A Practical Guide to 3-D Simulation

Digital Object Identifier 10.1109/MMM.2008.929772

62 December 20081527-3342/08/$25.00©2008 IEEE

©DIGITAL VISION

Thomas Weiland,Martin Timm, and

Irina Munteanu

This article is intended to give design engineers an overviewover some properties of numerical methods used in today’smost relevant commercial electromagnetic (EM) simulationtools. It cannot and does not want to be a rigorous analysisof the methods themselves nor a concise description of their

history. For an extensive overview, we would recommend textbookssuch as [1] and [2]. The authors have experience in not only the researchand development (R&D) of numerical methods but also in the supportof users in their daily work with commercial simulation software.

Designing passive components, whether it is obvious or not, is allabout solving Maxwell’s equations. From university, we know that thepen and paper approach for finding appropriate solutions is very limited:in complex systems, complicated differential equations can not be solved

Thomas Weiland is with TechnischeUniversität Darmstadt Institut für TheorieElektromagnetischer Felder Schloßgartenstr, 8 D-64289 Darmstadt. Martin Timm andIrina Munteanu are with CST AG, BadNauheimer Str.19, 64289 Darmstadt,

Germany, www.cst.com.

Authorized licensed use limited to: Tsinghua University Library. Downloaded on July 28, 2009 at 03:05 from IEEE Xplore. Restrictions apply.

by analytical methods. Designers typically circumvent thisproblem by simplification. Often empirical models areused that replace reality, for example, by introducing cir-cuit elements. These models typically have a limited rangeof validity, which is easily disregarded while using them.In this article, we will deal with three-dimensional (3-D)volume based numerical methods. Their advantage is thatthey have very little physical constraints in their applica-tion range. Even though a single numerical field simula-tion requires much more time and computing resourcesthan a circuit simulation, this additional deployment ofresources might be well invested.

It is widely accepted that 3-D numerical simulations ofEM fields are essential to the success of an R&D depart-ment working predominantly on passive components.Obviously, simulating a virtual prototype is much cheap-er than building hardware and measuring it, in particularif the design cycle time is considered as well. Looking atmodern optimized antenna designs, for example, it isarguable whether this design would have been possibleat all without EM field simulation tools, without auto-matic optimization, without the possibility to visualizethe previously invisible. But saying, “all right, let’s go andbuy a 3-D EM field simulator and everything will be fine”is probably not sufficient. We want to discuss the prosand cons of different methods here, as well as give somehints on how to use such simulators.

Solving Maxwell’s Equations All numerical approaches to solve Maxwell’s equationspartition space into subdomains, where solutions can befound more easily. A mode-matching code, in its simplestapplication, composes a waveguide system from sectionswith known behavior by performing a modal expansionand matching the fields at the intersection areas. Amethod-of-moments (MoM) code synthesizes the far field of anantenna by integrating the Green’s functions of singlemetallic surface patches. Volume discretization methodswork with even more brute force. They subdivide spaceinto small cells and apply Maxwell’s equations on eachsuch entity. To solve the full problem, all single-entity solu-tions are summed up in a usually large system of equa-tions, which needs to be tackled in one way or another.

When discussing the properties of the differentmethods, it is necessary to classify them. A major pointof difference is the domain they are working in, whichis either time domain or frequency domain. Concen-trating on the methods that are most relevant commer-cially, we find on the time domain side the finite inte-gration technique (FIT) (see “Finite Integration Tech-nique” [3], [4]); finite difference time domain (FDTD) inits explicit [5], [6] or implicit [7], [8] variants; and thetransmission line matrix (TLM) method [9], [10]. Thefrequency domain is represented by the finite elementmethod [11], [12]; FIT; and the MoM [13]. All methodsare volume discretization methods, except for theMoM, which is a surface discretization method.

A Typical Model Set-UpIn setting up a computer model for a real device, thereare several steps, which are common to all discussedsimulation methods. All of them bear risks to introduceerrors into the simulation, i.e., discrepancies betweensimulation results and measurements. A typical modelsetup is demonstrated in the following, using the exam-ple of a rather simple 90◦ coaxial connector (Figure 1).

First of all, a geometrical model needs to be created.This can be either done by using a modeler built into thesimulation software or by importing the geometricaldata from a mechanical CAD tool. Importing from CADtools is not as easy as it sounds, and the quality ofimport filters varies significantly; but this is beyond thescope of this article. If comparing to an existing device,the exact same dimensions have to be used. Sounds sim-ple? Besides obvious errors, there are always tolerances,and sometimes details are neglected, which are relevantat microwave frequencies and radiofrequency (RF).

The considered connector is assembled from differ-ent materials, like polytetrafluoroethylene (PTFE,Teflon), copper, etc. Knowledge of the exact materialproperties is essential for an accurate simulation, butthis is normally not available.

The computational effort for volume-based methodsdepends also on the volume size, and the size of thesimulation model must always be finite, even if in real-ity the component is placed in an infinite surrounding.In order to reduce surrounding space, boundary condi-tions need to be introduced that represent, for example,electric walls, free space, symmetry, or periodicity. Forour connector, this is not relevant, because we can sim-ply assume that the space surrounding it is a perfectconductor, as we know that there will be virtually nofield penetrating the conductor shielding.

Finally, we have to define ports in the model to excitethe structure and to monitor simulation results such astransmission and reflection. Ideally, these ports shouldnot have an impact on the simulation results.

Performing a Simulation Having set up our geometrical representation of the realstructure in the software environment, we can now startthe steps towards the final results. The first step is theactual space discretization—the mesh setup—which isautomated to a large extent in modern commercial soft-ware. Despite the high degree of automation, the pro-posed mesh might need to be checked or influenced man-ually in order to obtain accurate results. In a second step,the software creates the system matrices based on thegeometrical information from this grid and the methodchosen for approximating Maxwell’s equations. After allthe required matrices are created and assembled, thethird step starts; namely the solution of the finite algebra-ic system. Here we want to calculate the S-parameters forour connector, since they are the most often requestedresult for passive component characterization.

