on the algorithms of the method of generalized scattering matrices. Due to these approaches, the complicated structures, whose elements are situated within a regular interaction domain of a finite length, can be efficiently analyzed, just as their separate parts. The interaction is accounted for by solving an additional, elementary, from the point of view computational complexity, problem—the canonical Fredholm set of equations with entries of the matrix operator exponentially decreasing along both the lines and columns.

[4] Method of generalized scattering matrix is also well-known in the diffraction theory, its advantages are pronounced most well only if they are thoroughly matched with the algorithms of rigorous analytical-numerical frequency domain methods, particularly, with the algorithms of the analytical regularization methods that are employed to create libraries of “elementary” units in the advanced modeling systems.

[5] Suggesting direct analogues of these powerful methods for solving problems in time domain, we hope that here in TD they will acquire same significance as their prototypes in FD. For being specific we consider as examples initial boundary-value problems for circular and coaxial waveguides and restrain the study to the simplest case of excitation of such structures by symmetrical TE0p electromagnetic waves. Thus, the study of numerical implementation of the solution and several tests of the algorithm that are of principal importance for generic structures, are discussed in the framework of the most popular canonic problem that is a waveguide bifurcation.

2. Evolutionary Basis of the Signal and Transform Operators

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

2.1. Essential Definitions

[6] Let us first consider a model initial boundary-value problem for a certain type of discontinuity in waveguide of arbitrary cross section. (see Figure 1a). Formally this 3D scalar problem corresponds to the propagation of acoustical waves. The results derived in course of its solving can be easily extended for the case of electromagnetic waves propagation. We have chosen it as a model problem only because the 3D vector electromagnetic problems are quite cumbersome for the description of rather general principal questions. In the same time the description of 2D scalar problems would restrict the class of problems that are planned for consideration significantly.

image

Figure 1. Modeled problems: (a) certain discontinuity in arbitrary waveguide and (b) compound waveguide discontinuity that is connection of generic structure via regular waveguide section.

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[7] Function U (g, t) describing wave scattering process is subject to the equations

  • equation image

We assume that the finite in the domain equation image functions φ(g), F(g, t), Ψ(g), ε(g) − 1 and σ(g) satisfy the conditions of the theorem about single-valued solvability of (1) in the energy space (Sololev's space) W21 (QT), QT = Q × (0, T), T < ∞ [see Ladyzhenskaya, 1985]. Here L is the differential elliptical operator of second order that takes in the local cylindrical coordinates xj, yj, zj the form ∂2/∂zj2 + equation image; M is the differential operator not higher than of the first order; Sz is the boundary of domain Q that is constituted by two semi-infinite regular (by zj > 0, j = 1, 2) waveguides that are bound by a compact discontinuity; g = {x, y, z} ; time t has the length dimension; the real functions ε(g) ≥ 1 and σ(g) ≥ 0 determine the influence of the discontinuity on the excitation propagation velocity and its dissipative characteristics.

[8] In a particular case of an infinite regular waveguide (section int S of the domain Q by the planes zj = const is constant along the axis z, and ε(g) − 1 = σ(g) ≡ 0), let us write the solution to (1) like

  • equation image

where the sequence of functions v(z, t) = {vn (z, t)} satisfy the equations

  • equation image

and the initial conditions

  • equation image

and {μn (ρ, ϕ)} and {λn} are sets of the eigen functions and eigen values of the homogeneous boundary-value problem

  • equation image

that is derived from (1) by separating the cross coordinates x, y (S is the boundary of the domain intS and the supposed properties of operators Lx, y and M are so that the function system {μn (x, y)} constitute an orthonormal basis in the corresponding plane domain). Here an (z, t), bn (z), cn (z) are the Fourier coefficients of functions F (g, t), φ (g), Ψ(g), by expanding the latter into series over the basis system {μn (x, y)}; and {n} is an ordered set of numbers n.

