Efficient and accurate computation of Green's function for the Poisson equation in rectangular waveguides

Authors


Abstract

[1] In this paper, a new algorithm for the fast and precise computation of Green's function for the 2-D Poisson equation in rectangular waveguides is presented. For this purpose, Green's function is written in terms of Jacobian elliptic functions involving complex arguments. A new algorithm for the fast and accurate evaluation of such Green's function is detailed. The main benefit of this algorithm is successfully shown within the frame of the Boundary Integral Resonant Mode Expansion method, where a substantial reduction of the computational effort related to the evaluation of the cited Green's function is obtained.

1. Introduction

[2] Many numerical techniques in electromagnetics, such as moment methods, integral-equation techniques, boundary element methods, and so on, make use of Green's functions for particular geometries. Recent advances in these methods have been due to the improvements in terms of accuracy and efficiency related to the computation of the cited Green's functions. To obtain higher rates of convergence, a bunch of series acceleration techniques has been proposed in the technical literature. For instance, the Poisson summation formula for free-space periodic Green's functions was the preferred method in the 1980s [Lampe et al., 1985], where it was applied to 2-D and 3-D Green's functions for the Laplace and the Helmholtz equations. The Ewald summation technique, originally proposed by Ewald in 1921, was used to accelerate the convergence rate of 3D periodic Green's functions [Jordan et al., 1986]. Recently, this technique has been applied to solve the radiation of a line of sources in 2-D and 3-D (see Capolino et al. [2005] and Capolino et al. [2007], respectively).

[3] The rectangular waveguide geometry has been widely used in many practical applications. Therefore, the efficient and accurate evaluation of Green's functions for such a topology has been profoundly investigated. For instance, the Ewald method was successfully proposed to evaluate such functions for both rectangular cavities [Park et al., 1998] and waveguides [Park and Nam, 1998]. More recently, in the work of Quesada-Pereira et al. [2007], Green's functions for the parallel-plate waveguide have been further accelerated using Kummer's transformation: this has been advantageous for the efficient study of inductive obstacles in rectangular waveguides. In this paper, we focus on improving the numerical efficiency related to the precise computation of 2-D Green's function for the Poisson equation in rectangular waveguides.

[4] The scalar Green's function considered in this paper is widely used within the frame of several numerical techniques, such as the well known Boundary Integral Resonant Mode Expansion (BI-RME) method [see Conciauro et al., 1984]. This hybrid method allows one to obtain the modal chart of arbitrarily shaped waveguides, i.e., waveguides with metal perturbations in their rectangular cross section, as well as the full-wave analysis of arbitrary H and E plane waveguide components and junctions [Conciauro et al., 1996a, 1996b; Arcioni et al., 1997; Arcioni et al., 1999; Conciauro et al., 2000]. Therefore, the aim of this paper is to accelerate the computation of the involved scalar Green's function by preserving the prescribed accuracy. For such purposes, a closed-form expression written in terms of Jacobian elliptic functions with complex argument will be considered. A good tutorial on the properties of elliptic functions can be found in the work of Orchard and Willson [1997], in which the problem of rapid evaluation of such functions with real argument has been discussed. In this paper, the required elliptic functions have been accurately evaluated using the fast recursive algorithm, which will be thoroughly described.

[5] The first closed expression for the scalar Green's function in a rectangular waveguide domain was written in 1924, through the works carried out in the field of mathematical physics. Such analytical expression (without summations) is defined in terms of a special function called Weierstrass σ function, which is also related to the elliptic functions theory [Courant and Hilbert, 1989]. There is an alternative closed solution, also based on the elliptic functions, where the source and the field point contributions are explicitly separated [Morse and Feshbach, 1953]. Such solution obtained after applying a conformal mapping to the original rectangular domain is advantageous for multiple evaluations of Green's function in a grid of point values. However, the lack of knowledge of efficient algorithms for evaluating the involved elliptic functions has prevented their direct use up until now. This paper is intended to fill this gap by providing easy-to-program algorithms for the fast and accurate computation of this closed expression for the scalar Green's function, either individually or in a double grid of source and field points. The singular behavior of Green's function will also be extracted, with the aim of preserving computational stability when used in practice. A set of numerical examples that successfully confirms the wide reduction obtained in terms of computational effort when preserving the accuracy degree has been included.

