Research Article | Open Access
The Exact Absorbing Conditions Method in the Analysis of Open Electrodynamic Structures: Circular and Coaxial Waveguides
The exact absorbing conditions (EAC) have been constructed and used for truncating an unbounded domain of computation in open initial boundary value problems, which describe space-time transformations of electromagnetic waves in axially symmetrical waveguides. The equivalence theorem is proved that gives grounds for rigorous theoretical justification of the EAC-method.
Exact absorbing conditions (EAC) are used in computational electrodynamics of nonsine waves for truncating the domain of computation when replacing the original open initial boundary value problem by a modified problem formulated in a bounded domain [1–9]. In the present work, we construct and analyze the EAC as applied to the axially symmetrical waveguide structures illuminated by symmetrical pulsed TE- and TM-waves (or - and -waves) and prove the equivalency of the original (open) and modified (closed) initial boundary value problems. The following results are presented: the EAC for virtual boundaries in a cross-section of regular circular and coaxial waveguides and the theorem about the one-valued solvability of the modified closed problem and its equivalency to the initial open problem giving grounds for rigorous theoretical justification of the EAC-method.
The efficient limitation of the computational space in open initial boundary value problems (i.e., the problems whose domain of analysis is infinite in one or more directions) is a vital issue in computational electrodynamics as well as in other physical disciplines dealing with mathematical simulation and numerical experiments. Most of the well-known and extensively used heuristic and approximate solutions to this problem are based on the so-called absorbing boundary conditions (ABC) [10–13] and perfectly matched layers (PML) [14–16]. The use of various modifications and improving techniques for respective methods yield good results in various specific physical situations. However, it appears that, for certain problems associated with the resonant wave scattering, the numerical implementation of these methods may cause unpredictable growth of the computational error for large observation times [17–19].
The method utilizing the exact absorbing conditions for the artificial boundaries that truncate an unbounded domain of computation [1–9, 20, 21] is outnumbered by the classical approximate approaches. However, testing numerical experiments as well as a series of physical and applied results obtained with the help of this method (see, e.g., [3, 5, 8, 21–30]) shows its evident potential, especially, for obtaining reliable numerical data on space-time and space-frequency electromagnetic field transformations in open waveguide, periodic, and compact resonators.
The essence of the method for open scalar problems, which are formulated in the part of the space ( is -dimensional Euclidean space), is as follows. Assume that the excitation sources and inhomogeneities of the medium are located in a bounded region of the unbounded analysis domain . The propagation velocity of the electromagnetic wave , , generated by these sources and obstacles is finite. Therefore, in time , the signal will not go outside the boundaries of some bounded domain : Here, is the closed interval and is the complement of the domain with respect to ; that is, .
Formula (1) gives us the well-known exact radiation condition for outgoing (from the domain where all sources and obstacles are located) pulsed wave . The only but rather essential limitation of this simple condition is that, with growing , the domain is expanding, and the boundary is moving farther away from the domain . That is why the condition in the form of (1) is not used for truncating the computational space of open electrodynamic problems. In EAC-method, the condition (1) is transferred from the field-free points onto some artificial boundary located in the region, where the intensity of space-time field transformations can be arbitrary in magnitude and takes the form The electromagnetic wave must be outgoing in this case as well, or, in other words, it is bound to intersect the boundary only in one direction, moving away from the sources and obstacles. Here, as above, (, ) is a scalar or vector field function, while is some integrodifferential operator on , . Symbol “” denotes direct product of two sets.
The boundary divides the unbounded domain into two domains and such that . In the first one (bounded), we can formulate the initial boundary value problem with respect to the function using the boundary condition in (2). We will further call this problem the modified problem as distinct from the original initial boundary value problem formulated in the unbounded domain with the radiation condition (1) involved. In the domain , the desired function can be determined by using standard finite-difference  or finite element  algorithms. For the domain , EAC-method allows us to construct and use the so-called “transport operators” [3, 8, 33, 34] to calculate the function at the points from its values on the boundary .
The analytical forms of the operators and depend on the geometry of the domain , and, evidently, on the problem dimensions and the coordinate system. However, in all cases, the derivation of these operators is based on the common sequence of transformations widely used in the theory of hyperbolic equations [3, 35]: (i) the isolation of the regular domain where the wave propagates freely moving away from the domain enveloping all sources and scattering objects, (ii) incomplete separation of variables in the original initial boundary value problem for the domain resulting in the problem for the one-dimensional Klein-Gordon equation with respect to the space-time amplitudes of the field , (iii) integral transformation (image original function) of the problem for one-dimensional Klein-Gordon equation, (iv) solution of auxiliary boundary value problems for ordinary differential equations with respect to the images of amplitudes of the field , and (v) inverse integral transformation.
