eigenvalue extraction from time domain computations · acquired time domain simulation results in...

Adv. Radio Sci., 11, 23–29, 2013www.adv-radio-sci.net/11/23/2013/doi:10.5194/ars-11-23-2013© Author(s) 2013. CC Attribution 3.0 License.

Advances inRadio Science

Eigenvalue extraction from time domain computations

T. Banova1,2, W. Ackermann2, and T. Weiland2

1Technische Universitat Darmstadt, Graduate School of Computational Engineering, Dolivostraße 15, 64293 Darmstadt,Germany2Technische Universitat Darmstadt, Institut fur Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstraße 8,64289 Darmstadt, Germany

Correspondence to:T. Banova ([email protected])

Abstract. In this paper we address a fast approach for anaccurate eigenfrequency extraction, taken into considerationthe evaluated electric field computations in time domain of asuperconducting resonant structure. Upon excitation of thecavity, the electric field intensity is recorded at differentdetection probes inside the cavity. Thereafter, we performFourier analysis of the recorded signals and by means of fit-ting techniques with the theoretical cavity response model (insupport of the applied excitation) we extract the requestedeigenfrequencies by finding the optimal model parametersin least square sense. The major challenges posed by ourwork are: first, the ability of the approach to tackle the largescale eigenvalue problem and second, the capability to ex-tract many, i.e. order of thousands, eigenfrequencies for theconsidered cavity. At this point, we demonstrate that the pro-posed approach is able to extract many eigenfrequencies ofa closed resonator in a relatively short time. In addition tothe need to ensure a high precision of the calculated eigen-frequencies, we compare them side by side with the refer-ence data available from CEM3D eigenmode solver. Further-more, the simulations have shown high accuracy of this tech-nique and good agreement with the reference data. Finally,all of the results indicate that the suggested technique can beused for precise extraction of many eigenfrequencies basedon time domain field computations.

1 Introduction

The field of quantum chaos encompasses the study of themanifestations of classical chaos in the properties of thecorresponding quantum or more generally, wave-dynamicalsystem (nuclei, atoms, quantum dots, and electromagneticor acoustic resonators). Prototypes are billiards of arbitraryshape. In its interior a point-like particle moves freely and

is reflected specularly at the boundaries. Depending on theshape its properties could exhibit chaotic dynamics.

Within this work, we investigate quantum billiards withits statistical eigenvalue properties, which reveal the periodicorbits in the quantum spectra and give the quantum chaoticscattering (Dembowski et al., 2005). Specifically, we sim-ulate microwave resonator with chaotic characteristics, seeFig. 1, and we compute the eigenfrequencies that are neededfor its level spacing analysis (Dembowski et al., 2002). Ac-cordingly, the eigenfrequency level spacing analysis for de-termining the statistical properties requires many (in order ofthousands) eigenfrequencies to be calculated and the accu-rate determination of the eigenfrequencies has a crucial sig-nificance. Moreover, considering that the problem is to com-pute a large number of eigenfrequencies, they can be oftenlocated in different ranges, i.e. left-most, right-most or inte-rior portions of the spectrum could be sought.

In many scientific and engineering applications, as wellas, in computational science, this results in solving one ofthe fundamental problems, the large scale eigenvalue prob-lem. We list below just a few of the applications areas(Saad, 2011) where eigenvalue calculations arise: acceler-ation of charged particles, structural dynamics, electricalnetworks, Markov chain techniques, combustion processes,chemical reactions, macro-economics, control theory, etc.The above-mentioned realistic applications frequently chal-lenge the limit of both computer hardware and numerical al-gorithms, as the involved matrices are of large scale.

The frequency domain methods (Jacobi-Davidson,Arnoldi, Lanczos, Krylov-Schur etc.) for eigenvalue cal-culation for some cavity structures might result in anextremely time consuming simulation, along with slowconvergence and huge storage space. Particularly, structureswith complicated geometry require a large number of gridpoints to achieve accurate simulation results. An additional

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

24 T. Banova et al.: Eigenvalue extraction from TD computations

challenge from our application is that the dimension of thedesired eigen subspace is large, namely, one might possiblyneed thousands of eigen pairs located in a specified range,also referred to as a “window”, of matrices with dimensionin excess of several millions. Despite the fact that manyalgorithms for eigenvalue determination exist, accompaniedby the numerical models that are becoming increasinglymore sophisticated, not as many are specifically adaptedfor computing a large number of interior eigen pairs. Theycan usually calculate limited number of extreme or interioreigenfrequencies, and relatively few are designed for effec-tively reusing a large number of good initial presumptions,when they are available. Finally, computing a large numberof interior eigenvalues remains one of the most difficultproblems in computational linear algebra today.