2. Green's Function

[6] Green's function for the Poisson equation in the rectangular domain was studied many years ago, and appeared in many classical books on electromagnetics [see Collin, 1991; Balanis, 1989]. This problem has a simple formulation

equation image

with Dirichlet boundary conditions

equation image

where (x, y) is a generic field point, and there is a source point located at (x′, y′) given by Dirac's delta. The rectangular domain is called Ω and its boundary is denoted by ∂Ω. Let us assume that the rectangle side dimensions are a in the x axis and b in the y axis, with the origin of coordinates coincident with the lower left corner. The problem is equivalent to obtain a Green's function of a line charge inside a waveguide along the direction of propagation.

[7] Many expressions have been worked out because of the poor convergence of the eigenfunction expansion, which is the simplest one [Balanis, 1989]

equation image

This expression is exceedingly slow to converge, and makes its numerical evaluation by truncating the series difficult and computationally expensive. In fact, the double series ∑ ∑ (m2 + n2)−1 diverges [Weinberger, 1995], and therefore it is even difficult to find a good upper bound for the error.

[8] Using the Poisson summation formula, (3) can be accelerated to obtain a fast converging expression depending only on one index. Using the well-known method of images, an equivalent expression is obtained [Conciauro et al., 2000]

equation image

where

equation image

It should be stressed that if ∣m∣ increases, Tm+ tends to Tm exponentially and the logarithm rapidly tends to zero, thus making (4) very convenient for the fast computation of Green's function. This series converges faster as b/a becomes smaller. If our problem has a ratio b/a > 1, we simply exchange the roles of x, y by x′, y′ and a by b to obtain a faster convergence. The main question is to determine the number of terms that should be added in (4) to achieve the desired accuracy, because the rate of convergence depends on the location of the source and the field points, as well as on the dimensions of the rectangular domain.

[9] To overcome this problem and to decrease the related computation time, a different expression for the scalar Green's function (reported by Morse and Feshbach [1953] as a potential created by a charge q inside a rectangular domain that is equivalent to the line charge inside a waveguide) will be used instead of (4). This expression uses conformal mapping in the complex variable, which has been widely used for solving many problems in electromagnetics [Schinzinger and Laura, 2003]. This technique has provided analytical expressions for 2-D Green's functions associated with the Poisson equation [Mosig, 2003].

2.1. Geometry of the Problem in the Complex Plane

[10] Let a waveguide be invariant along the z axis (i.e., a 2-D problem), and therefore we can use the complex plane to draw its rectangular cross section. Without loss of generality, we consider a > b, where a and b are the waveguide width and height, respectively. The rectangle is allocated in the first quadrant, with one vertex at the origin of the z plane (see Figure 1). The vertices are written in capitals, and j = equation image is the imaginary unit. From now on, variable z will be used for the name of the complex plane because we are dealing with a 2-D problem and the z axis is not used. In fact, this problem in 3D is equivalent to obtain the potential given by a line charge along the z direction.

Figure 1.

Geometry of the problem. The mapping will apply to the interior of the rectangle.

[11] The basic idea is to find the required Green's function by following the three steps given below: (1) Use of the mapping to transform the boundary problem for the region Ω (the rectangle) onto another boundary problem (upper half complex plane). (2) Solve the problem in the upper half plane. (3) Transform the solution of step 2 using an inverse mapping.

[12] The mapping will transform the interior of the rectangle in the z plane onto the upper half of the w plane (see Figure 2). All mapped vertices (primed capitals) lie on the u axis, and point A′ (corresponding to the mapped vertex A) is located at u = −∞, v = 0.

Figure 2.

The rectangle is mapped onto the upper half plane. All vertices are located in the u axis.

[13] The transformation in both directions is completely defined following the theory of the Schwarz-Christoffel transformation [Spiegel, 1964] applied to the rectangular boundary

equation image
equation image

where sn is one of the Jacobian elliptic functions and K(k) is the complete elliptic integral of the first kind. Definitions and related identities are found in the work of Abramowitz and Stegun [1965]. Here K(k) is simply called K to avoid cumbersome notation. In addition, K(k) must fulfill an additional condition

equation image

which can be solved either by numerical methods or by using suitable tables [see Abramowitz and Stegun, 1965].

2.2. Scalar Green's Function Expression

[14] Green's function for the Poisson equation in 2-D can be easily obtained for free space [Hanson and Yakovlev, 2002]

equation image

where r and r′ are, respectively, the field and the source point position vectors in the cylindrical coordinates. In the w plane, by using a complex notation

equation image

where the primed and unprimed coordinates are related to the source and the field points respectively.