As a result, the nonlocal (in space and time) exact absorbing conditions on the artificial boundary are derived. In some cases, these nonlocal conditions can be reduced to the local conditions by replacing certain integral forms with the differential forms and defining an additional initial boundary value problem with respect to some auxiliary function of time and transverse coordinates [2, 3, 8]. The exact EAC (2) can be then included into a standard finite-difference algorithms or into algorithms of finite element method with the domain of calculation reduced down to . However, one can confidently assert that relevant computational schemes are stable and convergent only when the modified problem is uniquely-solvable and equivalent to the original problem . Although the corresponding assertions have been formulated in some works on the subject (see, e.g., [1, 3, 8]), they have been proved analytically just only in  for the initial boundary value problems describing - and -pulsed wave scattering on compact open axially symmetrical structures. In Section 6, we replicate the proof scheme for the initial boundary value problem associated with compact discontinuities in circular and coaxial waveguides and thus demonstrate that there is reason to believe that it can be used for the other types of EAC as well.
2. Formulation of the Model Problem
The two-dimensional initial boundary value problem describing the transformation of pulsed symmetrical () - () and -waves () in the open axially symmetrical waveguide units (Figure 1) is given (see [3, 8]) by Here, and are the electric and magnetic field vectors, are the cylindrical coordinates, for -waves, and for -waves. The function determines other nonzero field components as well: , in the case of -waves and , in the case of -waves (see, e.g., formulas (1.13) and (1.15) in ). By , we denote perfectly conducting surfaces obtained by rotating the piecewise smooth curve about the -axis. The relative permittivity () and specific conductivity are smooth enough nonnegative functions inside and take free space values outside (in case of -waves) or and (in -case), is the impedance of free space, and are the electric and magnetic constants of vacuum. We use the SI system of units. The “time” variable is the product of the real time by the velocity of light in free space and so is measured in meters.
The domain of analysis is the part of the half-plane bounded by the contours . The regions and (regular semiinfinite circular and coaxial waveguides) are separated by the virtual boundaries and , . Here, is the local coordinate system associated with the waveguide (Figure 1).
The functions , , , , and which are finite in the closure of are supposed to satisfy the theorem on the unique solvability of problem (4) in the Sobolev space , , (see Statement 2 and [3, 36]). The “current” and “instantaneous” sources given by the functions and , as well as all scattering elements given by the functions , , and by the contours are located in the region . In axially symmetrical problems, at the points such that , only - or -field components are nonzero [2, 3, 8]. Hence, it follows that for , in (4).
Statement 2. Let , (for -waves) or (for -waves) and . Then, problem (4) has a generalized solution from , and the uniqueness theorem is true in this space.
Here, the following notation is used: is the partial derivative of with respect to or , is the space of functions (where ) for which the function is integrable in , is the set of all the elements from having generalized derivatives up to the order inclusive from , is the space containing all elements with finite norm , and is the subspace of space , in which the set of finitary and infinitely differentiable in functions is a dense set.
3. Exact Radiation Conditions for Outgoing Pulsed Waves
Separation of variables in (5) results in where the orthonormal (with the weight factor ) basic sets (and the sets of the transversal eigenvalues ) are obtained from the homogeneous Sturm-Liouville problems (if is the circular waveguide with ) or (if is the coaxial waveguide with ). Hereafter, only in the case of -waves and only for coaxial waveguide . In all other cases, . The form of the boundary conditions on waveguide walls for the case of -waves in (7) and (8) is determined by the following relationships: , , and .
The spatial-temporal amplitudes of the wave , are obtained by solving the initial boundary value problems (it is assumed that the excitation , being generated by the sources , , and located in , has not yet reached the boundary separating the regions and by the time ).
Analytical representations for solutions and to the problems (7) and (8) are well-known [3, 8] and, for -waves, take the form: for coaxial waveguide and for circular waveguide . For -waves, we have (circular waveguide ) and (coaxial waveguide ). Here, and are the Bessel and Neumann cylindrical functions.
By applying to (9) the cosine Fourier transform and taking into account that , we are led to the following Cauchy problems for the images of the functions : where . The fact that the wave () is an outgoing one and thus the amplitudes of all its partial components are zero at any time for the values of large enough played an important part in derivation of (16).
Continuing the functions and by zero on the semiaxis , we pass on to generalized formulation of the Cauchy problems (16) : where and are the Dirac delta-function and its generalized derivative of the first order.
A convolution of the fundamental solution ( is the Heaviside step function) of the operator with the right-hand side of the equation gives its solution [3, 38]. Therefore, the unknown functions can be represented as Applying to (18) the inverse transform (14), we get for the originals : On multiplication of (19) by and summation over all , we obtain the exact radiation condition (ERC) for the pulsed waves outgoing towards : In (20), one should set and for a circular waveguide or and for a coaxial waveguide.
Equations (19) and (20) specify the diagonal transport operator , which operates in the space of amplitudes according to the rule and the operator accounting for all transformations of the wave outgoing through the boundary into the hollow regular waveguide .
4. Nonlocal Absorbing Conditions
By dropping the observation point in (19) onto the artificial boundary (), we obtain Differentiating (23) with respect to , we can write Here, the well-known relationships , , and , where stands for the generalized derivative of , have been used.
Let us now apply to (24) the Laplace transform Taking into account the formulas (the convolution theorem), , and , we pass on to the following expression in the space of images : Equation (26) can be transformed to By applying the inverse Laplace transform to (27), we turn back to the originals: Here, the formula  is used.