December 2008 63


64 December 2008

In the frequency domain, this process is straight-forward [Figure 2(b)]: one simulation delivers the S-parameters at one frequency point. However, thebehavior is usually relevant in a specified frequencyrange, so looking at a single frequency is notsufficient. Therefore, a number of simulations in thefrequency band of interest have to be performed. Spe-cial algorithms are used to minimize the number ofsimulations required to achieve a predefined accura-cy by interpolating the S-parameters in between the

simulated frequency points. In the case of our con-nector, ten simulations are necessary to cover therange of 0–8 GHz with a predefined accuracy of 1%over the entire frequency band.

In time domain, the approach is quite different [Fig-ure 2(a)]. The user specifies the frequency range ofinterest (e.g., 0–8 GHz). A Gaussian signal X( f ) cover-ing this frequency range is defined. This spectrum isthen transformed into time domain by using an inverseFourier transformation, resulting in a time signal x(t)

The Finite Integration Technique gets its name from thefact that it discretizes the integral rather than the differ-ential form of Maxwell’s equations. The unknowns arethe electric voltages, denotedby e, on the edges of thediscretization mesh and themagnetic fluxes, denoted byb, on the mesh faces.

For discretizing Faraday’slaw (1) on a mesh face, forinstance, we note that theleft hand side of (1) is a lineintegral of the electric field(i.e., an electric voltage)along the border of the face.This integral can be simplywritten as an algebraic sumof the edge unknowns. Theright hand side is nothingelse than the time derivative(denoted by a dot) of themagnetic flux through theface. Note that, for any fixedmesh (which alreadyincludes a space discretiza-tion error), no supplemen-tary equation discretizationerror is involved, when pass-ing from the continuous tothe discrete form. This isbecause, with this choice ofunknowns, the passage from(1) to (2) is based solely onthe mathematical propertiesof the integral. On the otherhand, an equation discretization error will occur whendiscretizing the material property relations.

By grouping the +1 and −1 coefficients of the alge-braic sum into a matrix C (the discrete counterpart of thecurl operator), and the electric and magnetic unknownsin vectors e and b, a compact matrix form results, whichlooks strikingly similar to the continuous differential formof Faraday’s law curl �E = − �̇B.

In a similar way, all Maxwell’s equations can be dis-cretized with the FIT to yield their discrete counterparts,with a compact and elegant matrix form [3]. The matrix

operators C, C̃ (the discrete curl operators) and S, S̃ (thediscrete divergence operators) are topological matrices,containing only 1,−1 and 0 entries. On a Cartesian grid,FDTD is equivalent to the FIT [4]. Even the modern viewon the FEM method uses exactly the same form (6) ofdiscretized Maxwell’s equations [11]. The differencebetween modern FEM and FIT is only in the discretiza-tion of the material property relations.

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Finite Integration Technique


December 2008 65

with a Gaussian envelope. The mode pattern at theinput port is then excited with this time pattern andpropagated through the structure. Reflected and trans-mitted time signals, denoted generically by y(t), aremonitored and after the simulation is ready, a Fouriertransform is applied to yield the respective spectraY( f ). These spectra are eventually divided by the exci-tation spectrum, et voilà: the S-parameters for the entirefrequency range in one single go!

The accuracy of a simulation, namely the accordanceof simulation results and the behavior in reality, is usu-ally limited due to simplifications in the simulationmodel. Having the simulation results in front of us, wemay wonder whether these are the true S-parameter ofour device. All numerical methods promise that the sim-ulation results will eventually converge against the actu-al solution, if only the mesh is fine enough and alldetails and effects are represented in the numericalmodel. If the results of interest do not change signifi-cantly anymore after several mesh refinement steps, theconverged solution has been reached. Cross verificationof the results by applying two different numericalapproaches to the same problem gives even more confi-dence, e.g., by comparing the time domain solution andfrequency domain solutions (Figure 3). This reassurance

is even more convenient to reach, if the simulation soft-ware offers the possibility to switch between numericalapproaches without changing the interface.

As we can see in Figure 3, both approaches, frequen-cy and time domain, deliver the same results. There isjust another constraint, which has not yet been

Figure 2. Schematic of the simulation procedure to derive the S-parameters of a passive component in (a) time and (b) fre-quency domain. The time signals can also be used to perform a time domain reflectrometry (TDR) analysis of the structure.

Time Domain

In Out In

In

Out

Out

Time Domain

Input Signal

Reflected Signal

Z 1 TDR

TDR S-Parameter S-Parameter

S 11 S 11

Reflected Spectrum

x(t)

y(t) Y(f)

x(f) x(f)Input Spectrum Input Spectrum

Frequency Domain Frequency Domain

Frequency Domain

10.80.60.40.2

00

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2 4 6 8

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1

00 2 4 6 8

(a) (b)

Figure 1. Simulation model of a 90◦ coaxial connector.The white space around the connector is perfectly electricconducting. The different colors denote different materials(blue: air, orange: Teflon, yellow: rubber). The ports arealready attached (red faces).


66 December 2008

considered—the simulation performance. It is definedby the time required for a simulator to reach apredefined accuracy. For our connector, the simulationtime does not differ much between a FIT transient solver(1 min) versus a FIT-FEM frequency domain solver (1.5min). However, for other applications, the difference incomputing time may be significant.

In order to find the most efficient numerical solutionfor a certain application, it is essential to understand themethods in more detail.