[15] The scalar Green's function with Dirichlet boundary conditions (G = 0 on ∂Ω) is achieved by forcing the same boundary condition along the u axis. To get the required boundary condition, that is G = 0 along the u axis, the method of images can be employed in directly. Placing an image source with opposite charge at the same coordinate u, but opposite coordinate v, leads to

equation image

where equation image stands for the complex conjugate of w′. By applying the transformation (6), the final expression

equation image

is achieved.

[16] The main advantage of this solution is that Green's function is not given in a series form, and hence no convergence problems will arise. However, there will be few drawbacks which are listed below: (1) The computation of the Jacobian elliptic function sn with complex argument is required. (2) A suitable value for k satisfying equation (7) is needed. (3) The numerical evaluation of K(k) is also required.

[17] Therefore, the above mentioned drawbacks most likely lead the developers of the BI-RME method to use (4) instead of (11) to obtain the numerical evaluation of the required scalar Green's function [Conciauro et al., 2000]. In the following sections, the problems outlined in the previous list will be discussed in detail.

3. Fast Computation of the Jacobian Elliptic Function sn

[18] To efficiently compute the Jacobian elliptic function sn, two main possibilities are available: the arithmetic-geometric mean and a “suitable” Landen transformation. The former option is not suitable when complex arguments are involved, since square roots of complex numbers must be computed, and it is also not easy to decide which root is the correct one because of its multivalued nature. Real and imaginary parts can be split and computed separately [Abramowitz and Stegun, 1965], but the number of operations increases dramatically.

[19] Therefore, in this work, we follow the descending Landen transformation (the ascending transformation produces a higher number of operations), and is given by

equation image

where the notation has been adapted to work with argument and modulus, instead of using the elliptic function parameters (which is the preferred choice by Abramowitz and Stegun [1965]). The iterative sequences of modulus and argument have the following expressions

equation image
equation image

and by writing kn+1 in terms of kn we finally have

equation image

[20] Taking into account that 0 < k < 1, we find out that the sequence of the modulus (15) converges to zero quadratically. Roughly speaking, quadratic convergence doubles the number-of-digits agreement between successive iterations and the limit. But in practical cases, there are less than six iterations typically needed to achieve the required accuracy.

[21] The basic idea of the descending Landen transformation is to decrease the modulus value by using iteration (15), and starting with k0 = k. The iteration is successively applied until k becomes negligible: let us say that in the Nth iteration kN ≈ 0. The expansion of sn for small values of k can be approached as follows [Abramowitz and Stegun, 1965]

equation image

and when k is small enough, this last equation tends quadratically to sin u. Finally, if we repeatedly apply (12), starting with sn(u, kN) = sin u, then the intermediate values for the sn(u, ki) terms of the recurrence formula will be obtained, until the final desired value of sn(u, k0) is reached. To reduce the number of operations, we will use the inverse of sn

equation image

[22] After performing a suitable reindexation, the reciprocal of equation (12) can be written as

equation image

[23] The previous equation can be rewritten if we take into account that the argument of ns functions is of type Kz/a. According to (14), if we define un−1 = Kn−1z/a, then the updated un can be derived as

equation image

[24] Now, to write un in terms of Kn (instead of Kn−1), we should know the expression of Kn after applying the Landen transformation. Such an expression is reported by Borwein and Borwein [1998] as follows

equation image

in which (13) has been used to derive the final expression. Finally, we can use (20) to express (19) in the following way

equation image

which could have been expected intuitively, but it must be proved rigorously. Hence, by using this definition for un in terms of Kn, (18) has the following expression

equation image

[25] To sum up, the computation of sn(Kz/a, k) will be performed by using the following steps: (1) Compute the values of kn by using (15) with k0 = k until kN becomes negligible. (2) Use the following limit (derived from basic properties) to compute ns(KNz/a)

equation image

(3) Use (22) to compute the intermediate values of ns(Kiz/a, ki) until the ns(Koz/a, ko) is reached. (4) Compute the inverse to obtain sn(Kz/a, k).

[26] A detailed algorithm is listed in Figure 3. It is worth noting that for computing sn(, k), the evaluation of K(k) is not carried out, thus avoiding many unnecessary operations.

Figure 3.

Algorithm to compute sn(, k) for complex α and given tolerance.

4. Computing the Modulus k

[27] The value for modulus k must be computed once to hold (7), and will remain constant for the evaluation of Green's function, no matter how many points are required. If the b/a ratio of the rectangular domain is not changed, then there is no need to find a new value for k.