To justify the use of the Laplace transform (25) when passing from (24) to (28), we refer to the estimates performed in the book . From these estimates, it follows that, at points of any bounded subdomain of the domain , the field generated by a system of sources with compact supports cannot grow faster than for , where is some constant. The estimates are valid for all electrodynamic structures whose spectrum of complex-valued eigenfrequencies does not contain the points from the upper half-plane of the first (physical) sheet, which constitutes a natural domain of variation of the complex frequency parameter . All the open structures considered in this section fall into this category .
Relationships (23), (24), (28), and (29)–(31) are valid for all the waves outgoing into the waveguide . Equations (23), (24), and (28) relate space-time amplitudes of all modes guided by the regular structure towards on the boundary , while (29)–(31) relate space-time amplitudes of the waves integrally. Therefore, the open problem (4) and the problem with a bounded domain of analysis given by (4) together with any boundary condition like (29)–(31) on the virtual boundaries , , are equivalent (see Section 6). For the same reason, the boundary conditions given by (29)–(31) can be considered as exact boundary conditions: the wave is not distorted when crossing the boundary ; there is no reflection into the domain ; the wave passes into the domain completely, as if it is absorbed by that domain or by its boundary . For the first time, (23) and (29) were used as exact absorbing boundary conditions in .
Conditions (29)–(31) are nonlocal in the spatial variable () and in time as well; the function and its derivatives at each point of the boundary and at each time point are related to the values of this function or its derivatives at all points of the boundary and at all time points .
5. Local Absorbing Conditions
By using the relation , rewrite (23) as Denote Then, and we have from (33) that The integral form (34) is equivalent to the following differential form: Indeed, by passing on to the generalized formulation of the corresponding Cauchy problem and using the fundamental solution of the operator (see ), we can easily verify that formulas (34) and (37) determine the same function .
Let us now multiply (36) and (37) by and sum over all . As a result, taking into consideration that for (see problems (7) and (8)), we obtainHere, is an auxiliary function. It can be determined by solving the initial boundary value problem (40b), which is the inner problem with respect to the local EAC (40a); is a numerical parameter.
Transforming (24) and (28) in a similar way, we obtain the following exact local boundary conditions, differing from (40a) and (40b): When deriving (41a) and (41b), the following formula was used: , along with the substitutions while, in derivation of (42a) and (42b), we applied the integral Poisson formula  and
One should supplement the initial boundary value problems given by (40b), (41b), and (42b) in (40a)–(42b) and posed relative to the auxiliary functions , by the following boundary conditions for all time points : (for the domain corresponding to a circular waveguide) and (for the domain corresponding to a coaxial waveguide).
Formulas (40a)–(42b) are derived from the same equations (23), (24), and (28) as formulas (29)–(31). They, as well as formulas (29)–(31), faithfully describe behavior of the wave on the boundary , which is why they also can be used as the exact absorbing conditions for the outgoing pulsed waves formed by the unit. The conditions given by (40a)–(42b) differ from those given by (29)–(31) in that they are local: in (40a), (41a), and (42a), the function and its derivatives at each fixed point of the boundary and at each fixed time point are determined by the auxiliary function at the same point and the same time.
6. Equivalence Theorem
The exact absorbing conditions derived in Sections 4 and 5 allow us to replace the original open problem (4) by the closed (modified) problem together with the total set () of the exact radiation conditions (20) for the points . Below, we prove that this replacement is equivalent, that is, any solution to the problem (4) is at the same time the solution to the problem (49), (20), and vice versa. By , , we denote the total set () of the EAC including any local or/and nonlocal boundary conditions from (29)–(31) and (40a)–(42b).
Problem (4) is uniquely solvable in the space of generalized functions . Its unique solution is at the same time a solution to problem (49) from the Sobolev space , where . This direct inclusion is trivial. It is proved by the constructions from Sections 3–5. The inverse inclusion will also be true if only the generalized solution of problem (49) belonging to the space is unique. Let us prove the uniqueness.
According to , the generalized solution of problem (49) is an element of the space being equal to at and satisfying the identity for any function from that is zero at . Here, is a lateral surface of the cylinder (; is the boundary of the domain ); and are cosines of the angles between the outer normal to the surface and the axes and , respectively. An element of the end surfaces of the cylinder is . Identity (50) is derived by multiplying the telegraph equation from (49) by and by integrating the result by parts in [36, 44].
Suppose there exist two solutions of problem (49) belonging to the space and . The difference of these solutions is the solution of the homogeneous problem (49); therefore it satisfies the identity (see formula (50))
Let us introduce an arbitrary and consider the following function: It can be easily verified that has (in ) the generalized derivatives  At the same time, we have . Substituting the function into identity (51), we obtain Since  then
By performing partial integration and taking into consideration that and , we also obtain
Thus, all the volume integrals entering identity (54) are nonnegative. Show that the integral is nonnegative as well. To this end, let us estimate the integral for the case of -waves, when (see [3, 8]) (the case of -waves can be considered similarly). Thus,