Finite Element Analysis of Rectangular Waveguide with ......The Finite Element Method (FEM) is a versatile numerical procedure to ﬁnd approximate solutions to partial differential

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Finite Element Analysis of Rectangular Waveguidewith Stochastic Geometric Variations

Xu Chen, Jose E. Schutt-Aine, and Andreas C. CangellarisDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, IL 61801

Email: xuchen1, jesa, [email protected]

Abstract—The analysis of rectangular PEC-wall waveguides isa classical boundary value problem with well-known analyticalsolutions. The modal cutoff frequencies and dispersion charac-teristics of any waveguide geometry can be easily computed usingthe Finite Element Method. This paper focuses on the analysis ofrectangular waveguides with geometric uncertainties. By lever-aging coordinate transformation and stochastic collocation, thestatistics of the waveguide characteristics can be quickly derivedwithout the need to perform lengthy Monte Carlo analysis.Results are verified against analytical and Monte Carlo solutions.

Index Terms—Stochastic, waveguide, finite element, stochasticcollocation, geometric uncertainty

I. INTRODUCTION

The Finite Element Method (FEM) is a versatile numericalprocedure to find approximate solutions to partial differentialequations in engineering applications [?], [?]. Over thepast two decades, FEM has been gaining popularity in thecomputational electromagnetics community as an effectivemethod for solving many types of boundary value problems.Many of today’s industry standard electromagnetic solversare based on FEM.

A major advantage of FEM is its utilization of geometricmeshes to discretize the solution domains. This allowsproblems with various complicated geometries to be solvedeasily using the same formulation. The shapes and densitiesof the meshes can also be customized to suit the problemsunder consideration, so a balance between accuracy and speedcan be achieved.

This chapter studies the stochastic dispersion characteristicsof rectangular waveguides with geometric uncertainties.Rectangular waveguide problems can be easily formulatedas closed-domain boundary value problems using FEM. Thevarious propagation modes of a given waveguide correspondto the eigenvalues and eigenvectors of the resulting set ofalgebraic equations from FEM discretization. Although thedeterministic solutions to rectangular waveguide problemscan be found analytically, the solutions for random cross-sections will be more involved. This problem is interesting toconsider since we expect waveguides that are manufactured toexhibit some uncertainty in its shape due to variations in themanufacturing process. With uncertainty quantifications of the

geometry, it is useful to perform classical waveguide analysisto obtain characteristics such as modal cutoff frequenciesand dispersion diagrams as random variables, so we canunderstand the probabilistic distributions of these parameters.

FEM is the ideal method to study this problem since itsformulation for closed waveguides is the same regardless ofgeometry. In this manner, randomness in the cross-sectiongeometry can be contained in only the mesh. Furthermore, thetriangular finite elements of the mesh can be projected ontoa standardized triangle using Jacobian matrices, and henceevery random instantiation of geometry can be modeled usingthe same standard mesh, preventing numerical discretizationerrors from polluting the statistical analysis. Other arbitrarywaveguide cross-sections can also be analyzed by generatingnew meshes. In this chapter we consider the standard hollowrectangular waveguide since our goal is to verify the validityof the proposed stochastic analysis scheme, and we wishto compare our results against the well-known analyticalsolutions of these types of waveguides. The FEM formulationand stochastic collocation method will be presented in SectionII. The results will be summarized and compared againstanalytical solutions and Monte Carlo results in Section III.

II. FORMULATION

A. Homogeneously-filled WaveguidesElectromagnetic waves can propagate in a PEC-walled

waveguide if the wave shape corresponds to a propagatingmode of the waveguide, and the frequency of the wave isabove a certain cutoff threshold. An arbitrary wave can bedescribed as a linear combination of many guided wavemodes. Waves for each mode below its respective cutofffrequency will be attenuated and cannot propagate. The cutofffrequencies and dispersion characteristics of an infinitely longwaveguide are determined by its cross-section. Propagatingmodes of a homogeneously-filled waveguide can be dividedinto two categories: Transverse Magnetic (TM) modes, wherethe magnetic field has only transverse components (Hz = 0),and Transverse Electric (TE) modes, where the electric fieldonly has transverse components (Ez = 0). The two modescan be described by the Helmholtz wave equations:

∇2tEz + k2

tEz = 0 in Ω (1)

1

kusum

Typewritten Text

*This use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author.*


Fig. 1. Affine transformation F that projects arbitrary finite element ei onto standardized triangle element S [?].