[28] In rectangular waveguides, the most common value for b/a equals 1/2, hence we obtain

equation image
equation image
equation image

[29] To compute the previous values, a zero finder algorithm included in MATLAB has been followed initially. Such an algorithm uses bisection/interpolation techniques, and double precision accuracy is obtained after seven iterations. For each iteration, the algorithm needs to evaluate the complete elliptic integrals K and K′, which can be unacceptable (from the computational point of view) if modulus k for several rectangular waveguides of different ratios b/a needs to be solved.

[30] To overcome this problem, the theory of Theta functions provides few expressions which are really convenient. First, let the definition of the Nome q be

equation image

and, fortunately, q is a positive real value (being q < 1) that can be computed from the very beginning, as the ratio K′/K must be equal to the ratio b/a. By using the definition of two of the basic Theta functions [Borwein and Borwein, 1998] in terms of the Nome

equation image
equation image

the following relationship holds (for proof, refer to Borwein and Borwein [1998])

equation image

[31] Note that both series for the Theta functions converge faster than the exponential raise for n (superexponential convergence). To achieve enough precision, we add elements to both series until their Nth term is less than the prescribed tolerance TOL. Hence, consider the general term of the series θ3 (the slowest convergence)

equation image

where ⌈x⌉ stands for the rounding up of x with the nearest integer toward infinity.

[32] The proposed algorithm to obtain modulus k, which is listed in Figure 4, becomes highly efficient and accurate. As an example, to get the value of k given in (24) with TOL = 10−14, only N = 5 is required. Following this algorithm, we again avoid the computation of the complete elliptic integrals K(k) and K′(k).

Figure 4.

Algorithm to compute k such that K′/K = b/a with given tolerance.

5. Computational Considerations

[33] In the context of many integral-equation techniques, such as the previously cited BI-RME method, it is needed to compute the scalar Green's function for a single rectangular waveguide at M × N different points, i.e., M field points {(xi, yi), i = 1, …, M} and N source points {(xj, yj), j = 1, …, N}. Our aim is to present an efficient procedure to determine matrix G whose element values are Gij = G(xi, yixj,yj).

[34] To obtain every element of the matrix G by following (11), two evaluations of the Jacobian elliptic function sn are required. However, taking a closer look at the algorithm in Figure 3, many computations do not depend either on the field or on the source point coordinates. In particular, steps 1–6 depend only on the value of k, which is computed once (see previous section). Only steps 7–11 involving few operations have to be repeated for different arguments (points).

[35] A further computational saving can be obtained if some points are repeated. If one field point (xk, yk) is fixed, while some different source points (xj, yj) are used in the argument of Green's function, then the sn function related to the field point does not need to be recomputed. This situation is really frequent when twofold numerical integrations occur in matrix evaluations. For instance, in the BI-RME method [Conciauro et al., 1984], one can find matrices whose elements are defined as follows

equation image

where ui(s) and uj(s′) are, respectively, weighting and basis functions arising from the application of the method of moments; σ defines an arbitrarily shaped contour placed within a rectangular waveguide domain and s and s′ are the variables used to parameterize such a contour.

[36] By following the double Nth-order Gaussian numerical method for computing the previous integral, N2 evaluations of the integrand are required. However, for every inner integration in (30), we can fix the field point, and then evaluate the integrand in the whole set of N source points. The same procedure can be repeated for every different field point, and since the arguments of the sn functions in (11) related to the source and field points are separated, only 2N evaluations of such functions (in fact, steps 7–11 in Figure 3) are required. The complete description of this algorithm is included in Figure 5.

Figure 5.

Algorithm to efficiently compute the matrix G.

6. Singular Behavior of Green's Function

[37] According to (11), the newly derived scalar Green's function has a singular behavior when the field (z) and source (z′) points approach. This singularity has the same behavior as the one related to the free-space Green's function, as it can be seen from (8) and as already explained by Haberman [1998]. During the computation of some of the matrix elements related to the numerical solution of integral equations, the cited singularities typically appear. For instance, when the diagonal elements Lii of the BI-RME method are evaluated following (30), the integrand has a singularity whenever the source and field points coincide. Such a singularity can be integrated, but not numerically since the integrand becomes infinity.

