Generation of FDTD subcell equations by means of reduced order modeling

1806 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 8, AUGUST 2003

Generation of FDTD Subcell Equations by Means ofReduced Order Modeling

Bart Denecker, Frank Olyslager, Senior Member, IEEE, Luc Knockaert, Senior Member, IEEE, andDaniël De Zutter, Fellow, IEEE

Abstract—Adapted finite-difference time-domain (FDTD)update equations exist for a number of objects that are smallerthan the grid step, such as wires and thin slots. In this contributionwe provide a technique that automatically generates new FDTDupdate equations for small objects. Our presentation will befocussed on 2-D-FDTD. We start from the FDTD equations in afine grid where the time derivative is not discretised. This yields alarge state-space model that is drastically reduced with a reducedorder modeling technique. The reduced state-space model is thentranslated into new FDTD update equations that can be used in anFDTD simulation in the same way as the existing update equationsfor wires and thin slots. This technique is applied to a number ofnumerical problems showing the accuracy and versatility of theproposed method.

Index Terms—Finite-difference time-domain (FDTD) methods,finite-difference methods, reduced order systems, transferfunctions.

I. INTRODUCTION

OVER the years the finite-difference time-domain method(FDTD) has become a powerful numerical analysis tool

in computational electromagnetics. A problem concerning theFDTD-method that has received a lot of attention in a numberof ways, is the incorporation of electrically small obstaclesin the simulation domain. Since small objects require a smallspace step for a correct discretization, often a lot smaller thanwhat would be dictated by the smallest wavelength present,the memory requirements rise excessively if the standardFDTD-method is employed. Corresponding with a smallerspace step a smaller time step needs to be used as well, leadingto excessive computing time.

One way to circumvent this problem is the use of subcellmodels. Based on the knowledge of the analytic local field be-havior around certain specific, widely used, small objects, onelocally adapts the FDTD time stepping equation. Some twentyyears ago this has already been done for a thin perfect electri-cally conducting (PEC) wire [1], where the in-cell inductance ofthe wire was the key factor linking the charge and current on thewire to the fields. In [2] the field assumption of the tangen-tial magnetic and the radial electric field near a thin wire were in-corporated in the update equations. Later, these subcell modelswere improved for wires with dielectric coatings [3], for wireend effects [4] and for bundles of wires [5]. A lot of researchers

Manuscript received July 4, 2001; revised April 26, 2002.The authors are with the Department of Information Technology (INTEC),

Ghent University, B-9000 Gent, Belgium (e-mail: [email protected]).Digital Object Identifier 10.1109/TSP.2003.815439

have also investigated the development of subcell models forthin slots. In [6] a thin-slot formalism was introduced that al-lowed modeling of arbitrarily narrow slots, based on the qua-sistatic behavior incorporated in an in-cell slot capacitance. In[7], a thin-slot formalism was introduced that allowed modelinga slot with a depth of several cells. The update equations werederived using a Faraday’s law contour integral approach. In [8],an integral-equation based thin-slot algorithm allowed to modelslots with a very small depth by using an equivalent antenna.Subcell models were also developed for thin sheets: in [9] sev-eral methods for modeling thin dielectric sheets are compared.In [8] subcell models for thin conducting sheets are also com-pared, where the thickness of the sheet has to be smaller than theskin depth. A method for modeling good but not perfectly con-ducting sheets, for sheets thicker than skin depth, is proposedin [10]. All these techniques tackle very specific and geometri-cally simple objects. More general approaches have also beenproposed in [11] and for the acoustical FDTD method in [12].There the FDTD update equations are adapted using correctionfactors which are obtained from the known static behavior orfrom a quasistationary solution of the subwavelength region ofinterest. All the subcell models developed over the years orig-inate from known or calculated static behavior that is incorpo-rated into the FDTD-algorithm, but as far as we know a generalmethod has not yet been proposed.