In this context, our investigations comprise efficient, ro-bust and accurate computations of many desired eigenfre-quencies in a reasonable time, which also constitutes themain aim of our study. Within this work, we cover an ap-proach for extraction of resonant frequencies given the out-put from time domain computations of closed resonators.The proposed approach uses the advantage that one singletime domain simulation can provide the whole response of anelectromagnetic system in a wide frequency band, whereasa frequency domain formulation uses one computation foreach individual frequency. In addition, due to the fact thatthe time domain computations in the field of electromagnet-ics are already highly developed and considerably more ef-ficient, as well as the fact that the transient solver containedin CST Microwave Studio (CST, 2012) uses a high degreeof parallelization provided with modern graphics processingunits (GPUs) feature the simulation can be dramatically ac-celerated. Therefore, we can easily and quickly get the timedomain responses for a wide frequency band. In this way, sig-nificant reduction in computation time can be achieved andtherefore, our high interest leads to time domain computa-tions for electromagnetic problems.

The paper proceeds as follows. In Sect.2 we brieflypresent the proposed approach for high precision eigenfre-quency extraction given the available electric field computa-tions. Here, we describe the fundamental modeling of the an-alyzed structure, followed by a description of the used signalpostprocessing techniques. Section3 investigates the simula-tion scenarios together with the obtained results. Lastly, con-clusions are addressed in Sect.4.

2 Time domain approach for eigenfrequency extraction

This section provides a brief overview of a precise time-domain (TD) approach for eigenfrequency extraction, whichis applicable for different cavity structures. In a two-step pro-cess the modeling and simulation of a specific cavity struc-ture is initially done, and afterward a postprocessing of theacquired time domain simulation results in MATLAB (MAT-

Fig. 1.Left: desymmetrized version of the three-dimensional gener-alized stadium billiard, consisted of two quarter cylinders with radiir1 = 200.0 mm andr2 = 141.4 mm. Right: CST setup of the bil-liard cavity with an exciting antenna and detection probes placed atvarious positions.

LAB, 2011) is conducted. A descriptive sketch of the pro-posed TD approach is given in Algorithm 1. Additionally,the critical implementation points and details are covered, aswell as, discussed within this section.

2.1 Field simulation in time domain

The proposed approach for eigenfrequency extraction (givenin the following) is utilized within the project for determin-ing the statistical properties of a cavity with chaotic char-acteristics. The requirements for chaotic characteristics aremet by using a superconducting resonator with the shapeof a desymmetrized three-dimensional stadium billiard (seeFig. 1-left). The billiard consists of two quarter cylinderswith radii r1 = 200.0 mm andr2 = 141.4 mm, respectively,and is made of niobium, which becomes superconducting attemperatures below 9.2 K.

The cavity of interest is modeled in CST Microwave Stu-dio (CST MWS) and a small tiny exciting antenna (as usedin a physical model) is put properly in a way that the modeswithin a specific frequency range would be excited (seeFig.1-right). Intentionally, the excitation signal applied at theantenna input is wide band signal, i.e. a Gaussian-modulatedsinusoidal signal is chosen, which certainly covers the rangeof eigenfrequencies being sought. The field simulation withhexahedron discretization mesh in time domain is carried outwith the transient solver from CST MWS and it detects andrecords the electric field intensity at specific field detectionprobes placed at various positions inside the cavity (see lines1–6 in Algorithm 1). Later, the obtained time domain sig-nals are used for further postprocessing in MATLAB, basedon fitting techniques with a proposed model of the cavity re-sponse, as follows below.