[38] To avoid the singularity problems related to some integrals previously mentioned, as well as to preserve a certain degree of accuracy, Green's function can be split in a regular and a singular part. As proposed by Conciauro et al. [1984] for the BI-RME method, the regular part can be integrated numerically, whereas the singular part must be treated analytically. The analytical treatment of the integrals involving singularities within the BI-RME method has been exhaustively reported by Conciauro et al. [1984] and Cogollos et al. [2003] for different kinds of integration domains.

[39] For identifying the singular behavior of the new scalar Green's function, we can rewrite the expression (11) letting u = Kz/a, u′ = Kz′/a, sn = sn(Kz/a, k) and sn′ = sn(Kz′/a, k), thus obtaining

equation image

where the singularity arises when u′ → u and then sn′ → sn. Therefore, the second term of (31) contains the singular behavior. Now adding and subtracting ln ∣uu′∣ to such second term, we can obtain

equation image

[40] Now by taking the limit when u′ → u in the second term of the previous expression

equation image

and by making use of the following property collected by Abramowitz and Stegun [1965]

equation image

where

equation image
equation image

we can finally write

equation image

[41] The first term on the right-hand side of (36) never gives a singular value inside the waveguide. Let us discuss the potential problematic situations (1) The numerator ℜ{sn}equation image{sn} can only be canceled at some points of the boundary or outside (as it can be concluded from the decomposition of sn in its real and imaginary parts [Abramowitz and Stegun, 1965]). (2) sn = 0 if u = 0. The origin belongs to the boundary and therefore Green's function is zero there. (3) cn = 0 if u = K, but u = Kz/a, and this can happen when x = a, y = 0 (the lower right corner of the waveguide), where Green's function is also zero. (4) dn = 0 if u = K + jK′, but u = Kz/a, and this value corresponds to the upper right corner of the waveguide, where Green's function is zero again.

[42] Finally, the second (and singular) term of expression (36) can be rewritten as

equation image

where R = equation image. The singularity of the type ln R is exactly the one considered by Conciauro et al. [1984] and Cogollos et al. [2003]. Therefore, the analytical treatment of such a singularity reported in these references can be completely followed.

7. Results

7.1. Example 1: Single Point Evaluation

[43] First, let us see the convergence properties of the new algorithm proposed to evaluate the desired Green's function following the expression (11). We will compare the convergency of such results with those provided by the evaluation of the original expression (4). For such purposes, two regular points (not too close to the rectangular contour) have been chosen to avoid problems with the convergence rate of (4). The field and source points are, respectively, (x, y) = (a/2, b/4) and (x′, y′) = (a/3, b/5), where a = 19.05 mm and b = 9.525 mm (standard WR-75 rectangular waveguide). The partial sums related to equation (4) are denoted as follows

equation image

To achieve a good accuracy in this example, we have chosen the stop criterion as ∣SnSn−1∣ < TOL with TOL = 10−14. The results for the partial sums and the absolute differences between consecutive iterations are reported in Table 1. The exact value up to 16 significant figures is given as G = 6.743294670343186 × 10−2, and to achieve this accuracy we need to add 11 terms, i.e., n = 5 in (38).

Table 1. Partial Sums Sequence and Convergence Criterion
nSnSnSn−1
06.822656641675849E-002
16.743147279521008E-0027.950936215484206E-004
26.743294945584274E-0021.476660632659943E-006
36.743294669829188E-0022.757550868204284E-009
46.743294670344145E-0025.149575210694479E-012
56.743294670343186E-0029.589551375199790E-015

[44] Following the descending Landen transformation proposed in this paper, we obtain the sequence of values for kn and ns(K(x + jy)/a) shown in Table 2. For the sake of space, only 10 of 14 significant digits have been included in Table 2. Adjacent to the smallest value of the modulus k in Table 2 (i.e., k6), we find the value of csc π(x + jy)/2 (the only evaluation of an ns function for each source/field point), which is the starting value for the downward recurrence.

Table 2. Sequence of Modulus k and ns Function
nknns
00.9851714311.053224272 − j6.436644200E − 002
10.7071067811.243383748 − j0.198349670
20.1715728751.332551292 − j0.256258543
37.469666729E − 031.337120970 − j0.259216467
41.394936942E − 051.337129567 − j0.259222032
54.864622683E − 111.337129567 − j0.259222032
65.916138463E − 221.337129567 − j0.259222032

7.2. Example 2: Matrix Computation

[45] To offer a fair comparison in terms of computational time, we have considered a practical situation where the scalar Green's function has to be evaluated in a dense grid of points. For such purposes, the rectangular cross section of a standard WR-75 waveguide has been meshed with 5000 points. Therefore, the G matrix to be computed following the algorithm of Figure 5 will have 25 million of entries. To avoid singularities in the computation of Green's function, two interlaced meshes of 100 × 50 points have been generated for the source and field points. Throughout this example, a precision of 14 decimal digits has been required.