Small details, on the other hand, can also be modeled in acoarse grid domain by using a subgridding or multigrid tech-nique [13]–[16]. A subgridding technique divides the problemspace into a general coarse grid and a local fine grid aroundsmall details. The fine grid is then stepped in time at a higherpace than the coarse grid. A difficult problem in subgriddingtechniques is the time coupling of both grids, since mostmethods in one way or another use time extrapolation [15], [16]or a discretized wave equation [13], [14], which deteriorate theresults. The higher the refinement ratio or the ratio of coarse tofine grid space step is, the more difficult it becomes to couplethe coarse grid to the fine grid. Nevertheless subgriddingtechniques are more versatile than the use of subcell modelssince they can be used for small objects of any shape.

In this paper, we present a technique that automaticallygenerates subcell models in FDTD. As with subgridding, thetechnique can be used for arbitrarily shaped objects. Thetechnique works as follows: around a small object one considersa fine FDTD grid that resolves the object. In this fine FDTD gridone writes down the normal FDTD equations while explicitlymaintaining the time derivatives. This set of equations isthen interpreted as a state-space model with the electric field

0018-926X/03$17.00 © 2003 IEEE

DENECKERet al.: GENERATION OF FDTD SUBCELL EQUATIONS BY MEANS OF REDUCED ORDER MODELING. 1807

Fig. 1. Sample of a 2-D FDTD mesh, the bold solid line is the contourC.

components at the boundary of the fine grid as input variablesand the magnetic field components at the boundary as outputvariables or vice versa. The internal variables are all the otherelectric and magnetic field components inside the fine grid.The crux of the method is that with a reduced order modelingtechnique (ROM) the size, i.e., the number of internal variables,of this state-space model is drastically reduced without affectingthe behavior up to a certain frequency. In this reduced modelone then discretizes the time and links it to a coarse FDTDgrid. The main advantage of this technique is that for eachgeometrically small object, whatever electric properties it has,this reduced state-space model can be generated once and forall before the actual FDTD simulations. Other advantages are:the memory requirements are smaller than with subgridding,time coupling is no problem and the time step is mainly dictatedby the coarse grid.

In a previous paper [17], the idea of using a ROM-techniqueto generate a small model that can be used in time-domain sim-ulations was already established. Contrary to [17], in this paperwe combine a reduced state-space model with the powerful reg-ular FDTD-equations. The size of the reduced state-space modelis not only minimized by using a ROM-technique, but is alsobased on subgridding techniques. In [17], no subgridding wasused, but there essentially a 1D-model for each subdomain wasgenerated, which then allowed fast modeling of the problem.The examples presented here are much more versatile since wedon’t restrict ourselves to perfect electric conductors in vacuumbut also use dielectric and lossy materials. Also, in the presentpaper we provide deeper theoretical grounds for our approach.

For the reduction of the state-space model, we use a techniquethat has been recently developed in [18], [19]. This reductiontechnique creates a Krylov-based Laguerre approximant of the

transfer matrix, and has the property to yield a reduced ordermodel that is valid from DC up to a certain frequency . Thehigher one chooses , the lesser the model can be reduced.This method has been used very successfully for the reductionof circuits [19] originating from the partial element equivalentcircuit (PEEC) technique and lossy coupled transmission lines.

In this paper, we will restrict ourselves to a 2-D situation. Themethod is introduced in three phases. In phase 1 we derive thestate-space model for the fine grid surrounding a small object.In phase 2 the reduced order modeling technique is applied toreduce the size of this model. Finally in phase 3 we take care ofthe coupling of the reduced state-space model with the coarseFDTD grid. The last part of the article demonstrates the powerof the method in a number of examples.