2.2 Postprocessing of the CST signals

2.2.1 Limitations from finite simulation time

In view of the fact that we are dealing with superconductingwalls, the response of a cavity could stay for a long time as

Adv. Radio Sci., 11, 23–29, 2013 www.adv-radio-sci.net/11/23/2013/

T. Banova et al.: Eigenvalue extraction from TD computations 25

0 0.5 1 1.5 2x 10

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Am

plitu

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0 5 10 15x 10

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100

d) Time/s

Amplitude spectrum

Gaussian pulse Gaussian modulated signal

Fig. 2. Operating signals during the process of eigenfrequency extraction, listed in the respective sub-figures.(a) Acquired time domainsignal and Gaussian windowed time domain signal.(b) Amplitude spectrum of the Gaussian windowed time domain signal.(c) IsolatedGaussian pulse within the eigenmode spectrum.(d) Gaussian modulated signal.

T. Banova et al.: Eigenvalue Extraction from TD Computations 3

0 0.5 1 1.5 2x 10

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Gaussian pulse Gaussian modulated signal

Fig. 2. Operating signals during the process of eigenfrequency extraction, listed in the respective sub-figures. a) Acquired time domain signaland Gaussian windowed time domain signal. b) Amplitude spectrum of the Gaussian windowed time domain signal. c) Isolated Gaussianpulse within the eigenmode spectrum. d) Gaussian modulated signal.

Input: Given a cavity with a specified probe P(x,y,z)./* CST simulation */Data: run transient simulation with the predefined simulation

time.Data: export a TD sig for the electric field intensity in

P(x,y,z).

/* Post processing and frequencyextraction in MATLAB */

TD sig = TD sig∗gausswin(size(TD sig));fft data = fft(TD sig);amp spec = abs(fft data);forall the Gaussian pulses in amp spec do

gauss pulse = locate gauss pulse();/* ind start=starting index

ind end=ending index */GoF1 = cftool(gauss pulse,gauss model());/* GoF = Goodness of Fitting */if GoF1� 0 then

gausswin fft data(ind start : ind end) =fft data∗gausswin(ind end− ind start);forall the ind /∈ gauss pulse do

gausswin fft data(ind) = 0;endgauss mod sig = real(ifft(gausswin fft data));GoF2 =cftool(gauss mod sig,gauss mod model());if GoF2' 100 then

extract the eigenfrequency;end

endend

Algorithm 1: A sketch of the time domain approach forextracting eigenfrequencies.

the power losses in the walls are negligible. Theoretically,the response of an ideal-conducting cavity is Dirac impulsesequence in frequency domain, i.e. a summation of sinu-soidal signals with the associated eigenfrequencies in timedomain. Nevertheless, due to the limited simulated time160

interval the ideal Dirac impulses of the true spectrum arewidened. Moreover, due to the finite conductivity and the in-serted antenna, the amplitude spectrum of the signal does notcontain Dirac delta pulses, but it has pulses of finite width.

An important issue coming from the limitation in time is165

the discontinuities at the edges of the measured time (Mandaland Asif, 2010). Given that sharp discontinuities have broadfrequency spectra, these will lead to a higher side lobes leveland each spectral line of the signal’s frequency spectrum willbe spread out in the same way. In other words, the spreading170

means that signal energy, which should be concentrated onlyat one eigenfrequency, leaks instead into other frequencies,the so called ’spectral leakage’. Consequently, the wholespectrum is distorted and some weak impulses, i.e., eigen-frequencies, can be masked by the resulting convolution with175

neighboring strong pulses. This leads to the idea of multiply-ing the original signal within the measurement time by Gaus-sian function (see Fig. 2a) that smoothly reduces the signalto zero at the end points of the measurement time: thereforeavoiding discontinuities overall (see line 9 in Algorithm 1).180

As a result, in case of an ideal cavity, whose spectrum theo-retically is constituted of Dirac impulses located at the eigen-frequencies, we would have Gaussian pulses instead (see Fig.2b).

Furthermore, manifestation of a spectral distortion occurs185

as result of a reduced spectral resolution. Namely, this isan important issue, and the minimum separation needed be-

the power losses in the walls are negligible. Theoretically, theresponse of an ideal-conducting cavity is Dirac impulse se-quence in frequency domain, i.e. a summation of sinusoidalsignals with the associated eigenfrequencies in time domain.Nevertheless, due to the limited simulated time interval theideal Dirac impulses of the true spectrum are widened. More-over, due to the finite conductivity and the inserted antenna,the amplitude spectrum of the signal does not contain Diracdelta pulses, but it has pulses of finite width.