[9] Going to the generalized definition of the Cauchy problems (3), (4), see [Vladimirov, 1971] and using the fundamental solution G(λ; z, t) = (−1/2) χ (t − ∣z∣) J0 [λ (t2z2)1/2] of operator D(λ) (by definition D(λ)[G(λ)] = δ(z)δ(t)), we yield

  • equation image

Here the asterisk stands for the convolution operation, fn (z, t) = an(z, t) − δ(1) (t)bn(z) − δ(t)cn(z), χ is the Heaviside function, Jm—Bessel function, δ(m) is the generalized derivative of the delta-function of the order m. Equations (2) and (5) demonstrate the general and specific, corresponding to the given sources, form of wave propagating in a regular waveguide. Its alteration in space and time are determined exclusively by the system v (z, t) (or {Gn) * fn}) whose properties allow it to serve as a universal evolutionary basis of any signal on any finite section of a regular guiding structure.

[10] Let the wave of the type (2) serve as the excitation wave to the open waveguide resonator shown in Figure 1a. The excitation field equation image assumed to be non-zero only in the waveguide A that is regular for all z1 > 0. The field reaches the left boundary of the discontinuity lying in the plane z1 = 0 after the time t = 0. The secondary field that is induced in the regular semi-infinite waveguides A and B and propagates toward the increasing values of z1 and z2 we present like

  • equation image

Through the relations

  • equation image

we introduce the boundary (at the boundaries of the discontinuity zj = 0; δmn is the Kronecker delta) transform operators RAA and TBA of the evolutionary basis of the non harmonic wave arriving from the left, from the waveguide A:

  • equation image

The entries RnmAA (t − τ) and TnmBA (t − τ) of these operators define the space-time distribution of the incident wave energy Ui (g, t) in the reflected and the passed through the waveguide resonator field. The connection of the domains is defined in following way: the upper index from the domain whose identifier stands on the right to the domain with the identifier on the left. The connection of the modes: inferior index; on the right is the mode number of the incident wave, on the left is the mode number of the secondary field.

[11] Obvious that RAA and TBA operating in the domain of the signal's evolutionary basis are the characteristics of the waveguide unit itself and just sum up properly the results of a multiple elementary disturbances that can constitute any primary signal Ui (g, t). Thus, if vn1 (0, t) = δnp δ (t − η), where p is an integer and η > 0, we have wn1′ (0, t) = RnpAA (t − η) and wn2′ (0, t) = TnpBA (t − η). The use of such an abstract, physically impossible signal is justified by the methodological backgrounds, namely through this signal the structure gets elementary “excited” that enables one to extract from the generated field the “pure” components RnpAA (t − τ) and TnpBA (t − τ) of the transform operators. The elements of the matrix-functions RAA and TBA can be certainly computed in a different way by searching the sequence of values p in the excitations waves with Ui (g, t), with vn1 (z1, t) = δnpvn1 (z1, t), n ∈ {n(1)}, where vn1 (z1, t) are now sufficiently arbitrary functions. From (7) follows the relation between the amplitudes wnj′(0, t) of the secondary field corresponding to the fixed value of p and the sought values:

  • equation image

After its inversion using the operation method we obtain

  • equation image

where L and L−1 are the direct and inverse Laplace transforms. There are many ways to implement the corresponding presentation with the minimal error and acceptable computational efforts. Properly chosen function vp1 (0, t) is a crucial factor here. In an ideal case vp1 (0, t) has to have such a form that the necessary integral transformations could be done analytically.