Time DomainAll time domain methods that we are discussing here—FIT, FDTD, and TLM—feature a Cartesian (or cuboidhexahedral or circular cylindrical coordinate) grid andan explicit time integration scheme. These two facts areclosely related. The coordinate grid implicates a simpleband structure of the system matrix on which the leapfrog algorithm can be applied [5]. The fields are propa-gated through the structure by matrix vector multipli-cations with a specific time step. The larger the timestep, the shorter the simulation time. The maximum

possible time step is determined by the Courant-Friedrich-Lewy (CFL)-criterion [14], [27]. It is basicallythe time required for light to pass the smallest mesh cellin the calculation domain. It might be more illustrativeto think of the CFL-criterion as a way to force informa-tion from a mesh cell to touch every neighbored meshcell in every time step. The memory requirements andthe simulation time increase linearly with the numberof mesh points. Because of these properties, timedomain simulators are well suited to solve electricallylarge and detail-rich structures. Billions of unknownshave been practically demonstrated.

There are other time domain approaches that usenonorthogonal grids [28] and/or implicit time integra-tion schemes [7], [8]. In the area of microwave and RF,there are currently no commercial implementationsavailable. An implicit algorithm always has to solve asystem of equations for every time step, but then thetime step size may be chosen somewhat larger.

As we have seen, it is possible to derive frequencydomain data by applying Fourier transforms to the timedomain signals. Steady-state 3-D EM fields can also veryeasily be extracted from the transient broadband simula-tion. Since the excitation signal is broadband, it is possi-ble to obtain fields for various frequencies in one simula-tion run. Two typical applications shall be mentionedbriefly. The first one is a wideband dual ridged hornantenna. Farfields at 100 different frequencies are calcu-lated in one single simulation run to evaluate the broad-band gain (Figure 4). The second one is a multibandmobile phone antenna next to a human head model.Here it is also important to model the frequency depen-dent behavior of the biological tissues correctly.

Time domain naturally offers the possibility to studythe transient behavior of EM structures. For this pur-pose it is not necessary to stick to the Gaussian pulsethat has been introduced earlier. Arbitrary signals canbe fed into the simulator. (The approach to excite a sinu-soidal signal in time domain in order to obtain the har-monic results at the specified frequency is somewhatoutdated.) In addition to being used as virtual network

analyzer, the simulator can alsowork as virtual time domainreflectometer (TDR), Figure 2.Delay times and signal degra-dation on signal lines can bedirectly simulated.

Not only the signals, but alsofields can be studied in timedomain: e.g., transient farfieldsbecome increasingly importantin ultrawide-band (UWB) appli-cations. In multiport devices,every port can be excited indi-vidually with a different timesignal, and the fields can bemonitored accordingly.

Figure 4. Broadband simulation of a dual ridged horn antenna [22]. Farfields at 100frequencies are extracted in one single simulation run by applying broadband timedomain technique.

(a)

(b)

(c) 15141312111098765

0.8 1.8 2.8

Frequency (GHz)

dBi

MeasuredSimulated

3.8 4.8 5.8

Figure 3. S-parameters of the connector example derivedwith different solution methods: 1. Frequency domainsolver on a tetrahedral grid with 0.150 million tetrahedra(FD-TET). 2. Time domain solver with PBA on a hexahe-dral grid with 0.7 million mesh cells (TD-PBA). 3. Timedomain solver/staircase on a hexahedral grid with 17 mil-lion cells (TD-Staircase). The comparison shows goodagreement between cases 1 and 2.

S1, 1Converged Solution

Frequency (GHz)

S1,

1 (

dB)

S1, FD-TETS1,1 TD-PBAS1,1 TD-Staircase

−20

−25

−30

−35

−400 2 4 6 8


December 2008 67

Geometry Approximation in Time Domain Methods In traditional FDTD and TLM methods, every hexahe-dral mesh cell is filled entirely with one material. Thisleads to the so-called staircase approximation of thegeometry. Obviously, such a discretization can makethe accurate geometrical representation of many prac-tical devices very difficult, since most componentscontain rounded features. In order to increase the accu-racy in such cases, very fine meshing needs to beapplied. Conformal methods, such as the PerfectBoundary Approximation (PBA) [20], can improve thegeometry description without compromising the mem-ory efficiency of standard FDTD [6]. The performanceincrease through such a method is remarkable, as wecan also see in our connector example (Figure 5). Notonly is a smaller number of mesh cells needed, but thelarger mesh cells additionally entail a larger time step.Finally, it is interesting to see how the results convergeto a final solution when the mesh is refined. The PBAconvergence process is very smooth and extraordinar-ily fast [Figure 5(a)]; it can be confidently assumed that

every increase in mesh density will improve theresult’s accuracy. This statement is not true for staircaseapproximations where convergence is slow and notsteady [Figure 5(b)].

Endeavors have been made to improve the geome-try approximation inside the Cartesian grid. PBA, forexample, can also be used to model finite-thicknessmetallization within one mesh cell. This would resultin tiny mesh cells in traditional FDTD and vice versa tovery small timesteps and long simulation times. InTLM, however, even more advanced compact modelscan be found. Fine structured elements like slots, vents,or cables are replaced by specific macromodels in orderto avoid the sampling of all details by the grid. Thisapproach has been proven particularly useful in EMCapplications (Figure 6).

The standard FDTD grid is structured. This meansthat every mesh line starts on one side of the calculationdomain and ends on the other side. In order to avoid theincrease of mesh cells in the outer regions, subgriddingalgorithms have been introduced, which allow locallysmaller mesh cells. The mesh cells need only to be small

Figure 5. Solution convergence for the connector example. When making the mesh finer and finer, the S-parameter results inthe PBA case get closer and closer to the final solution. For the staircase mesh, the convergence is not as smooth as in the PBAcase. It takes the staircase model 15 times longer on the same computer to reach the same convergence goal. For a comparisonof the converged results, refer to Figure 3.

