Radio Science

Self-adaptive fast frequency sweep for the finite element analysis of waveguide modes



[1] This paper presents a self-adaptive frequency sweep technique for the modal analysis of inhomogeneous electromagnetic waveguides over wide frequency bands. It rests on a multipoint order reduction approach for the finite element method and employs an adaptive scheme based on successive bisection to control the error in the propagation constants of the considered modes. The suggested approach is highly accurate and significantly faster than full finite element computations. Numerical examples are presented to validate the new method and to demonstrate the exponential convergence rate of the reduced-order model.

1. Introduction

[2] The finite element (FE) method is a very powerful technique for the modal analysis of electromagnetic (EM) waveguides (WG) [Rahman and Davies, 1984; Lee et al., 1991; Dillon et al., 1993]. Since it is just the WG cross sections that need to be discretized, single solutions are computationally inexpensive. In many applications, however, the characteristics of WG modes are to be determined over wide frequency bands. Since modal field patterns may be frequency dependent and the corresponding propagation coefficients highly dispersive, and because dispersion curves may feature bifurcations, crossover points, and coupled-mode sections, the broadband analysis of EM waveguides typically requires a large number of FE computations, at different frequency points. In such cases, computer runtime is still a limiting factor. Methods of model order reduction (MOR) [Polstyanko et al., 1997; Bertazzi et al., 2002; Lee et al., 2005; Schultschik et al., 2008; Ahmadloo and Dounavis, 2008] provide a means to speed up the solution times of frequency sweeps very significantly, at little additional error. Most MOR methods are based on projection [Antoulas, 2005]. In the FE context, they can be viewed as restrictions of the trial and test spaces to low dimensional subspaces of global shape functions. Since the dimension of a reduced-order model (ROM) is typically in the tens, solution rates are very high: some hundreds to thousands per second, on a personal computer.

[3] Depending on the way the basis functions are constructed, MOR methods can be categorized as either single-point or multipoint approaches. Single-point methods [Polstyanko et al., 1997; Lee et al., 2005; Ahmadloo and Dounavis, 2008] match higher order derivatives of the considered eigenvalues and eigenvectors about a single expansion point. Hence, they need to factorize the FE matrix only once to set up the shift-and-invert preconditioner for the eigenvalue solver, which makes them computationally efficient. In case of very broad frequency ranges, however, single-point methods may suffer from numerical cancelation [Polstyanko et al., 1997; Lee et al., 2005; Ahmadloo and Dounavis, 2008]. Multipoint methods [Bertazzi et al., 2002; Schultschik et al., 2008], on the other hand, employ several expansion points. They are numerically robust and very well suited for broadband applications. Numerical tests for the driven case [Schultschik et al., 2009] have shown that multipoint methods require fewer shape functions than single-point methods to meet a given error limit.

[4] This paper introduces a multipoint MOR method employing an adaptive point placement scheme to control the error. To the authors knowledge, this is the first fully automatic multipoint MOR method for the modal analysis of EM waveguides. It extends the adaptive framework of Schultschik et al. [2009] from the driven case to parameterized eigenvalue problems and improves on the FE formulation of Schultschik et al. [2008]. The proposed strategy is based on successively bisecting the interval of worst error measure. To guide the adaptive process, an incremental error indicator for the propagation coefficient is provided. The numerical results of section 6 show that the suggested approach achieves exponential convergence rates, in accordance with the best theoretical estimates available [Binev et al., 2011; Buffa et al., 2011].

[5] The paper is organized as follows. Section 2 gives the problem setting and defines commonly used abbreviations. The FE formulation used in this work is summarized in section 3. Section 4 derives the MOR method and comments on the special case of homogeneous material properties. The adaptive point placement strategy, including the error indicator, is presented in section 5. The numerical examples of section 6 demonstrate the accuracy and speed of the suggested approach.

2. Preliminaries

[6] In the following, the abbreviations equation image and equation image stand for the electric field strength and the magnetic flux density, equation imager and μr for the relative electric permittivity and magnetic permeability, c0 and η0 for the vacuum velocity of light and characteristic impedance, and k for the wave number. The unit matrix is denoted by I and the unit vector by equation image.