II. PHASE 1: FINE GRID STATE-SPACE MODEL

Fig. 1 shows a part of a 2-D FDTD mesh. Let us assume thatwe work in a TM situation (the TE-case is completely dual).This means that the electric field is directed along the-axisand that the magnetic field is located in the-plane. Let usnow consider a part of the mesh surrounded by the boundary

. In principle this boundary can have any shape but for sim-plicity assume it to be a rectangle. The electric fields just out-side the boundary are grouped in a vector. On the figurethis vector has a dimension equal to 14. The magnetic fields onthe boundary are grouped in a vector. Also this vector has adimension equal to 14 for the case of the figure. The electric andmagnetic fields inside and on the boundaryare grouped in thevector , which has dimension 43 on Fig. 1. Here the vectoris a subset of .


In the next step, we write the FDTD-equations for the electricand magnetic fields without differencing the time

(1)

(2)

(3)

If we now write down these equations for all the field com-ponents in the vector then one obtains a large system offirst-order differential equations of the form:

(4)

The matrix is a sparse matrix with nonzero elements equalto or . The matrix is a diagonal matrix withelements and . The matrix is also sparse with nonzeroelements equal to or . The matrix containsa few elements equal to 1 in order to pick the subsetfrom

. Now we consider the following alternative FDTD scheme.Suppose everything is known until timestep .

• First, we update the electric fields outsidewith the reg-ular equation

(5)

Note that when that some of the mag-netic fields on the right hand side are extracted from

.• Second, we update the magnetic fields outsidewith the

regular equations

(6)

and similar for the -components.• Third, we use a finite-difference scheme to calculate the

magnetic fields on , i.e., the elements of, at timeusing (4), we have (7) at the bottom of this

page.• Last, then the first step is repeated at

There are some remarks that need to be made.

• At first sight, this is a rather costly update scheme withrespect to memory and CPU time considerations. Indeedthe matrices , , , are large and in (7) the inversesand products of these matrices need to be calculated. Thisproblem is addressed in phase 2.

• In Appendix A, it is shown that the update (7) is uncondi-tionally stable. This means that the time step is determinedby the regular update (5) and (6), i.e.,

(8)

• It is clear that inside the medium can be inhomoge-neous, dispersive, or whatever as long as the FDTD equa-tions (without differentiation with respect to time) can bewritten under the form (4).

III. PHASE 2: REDUCED ORDER MODELING

As noted before, the scheme developed in phase 1 is not veryadvantageous with respect to CPU-time and memory. We willremedy this now. The state-space (4) are a typical description ofa linear system with internal variables, inputs and outputs

. In the update process we only need the variables inand, those in are not important. This means that another linear

system with the same transfer function betweenand but withdifferent internal variables is equally suitable. We will use amodel order reduction algorithm as described in Appendix Band in [19] to change and to reduce the size of. The methodis based on an expansion of the impulse response in terms ofscaled Laguerre functions ( ).The advantage of this algorithm is that it allows the specifica-tion of a frequency up to which the transfer function be-tween and in the reduced order model (ROM) is a faithfulrepresentation of the transfer function in the original system. In[19] is proposed as a rule of the thumb. The linearsystem for the reduced model is written as

(9)

where is the reduced set of internal variables. The dimensionof is and that of and is . The dimension of is de-noted as with theorder of approximation. In the reductionalgorithm this can be chosen. Now we can repeat the updatescheme of phase 1 with the reduced system (9).

We can make the following remarks.

• Since the size of and hence of , , and is muchsmaller (the examples will demonstrate this) than that of

, , , and , the CPU and memory requirements aremuch lower. Of course there is some CPU-time involved in

(7)


Fig. 2. Mesh containing the region insideC several times.

Fig. 3. The contourC (solid fine line) surrounding a local mesh with higherresolution. The contourC is the solid bold line. The bold lines correspond tothe surrounding coarser grid.

generating the reduced system. However, this can be doneonce and for all before initiating the iterative process. Thesame holds for the matrix inversions and multiplications in(7). This becomes even more advantageous if we can reusethe region inside a few times in the overall structure.This is illustrated in Fig. 2.