∇2tHz + k2

tHz = 0 in Ω (2)

where kt is the transverse wavenumber, Ω is the cross-sectionof the waveguide, and ∇t is the transverse del operator:

∇t = x∂

∂x+ y

∂

∂y(3)

∇2t =

∂2

∂x2+

∂2

∂y2(4)

The PEC wall of the waveguide can be described by a setof boundary conditions for the Helmholtz equations:

Ez = 0 on Γ (5)

∂Hz

∂n= 0 on Γ (6)

where Γ is the PEC boundary of the waveguide.

The boundary value problem described by equations (1)-(6)has a countably infinite number of solutions. For each solutionto kt, we can find the corresponding propagation constant:

k2z = k2 − k2

t , k = ω√µε (7)

where ω is the angular frequency of the wave, and µ andε are the permittivity and permeability of waveguide filling,respectively. For a hollow waveguide, we will use free-spacevalues µ0 and ε0. We can easily see that in order for thepropagation constant to have a real value, kt must be smaller

than k. For these cases, the wave mode can propagate inthe waveguide. If the propagation constant is imaginary, thenthe mode will be attenuated. Therefore the solutions kt arethe cutoff wavenumbers of respective modes. The angularfrequency at which k = ω

√µ0ε0 = kt is the cutoff frequency.

The propagation constants at each frequency can also bedetermined.

B. Finite Element Formulation

The boundary value problem described in the section abovecan be converted into a system of algebraic equations. Wefirst consider the TM case, where the only z-direction fieldcomponent is the electric field Ez . By applying Galerkin’smethod using nodal linear testing and basis functions toequation (1) and the boundary condition in equation (5), weobtain:

[A]Ez = k2c [B]Ez (8)

Aij =

∫ ∫Ω

∇tNi · ∇tNj dΩ, i, j = 1, 2, ..., N (9)

Bij =

∫ ∫Ω

Ni ·Nj dΩ, i, j = 1, 2, ..., N (10)

where N is the total number of nodes not on the Dirichletboundary, and the basis functions Ni are first order Lagrangepolynomials with value 1 at node i and 0 elsewhere.

To simplify the evaluation of the integrals, we can projecteach triangular finite element in the domain onto a unit right

2


Fig. 2. Full tensor grid and Clenshaw-Curtis sparse grid in 2 dimensional random space. The sparse grid is plotted for level k = 5 and has 145 nodes. Thetensor grid has 1089 nodes. [?]

triangle (Figure 1) through an affine transformation F [?].The integrals can then be performed over the area of the righttriangle:

Ae =

∫ ∫E

∇tNr · ∇tNs dE

= (J−T∇λr)T (J−T∇λs) |J |∫S

dS

(11)

Be =

∫ ∫E

Nr ·Ns dE = |J |∫S

λr · λs dS (12)

J =

[(xj − xi) (xk − xi)(yj − yi) (yk − yi)

](13)

∇λ =

[−1 1 0−1 0 1

](14)

∫S

dS =1

2,

∫S

λr · λs dS =

1/12 1/24 1/241/24 1/12 1/241/24 1/24 1/12

(15)

where E is the domain of the triangular element, S isthe domain of the reference triangle, J is the Jacobian ofthe affine transformation, and λ is the basis function overthe reference triangle. Ae and Be will be 3 × 3 matrices,and each entry can be assembled into the global A and Bmatrices based on the nodal connectivity array. For eachnode i on the Dirichlet boundaries, Aii should be set to 1,while all other entries in ith row and column should be set to 0.

Likewise, for TE cases, the boundary value problem can beformulated as:

[A]Hz = k2c [B]Hz (16)

Unlike in the TM case, the PEC walls of the waveguideact as Neumann boundaries for the magnetic field; therefore,the A and B matrices need to contain entries for all nodes inthe mesh, including those on the boundaries.

The algebraic system of equations (8) and (16) is in theform of generalized eigenvalue problems. These equations canbe solved using readily available eigensolvers. The resultingeigenvalues represent squares of the cutoff frequenciesof various propagating modes, and the correspondingeigenvectors represent field distributions in the waveguidecross-section for the corresponding mode. The fielddistribution of each mode can be visualized by plottingthe eigenvectors. Since the matrices A and B have veryfew non-zero entries, the computational efficiency of theeigendecomposition process can be improved drastically ifthese matrices are stored and processed as sparse matrices.This is critical for dense meshes since the computationalcomplexity of a non-sparse eigendecomposition process isO(n3) while sparse algorithms can achieve O(n log n)performance.