[7] The waveguide is assumed to be bounded by electric and magnetic walls, and to possess material properties that are scalar valued and uniform along the waveguide axis z but nonuniform in the transverse plane t. In consequence, the axial behavior of the modal fields is given by exp (−γz), wherein γ denotes the propagation coefficient.

3. Finite Element Formulation

[8] As detailed by Lee et al. [2003] and Farle et al. [2004], conventional field formulations for the modal analysis of EM waveguides suffer from low frequency instabilities, because the electric flux balance enters the eigenvalue problem in frequency dependent form. As a result, the underlying differential equations become linearly dependent when the frequency approaches zero, so that the formulation breaks down in the static limit. The formulation used in this work [Farle et al., 2004] overcomes this deficiency by imposing the flux balance explicitly, with the help of a magnetic vector potential equation image and a scaled electric scalar potential equation image. Specifically, it employs the gauge Az = 0 and a tree-cotree splitting that decomposes equation imaget into components of nonvanishing circulation equation imagetc plus the transverse gradient equation imaget of a scalar field ψ. Hence we have

equation image
equation image

and the EM fields are represented by

equation image
equation image

By plugging (1) and (2) into the time harmonic Maxwell equations, we arrive at the eigenvalue problem (EVP)

equation image
equation image

FE discretization results in the algebraic EVP

equation image

wherein xA, xψ, and xV denote the component vectors for equation imagetc, ψ, and (jV), respectively, and S0, S1, S2, and T are sparse symmetric matrices, whose structure can be found in the work by Farle et al. [2004]. Note that equations (5a) and (5b) are satisfied not only by physical waveguide modes but also by a set of null-field solutions, i.e., nontrivial solutions with equation image = 0 and equation image = 0:

equation image

In the FE context (6), the null-field solutions n read

equation image

Equation (6) implies the generalized orthogonality equation

equation image

Hence any superposition of physical modes p(k) satisfies

equation image

which enables us to reconstruct p from given components pA and pψ. The resulting equation takes the form

equation image

The structure of the matrices P0 and P1 can be found in the work by Farle et al. [2004].

4. Multipoint Model Order Reduction

[9] The main idea behind the MOR approach of this paper is to restrict the trial and test spaces in the FE system (6) to suitable subspaces whose dimension q is much smaller than that of the original system, p. For this purpose, we construct projection matrices V(k), W(k) and apply a Petrov-Galerkin process [Ihlenburg, 1998] to (6). The resulting reduced-order EVP is of the form

equation image


equation image
equation image

and the corresponding approximations xm to the eigenvectors of (6) are given by

equation image

The matrices V(k) and W(k) are constructed as follows: we first compute the dominant M eigenpairs of (6) at N expansion wave numbers kn and assemble their components in equation imagetc and ψ to a matrix Ξ:

equation image

To provide a stable basis, we next compute the QR factorization of Ξ. In view of (11), we then construct V(k) by

equation image

Hence the trial space of the ROM, colsp V(k), contains superpositions of physical modes only. According to (14), the approximate eigenvectors xm satisfy (10), which assures that the ROM will not lead to null-space solutions.

[10] Following an idea from Schultschik et al. [2008], W(k) is taken to be

equation image

By plugging equations (16a) and (16b) and equations (13a) and (13b) into equation (12) and collecting terms of equal power in k, we arrive at the final form of the ROM:

equation image

Equation (17) features explicit k dependence, and all matrices are in equation imageq×q. Since qp, the EVP (17) can be solved much more efficiently than the underlying FE system (6).

5. Self-Adaptive Point Placement Strategy

[11] For practical reasons, the following algorithm is formulated in terms of the operating frequency f rather than the wave number. The goal is to compute the dispersion characteristics of M dominant modes on a set equation image0 of L equidistant evaluation frequencies fl within user-defined bounds fmin and fmax.