• In practice, usually some small losses are introduced inthe system, such that no eigenvalues of are on theimaginary axis and this in order to guarantee that no eigen-values of are in the left half plane after the reduc-tion algorithm. These losses can be taken very small andhave insignificant influence on the results.

• The reduction algorithm guarantees that a stable linearsystem is transformed into a stable reduced linear system.

Fig. 4. Subdomain used for the derivation of the free space subcell model, therefinement ratio is 15.

Hence, the poles of both the original and reduced systemare located in the right half plane. From the result in Ap-pendix A, it then follows that the finite difference versionof the reduced system is also unconditionally stable.

IV. PHASE 3: COARSEGRID COUPLING

In this section, we will demonstrate how to use the previoustechnique to generate automatic subcell formalisms in FDTD.Consider a structural element that cannot be resolved in a coarsemesh with cell size as is shown in Fig. 3. A gen-eralization where is also feasible but not consid-ered here. Inside a contour around the considered structuralelement, we subdivide the coarse mesh into a fine mesh, fineenough to resolve the element in a sufficiently accurate way. Thecell size of this finer mesh is . The use of local finermeshes, so-called subgridding, has been studied abundantly inthe past. However, our objectives are different.

Let denote the electric fields in the fine mesh just outside. The contour is chosen such that some of the electric fields

in coincide with the electric fields in the coarse mesh (Fig. 3,e.g., and ). This is only possible if is odd. Thisis not too much of a restriction. The electric fields in the coarsemesh just outside , i.e., the electric fields in the coarse meshthat coincide with the fields in , are denoted as .

For the vector we use a slightly different definition than inthe previous phase. In we now store the magnetic fields on aboundary inside that coincides with the coarse mesh (seeFig. 3). is a vector with the magnetic fields of the coarse meshlocated on that contour .

Using the FDTD-equations in the fine mesh we can again setup some linear system with inputand output as in (4). Nowwe have to connect the input and output variablesand of the


Fig. 5. Maximum spurious scattered electric field as a function of frequency, for an angle of incidence of 36.9.

Fig. 6. Maximum spurious scattered electric field as a function of the angle of incidence atf = 23:7 MHz.

fine mesh to those of the coarse mesh. To relateto we useinterpolation. For the example of Fig. 3 this means

(10)

(11)

(12)

In general, we can write , where is again a rathersparse matrix. To calculate from we take the average of allthe that are closest to . For example

in Fig. 3. In general we write this as . The system (4)can hence be written as

(13)

with and . The vectors and are nolonger of the same dimension: ,and . In the last step, we reduce the size oftoobtain:

(14)


Fig. 7. Subdomain used for the derivation of the PEC wire subcell model, the refinement ratio is 17.

The size of is still , the number of input fields ( ) timesthe order of approximation ().

These equations can then be used in the FDTD update schemeexplained in phase 1. In this way, we obtain a subcell formalismfor the small structural element.

• Again we remark that this subcell formalism can be gen-erated once and for all in advance of the FDTD iterativeprocess.

• Since we use the transition between a fine and a coarsemesh, we lose the absolute stability we had in phase 1and 2. As a consequence we need to use a time step thatis smaller than that defined by the Courant limit of thecoarse mesh. Proving stability in a general and rigorousmanner is not straightforward. However, it turns out thatthe maximum time step which retains stable results is notmuch lower than the coarse mesh Courant limit and cer-tainly much larger than the fine mesh Courant limit. Theexamples demonstrate this.

• Subcell models generated for lossless dielectric or mag-netic materials are scalable. An timeslarger grid can make use of the same old subcell model,yet keeping in mind that the reduced order modeling willhave been done around instead of ( ).

• The universality of the state-space description (13) allowsto add one or more extra input variables, which can serveas external sources inside the fine mesh, increasing thedimension of vector , or add one or more extra output

Fig. 8. Test configuration for validation of generated PEC wire subcell models.

variables, which can serve to record the fields inside thefine mesh, increasing the dimension of vector.