An important issue coming from the limitation in time isthe discontinuities at the edges of the measured time (Mandaland Asif, 2010). Given that sharp discontinuities have broadfrequency spectra, these will lead to a higher side lobes leveland each spectral line of the signal’s frequency spectrum willbe spread out in the same way. In other words, the spreadingmeans that signal energy, which should be concentrated onlyat one eigenfrequency, leaks instead into other frequencies,the so called “spectral leakage”. Consequently, the wholespectrum is distorted and some weak impulses, i.e., eigenfre-quencies, can be masked by the resulting convolution withneighboring strong pulses. This leads to the idea of multi-plying the original signal within the measurement time byGaussian function (see Fig.2a) that smoothly reduces thesignal to zero at the end points of the measurement time:therefore avoiding discontinuities overall (see line 9 in Algo-rithm 1). As a result, in case of an ideal cavity, whose spec-trum theoretically is constituted of Dirac impulses located atthe eigenfrequencies, we would have Gaussian pulses instead(see Fig.2b).

Furthermore, manifestation of a spectral distortion occursas result of a reduced spectral resolution. Namely, this isan important issue, and the minimum separation needed be-tween two frequency components must be determined, so

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that they can be resolved. In our approach, we choose thefrequency resolution good enough to recover the sought fre-quency data. As a result, a large number of simulation timesamples might be needed for a reliable characterization ofthe resonance frequency of the structure, given that betterfrequency resolution unescapably requires a longer simula-tion time. In spite of this, the modern GPUs feature a largenumber of processing cores and the simulation is speeded upsignificantly in comparison with a simulation running on sin-gle CPU.

2.2.2 Fast Fourier transformation

As soon as the limitations that come from finite simulationtime or reduced frequency resolution are overcame, the stepfrom the recorded time domain response to the frequency do-main response is computed using the fast Fourier transforma-tion (see lines 10–11 in Algorithm 1, see Fig.2b).

The very classical approach in finding eigenfrequencies isto look for local maxima of the eigenmode spectrum. How-ever, this way is not efficient when the interest is in pre-cise determination of the resonant frequencies and has somedrawbacks. Firstly, the spectrum is discrete with a certain res-olution and the peak value may not be entirely located on asample point. The eigenfrequency could be a value that issomewhere in the range between two samples given withthe frequency resolution for certain discrete frequency val-ues (“bins”), implying that the local maximum is not alwaysthe frequency that is sought. Secondly, a more serious focuswas placed that the neighboring modes contribute a certainamount to the total response at the resonance of the modebeing analyzed and affect slightly the resonant frequency. Todeal with these problems, refined modal extraction methodsbased on signal processing techniques have been developed(see lines 12–31 in Algorithm 1).

2.2.3 Technique for locating a Gaussian pulse

For further analysis, local Gaussian pulses (see Fig.2c)within the spectrum should be located properly (see line 13in Algorithm 1). The location process is divided into severalsteps. Primarily, a local Gaussian pulse is found as a set ofsamples with a local maximum. Thereafter, supplementarycheck is conducted if some other samples might be added tothe right-most/left-most side of the current Gaussian pulse.Namely, if the average amplitude of the succeeding right/leftouter triple of samples is less than the average amplitude ofthe right-most/left-most triple of samples, the outer triple ofsamples is added appropriately to the right/left part of theGaussian pulse. After that, the “empirical rule” for the cur-rent located Gaussian pulse is applied. For this purpose, thestandard deviation for the pulse is estimated and four stan-dard deviations are accounted for the resulting pulse. At theend, the distance from the both ends of the Gaussian pulse toits maximum is equally adjusted.