S1,1 Versus No. of Mesh Cells

(a)

(57s) CPU Time

Frequency (GHz)

dB

−20

−25

−30

−35

−400 2 4 6 8

12e3 Cells

120e3 Cells53e3 Cells30e3 Cells

PBA

S1,1 Versus No. of Mesh Cells

(15 min) CPU Time

(b)

dB

−10

−20

−30

−40

−50

−600 2 4 6 8

12e3 Cells

891e3 Cells

113e3 Cells

1832e3 Cells

182e3 Cells

50e3 Cells

226e3 Cells

1485e3 Cells

325e3 Cells507e3 Cells699e3 Cells

27e3 Cells

1192e3 Cells

Frequency (GHz)


68 December 2008

in regions where small details are present. Additionalexceleration of the simulation can be achieved by usingdifferent time steps at different mesh levels. Althoughmany different subgridding algorithms have been pro-posed, many of them exhibit the so-called long timeinstability (see, e.g., [15] for references and an explana-tion of the phenomenon). The example in Figure 7 illus-trates the impact of a mesh with hierarchical subgrids. Itwas solved with a subgridding algorithm with mathe-matically proven stability [16]. The computing time isreduced significantly by a factor of 9.5.

Frequency DomainA characteristic of frequency domain solvers is the implic-itness of this approach; the resulting system is typically alarge linear system of equations. Thus, a matrix inversion

is needed in order to obtain the solution for one frequen-cy, no matter whether the grid is structured or not. In com-mercial applications, FEM on tetrahedral grids [12] istherefore the most popular general purpose numericalmethod. Tetrahedrons are the simplest volume entities,and their flexibility in approximating arbitrary geometrieshas many benefits. However tetrahedron quality is cru-cial: very flat tetrahedrons may compromise solutionspeed and accuracy as they make it more difficult for thealgebraic solution method to solve the system.

There are two distinct methods of solving the linearsystems of equations resulting from FEM discretization:direct and iterative solvers. A direct solver works direct-ly on the system of equations derived from the dis-cretization. Its key advantage is that it can solve for sev-eral port excitations at the same time in parallel. On the

other hand, the memory requirements arequite high. Typically, the memory require-ments increase quadratically with thenumber of tetrahedrons. Iterative solverstransfer the original system of equationsinto another one that can be solved byrepeated application of operations accord-ing to the specific algorithm. The iterativealgorithm has to be executed for each exci-tation individually. In compensation, thememory requirements are much less com-pared to a direct solver. Similarly to thetime domain methods, where small timesteps lead to many steps to be simulated,the overall computing time of a frequencydomain method also depends, for bothtypes of solvers, on the sampling granular-ity of the frequency range of interest.

Frequency domain solvers are well suit-ed to solve infinite periodic problems, suchas phased arrays, frequency selective sur-faces (FSS), photonic band gap structure(PBG), etc. Periodic boundaries can be set upeither with a phase difference between themor, more practically, with a certain scanangle. A Floquet mode port is a useful addi-tion to this capability. It enables the usage ofplane waves to monitor polarization or RCS,as well as the determination of main andgrating lobes of a phased array.

Special Filter Solvers—MOR and Modal AnalysisWhile frequency domain solvers are most-ly well suited to tackle resonant problems,and some of them are especially suited forfilter simulation. For example, a modelorder reduction (MOR) solver [17] workson both tetrahedral and hexahedral gridsand can also use PBA. It does not calculatethe EM fields, but directly accesses the

Figure 7. Subgridding mechanisms reduce the number of mesh points in asimulation. In this example (a) the full grid is 20 times larger (35e6 meshnodes) than (b) the subgridded version (1.75e6 mesh nodes).

(a) (b)

Figure 6. (a) Compact model applied to a vent in a computer housing. Com-parison of the mesh (b) for the full 3d structure and (c) for the compact modeldescription. The use of a compact model can reduce the number of mesh cellssignificantly, in this example by a factor of 10.

Vent

(a)

(b)

(c)

Full 3-D Representation of Vent

Compact Model Representation of Vent


December 2008 69

dominant modes of the matrix and introducesa reduced order replacement. In other words,it compresses the matrix to a much smallersize while keeping the information of interestin it. With this smaller system matrix, the S-parameters of the original device can bederived in an extremely short time. Whereapplicable, this approach may be 100 timesfaster than the others presented previously.

Another special approach is also particu-larly well suited for resonant devices. Amodal analysis solver [18] calculates theeigenmodes of the device and uses them tointerpolate fields in the interesting frequencyrange. Compared to a general purpose fre-quency domain method, this approach maybe an order of magnitude faster.

MoM-MLFMMThe MoM [13] only discretizes the surface ofthe devices rather than the entire volume. It showsadvantages if the structure is predominantly metallic,electrically small, and preferably also small comparedto the calculation domain, since the free space needsnot to be modeled.

Typically, it is a very memory intensive methodbecause the system matrix is not banded but fully pop-ulated. Since all these elements need to be stored, therange of practical application is typically limited togeometrically simple structures.

One important extension of the MoM is the multi-level fast multipole method (MLFMM) [19], whichenables the simulation of electrically very large prob-lems, such as the RCS of airplanes or antenna place-ment on ships (Figure 8). Using the same discretizationas the MoM, this extension saves storage by groupingelements together. However, this method is onlyadvantageous for very high frequencies.

Tips and TricksProbably the two most important questions that a userof a simulation tool is asking are:

• How accurate is my simulation?• How long will it take to achieve an accurate

solution? To obtain an accurate solution after the simulation,

there are quite a few ingredients:• model the reality correctly• ensure that the mesh is fine enough• ensure that the solution of the discretized system

of equations is numerically accurate.

Modeling the Reality

ExcitationsFor exciting the desired modes in the device, portsneed to be defined at the locations at which, in real-ity, the sources will be connected. This is usually atsome point along a transmission line (waveguide,microstrip, etc.).

A discrete port is simply a lumped voltage or currentsource, possibly with nonzero internal impedance/admittance. The source is connected by perfectly con-ducting wires to two points of the device [Figure 9(a)].

Figure 8. Electrically large problems can often only be tackled with effi-cient integral methods such as MLFMM. The ship model is about 130 mlong. It is illuminated with a plane wave at 1.5 GHz. The electrical sizeof the problem is therefore 650 wavelengths. Shown are the currents onthe metallic surface.

Figure 9. (a) Discrete port: two wires with a source in the middle. (b) Face port: the source is distributed along the red line.