[12] The operators RAA and TBA are the boundary ones, they determine all the characteristics of the transient processes directly at the discontinuity boundary of a regular guiding structure. The secondary field, departing from this boundary, propagates “freely” in the semi-infinite regular channels and in course of it becomes deformed. The space-time amplitudes of the modes {wnj (zj, t)} (signal's evolutionary basis) varies in time and distance in different way for various values of n and j. These changes can be described by using the diagonal transport operators ZA (z1) and ZB (z2) that are subject to the rule

  • equation image

The structure of these operators is detailed by the formula

  • equation image

that represent the common feature of the solution to the homogeneous equations of the type (3) at the semi-axes zj ≥ 0. The solutions that satisfy the zero initial conditions and are free from any components propagating along the decreasing zj. This formula is derived through the integral Laplace transform over t or the Fourier cosine transform over zj [Maikov et al., 1986]. Actually, on the assumption that at the initial moment of time t = 0 the excitation wave Uj (g, t) has not yet reached the discontinuity boundary in the plane z1 = 0, we obtain for the amplitudes {wnj (zj, t)} a sequence of homogeneous initial boundary value problems:

  • equation image

By using in (11) the Fourier cosine transform over zj at the semi-axes zj ≥ 0 (image [LEFT RIGHT ARROW] original)

  • equation image

we obtain the following Caushy problems for the images equation imagenj (ω, t):

  • equation image

Here we had taken into account that the waves Ujs (zj, t) are free of components propagating along the decreasing zj. The components traveling toward zj = ∞ at any finite moment of time t = T are zero for sufficiently large zj. The convolution of the fundamental solution equation image (λ; t) = χ (t−1 sin λt of operator equation image(λ) with the right-hand part of (12) enables us to write equation imagenj (ω, t) in the form

  • equation image

Formula (10) that defines the diagonal transport operators describing the field variation by a “free” propagation of the wave Ujs (g, t) in the finite sections of the regular waveguides A and B we obtain after transforming (13) to the originals wnj(zj, t)).

2.2. The Method of Transform Operators For Evolutionary Basis of Signal

[13] The above-introduced operators describe on the whole the scattering properties of discontinuities that are being excited from the channel A. Let us determine now, similarly to RAA and TBA, the transformation operators RBB and TAB of the evolutionary basis of the wave equation image arriving from the right, through the waveguide B, at the boundary z2 = 0:

  • equation image

or in operator form

  • equation image

Assuming the sets of operators R and T to be known for separate simple discontinuities, we develop an algorithm for analyzing the scattering properties of a compound unit constituted by such discontinuities. In the situation modeled that is shown in Figure 1b, the unit contains two sequent discontinuities I and II connected by a section B of the regular waveguide of a finite length L, and is excited by a wave of the type (2) arriving at the boundary z1(I) = 0 from the waveguide A. Remaining in the frameworks of the above accepted terms, (the obvious changes are due to two different types of discontinuities I and II), we present the solution of the corresponding initial boundary value problem in the regular domains A, B and C in the form

  • equation image

The first group of components here corresponds to the waves propagating from the left to the right, the second groups - to the wave moving from the right to the left (see Figure 1b). Following the expressions (7)(10), (14), and (15), and using the denotations

  • equation image

we obtain the following set of operator equations:

  • equation image

This set clearly demonstrates all the stages of forming a feedback of a complex structure to the excitation by the signal equation image with the evolutionary basis v1(z1(I), t) = {vn1(z1(I), t)}, or simply the signal ν1(I). Thus, e.g. the first equation of the set is a sum of signals first of which appears as a result of the reflection of the primary signal ν1(I) from the discontinuity I, and the second one is produced by the signal w1(II) that has passed through this discontinuity and the regular space B.

[14] By eliminating from (16) one unknown function, we arrive to an operator equation of the second kind with respect to the unknown vector-function w2′(I)

  • equation image

and to the “recalculating” formulas determining all the components of the field generated by the unit. The operator in the right-hand part of (17) influences the sought vector-functions w2′(I) whose argument τ is strictly less than the time t in the argument of the same function standing on the left (the lag effect that is conditioned by the finite velocity of the signal propagation). Hence, the numerical solution of the final equation can be obtained within the standard scheme of marching over time layers, thus, the initial “complicated” problem is equivalently re-formulated to the form allowing the direct inversion by using the conventional methods of computational mathematics. The complex unit is reduced to the category of elementary basic blocks after calculating the elements of boundary operators by the formulas (7) and (14).