V. NUMERICAL RESULTS

A. Lossless Free Space

In the first example we consider a subcell model for a pieceof empty space and we consider the spurious scattered fieldcaused by this subcell model. As plane wave excitation we usethe method established for a long time in [20], and which hasproven to be very accurate when applying some modifications[21]. Here higher order cubic interpolation and the numericalphase velocity were used to determine the incident values of theplane wave, the sinusoidal signal was switched on using a Han-ning window.

The subcell model is one coarse cell in size, the correspondingfine grid subdomain is shown in Fig. 4 and contains no special


Fig. 9. Recorded electric field: time domain results.

Fig. 10. Ratio of electric field to line source current: frequency domain results.

features. The bold lines correspond to the coarse grid and theother lines to the fine grid. In the figure the contoursandare indicated by solid lines. The refinement ratio, is 15.The size of the different vectors is 2581 for, 8 for and 4for . Based on this subdomain and using an order of approxi-mation , some subcell models were generated for ,2,4,7and . From this the dimension of the newvector becomes . The resulting subcell model is thendescribed by and (dimensions ), (dimensions

) and by (dimensions ). No losses were addedfor this example.

The spurious scattered field caused by the reduced order mod-eling and by the transition between the coarse and the fine gridhas been calculated. The space step was , thecorresponding Courant time step is . To assure sta-

bility, the time step had to be chosen smaller:or about half the maximum Courant time step. The total field re-gion was 3 3 cells in size. Two plots were generated (Figs. 5and 6). In each plot the maximum recorded scattered field wasplotted, where the incident field has amplitude of 1. In Fig. 5the maximum scattered field has been plotted for each gener-ated subcell model as a function of frequency, the angle of inci-dence (angle between-axis and direction of propagation) waskept constant at 36.9. The spurious scattered field in the classicFDTD simulation was also added to illustrate the accuracy ofthe plane wave formalism. When , the spurious scatteredfield is small and the accuracy of the method can be observed.In Fig. 6 the maximum scattered field was plotted as a functionof the angle of incidence, the frequency was kept constant at23.7 MHz. Since the problem is symmetrical, only the interval


between 0 – 45 was considered, the other angles can be de-rived from this. The case was not included in the figure,since its accuracy is too poor. The angle of incidence does in-fluence the accuracy, but for all angles, accuracy is good as longas . In both figures the bold vertical line shows the angleor the frequency that was kept constant in the other figure.

B. Perfectly Conducting Thin Wire

In a second example, we will generate a subcell model of aperfectly conducting wire. This can then be compared to sub-cell models previously derived in the literature [2], together withnormal but finely meshed FDTD simulations. The radius of thewire we will consider is 0.4 mm, the space stepof the mesh is1.7 mm, so the wire can easily be accomodated in one cell. Thefine grid subdomain, containing the staircase approximation ofthe circular wire, that was used to generate the subcell model,is shown in Fig. 7. The wire was positioned in the middle ofthe cell. This is not necessary, but this was done to allow com-parison with the method developed in [2]. The refinement ratio( ) is 17, so . The original number of in-ternal variables, i.e., the dimension of, was 3144 (known zerofields inside the wire are not part of). The dimension of the re-duced order subcell model (14) is, with , , and thedimensions of the different matrices defining the subcell modelare equal to those in the previous example. The factorwaschosen with . Some very small losses wereadded to force eigenvalues away from the imaginary axis intothe right half plane. This was done to prevent numerical noisein the ROM-algorithm from creating small unstable eigenvalues.These losses are: and .

Fig. 8 shows the configuration consisting of the wire, withat opposite sides the excitation and the electric field recordingpoint. The line source was excited with a Gaussian pulse modu-lated at 1.25 GHz. This problem was simulated in four differentways:

i) the coarse grid ( ) and the subcell modelintroduced in [2];

ii) the coarse grid ( ) and the subcell model, asproposed in this paper;

iii) normal FDTD using a fine grid ( ) and stair-case approximation of the PEC wire;

iv) normal FDTD using a very fine grid ( ) toget a better staircase approximation of the wire.