(a) (b)


70 December 2008

Long connection wires may strongly influence the solu-tion. This is because of the wire inductance, whichgrows linearly with the wire length. To reduce the par-asitic inductance of the discrete port, the so-called faceport (or delta-gap port) has been proposed. Here, thevoltage source is distributed along a small gap in ametallic face [Figure 9(b)]. The face port has a muchsmaller self inductance (imagine it as a parallel connec-tion of many wire inductances).

The discrete or face ports will always introduce asmall perturbation of the numerically calculated fieldat the location where they are placed in the model. Tocompletely eliminate this perturbation, one canimagine extending the excited transmission line toinfinity, thus preventing any reflections fromappearing. Of course, no infinite structure can everbe modeled numerically, so a special type of port, theso-called waveguide port was introduced as a meansof truncating the infinite line, without introducingany perturbations.

A waveguide port is a surface perpendicular to atransmission line on which the modes that can propagatealong the transmission line are calculated. The field pat-terns corresponding to these modes are then used asexcitation during the simulation. To ensure accuratemode calculation for arbitrary line configurations, themodes are typically calculated by solving a two-dimen-sion eigenmode problem on the port’s surface.

The size of the waveguide port is of utmost impor-tance for the accuracy of the solution. Whereas for a hol-

low or coaxial waveguide the port size is clear—it shouldbe as large as the waveguide’s cross section—for othertypes of transmission lines (microstrip, stripline, etc.), theuser might often have difficulty in guessing just howlarge the waveguide port should be. These transmissionlines allow the propagation of static-type TEM or quasi-TEM modes, whose fields become zero, theoretically, atinfinity. A few rules of thumb are given in Figure 10.

It is recommended, however, to make a few testswith the port size before starting the longer-lasting 3-Dsimulation. Just let the program calculate the portmodes and have a look at the fields at the port, espe-cially at the port’s boundary. You should see no fields atthe boundary. If there are fields at the boundary, theport size needs to be increased (Figure 11).

The first golden rule of an accurate simulation:never start the simulation before checking if the portmodes are the expected ones!

Boundary ConditionsAs already mentioned, the simulation domain, infinitein reality, has to be truncated for the purpose of simu-lating it on a computer. At the boundary, special bound-ary conditions need to be imposed, depending on thereal operating conditions of the device.

For example, if the device that needs to be simulatedis in reality placed within a metallic box, then electricboundary conditions (which impose zero tangentialelectric field, just as for a perfect metallic object) can beused on all sides of the boundary. An infinitely extend-

ed groundplane can be modeledby an electric boundary condi-tion as well.

If the structure is placed inopen space, such as an antenna,then a so-called radiation orabsorbing boundary condition isthe right one. It simulates theunperturbed propagation of EMwaves through this boundary.When choosing the domain’struncation, do not forget to leavesome free space around theantenna! A perfectly matchedlayer (PML) [21] boundaryrequires just a fraction of a wave-length of additional space,

Figure 10. Port size rules of thumb for: (a) microstrip, (b) ungrounded, and (c) grounded coplanar line.

(a)

5h

hh

6-10 ∗ w

(b) (c)

w

3w1.5w

3w3w

w w

Figure 11. Absolute value of the electric field (represented in logarithmic scale) at amicrostrip port. (a) Port size is too small; electric field has considerable magnitude atport’s border and will negatively impact on the solution’s accuracy. (b) Port size hasbeen increased laterally and above the microstrip and fields are practically zero (greencolour) at port’s border.

(a) (b)


December 2008 71

whereas other absorbing boundary conditions oftenneed more than one wavelength. Absorbing boundaryconditions should be used only when necessary sincethey typically require more computing resources than,e.g., the electric ones.

For infinite periodic structures, the periodic bound-ary conditions are available. Whenever both the simu-lated structure and the excitation are symmetric, theusage of symmetry conditions can reduce the numberof unknowns by half for each symmetry plane andtherefore shorten the simulation time.

Material PropertiesThe permittivity, permeability, and conductivity valuesfor all materials present in the model naturally play animportant role in the solution’s accuracy. Often, thesevalues are frequency-dependent (dispersive materials),and the more accurately this frequency dependence isknown, the more accurate the solution can be. Frequen-cy-domain solvers, as well as the advanced time-domain simulators, can take this frequency dependenceeasily into account.

Please do not forget that the often used constanttangent delta material model is actually fiction. Nomaterial can have a constant loss tangent from dc toseveral GHz. Even the most common materials, such asFlame Retardant 4 (FR4), exhibit a more complicateddispersion—in the case of FR4 the first order Debyemodel appears to be sufficiently accurate.

Meshing the StructureHow fine does the mesh need to be? First, it should befine enough to correctly represent the geometry. Sec-ond, it should be fine enough to represent the possiblysharp field variations within the device.

With most time-domain simulators, the used hexahe-dral mesh leads to a staircase representation of theboundaries, so the mesh needs to be made quite fine justto obtain a good representation of the geometry [Figure12(a)]. The advanced conformal meshing eliminates thisdrawback by using the memory-efficient hexahedral

mesh almost everywhere in the model and special algo-rithms conforming to the curved boundaries at materialinterfaces [Figure 12(b)]. This way, a considerable savingin terms of mesh cells can be achieved. Although tetra-hedral meshes can, in principle, offer a good geometryapproximation, this is only true if the real structure ismeshed; some meshers require the segmentation ofround structures and finally lead to polygonal approxi-mation of curvatures [Figure 12(c)].

Representing the field variations in the mesh is aneven more complicated issue. The first rule of thumbthat can be applied a-priori, before any simulation isstarted, is that in a time-domain simulation with hexa-hedral mesh, the size of a mesh cell should never belarger than λ/8, where λ is the wavelength correspond-ing to the upper limit of the interesting frequencyrange. Mesh cell sizes of λ/10 or λ/15 are often success-fully used in practice. For frequency domain FEMsolvers based on second-order finite elements, λ/4 is agood value to start with. Of course, a model is typical-ly made of several materials. Since the wavelength isdependent on the material properties, the size of everyindividual mesh cell depends on the material it is filledwith. That is why programs that only allow a uniformmesh (the same mesh cell size everywhere) may lead toan unnecessarily large number of mesh cells.

