Simulation and Analysis of Transient Processes in Open Axially-symmetrical Structures: Method of Exact Absorbing Boundary Conditions (Electromagnetic Waves) Part 2

Far-field zone problem. Extended and remote sources

As we have already mentioned, in contrast to approximate methods based on the use of the Absorbing Boundary Conditions or Perfectly Matched Layers, our approach to the effective truncation of the computational domain is rigorous, which is to say that the original open problem and the modified closed problem are equivalent. This allows one, in particular, to monitor a computational error and obtain reliable information about resonant wave scattering. It is noteworthy that within the limits of this rigorous approach we also obtain, without any additional effort, the solution to the far-field zone problem, namely, of finding the field U (g, t) at arbitrary point in Qext from the magnitudes of U (g, t) on any arc r = M < L, 0 lying entirely in Qint and retaining all characteristics of the arc T .

Thus in the case considered here, equation (39) defines the diagonal operator SL^r (t) such that it operates on the space of amplitudes u (r, t) = {un (r, t)} of the outgoing wave (19) according the rule

and allows one to follow all variations of these amplitudes in an arbitrary region of Qext. The operator

given by (40) , in turn, enables the variations of the field U (g, t), g e Qext, to be followed. It is obvious that the efficiency of the numerical algorithm based on (42) reduces if the support of the function F(g,t) and/or the functions p(g) and g) is extended substantially or removed far from the region where the scatterers are located. The arising problem (the far-field zone problem or the problem of extended and remote sources) can be resolved by the following straightforward way. Let us consider the problem

 

Let the relevant sources generate a field U’ (g,t) in the half-plane Q0 = {g: p > 0, |z| < <»j . In other words, let the function Ul (g, t) be a solution of the following Cauchy problem:

It follows from (47), (48) that in the domain Qext the function Us (g,t) = U(g, t)- U (g,t) satisfies the equations

and determines there the pulsed electromagnetic wave crossing the artificial boundary T in one direction only, namely, from Qint into Qext.

The problems (49) and (14) are qualitatively the same. Therefore, by repeating the transformations of Section 4, we obtain

or, in the operator notations,

the exact absorbing condition allowing one to replace open problem (47) with the equivalent closed problem

Determination of the incident fields

To implement the algorithms based on the solution of the closed problems (42), (51), the values of the functions Ui(1)(g, t) and U (g, t) as well as their normal derivatives on the boundaries T1 and T are required (see formulas (3), (5), (50)). Let us start from the function Ui(1)(g, t). In the feeding wave guide Q1, the field Ui(1)(g, t) incoming on the boundary T1 can be represented (Sirenko et al., 2007) as

Here (see also Section 3), n = 0,1,2,… only in the case of TM0n -waves and only for a coaxial waveguide Q1. In all other cases n = 1,2,3,…. On the boundary T1, the wave can be given by a set of its amplitudes. The choice of the functions which are nonzero on the finite interval 0 < T1 < t < T2 < T, is arbitrary to a large degree and dependsgenerally upon the conditions of a numerical experiment. As for the set , which determines the derivative of the function| on T1, it should be selected with consideration for the causality principle. Each pair

is determined by the pulsed eigenmodepropagating in the waveguidein the sense of increasing z . This condition is met if the functionsare related by the following equation (Sirenko et al., 2007):

The function Ul (g, t) generated by the sources F (g, t), cp (g), and \j/ (g) is the solution to the Cauchy problem (48). Let us separate the transverse variable p in this problem and represent its solution in the form (Korn & Korn, 1961):

In order to find the functions vx( z, t), one has to invert the following Cauchy problems for one-dimensional Klein-Gordon equations:

Here,  are the amplitude coefficients in the integral presentations (54) for the functions

Now, by extending the functions FX( z, t) and vX( z,t) with zero on the interval t < 0, we pass on to a generalized version of problems (56) (Vladimirov, 1971)

(S( ) (t) is the generalized derivative of the function S(t)). Their solutions can be written by using the fundamental solution

of the operator B(X) as follows:

Equations (54) and (58) completely determine the desired function Ul (g, t).

Conclusion

In this paper, a problem of efficient truncation of the computational domain in finite-difference methods is discussed for axially-symmetrical open electrodynamic structures. The original problem describing electromagnetic wave scattering on a compact axially-symmetric structure with feeding waveguides is an initial boundary-value problem formulated in an unbounded domain. The exact absorbing conditions have been derived for a spherical artificial boundary enveloping all sources and scatterers in order to truncate the computational domain and replace the original open problem by an equivalent closed one. The constructed solution has been generalized to the case of extended and remote field sources. The analytical representation for the operators converting the near-zone fields into the far-zone fields has been also derived.

We would like to make the following observation about our approach.

• In our description, the waveguide Q1 serves as a feeding waveguide. However, both of the waveguides can be feeding or serve to withdraw the energy; also both of them may be absent in the structure.

• The choice of the parameters a(ro) and P(ro) determining Zy(ro, r) (see Section 4) affects substantially the final analytical expression for the exact absorbing condition on the spherical boundary T . When constructing boundary conditions (41), (50), we assumed that a(ro) = 1 and P(ro) = 0 . In (Sirenko et al., 2007), for a similar situation, the exact absorbing conditions for outgoing pulsed waves were constructed with the assumption that a(ro) = -Ny(roL) and P(ra) = JT(roL) . With such a(ro) and P(co), equation (21) is the Weber-Orr transform (Bateman & Erdelyi, 1953). However, the final formulas corresponding to (39), (40) for this case turn into identities as r ^ L , which present a considerable challenge for using them as absorbing conditions. In addition, the analytical expressions with the use of Weber-Orr transform are rather complicated to implement numerically.

• The function U (g, t) (see Section 7) can be found in spherical coordinates as well. In this situation, we arrive (see Section 4) at the expansions like (19) with the amplitude coefficients vn (r,t) determined by the Cauchy problems

The range of the integers j = 0,1,…, J, k = 0,1,…K, and m = 0,1,…M depends both on the size of the Qint domains and on the length of the interval [0,T] of the observation time t. The condition providing uniform boundedness of the approximate solutions U (j, k, m) with decreasing h and l is met (see, for example, formula (1.50) in (Sirenko et al., 2007)). Hence the finite-difference computational schemes are stable, and the mesh functions U (j, k, m) converge to the solutions U (pj, zk ,tm) of the original problems (42), (51).

As opposed to the well-known approximate boundary conditions standardly utilized by finite-difference methods, the conditions derived in this paper are exact by construction and do not introduce an additional error into the finite-difference algorithm. This advantage is especially valuable in resonant situations, where numerical simulation requires large running time and the computational errors may grow unpredictably if an open problem is replaced by an insufficiently accurate closed problem.