[15] Let us return once more to the expressions (7)(10), (14), and (15) to prevent the eventual questions concerning the definition of boundary transformation operators. The point is that these operator act in a different way if compared with their analogues in the frequency domain. Theoretically, one can choose the traditional way in the time domain establishing relations by the boundary transformation operators, the pair “field [RIGHTWARDS ARROW] field” and not the pair “field [RIGHTWARDS ARROW] the field derivative toward the propagation”, as it was done in (7) and (14). Naturally, in such case we would have obtained RppAA (t − τ) = −δ (t − τ) instead of, e.g., quite a complicated expression [see Shestopalov et al., 1986a, 1986b] describing the characteristics of the simplest discontinuity “waveguide end cap”. Let us consider now more detailed the structure of the transport operators ZA(z1) and ZB(z2) (formulas (9) and (10)). It was the optimal, in terms of the computational costs, scheme of including them into the algorithm of analyzing the complex unit (see (16) and (17)), that predetermined the quite not physical choice made in (7) and (14). The transformation “field [RIGHTWARDS ARROW] field” is implemented by the product of the operators ZR and ZT directly, without any intermediate differentiating operator toward the signal propagation direction.

3. Canonical TD Problems

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[16] The separation of the evolutionary basis of the signal that is qualitatively the same for all guiding structures has determined the domain where the scattering properties of the waveguide discontinuities can be described in terms of corresponding boundary and transport transform operators. These operators allow studying the details of the general physical situation on the level of partial signal components, and, hence, they prove to be an efficient tool of the qualitative and quantitative analysis of the resonance (irregular) wave scattering processes. The operators considerably simplify the algorithmization of the problems of increasing complexity, which may encase simpler and already solved ones. In addition, they enable one, as it has been done in FD, to construct, using the sets of standard basic units, the systems of electromagnetic unit modeling whatever complicated they might be. The approach can obviously be recognized as an advantageous one; the operation principle and the formal mathematical description can be easily implemented in numerical form and remain unchanged by essential wide-range variations of the waveguiding structures' configuration and the conditions on their boundaries, as well as by going from the scalar problems to the vector ones. There is only one principal limitation on the applicability of these approaches, namely the type of operators L and M in (1): the basis character of the system {μn} of the eigen cross functions is required. One more serious problem of this approach is associated with the calculation of the values of operators R and T for quite a wide set of simple discontinuities. Obviously this set should be in line with the general objectives of the study and can be substantially different depending on the purpose and the level of complexity of this very model system. Here we consider in details only one of such problems, comprising almost all peculiarities of the approach.

3.1. Statements of the Problem

[17] The study of the TE0p symmetric (∂/∂ϕ ≡ 0) electromagnetic waves in cylindrical structures is reduced to solving the following initial boundary-value problems of the type (1)

  • equation image

Here ε(z) is the relative dielectric permittivity of the filling material, σ(z) = η0σ0(z), η0 = (μ00)1/2 and σ0(z) is the impedance of free space and the specific conductivity, U(z, ρ t) = Eϕ is the only non-zero component of the electrical field strength vector equation image, Ez = Eρ = Hϕ = 0, and the non-zero components of the magnetic field strength vector equation image are determined by the expressions equation image.