For lossy materials, the rule of thumb is to ensure twoto three mesh lines within the skin depth. This can provedisadvantageous for good conductors at high frequen-cies since the tiny skin depth would lead to tiny meshcells and considerably increase the simulation time.Advanced simulators apply special surface impedancemodels for metals during the simulation process andeliminate completely the need to mesh the skin depth.

How about sharp field variations—field singularities—due to geometrical features, such as edges, corners, etc.?There are two ways to represent them in the mesh. The a-priori solution used by some field simulation programsis to automatically detect these features and to useadvanced edge/corner correction algorithms during thesimulation. A second way is increasing the mesh density

Figure 12. Hexahedral and tetrahedral mesh for a piece of coaxial cable. (a) The staircase mesh provides a poor description ofcurved surfaces, unless a very fine mesh is used. (b) Conformal boundary approximation ensures the required geometric accu-racy, with a minimum of mesh cells. (c) Tetrahedral mesh generators often require a segmentation of round structures, leadingto a poor geometrical approximation.

(a) (b) (c)


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a-posteriori by the so-called adaptive meshing, whichwill be discussed later. These approaches are applicablein both time and frequency domain.

All this seems very complicated, and it appearsalmost impossible for a normal user to achieve amesh of good quality. Don’t worry, any advancedsimulation program should be able to do all thesesteps for you: generate an initial mesh according to aminimal user input (e.g., number of steps per wave-length), taking into account geometry, material prop-erties, and sharp corners and edges and refine themesh through an automatic adaptive meshing, untilconvergence is reached.

With a minimum of user input however, the size ofthe final mesh can be optimized and thus reduce theoverall computing time. For instance, you should ensurethat there are at least two mesh cells within the substratethickness, two to four mesh cells across a strip, and twomesh lines in a radiating gap (Figure 13). The mesh cellsshould be smaller close to metallic edges to sample thefringing fields. The height of the microstrip only has tobe discretized if the used software does not supportfinite thickness metallization.

Accuracy

Accurate Numerical SolutionIn frequency domain simulations, a linear system ofequations needs to be solved at each frequency point. Ifthe condition number of the system matrix is large, forexample, because some tetrahedra in the mesh areexcessively flat, the solution may be quite inaccurate.When using an iterative solver, a large number of itera-tions may be necessary.

In a time-domain simulation there is no system tosolve. Issues like matrix conditioning number are irrel-evant here. The main accuracy issue in the time domainsimulation is related to the inverse Fourier transforma-tion of the time signals, in order to obtain frequency-domain parameters. Namely, all signals should start atzero and end at zero in order to ensure accurate fre-quency-domain values. Especially for high-Q struc-tures, the output time signals continue to oscillate along time even after the excitation has been turned off(Figure 14). A good time-domain simulator will howev-er provide the user automatically with a criterion onwhen to stop the simulation.

The position of a resonancewill settle quite quickly andwill be quite accurate even ifthe simulation is stopped toosoon. The amplitude of the S-parameters however will onlybe accurate if the time signalshave sufficiently decayed. Topredict the correct ampli-tude, algorithms such as anautoregressive filter canapplied on nondecayed sig-nals and considerably savesimulation time.

Mesh Adaptationand ConvergenceThe accuracy of a simulationresult has to be tested by per-forming a convergence study. Ina convergence study the num-ber of mesh cells is continuous-ly increased until the results ofinterest, usually S-parameters, donot change anymore, at least notsignificantly. A convergencestudy is thus an essential part ofany simulation project.

Many software tools fea-ture automatic mesh adapta-tion schemes. Typically fieldsare evaluated after a simula-tion run. Wherever strongfield variations occur, the

Figure 14. Time signal examples. (a) The output signal still oscillates with a relativelyhigh amplitude at the moment when the simulation was stopped. The frequency-domainresults (e.g., S-parameters) are most likely inaccurate. (b) A time signal which is sufficient-ly decayed when the simulation is stopped will provide accurate frequency-domain results.

Time Signals

Time (ns) Time (ns)

(a)

−1

−0.5

0.5

0.20 0 1 2 3 40.4 0.6 0.8

0

1

−1

−0.5

0.5

0

1

(b)

02,101,1Time Signals

Figure 13. Rules of thumb for meshing planar structures with hexahedral meshes (grey-metal, dusky pink-substrate). (a) Discretization of a microstrip. (b) Discretization of gapsinside a metal sheet.

At Least TwoMesh Lines

2–4 Mesh Lines

1–3 Mesh Lines(Dep. on Thickness)

(a) (b)


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mesh is refined and the simulation is restarted. Thisprocess is repeated until the results do not changesignificantly anymore.

Although convergence study and mesh adaptationappear to be very similar approaches to guaranteeaccurate results, in practice they are different. For aconvergence study, we assume that the geometryapproximation of the structure as well as the results inthe entire frequency range of interest are improvedcontinuously with the refined mesh.

Frequency domain solvers perform the mesh adap-tation typically only for one frequency, per default usu-ally the highest frequency in the band of interest. Thehighest frequency is, for example for filters, not neces-sarily the one that is relevant for the device functionality.A relevant frequency has to be chosen formesh adaptation. This information is reli-ably available only a posteriori. In addition,the field distribution might change signifi-cantly with frequency, e.g., for multiplexersor multiband antennas. One single adapta-tion frequency is not sufficient in such cases.Either the simulation has to be split up intoseveral separate frequency bands, or severaladaptations frequencies have to be used inone simulation over the entire band.