The time step was selected using the Courant limit in i), iii), andiv). In simulation ii), for reasons of stability, the time step wasset at 0.55 times the Courant limit, or . Evenafter a long time ( time steps) the tail of the results didn’tshow any sign of instability with this time step. In Fig. 9 the re-sults are shown. It can be seen that the new subcell model showsbetter agreement than the old model [2] compared to the classicFDTD results. Fig. 10 shows the ratio of the frequency trans-formed electric field to the injected current for frequencies upto 10 GHz. The same conclusions can be drawn: the generatedsubcell models show a better behavior for the investigated wireradius, but the influence of the staircase approximation must notbe underestimated. Another subcell model was also generated,with (twice as much internal variables), but no differencewas found between both subcell models.

Fig. 11. Subdomain used for the derivation of the dielectric wire subcellmodel,re�nement ratio = 13.

Fig. 12. A 3� 3 periodic structure test configuration for validation ofdielectric wire subcell models.

C. Dielectric Thin Wire

The third example is similar to the previous one, but insteadof studying the effects of a perfectly conducting wire, a dielec-tric wire is considered. In Fig. 11 the fine grid that was usedfor extracting the subcell model is shown. The different dimen-sions are: , and the wire radius is0.9 mm. The dielectric constant of the wire is: . The sizeof the subcell model does not cover just one cell but covers anarea of 2 2 cells. The dimensions of the original system (13)are: , and . Thenumber of internal variables ( ) is reduced to in-ternal variables ( ), and was used.The generated subcell model was validated in the configurationof Fig. 12 for several values of. The source was a line source,the field was recorded at the other side of a periodic dielectric



structure consisting of 9 dielectric wires. The source was ex-cited by a modulated Gaussian pulse. Fig. 13 shows the resultsfor 3 different subcell models (different orders of approximation) simulated in a grid with . These results can be

compared with the normal FDTD results, obtained using a finegrid ( ). The time step used in the simulations con-taining the subcell models was or about 0.45 timesthe Courant limit related to the coarse grid ( ). Withthis time step results showed no sign of instability even after

time steps. The curve with agrees with the FDTDresult only in a the small frequency region up to .When a better model is used, , the results agree up tomuch higher frequencies: . For both curves onlystart to deviate at about 11 GHz.

In Figs. 10 and 13 it can be seen that the frequency regionwhere we get a good approximation extends far larger than

. The parameter follows from theoretical grounds(see [18], [19]), but all numerical examples show that thisparameter is too conservative.

D. L-Shaped Lossy Dielectric Object

In a last example, a subcell model of a small nonsymmetriclossy dielectric object was generated. The L-shaped object canbe seen in Fig. 14. The parameters of the lossy dielectric are:

and . Cell sizes are ,and the subcell model is 22 cells in size. The

skin depth associated with the loss is about at the max-imum frequency of interest. The time step and dimensions ofmatrices and vectors are the same as for the previous example.The configuration used for the simulation is shown in Fig. 15. InFig. 16, the frequency results are shown, where the results basedon a fine FDTD grid are used as a reference. It can be observedthat with increasing order of approximation, the frequency do-main, where the model holds, becomes larger. For this isup to 1.5 GHz, for this is already 4.5 GHz and for

Fig. 14. Subdomain used for the derivation of a subcell model of a lossydielectric L-shaped object, the refinement ratio is 13.

Fig. 15. Test configuration for validation of lossy dielectric subcell model.



TABLE ICOMPARISON: COMPUTATIONAL COMPLEXITY REQUIREMENTS

TABLE IICOMPARISON: MEMORY SAVINGS

the model holds up to 6 GHz. An important remark is, whichcould also be seen in the previous examples, that every subcellmodel holds in the lower frequency domain, only at higher fre-quencies the generated subcell models become invalid.