[18] The excitation wave Ui (z, ρ, t) is assumed to be non-zero only in the regular semi-infinite waveguide (in the domain A for the structure presented in Figure 2, and in the domain B for those in Figure 3), from which the wave at the time t > 0 reaches the discontinuity. In this waveguide U(z, ρ, t) = Ui(z, ρ, t) + Us(z, ρ, t). The geometry and the variable coefficients in the equations (18) allow presenting their common solution in all the regular partial domains like

  • equation image

where the orthonormal in the corresponding plane domain intS (circular or ring-shaped) equation image is determined by the set of nontrivial solutions of homogeneous (spectral) problems

  • equation image

and the space-time amplitudes vn (z, t) (elements of the signal's evolutionary basis) satisfy the equation

  • equation image

The considered problems can be subdivided into three types of domains intS (see Figure 2); two circular ones (ρ < a and ρ < b, a > b) and one ring-shaped (b < ρ < a). The sets {μn}, {λn} that are solutions to the spectral problems (20) are known:

  • equation image

Here Jq and Nq are the Bessel and Neumann functions; ξn and γn, n = 1, 2… are the strict positive roots of equations J1(ξ) = 0 and G1(γ, 1) = 0, and

  • equation image
image

Figure 2. Canonic problem—bifurcation in circular waveguide.

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image

Figure 3. Waveguide open resonator, illustrating application of scattering matrix technique analog in TD.

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3.2. The Residue Calculus Methods and the Analytical Regularization Method for the Canonical Time Domain Problems

[19] Let us consider the classical for the frequency domain problem about a coaxial semi-infinite bifurcation of a circular waveguide (see Figure 2). The excitation wave Ui = vp(z, tp(ρ), propagating in the domain A (z ≥ 0), induces in the waveguides, B and C the field

  • equation image
  • equation image

Formulas (23) are derived from general expressions U (A) = ∑μn(ρ)wn(z, t) + Ui and U(B, C) = ∑μnj(ρ)wnj(z, t), j = (1, 2), substituting (10) into them and providing for the changing direction while taking the derivatives from the amplitudes {wnj (z, t)}. Functions μn(ρ), μnj(ρ) and related values λn, λnj are determined by the formulas (22). The index value j = 1 belongs to the domain B (z ≤ 0, the coaxial waveguide), all values with the index j = 2 belong to the domain C (z ≤ 0, the circular waveguide of the radius b). In case if vp (0, t) = δ (t − η), the amplitudes of the secondary fields determine explicitly the elements of the boundary operators RAA, TBA and TCA: wn′ (0, t) = − TnpBA (t − η), and wn2′(0, t) = TnpCA (t − η).

[20] The application of the continuity conditions of functions U and ∂U/∂z in the plane z = 0, that provide for the uniqueness of the extension of the solution to (18) from one partial domain (A), into two others (B and C), leads us to the functional equations that can be equivalently written in the terms of Fourier coefficients of matching functions in the form:

  • equation image
  • equation image
  • equation image

The relation (24) is a dual operator equation with respect to the set of unknown functions {wn′ (0, t)}, (25) is the “recalculating” formula to determine the amplitudes of transient fields in the first (B, j = 1) and in the second (C, j = 2) waveguiding channels of the domain z < 0. Differentiating (24) over t, we obtain

  • equation image

Obvious that elements of the unknown vector-function {wn′(0, τ)} contained in the right-hand part of (27)

  • equation image

determine its value at the moment t by their values at the moments τ < t (τ is strictly less than t). Hence, by a fixed t (e.g., on a certain step while passing the time layers), we can consider (27) as a coupled infinite set of linear algebraic equations of the first kind with respect to the set of unknowns {wn′ (0, t)} with a known right-hand part {fmj (t)}. The elements Fnmj do not depend on the time parameters, thus, the solution to the problem for various t reduces to a single inversion of the corresponding operator.

[21] Let us introduce new notations (see (22) and (26))

  • equation image

ωm2 = −fm2 θ/ξm, now (27) takes the form

  • equation image

To invert the system of (28) we use the residue calculus method [Mittra and Lee, 1971] that is based on the Mittag-Leffler's theorem about expanding the meromorphic function into series of its principal parts. Assume that in the plane of the complex variable w there are the meromorphic functions Qrj(w) with the following parameters: Qrj (w) have simple poles in the points w = ξn2, n = 1, 2,…; Qr1m2) = −δrm, Qr1m22) = 0, Qr2m2) = 0, Qr2m22) = −δrm, r, m = 1, 2,…; Qrj (w) decrement on a certain regular system of circuits Cw in the space ∣w∣ with their radius growing. Here we also assume that the elements ωrj decrement, with the index r being increased, sufficiently fast to provide quite a uniform convergence of the series equation image in the domain of analytical functions Qrj (w). On these assumptions,

  • equation image

Actually, as

  • equation image

for all m = 1, 2,… and j = 1,2, then {Rn} from (29) gives us the solution to (28).