[12] The MOR method of section 4 provides two degrees of freedom to control the error: the number of expansion points and their respective locations; see (15). The adaptive strategy we propose is based on successive bisection. It places a new expansion point in the middle of that subinterval equation image, for which the error indicator E is worst. The procedure is repeated until the error indicator on the whole of equation image0 falls below a user-defined threshold Etol.

[13] Our error indicator is in terms of equation image2. It is of incremental type and covers all modes and evaluation points. Specifically, we set

equation image

wherein the indices + and − denote the present and preceding iteration.

[14] Section 5.1 gives the details of the proposed technique. Line 2 and Line 8 show that the first two expansion points are always placed at the boundaries of the frequency range. The main loop starts at Line 10. Note that the dimension of the reduced-order EVP at Line 15 is larger than the number of sought modes, M. Hence there are two classes of eigenpairs: close approximations to the dominant modes and higher-order solutions without any merit. The for loop starting at Line 16 provides a simple filter for the propagation constants of the dominant modes. At Line 26 and Line 27, we detect the interval of worst error indicator equation image and set the new expansion point equation image at the evaluation frequency closest to its middle.

5.1. Algorithm: Waveguide MOR With Adaptive Point Placement

[15] PARAMETERS: frequency range [fmin, fmax],        number of evaluation frequencies L,        maximum number of expansion points jmax,        error threshold value Etol.     1: Compute P0, P1, equation image0(fmin, fmax, L)     2: Solve (∑ifmaxiSi)X = TX diag γm2    3: Q ← updateQR(Q = 0, X)     4: [equation imageSi, equation imageTi] ← updateROM(Q; P0, P1)     5: forl = 1 toLdo    6:Solve (∑ifliequation imageSi) equation image = (∑ifliequation imageTi)equation imagediag equation imagem−2(fl)     7: end for    8: equation image = fmin {Next expansion point}     9: equation image1 = equation image0 {First interval}     10: forj = 2 tojmaxdo    11:Solve (∑iequation imageiSi)X = TX diag γm2    12:Q ← updateQR(Q, X)     13:[equation imageSi, equation imageTi] ← updateROM(Q; P0, P1)     14:forl = 1 toLdo    15:Solve (∑ifliequation imageSi)equation image = (∑ifliequation imageTi)equation image diag equation image2    16:form = 1 toMdo     17:equation imagem+2(fl) = arg equation image(∣equation image2equation imagem2(fl)∣)     18:end for    19:end for    20:fork = 0 toj − 1 do    21:Compute E(equation imagek)     22:end for  23:   ifE(equation image0) < Etolthen     24:return converged     25:end if    26:equation image = arg equation imageE(equation imagek) {Interval of worst error}     27:equation image = arg equation imagefequation image∣ {Expansion point}     28:equation image ← [min equation image, equation image]     29:equation imagej = [equation image, max equation image]     30:for allm, ldo     31:equation imagem2(fl) ← γm+2(fl)     32:end for    33: end for

5.2. Simplification for Homogeneous Material Properties

[16] Waveguides with homogeneous material properties are known to support transverse electric (TE), transverse magnetic (TM), and possibly transverse electromagnetic (TEM) modes. They all have in common that the transverse field patterns are independent of frequency. Equation (3) implies that the modal patterns in terms of equation imagetc and ψ must be frequency independent, too. The V components do depend on frequency, but the proposed MOR method reconstructs them from equation imagetc and ψ via (11) and (16a), respectively. Hence, a single expansion point suffices to fully characterize any TE, TM or TEM mode over an arbitrary frequency range.

6. Numerical Examples

[18] In the following, errors in propagation constant are computed with respect to full FE solutions, using the same discretization as the MOR method. The termination criterion for the adaptive process is set to Etol = 10−6. An overview of all computational parameters and results is given in Table 1. Note that the number of modes computed in the underlying FE model has been chosen somewhat larger than that displayed in the frequency sweep, to account for the fact that the preconditioned Arnoldi method used for solving the FE system may sometimes produce higher-order modes first [Trefethen and Bau, 1997].

Table 1. Computational Dataa
 Model A (ɛ1 = 1)Model A (ɛ1 = 8.875)Model B
  • a

    For a single core of the Intel Core 2 Extreme 3 GHz processor.