As an illustration of the power of the proposed method wecompare in Table I the computational complexity (in flops) forthis example as opposed to what was needed for the regular finegrid simulation. In Table II the memory savings are considered.The size of the grid as shown in Fig. 15 was used.

Table I illustrates that, if the same amount of simulated timeis considered, even for the worst case , the computationalsavings are considerable: a factor 40 to 100. When memory isconsidered (see Table II), a significant gain can also be deter-mined: a factor 10. Moreover, if the same model is used several

times, as in the example with the thin dielectric wires, savings inmemory are even larger. This is a consequence of the fact thatfor each repeatedly introduced subcell model, only a vectorhas to be added since the matrices in the reduced counterpart of(7) remain the same and have to be stored only once.

VI. CONCLUSION

In this paper, a straightforward method has been introducedthat allows the automatic generation of subcell models for smallobjects. The accuracy and stability of the method was care-fully investigated. The method turns out to be stable at a timestep which is about one half of the Courant limit of the coarsegrid. The versatility of the method was demonstrated by thedifferent shapes and electrical properties of the simulated ob-jects: a PEC-wire, a dielectric wire and an L-shaped lossy di-electric. When the computational requirements for generatingthe reduced order model are not considered, the savings in com-putational memory and CPU time are considerable, especiallycompared to the normal FDTD technique with a fine mesh. Fi-nally we remark that the subcell models are scalable for losslessstructures.

Although only the 2-D situation was investigated here, it ispossible to apply this technique to 3-D problems. This is cur-


rently under investigation. The main problems that will remainare not theoretical but related to the size of the state-spacemodel.

APPENDIX A

EIGENVALUES

The update (7) is stable when the eigenvaluesofhave a modulus not ex-

ceeding 1. These eigenvalues are solutions of

(A.1)

With some simple manipulations one can show that this is equiv-alent to (when )

(A.2)

If are the eigenvalues of , then it follows that. If the eigenvalues of or

equivantly of are on the imaginary axis or in the righthalf plane then . Since represents a passivesystem this is always the case.

APPENDIX B

ROM ALGORITHM

The state-space model (4), with unit impulse excitations at theinputs and zero initial conditions, yields the impulse

response matrix with corresponding transfer matrix

(B.1)

where is the Laplace transform, and, are the originalstate-space matrices. In the ROM technique used in this

paper, the impulse response matrix is expanded in scaledLaguerre functions defined as

(B.2)

where is a scaling parameter and represents theLaguerre polynomial of order

(B.3)

The expansion of the impulse response matrix in scaled La-guerre functions then becomes

(B.4)

Considering the Laplace transform of the scaled Laguerre func-tions (B.2)

(B.5)

allows us to write the transfer matrix as

(B.6)

By utilizing the bilinear transformation

(B.7)

the -domain Laguerre expansion (B.6) can be mapped into the-domain, rendering a power expansion

(B.8)

In other words an -th order Laguerre approximation in the-domain is equivalent with an -th order Padé approximation

of the modified transfer matrix

(B.9)

in the -domain.Since the Laguerre expanion is transformed into a Padé ex-

pansion we can use a reduced order algorithm for Padé approx-imation. This algorithm is the PRIMA-algorithm described in[22]. It starts by introducing matrices and

(B.10)

(B.11)

Next the column-orthogonal matrix is obtained,which is associated with the block Arnoldi process as applied tothe Krylov matrix

(B.12)

where the factor is the order of reduction. This then yields areduced order system described by

(B.13)

such that the reduced order state-space model becomes (9).Instead of the matrix , associated with the block Arnoldi

process, it is shown in [18] that it is numerically more stable touse the left SVD column-orthogonal factorof the matrix .Hence the ROM algorithm eventually becomes the following.

• Select the values for and .• Solve .• Solve , for

.• Construct .• Calculate SVD: .• Construct new reduced order system matrices

(B.14)

It can be shown [19], that a suitable choice for the Laguerreparameter is , where is the bandwith of thesystem. Finally we remark that the reduction process conservespassivity of the system.