[22] Functions Qmj(w) that have all the above mentioned properties we present like:

  • equation image

Here equation image. The direct verification with the use of the properties of cylindrical functions shows that the functions Qmj (w) have simple zeros and poles in the required point of the domain and only there; the singularity in the point w = 0 (branching point of function w1/2 and the simple pole of function N1(w)) can be removed; the required normalization is provided by the factor introduced in (30) that does not depend on w; by large ∣w∣, n and m

  • equation image

It is evident that the study of the scattering properties of a discontinuity excited from the opposite site (from the domains B and C ), requires none of principal changes in the algorithm. As a result, we have to invert the same operator as in (28), but in the right-hand parts that should be not equal to ωmj.

[23] By constructing the solutions to the canonical problems discussed in this papers and before, in (4), the regularization was done through differentiating over t the integral convolution operators equation image and inverting part of the functional (see, e.g., the transfer (24) [RIGHTWARDS ARROW] (27)) or matrix ((28) [RIGHTWARDS ARROW] (29)) equations. As a result, the current states of w(t) have been strictly separated from the preceding ones (w(τ) with τ < t), and the related parts of operators have been explicitly inverted. The final operator equations of the second kind clearly demonstrate the whole history of transient processes, searching consecutively one time layer after another by using only direct computational operations. Note that operator F = {Fnmj} from (27), similarly to any other convolution operator occurring by re-expansion of the function, that is specified in one basis with respect to the other basis, can be easily converted by using the standard technique of the Fourier series theory. Applying the technique of the meromorphic functions theory in this case, we clearly demonstrate the methods that enable one to substantially extend the analyzing capabilities in even more complicated cases.

4. Numerical Tests of Algorithm

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[24] Sophisticated mathematical methods put certain requirements to their code implementation, just in order not to loose the advantages, gained by mathematical efforts applied during their development, but to benefit utterly from their advanced properties. To start with we present (29) in complete form:

  • equation image
  • equation image
  • equation image
  • equation image

Here Sjrm = ResQrjm2), Fjnm are the coefficients, that are the combinations of Bessel functions, specific of the configuration of waveguide cross section.

[25] Final equation we have to solve numerically (32) is Volterra equation of the second kind with all advantages that it has in numerical implementation. Almost all methods for solving integral equations numerically make use of quadrate rule [Press et al., 1992]. Though for Volterra the straightforward methods are generally satisfactory. As integral equations of this type have matrix analog (IK) R = B with K low triangular, with zero entries above the diagonal. Such matrix equations are simply soluble by forward substitution. Most algorithms for Volterra equations march out from, t = 0 building up solution as they go. The simplest and as turned out to be the most efficient way to proceed is to solve equation (32) on the mesh with uniform spacing t1 = ih, i = 0, 1, 2… N, in our case h = Δt. (For algorithms' convenience for our goals it was more suitable to fix maximum time we compute up to T and vary the number of sampling points N, so that Δt = TN−1). To do so we must choose a quadrature rule. For a uniform mesh the simplest schema is the trapezoidal rule, that gives the solution in O(N2) operations.

[26] Clear that the scheme of the solution becomes more complicated when we have to determine considerable numbers of entries in vector R (in general case of the size (Nmax × Nmax) for p = 1, 2…Nmax) and the kernel in such case is matrix K(Nmax × Nmax). Then we have for each i to solve (Nmax × Nmax) set of linear algebraic equations by Gaussian elimination. That is surely more time consuming.