  • b

    Including all evaluation points.

  • c

    Including adaptivity and all evaluation points.

Frequency (GHz)0 to 250 to 250 to 30
Evaluation points1,0011,0011,001
Number of modes101019
ResultsFigure 2Figure 3Figure 6
FE Model
Degrees of freedom25,55325,55312,612
Order of FE basis222
Number of modes111223
Total runtimeb(s)4,7215,0454,827
Expansion points157
Error threshold Etol10−610−6
ROM dimension1160161
Total runtimec(s)1195402

6.1. Shielded Microstrip Line

[19] Figure 1 shows a shielded microstrip line [Bertazzi et al., 2002]. The FE model represents one half of the structure and uses a magnetic wall for the middle plane. In our first test, the dielectric substrate is replaced by free space, so that the resulting WG has homogeneous material properties. To substantiate our claim of section 5.2, we here use a single expansion point, placed at 25 GHz. Figure 2 presents the results: as predicted, both the dominant TEM mode as well as the higher-order TE and TM modes are perfectly represented everywhere in the range 0–25 GHz.

Figure 1.

Shielded microstrip line. All dimensions are in millimeters.

Figure 2.

Dispersion curves and error plot for the microstrip line with homogeneous material properties (ɛ1 = 1).

[20] Dispersion curves and error plots for the first 10 modes of the inhomogeneous WG can be seen in Figure 3. Each point of the error plot corresponds to the error in one mode at one of the 1001 evaluation points. Note the occurrence of bifurcations and complex modes. The adaptive loop finishes after 5 expansion points. Again, the MOR solutions are in excellent agreement with reference results from FE computations. However, Table 1 shows that the MOR scheme is 53 times faster. Figure 4 presents the maximum error between the ROM and original FE solutions as a function of ROM dimension. The constant slope in the semilogarithmic plot indicates that the proposed adaptive strategy reaches exponential convergence with

equation image
Figure 3.

Dispersion curves and error plot for the microstrip line with dielectric substrate (ɛ1 = 8.875).

Figure 4.

Maximum of true error of adaptive ROM for the microstrip line with dielectric substrate.

6.2. Dielectric Loaded Waveguide

[21] Our second example is the dielectric loaded WG [Strube and Arndt, 1985] of Figure 5. We consider the dominant 19 modes in the range 0–30 GHz. In this case, the method terminates after 7 iterations. Figure 6 presents dispersion curves and error plots for the propagation coefficients. Note the highly nonuniform distribution of the expansion points. It can be seen that the MOR solutions are in excellent agreement with reference results from FE computations. Moreover, Figure 7 confirms that the adaptive strategy yields exponential convergence, with

equation image

This time, MOR is 12 times faster than conventional FE analysis.

Figure 5.

Dielectric loaded waveguide. All dimensions are in millimeters.

Figure 6.

Dispersion curves and error plot for the dielectric loaded waveguide.

Figure 7.

Maximum of true error of adaptive ROM for the dielectric loaded waveguide.

7. Conclusion and Outlook

[22] This paper has presented a multipoint order reduction method with a self-adaptive point placement strategy for the broadband FE analysis of EM waveguides. The underlying MOR method employs two-sided projections with null-field orthogonalization, and the adaptive scheme is based on successive bisection, guided by an incremental error indicator for the propagation constant.

[23] The numerical tests of section 6 confirm that the number of expansion points required by the adaptive scheme remains very small, even for very broadband applications. In consequence, the proposed method is significantly faster than traditional FE analysis. At the same time, errors in propagation coefficient are negligible over the whole frequency band. Moreover, the numerical examples give evidence for exponential convergence of the suggested adaptive scheme.

[24] The numerical results of section 6 are for closed waveguides only. In principle, the proposed method can be extended to open waveguides, with the help of perfectly matched layers (PMLs). However, PMLs require complex diagonal materials tensors [Sacks et al., 1995; Dyczij-Edlinger et al., 1996], which have not yet been implemented in the authors' FE codes.