In order to derive accurate simulationresults, the geometry representation on thegrid has to be as good as possible. Particu-larly tetrahedral grid based frequencydomain solvers do often not improve thegeometry approximation during meshadaptation. In the mesh adaptation processthe initial tetrahedrons are simply subdivid-ed in order to improve the fieldsampling (Figure 15). Thereforewe see a convergence ofresults, though not for theinput model but for the initial-ly approximated geometry.This effect is even more criticalif shapes are segmented beforesimulation (Figure 16).

In contrast to frequencydomain, the time domainapproaches can perform themesh adaptation broadband.Moreover, every refinementalso entails a better geometryapproximation, since the entiremeshing process is restarted atevery mesh adaptation step.

Finally it should be men-tioned that, unlike a tetrahedralmesh, a structured time domaingrid can be easily controlled bythe user, by manipulating mesh

lines or meshing densities. Thus the final mesh of anadaptation is nearly reproducible by the user withoutrerunning the adaptation.

Checklist for accuracy• Do the chosen ports correctly model the reality?• Are the port modes and line impedances the

expected ones?• Are the material parameters the right ones on the

whole frequency range? • Is the mesh reasonable and fine enough?• Have all output time signals decayed to almost

zero in a time domain simulation?• Was a convergence study performed?And last but not least:

Figure 15. Final mesh after the adaptation process for a piece of coaxialcable. (a) Mesh adaptation without snapping on the true geometry. (b)Snapping onto the geometry during the adaptation process leads to a goodapproximation of the geometry and hence to more accurate results.

(a) (b)

Figure 16. Mesh adaptation and convergence. The cylinders of the connector model aresegmented before meshing. The small connector pictures show the (a) 6 segment and the(b) 12 segment version. In all cases, a mesh adaptation was performed and the S-parame-ters were converged.

S1,1 Versus Cylinder Segmentation

Frequency (GHz)

S1,

1(dB

)

(a)

−400 2 4 6 8

−35

−30

−25

−20

(b)

6 Segments12 Segments20 SegmentsTrue Surface


74 December 2008

• Is the model just as large or small as in reality?(mm instead of inch, a cable which is 3.2 mm longinstead of 2.3 mm long, these things happen moreoften than one thinks!)

How Much Accuracy Do I Need?The answer to this question depends on the purpose ofthe simulation, as well as on the phase in the designprocess. In the beginning, the design is probably still farfrom the sought optimum, and it does not make senseto look for an excessively accurate solution for a designvariant that will most probably not be retained as final.You can set up the simulation with a coarse mesh andstop the simulation quite early.

Towards the end of the design process, when you areclose to the design goal, it is worth investing moreresources in a more accurate solution: make the mesh finerand allow the time signals to fully decay towards zero (ina time-domain simulation) or set the error criterion for thelinear equation system to a lower value (in a frequencydomain simulation).

The verification of measurements or analytic resultsrequires the highest accuracy settings

Time Domain Versus Frequency DomainAs we have seen, the simulation process in time domainseems to be more complicated as far as S-parameters areconcerned, but it has many advantages. Only one simu-lation is required for a broadband result. In addition wecan benefit from an interesting property of the Fouriertransform: the frequency resolution � f of the spectrumrelates to the signal length tmax in time domain by the

Nyquist-Shannon theorem for limited-bandwidth signals� f = 1/(2tmax) [25]. Since at the end of a time-domainsimulation all signals have, ideally, decayed to zero, wecan prolong the signal by just adding zeros at the end inorder to increase tmax. This means that we can calculatethe spectrum with an arbitrarily fine frequency resolu-tion, and, most importantly, without additional computa-tional effort. Unlike in frequency domain, in a transientsimulation, it is thus virtually impossible to miss sharpresonances inside the requested spectral range.

In a time domain simulation, a signal has to enter andleave the device under test. Our connector does not haveany resonances; it is supposed to behave like a broadbandtransmission line. Therefore, the simulation runs quickly.If there would be resonances, the transient signal wouldexcite them and after the excitation signal has vanished,these resonances would continue to ring with their energydecaying according to the device’s quality factor [Figure14(a)]. The transient simulation can be terminated when asteady state is reached, or when the signal can be predict-ed by using digital signal processing techniques. Frequen-cy domain solvers do not face this problem. To find theresonance frequency in high Q structures may requirenumerous simulations though.

Although the finite thickness of some metallization istechnically relevant, it is usually not considered in mostsolvers. In FEM, its inclusion leads to a large number oftetrahedrons at the edges, or to tetrahedrons with a poorquality. In standard time domain methods, the thicknesshas to be sampled by a mesh cell. This will not lead to alarge increase in mesh size. However, these cells will bevery small, which will in turn reduce the time step

because of the CFL criterion, andtherefore increase the simulationtime. Conformal methods such as FITwith PBA face no problem here,because the metallization thicknesscan be considered inside a mesh cellwithout compromising the time step.For thin dielectric sheets, similar effi-cient subcell models are available [26].

There is one other distinctive fea-ture, the electric size. Generally, thediscussed high-frequency solvers areeffective for electric structure sizeswithin the range of about 1/1,000 to1,000 wavelengths. The lower boundof this frequency range sees slightperformance advantages for the gen-eral purpose frequency domain.Towards the upper bound the memo-ry requirements become relevant.

On a typical workstation (8GBRAM) problem sizes of about 40 wave-lengths in each spatial direction can betackled with a transient solver, whereasFEM is restricted to about ten

Figure 17. A complete IBM package layout [23] used for full wave signal integrityanalysis. It consists of eight metallization layers and 40,000 geometrical entities.The full package, shown here in total and detailed view, was imported into CSTMWS for a full wave analysis. The benchmark fraction was solved by using thetransient approach and the FD solver (27 million mesh nodes and 5.3 million tetra-hedrons, respectively). The transient solver model of the full package had 640 mil-lion mesh cells and 3.7 billion of unknowns respectively [24]. This level of detailmade the usage of the FD solver for a simulation of the full package unfeasible.


December 2008 75

wavelengths. Beyond these problem sizes, the use of anMLFMM solver is advisable since it is specifically designedto efficiently tackle electrically very large problems.