2.2.4 Fitting models

By invoking the acquired knowledge of the eigenmode spec-trum of the cavity, the fitting model should be nonlinear inthe parameters, i.e. we should use Gaussian nonlinear modelto obtain the parameters. In this direction, the parametric fit-ting is involved as essential technique for precise determina-tion of the eigenfrequencies and reducing the amount of datarequired for a given resolution. Relying on the above discus-sion, a custom Gaussian model is modeled within the MAT-LAB curve fitting tool (see line 16 in Algorithm 1), whichsuits to the specific curve fitting needs, as shown below:

yi = aiε

−(f −fi )2

2σ2i , (1)

wherei goes from unity to the number of the located localGaussian pulses in the amplitude spectrum for the analyzedstructure. With this model all of the local Gaussian pulses arefitted and in eachi-th fitting the values for the found param-etersai , fi , andσi are saved. The parametersai andfi areused as initial reasonable starting values of the parametersin the further fit, whereas the parameterσi is included to in-crease the numerical robustness. The values of the parameterfi are good candidates for eigenfrequencies, since they givethe position of the maximum value of each Gaussian pulse.

However, following this way we are restricted to a limitednumber of samples, same as the number of samples whichconstitute the local Gaussian pulse. The limitation in the sam-ples causes that the coefficient, which represents the good-ness of fitting, is very low. That means that the accuracyof the eventual eigenfrequency cannot be high. Additionally,with this approach only the amplitude information of the sig-nal is used, which is not sufficient for precise extraction.

These disadvantages lead to extending our approach, in asense where we can get also the phase information of thesignal in a form that is suitable for implementation. Namely,after fitting a local Gaussian pulse in frequency domain thevalues of the parameters and the coefficients for the good-ness of the fit are obtained. If the goodness of the fit forsome Gaussian pulse is positive then we window this Gaus-sian pulse with Gaussian windowing function and we go backin time domain using Inverse Fast Fourier Transform (IFFT)(see lines 18–31 in Algorithm 1). Otherwise, we do not takeinto consideration this pulse, which is most probably noise.From signal processing is known that shifting in frequencydomain means modulation in time domain. So, the Gaussian-

modulated sinusoidal signalyi = ai sin(2πfi t − φi)ε

−(t−t0)2

2σ2i

is expected in time domain. The frequency of the modulationfi is exactly the frequency that is sought. Consequently, wefit the resulting signal in time domain with a custom Gaussianmodulated model of the cavity response and again by findingthe optimal model parameters in the least squares sense wedetermine the “true” eigenfrequency.



The main advantage comparing the proposed approachwith the classical approach for finding the peaks, is thatwithin this approach the phase information of the signal isused and parametric fitting with all of the data available inthe time domain representation is applied. Although, the fit-ting based only on the amplitude information of the signalresults in poor fit, now using the phase information of thesignal, very good value for the goodness of the fit can bereached. In this way, as shown in Sect.3, we gain a very highaccuracy in the eigenfrequency extraction from time domaincomputations.

2.2.5 Nonlinear parametric fitting

The fitting process uses the method of nonlinear leastsquares, presented inNash and Sofer(1996) and Lewis etal. (2006), which minimizes the summed square of residuals,identified as error associated with the data, given by

S =

n∑i=1

r2i =

n∑i=1

(yi − yi)2, (2)

where the residualri for the i-th data point is defined as thedifference between the observed response valueyi and thefitted response valueyi while n is the number of data pointsincluded in the fit. In case of nonlinear models the parame-ters cannot be estimated using simple matrix techniques andthese models are particularly sensitive to the starting points.This leads to difficult fit and the starting points are adjusted.Therefore, at the beginning initial reasonable starting valuesfor each parameter are provided. Then, the iterative approachfollows some steps until the fit reaches the specified conver-gence criteria. At the end, the direction and magnitude ofthe adjustment of the parameters depends on the Levenberg-Marquardt fitting algorithm proposed byLevenberg(1944),Marquardt(1963), andMore (1978).

3 Simulation results

The accuracy of the TD method for eigenfrequency ex-traction is tested for both analytically and non-analyticallyresolvable problems. Additionally, the proper functionalityof the method has been checked during its implementationprocess. In the numerical tests, several resonators are con-sidered. Namely, due to verification purposes, rectangular,cylindrical and spherical resonant structures are analyzed,whose exact solution can be analytically evaluated. Firstly,the eigenvalues of the above mentioned resonators are com-puted from their analytical expressions and following thisway, a logarithmic relative error is calculated as

relative error= log10

∣∣fanalytical− fnumerical∣∣

fanalytical, (3)

by considering the first computed mode eigenfrequency, un-less otherwise stated.