[46] The results have been obtained with a FORTRAN code running on a PC platform (Intel Core 2 series 6600 at 2.40 GHz) with 2 GB of RAM. Following equation (4), the G matrix is filled in 84.48 s, while the same result is reached in less than 0.01 s using the algorithm shown in Figure 5. It means that the new algorithm proposed in this paper is over 8000 times faster than the previous technique. The only drawback of the novel algorithm is related to the additional storage of the complex values for sni and sn′j (see steps 7 and 12 in Figure 5); in our case the 2 (M + N) real numbers, where M and N are the number of field and source points, respectively. In this example, M = 5000 and N = 5000, thus 104 complex numbers have been stored using 16 bytes each, meaning 156.25 KB of additional storage. Matrix G has 25 million real elements, meaning 190.73 MB of memory. The additional storage due to our new method, compared with the total size of the G matrix, can be practically considered negligible.

7.3. Example 3: Accuracy Breakdown

[47] Owing to the fact that a series like (4) have a convergence behavior depending on the location of the source and field points, as well as on their distance from the boundary, not every evaluation of such series requires the same number of terms. Furthermore, the ratio b/a has a strong influence in the required number of terms. For b/a = 1/2, the number of terms is equal to 11 (see Example 1), but if b/a = 1 (i.e., a square waveguide) the number of terms increases to 23. Even though the required number of terms in (4) changes with the ratio b/a, the accuracy of the final result is not jeopardized by this fact. In our case, the series appearing in (28) have a superexponential convergence giving modulus k. Furthermore, reduction formula (15) converges quadratically for any ratio b/a.

[48] Major problems arise when the distance between the field and source points decreases. In such a case, the solution consists of extracting the singularity as it was explained in section 6. Then, the main question is to find out the threshold distance that makes equations (4) and (11) provide inaccurate results. To check this, some evaluations have been performed with a decreasing distance between the field and source points. For this purpose, a square waveguide (a = b = 1) with the field point fixed to its center (xi = a/2, yi = b/2) has been used. The direction of approximation to such fixed point is the line xj = yj. The tolerance required for both the algorithms has been TOL = 10−15.

[49] The first column of Table 3 displays the distance range between the field and the source points (d ∈ [10−1, 10−8]). In the second column, we show the absolute difference between the values for Green's function provided by (4) and (11). The third column gives T0 term responsible for the singular behavior of (4), whereas the fourth column includes the value for the denominator of (11) (which can originate a singularity in such expression). Results in Table 3 show stronger disagreements, as the distance d between field and source points becomes less; this indicates an accuracy breakdown. The singular term T0 decreases two orders of magnitude per each order of magnitude of d. Surprisingly, the real part of the singular term of (11) decreases at the same rate of d, but the imaginary part follows the same behavior as T0. This fact means that the complex division in (11) is more stable, since the modulus of the denominator reaches the underflow value after T0 term.

Table 3. Accuracy Near the Singularity
dG1G2T0sn2Kz/a − sn2Kz′/a
1E − 1<1E − 154.93E − 2−0.36E + 0 + j6.65E − 2
1E − 21.33E − 154.93E − 4−3.71E − 2 + j6.87E − 4
1E − 31.89E − 124.93E − 6−3.71E − 3 + j6.88E − 6
1E − 45.88E − 124.93E − 8−3.71E − 4 + j6.88E − 8
1E − 55.90E − 94.93E − 10−3.71E − 5 + j6.88E − 10
1E − 61.34E − 64.93E − 12−3.71E − 6 + j6.88E − 12
1E − 78.73E − 54.93E − 14−3.71E − 7 + j6.81E − 14
1E − 88.39E − 34.44E − 16−3.71E − 8 + j0

8. Conclusions

[50] A new algorithm for the faster computation of the scalar Green's function in a rectangular waveguide domain has been proposed. Computationally convenient expressions have been developed from the theory of elliptic functions, showing a good adaption for numerical techniques used in applied electromagnetics. The singular behavior of the new expression for Green's function has been studied, resulting in the typical ln R form related to the solution of 2-D problems. This algorithm has been revealed to be extremely efficient when a high number of evaluations of Green's function is practically required.

Ancillary