[27] To make a choice for most efficient and reliable method, we tested the method of sequential iteration to solve this equation. Due to the properties of the equation (31) it turned out to converge rather fast and the solution obtained by method of iteration and with straightforward schema converged to the same result (12 digits of the solution coincide). But surely straightforward schema was much less time consuming and we made final choice for it. We have studied the convergence of numerical solution when sampling interval Δt has been changed. In fact we changed N and recalculated Δt for fixed values of T. All numerical experiments presented here had been performed for bifurcation with radii ratio θ = 0.5 and with excitation of the form equation image Observing the curves R11 (t) that are calculated for different number of sampling points one can clearly see the convergence: with N increasing the curve of R11(t) becomes smoother, and from visual point of view, it is enough to take Δt = TN−1 ≤ 0.01.

[28] The quantitative study of convergence may be done if we study numerically the behavior of the values of σn(N) = ∣R11NR11N − 1∣ that is the difference of the solutions, obtained for two consequent sampling numbers N and N − 1. Using trapezoidal rule for integration we have errorO ((h)3). If we assume that the relation εn(N) = ∣RnpRnpN∣ = C (N, n, p θ, t, T)h−ξ describes the convergence to the explicit solution, then it has to be for

  • equation image

In Figure 4 the curve for σ(N) in logarithmic scale is presented (solid line), for easy comprehension the line N−2 is also presented as a dashed line. We can see from the graph that for N ≥ 100 the line σ(N) acquire its asymptotic behavior and becomes practically parallel to the line prescribed from the estimate (33). So the quadrature method in numerical solution is correctly implemented and the trapezoidal rule convergence dominates being not “spoiled” anyhow by the properties of functions in (32). The numerical estimates obtained provide us with possibility to run the accuracy of the solution, as we have always the possibility to improve the solution calculated for small N with relation (33).

image

Figure 4. Convergence of numerical solution of (32) by means of quadrature scheme application.

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[29] Though it has been proved that all mathematical derivation preserved the equivalence of the solution of (29) to original problem it was interesting also to carry out several numerical tests of the algorithm from point of view of the adequacy of the numerical solution to original boundary value problem. In this course we have performed tests of edge conditions, as one of the most trustworthy condition in the issue. We have investigated the decrease rate for the Fourier coefficients of the field expansion Rnp with n increasing. Estimate (31) states that ∣Rnp∣ have to decrease for n ≫ 1 faster than n−1. The specific decreasing rule is defined by geometry of structure as described for FD in [Mittra and Lee, 1971] (and see references in it) ∣Rnp∣ = O(n−τ) n [RIGHTWARDS ARROW] ∞. This statement is naturally valid in TD. From the requirements on the singularities of the electromagnetic field at edges follows that in our case for the discontinuity containing conducting wedge with angle 2π we have ∣Rnp∣ ∼ O (n−3/2). In Figure 5 the normalized dependencies of reflected field amplitudes Rn1 on n are presented in logarithm scale in the moment t = 4, T = 10. It is clearly seen that values of coefficients for n ≥ 10 acquire “monotonically” oscillating character around asymptotic line n−3/2. Oscillations are caused by the presence of Bessel functions in the expression of Rnp.

image

Figure 5. To the numerical tests of algorithm: fulfilment of edge conditions.