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Bart Denecker was born in Roeselare, Belgium, in1975. He received the electrotechnical engineeringdegree and the Ph.D. degree from the Department ofInformation Technology, Ghent University, Belgium,in July, 1998 and 2003, respectively.

His primary research concerns the finite-differencetime-domain method.

Frank Olyslager (S’90–M’94–SM’99) was born in1966. He received the Electrical Engineering and thePh.D. degrees from Ghent University, Belgium, in1989 and 1993, respectively.

At present, he is a Full Professor of electromag-netics at Ghent University. His research concernsdifferent aspects of theoretical and numericalelectromagnetics. He is Assistant Secretary Generalof URSI and was Associate Editor ofRadio Science.He has authored or coauthored about 160 papers injournals and proceedings. He coauthored one book

Electromagnetic and Circuit Modeling of Multiconductor Transmission Linesand authored another bookElectromagnetic Waveguides and TransmissionLines(Oxford University Press, Oxford, U.K.). In 1994, he became laureate ofthe Royal Academy of Sciences, Literature and Fine Arts of Belgium.

Dr. Olyslager received the 1995 IEEE Microwave Prize for the bestpaper published in the 1993 IEEE TRANSACTIONS ON MICROWAVE THEORY

AND TECHNIQUES and the 2000 Best Transactions Paper award for the bestpaper published in the 1999 IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY . In 2002, he received the Issac Koga Gold Medal at the URSIGeneral Assembly.

Luc Knockaert (M’81–SM’00) received the M.Sc.degree in physical engineering, M.Sc. degree intelecommunications engineering, and Ph.D. degreein electrical engineering from Ghent University,Belgium, in 1974, 1977, and 1987, respectively.

From 1979 to 1984 and from 1988 to 1995, he wasworking in North-South cooperation and develop-ment projects at the Universities of Congo (formerlyZaire) and Burundi. He is presently a Guest Professorand Senior Researcher at INTEC-IMEC. His currentinterests are the application of statistical and linear

algebra methods in signal identification, sampling theory, matrix compression,reduced order modeling and the use of integral transforms in electromagneticproblem solving. As author or coauthor he has contributed to more than 40international peer-reviewed journal papers.

Dr. Knockaert is a Member of ACM and SIAM.

Daniël De Zutter (F’00) was born in 1953. Hereceived the M.Sc. degree in electrical engineeringfrom Ghent University, Belgium, in 1976. From 1976to 1984 he was a research and teaching assistant atthe same university. In 1981, he received the Ph.D.degree and in 1984, he completed a thesis leadingto a degree equivalent to the French Aggrégation orthe German Habilitation.

From 1984 to 1996, he was with the National Fundfor Scientific Research of Belgium. He is now a fullProfessor of electromagnetics. Most of his earlier sci-

entific work dealt with the electrodynamics of moving media. His research nowfocuses on all aspects of circuit and electromagnetic modeling of high-speedand high-frequency interconnections and packaging, on Electromagnetic Com-patibility (EMC) and numerical solutions of Maxwell’s equations. As author orcoauthor, he has contributed to more than 120 international journal papers and140 papers in conference proceedings. In 1993, he published a book entitledElectromagnetic and Circuit Modeling of Multiconductor Transmission Lines(with N. Faché and F. Olyslager) in the Oxford Engineering Science Series.

Dr. De Zutter received the 1990 Montefiore Prize of the University of Liègeand the 1995 IEEE Microwave Prize Award (with F. Olyslager and K. Blomme)from the IEEE Microwave Theory and Techniques Society for best publicationin the field of microwaves for 1993. In 1990, he was elected as a Member ofthe Electromagnetics Society. In 1999 he received the Transactions Prize PaperAward from the IEEE EMC Society.

Generation of FDTD subcell equations by means of reduced order modeling

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