Besides the analytically resolvable resonators, the rela-tive error is also measured for the chaotic billiard resonator,which has a clustered eigenvalue distribution (Dembowskiet al., 2002). A clustered distribution containing too closeeigenfrequencies might cause difficulties in the eigenfre-quency extraction. As reference data, eigenvalues calculatedwith CEM3D eigenmode solver for different degrees of free-dom (DOF) are computed. Here, CEM3D eigenmode solveris a parallel program, developed by (Ackermann and Wei-land, 2012) for the accurate calculations of eigenfrequenciesfor a given structure. It implements a FEM formulation bymeans of higher order curvilinear elements.

In all simulations, we first modeled and meshed the re-lated geometries in CST MWS with hexahedrons and savedthe electric field intensity during the transient simulation. Af-terward, the eigenvalue extraction is calculated in MATLAB.

In the simulation studies, we experienced that for the timedomain field computations, a single personal computer issuited for problems with a moderate number of degrees offreedom (say, up to 106 DOF). To be precise, a computer witha single core Pentium 3 GHz processor and 6 GB main mem-ory was used. The same PC configuration was also utilizedfor the TD method for eigenvalue extraction. On the otherhand, a more powerful computer was necessarily used to en-able the handling of meshes with hundreds of thousands upto several millions DOF. The larger-scale simulations wereperformed on a GPU computer, e.g. 2.00 GHz (quadcore)having 32 GB RAM memory and 4 nVIDIA Quadro GPUs.

3.1 Rectangular, cylindrical, and spherical cavity

As a first experiment, we consider a rectangular cavity withperfectly conducting walls, containing a perfect vacuum. Thedegeneracy is broken by making the side lengths different,i.e. rectangular resonator with dimensionsa = 20 cm, b =

10e cm, andc = 10π cm. The resonance frequency of therectangular microwave cavity for the TE011 mode (the modewith the lowest cutoff frequency for a rectangular waveg-uide wherec > b > a) is found by imposing boundary condi-tions on the electromagnetic field expressions in which all ofthe field components vary sinusoidally at a single frequency(Pozar, 1998).

Along this line, the fundamental mode in a cylindrical cav-ity (Reitz and Milford, 1960) with radius R = 20 cm andlengthL = 10π cm is calculated, too. For the analyzed cylin-drical cavity, since we do not have largeL fulfilling L >

2.03R, the TM010 mode constitutes the fundamental oscil-lation. The mode of interest has azimuthal symmetry and theelectric field has no longitudinal variation (δE/δz = 0).

Lastly, the first TM101 mode of a spherical resonator witha radiusR = 1m is computed from analytical expressionsgiven in (Gallagher and Gallagher, 1985) by employing aroot finding algorithm of transcendental equations which issimply explained in (Pozar, 1998).



0.005 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 −5

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f−f

f

∣ ∣ ∣

Rectangular cavity (1st mode)

Cylindrical cavity (1st mode)

Spherical cavity (1st mode)

O(log10(cells)−2/3)

Fig. 3. Relative deviation of the numerically obtained valuesf forthe lowest eigenfrequency to the analytical resultf as a functionof the degrees of freedom (DOF) for a rectangular, cylindrical, andspherical resonator.

Specifically, for the time domain field simulations severaldifferent discretization meshes have been used and the con-vergence study based on the calculation of a relative error,given with Eq. (3), is shown in Fig.3. As the number of dis-cretization mesh cells increases, the difference between theanalytical and the numerical solutions becomes smaller andabsolute error in order of 10−4 is present. So, fast conver-gence is observable and it should be emphasized. As sug-gested by the convergence study, good accordance of the nu-merical with the analytical results is evident.

3.2 Billiard cavity

Using the previosly described algorithm, about 900 eigenfre-quencies up to 7 GHz have been calculated for the billiardcavity structure. Unfortunately, for this shape of resonatoran analytical solution for the electromagnetic problem is notavailable and in order to validate the obtained results, ref-erence data from CEM3D eigenmode solver are used (seeFig. 4). The structures are modeled and meshed with curvi-linear tetrahedrons and the corresponding meshes are im-ported to CEM3D eigenvalue solver in order to compute therequested eigenfrequencies for the analyzed structure.