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5. Examples of Implementation of the Transform Operators' Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[30] Discontinuities, that are to be the generic structures for the further computer design of transmission lines, are not of resonant type. None of these discontinuities shows the properties required to implement the non-standard dispersion regularities, and hence, any interesting, from the physical point of view, irregular, or resonance, wave scattering modes [Shestopalov et al., 1986a, 1986b]. Situation changes for compound structures. The dynamics of the components of frequency spectra (the behavior of the complex eigen frequencies by varying parameters) of the open waveguide resonators shown in Figure 3 are rather complicated and diverse. As a result, the spectrum of the physical effects, that can be observed by scattering of monochromatic waves by such structures, is diversified, too [Shestopalov et al.,1986b; Pochanina et al., 1991]. Similarly, in the time domain the effects of existing super-high-Q-factor free oscillations, the effects of linear “interaction” of free oscillations in the areas of spectrum points concentration, etc. can stimulate the implementation of non-standard, never noticed before wave scattering regimes. This is confirmed, particularly, by the prognostic qualitative analysis (asymptotes of large t) undertaken in [Sirenko et al., 2000]. The detailed study of such phenomena, characteristic features of the space-time transformations of electromagnetic field is of great interest for both fundamental and applied science. There is no alternative to the numerical experiment here, however, the nature of the studied processes should be certainly taken into consideration while developing and estimating the actual features of the applied models and algorithms. Below we give some examples of deriving such algorithms by implementing for two concrete structures the general scheme described above.

5.1. Computational Algorithms for the Transient Characteristics of Resonant Discontinuities

[31] Let the discontinuity shown in Figure 3 be excited by the wave equation image from the coaxial waveguide B. The complete field U induced due to the scattering of the excitation wave Ui, we describe in each of the partial domain like

  • equation image
  • equation image

which reformulates (19) and gives the general solution to (18). The elements of the evolutionary basis of the waves propagating toward the growing values of z are denoted by wn, and those propagating toward the decreasing z - by vn. In these terms, by using the boundary and the transport operators, the problem can be reduced to the self-consistent complete set of operator equations similar to (19) or in integral form:

  • equation image

with kernel including straightforward triple integration reflecting the resonator's boundaries interaction:

  • equation image
  • equation image
  • equation image

boundaries in TD, and, for example, “recalculating” relation for region D has a form:

  • equation image

We arrived again to Volterra integral equation of the second kind. Now we are able to use algorithm already developed and tested for solving (32). The uniform type of equation that we have finally to solve numerically, makes the solution rather efficient for code implementation while the complexity of the problem increases.

6. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References

[32] Having developed the numerical algorithms for one canonical problem and for one modification of the structure into compound one (this step required the introduction of the analogue of FD's scattering matrix technique), we solved the principal methodological problem: we proved the validity an efficiency of the new method we have suggested.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Evolutionary Basis of the Signal and Transform Operators
  5. 3. Canonical TD Problems
  6. 4. Numerical Tests of Algorithm
  7. 5. Examples of Implementation of the Transform Operators' Method
  8. 6. Conclusion
  9. References
  • Ladyzhenskaya, O. A., The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985.
  • Maikov, A. R., A. G. Sveshnikov, and S. A. Yakunin, Difference scheme for nonstationary Maxwell equation in waveguide systems (in Russian), J. Comput. Math. Math. Phys., 26, 851863, 1986.
  • Mittra, R., and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, Macmillan, New York, 1971.
  • Pochanina, I. Y., V. P. Shestopalov, and N. P. Yashina, Interaction and degeneration of eigen oscillations of open waveguide resonators, Rep. Acad. Sci. USSR, 320, 9095, 1991.
  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, Cambridge Univ. Press, New York, 1992.
  • Shestopalov, V. P., A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, vol. 1, Diffraction Gratings, Naukova dumka, Kiev, 1986a.
  • Shestopalov, V. P., A. A. Kirilenko, and L. A. Rud', Resonance Wave Scattering, vol. 2, Waveguide Discontinuities, Naukova dumka, Kiev, 1986b.
  • Sirenko, Y. K., I. V. Sukharevsky, O. I. Sukharevsky, and N. P. Yashina, Fundamental and Applied Problems of the Theory of Electromagnetic Wave Scattering, Krok, Kharkov, 2000.
  • Vladimirov, V. S., Equations of Mathematical Physics, Marcel Dekker, New York, 1971.
  • Yashina, N. P., Accurate analysis of coaxial waveguide slot bridge, Microwave Opt. Technol. Lett., 20, 345349, 1999.