The lower memory requirements of the time domainmethods allow also the solution of very detail rich struc-tures (Figure 17). In the scope of this article we do notwant to discuss what can be done on a hardware level, toextend simulation speed or accessible model sizes.

ConclusionsHow can I select the best simulator? Most people wouldthink it is obvious. Some will take the most accurate.Others will take the quickest, or the cheapest. All theseselection criteria need to be looked at as a whole inorder to make an informed choice. A good overall crite-rion would be something like the simulator’s qualityfactor: Q = Accuracy/Effort.

Therefore, choose the program which gives youthe best accuracy for a given simulation duration, orsum of money, or RAM, or all three together (= Effort)and you won’t do anything wrong. Or, if it’s accuracythat is of utmost importance, choose the program thatachieves the desired accuracy with the least time andmemory effort. By the way, do not forget about thelabor costs while integrating the software into yourdesign flow. A program with a good user interfaceand a high degree of automation will save valuableengineer’s worktime and therefore money.

Beware of brute-force hardware arguments like “ona cluster, program X is also very quick.” An intelligentalgorithm is quick on any type of hardware and is evenquicker on a faster computer, clusters or graphics accel-eration card. It’s the combination of intelligent algo-rithms and best available hardware that will give theuser the optimal computing speed.

As we have seen, there is not one single solverapproach best suited for all types of applications. It isvery convenient if several solver types can be selectedjust from the one modeling interface. It would be evenmore convenient, if the software chose the best suitedsolver by itself.

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[2] F. Gustrau and D. Manteuffel, EM Modeling of Antennas and RF Compo-nents for Wireless Communication Systems. Berlin, Germany: Springer-Ver-lag, 2006.

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[4] T. Weiland, “Time domain electromagnetic field computation with finitedifference methods,” Int. J. Numer. Model., vol. 9, no. 4, pp. 295–319, 1996.

[5] K.S. Yee, “Numerical solution of initial boundary value problems involv-ing Maxwell’s equations in Isotropic media,” IEEE Antennas Propagat.,vol. 14, no. 5, pp. 302–307, 1966.

[6] A. Taflove and S.C. Hagness, Computational Electrodynamics: TheFinite-Difference Time-Domain Method, 3rd ed. Norwood, MA:Artech House, 2005.

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[8] E.P. Li, I. Ahmed, and R. Vahldieck, “Numerical dispersion analysis withimproved LOD-FDTD method,” IEEE Microwave Wireless ComponentsLett., vol. 17, no. 5, pp. 319–321, May 2007.

[9] P.B. Johns and R.L. Beurle, “Numerical solutions of 2-dimensional scat-tering problems using a transmission-line matrix,” Proc. Inst. Elect. Eng.,vol. 118, no. 9, pp. 1203–1208, 1971.

[10] C. Christopoulos, The Transmission-Line Modeling Method: TLM. Piscat-away, NJ: IEEE Press, 1995..

[11] A. Bossavit, Computational Electromagnetism Variational Formulations,Complementarity, Edge Elements. New York: Academic Press, 1998.

[12] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. Piscataway,NJ: IEEE Press, 2002.

[13] R.F. Harrington, Field Computation by Moment Methods. New York: Pis-cataway, NJ: IEEE Press, 1993.

[14] A. Taflove and M.E. Brodwin, “Numerical solution of steady-state elec-tromagnetic scattering problems using the time-dependent Maxwell’sequations,” IEEE Trans. Microwave Theory Tech., vol. 23, pp. 623–630, Aug.1975.

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[16] P. Thoma and T. Weiland, “A consistent subgridding scheme for thefinite difference time domain method,” Int. J. Numer. Model., Electron. Net-work, Devices Fields, vol. 9, no. 5, pp. 359–374, 1996.

[17] W.H. Schilders, H.A. van der Vorst, and J. Rommes, Model Order Reduc-tion: Theory, Research Aspects and Applications (Mathematics in Industry/TheEuropean Consortium for Mathem). New York: Springer-Verlag, 2008.

[18] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analy-sis. New York: Wiley, 2000.

[19] W.C. Chew, J.-M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algo-rithms in Computational Electromagnetics. Norwood, MA: Artech House,2001.

[20] B. Krietenstein, R. Schuhmann, P. Thoma, and T. Weiland, “The perfectboundary approximation technique facing the challenge of high preci-sion field computation,” in Proc. XIX Int. Linear Accelerator Conf.(LINAC’98), Chicago, 1998, pp. 860–862.

[21] J.P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994.

[22] L. Foged, A. Giacomini, L. Duchesne, E. Leroux, L. Sassi, and J. Mollet,“Advanced modelling and measurements of wideband horn antennas,”in Proc. 11th Int. Symp. Antenna Technology Apply Electromagnetics(ANTEM 2005), pp. 105-108.

[23] IBM, Thomas J. Watson Research Center, Yorktown Heights, NY, pri-vate communications, A. Deutsch.

[24] E. Gjonaj, M. Bender Perotoni, and T. Weiland, “Large Scale Simulationof an Integrated Circuit Package,” in Proc. 15th Topical Meeting on Electri-cal Performance of Electronic Packaging (EPEP 2006), Oct. 23, 2006, pp.291–294.

[25] C.E. Shannon, “Communication in the presence of noise,” in Proc. Insti-tute of Radio Engineers, Jan. 1949, vol. 37, no. 1, pp. 10–21. Reprint as clas-sic paper in: Proc. IEEE, vol. 86, no. 2, Feb. 1998.

[26] J.G. Maloney and G.S. Smith, “The efficient modeling of thin materialsheets in the FDTD method,” IEEE Trans. Antennas Propagat., vol. 40, no.3, pp. 323–330, 1992.

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[29] All simulation pictures created with CST MICROWAVE STUDIO,except Figure 6, created with CST MICROSTRIPES.


A Practical Guide to 3-D Simulation

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A Practical Guide to 3-D Simulation