In Fig. 4, a part of the results that are found with theTD approach compared to the reference data, is presented.On the abscissa are given the frequencies in an a priori se-lected frequency band, i.e. from 1.8 up to 2.4 GHz. Theordinate shows the eigenfrequencies obtained with the timedomain approach using four different discretization meshes,and the reference data calculated using a field simulation infrequency domain with two different tetrahedral meshes. Thetotal time for the transient simulations is 3.5× 10−5 s, whichresults in frequency resolution of 30 kHz.

The results in Fig.4 indicate that when fine mesh is usedthe number of eigenfrequencies found with the proposed TDapproach is exactly the same as the reference data, i.e. no ad-ditional frequency is added or no missed. In addition, suchcheck was conducted for all of the 900 calculated eigen-

Cells: 59,292

Cells: 461,214

Cells: 3,882,816

Cells: 23,356,890

Tets: 318,425

1.8 1.9 2 2.1 2.2 2.3 2.4x 10

9Frequency/Hz

Tets: 7,056

Fig. 4. Convergence study showing a comparison between theeigenfrequencies calculated with the proposed TD approach (redcolor) and the reference eigenfrequencies obtained with CEM3Deigenmode solver based on higher order curvilinear elements (greencolor). For the TD approach four different hexahedron discretiza-tion meshes are used. At the same time, the reference data are ob-tained using two different tetrahedral discretization meshes.

Cells: 59,292

Cells: 461,214

Cells: 3,882,816

Cells: 23,356,890

Tets: 5,568,907

Tets: 318,425

2.3 2.31 2.32 2.33 2.34 2.35 2.36 2.37x 10

9Frequency/Hz

Tets: 7,056

Fig. 5. Convergence study showing a comparison between theeigenfrequencies calculated with the proposed TD approach (redcolor) and the reference eigenfrequencies obtained with CEM3Deigenmode solver based on higher order curvilinear elements (greenand blue color). For the TD approach four different hexahedron dis-cretization meshes are used. At the same time, the reference dataare obtained using three different tetrahedral discretization meshes.

frequencies. Concerning the accuracy of the obtained data,slightly shifting of the frequencies can be observed in caseof coarse meshes. As the number of mesh cells increases, agood agreement with the reference data is clearly observed.For this purpose, a part of Fig.4 is enlarged and shown inFig. 5. According to Fig.5, in case of fine meshes a goodagreement of TD results with the reference data can be ob-tained. Furthermore, in Fig.5 an additional row is added rep-resenting extremely accurate reference data for the eigenval-ues in the range [2.30, 2.37] GHz. These results are obtainedwith almost 6 million of tetrahedral mesh cells and are usedto calculate the logarithmic relative error for the modes: 2.36and 2.37 GHz (see Fig.6). From this figure a second orderconvergence error is observed. Consequently, the proposed



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Mode 2.36 GHzMode 2.37 GHz

O(log10(cells)−2/3)

Fig. 6. Relative deviation of the numerically obtained valuesf tothe reference resultsf as a function of the degrees of freedom(DOF) for a billiard resonator. Two eigenfrequencies are consid-ered: 2.36 GHz and 2.37 GHz mode eigenfrequency.

approach can be used for precise extraction of eigenfrequen-cies from time domain computations.

4 Conclusions

In conclusion, we have presented an approach for eigenfre-quency extraction when the time domain response for a su-perconducting closed resonator is available and when plentyof eigenfrequencies are required. The approach is relativelyfast and very simple compared to others presented in litera-ture. In addition, we have inspected the robustness and con-vergence of the proposed technique in time domain with thereference data determined by the CEM3D eigenmode solverwith higher order curvilinear elements. Hereby, the conclu-sion shows that the proposed approach results in solutionswhich agree well with the reference data, gaining high accu-racy and efficiency in eigenvalue extraction. Finally, all of theresults indicate that the proposed TD technique can be usedin different areas of applications, where a precise extractionof plenty of eigenfrequencies takes a crucial role.

Acknowledgements.The work of Todorka Banova is supported bythe “Excellence Initiative” of the German Federal and State Gov-ernments and the Graduate School of Computational Engineeringat Technische Universitat Darmstadt.

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