C. Stochastic Computations

When there are uncertainties in the waveguide cross-section,we expect the cutoff frequencies and propagation constantsof each mode to also be uncertain. Our goal now is to findthe probability density function (PDF) of cutoff frequencies

3


and propagation constants for each mode, using PDFs of thegeometries as inputs. The classical approach for solving thistype of problem is by the Monte Carlo (MC) method, wherea large number of input samples are generated conforming tothe PDFs of the geometry, and the solution computed for eachinstantiation. PDFs of the cutoff frequencies, etc., can then beobtained statistically from the solutions. A crucial limitationto MC is its slow speed and convergence. According to theCentral Limit Theorem:

Sn − nµ√n

d.−→ N (0, σ2) as n→∞ (17)

which means that the expectation converges at the rate of1√N

for N numbers of samples. Therefore a large samplesize is needed to achieve convergence in distribution. Foreach sample, a full FEM analysis needs to be performed.

A much more efficient method for calculating the desiredPDFs is through stochastic collocation. Stochastic collocationis based on the idea that random variables can be expandedas the sum of orthogonal polynomials [?]:

X(ξ) =

P∑k=0

Xk · φk(ξ) (18)

By the orthogonality principle, for continuous randomvariable X with PDF ρ(ξ),

E[φm(X)φn(X)] =

∫φm(ξ)φn(ξ)ρ(ξ)dξ

=

0 for m 6= n

E[φ2m] for m = n

(19)

Similar to FEM method for electromagnetics, we can choosefirst-order Lagrange polynomials as the basis functions for theexpansion in (18) [?]. This is a linear interpolation approachto stochastic collocation:

w(Z) =

M∑j=1

u(Z(j))Nj(Z) (20)

where Z is the input random vector, and the function isevaluated at a set of nodes ΘM in the input random space.The weights u(Z(j)) are calculated at each node and used tointerpolate the response surface in random space. The PDFcan then be extracted by performing MC sampling on theinterpolated response surface.

The most straightforward nodal set for interpolation is a fulltensor product set in the d-dimensional random space:

ΘM = Θm11 × ...×Θmd

1 (21)

where each random dimension is discretized into m points.The total number of nodes is M = m1 × ... × md. We cansee that the total number of nodes grows exponentially with

the number of random dimensions, a phenomenon known asthe “curse of dimensionality”. A more efficient alternativeapproach is to use Smolyak sparse grids for interpolation:

ΘM =⋃

N−d+1≤|i|≤N

(Θi11 × ...×Θid

1 ) (22)

where N ≥ d is an integer denoting the level of the con-struction. The type of sparse grid we use in this chapteris Clenshaw-Curtis nodes, which are extrema of Chebyshevpolynomials defined by:

Z(j)i = −cos

π(j − 1)

mki − 1

, j = 1, ...,mki (23)

where k = N − d is the level of construction, and the totalnumber of nodes is mk

i = 2k−1 + 1. By using this sparsegrid, the response surface can be approximated with muchfewer number of nodes. Figure 2 shows a full tensor grid anda Clenshaw-Curtis sparse grid over the same 2-dimensionalspace.

III. RESULTS

A. Problem Set-up

We study a hollow rectangular waveguide with PEC wallsand a nominally 2 meters × 1 meter cross-section. Theanalytical solution for the cutoff wavenumber is:

kc =

√(mπA

)2

+(nπB

)2

(24)

for TMmn or TEmn modes of the waveguide. We denote thecross-section dimensions A and B using capital letters sincethey are random variables:

A ∼ N (2,√

0.06), B ∼ N (1,√

0.03) (25)

and we can normalize these random variables:

A = 2(1 + 0.03 ξ1),

B = 1(1 + 0.03 ξ2),

ξ1, ξ2 ∼ N (0, 1) i.i.d.

(26)

such that samples of the Gaussian random variables can begenerated readily using existing random number generator inpackages like MATLAB.

We generate a triangular mesh on the mean-dimensionrectangle, shown in Figure 3. The density is chosen so thathigher order modes can be calculated accurately. In general,the element size should be less than 1/20 of the wavelength.By utilizing the Jacobian formulation in (13), we can usethe same mesh for all random variations of the problem bysimply scaling the Jacobian with ξ1 and ξ2. In this sense, thestochastics is formulated into the FEM problem by randomstretchings of the mesh.

4


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Finite element mesh of a 2m × 1m rectangular waveguide cross-section. There are 2688 triangular elements and 1405 nodes.

B. Finite Element Solutions Of Waveguide

By solving the eigenvalue problems formulated in (8) and(16), we obtain a series of eigenvalues that correspond tothe square of the cutoff wavenumbers of various propagatingmodes. These values are checked against the analyticalvalues obtained from (24) to ensure that the FEM programis functioning correctly. A comparison is shown in TableI. The corresponding eigenvectors represent the Ez andHz distribution on the waveguide cross-section at eachnode. By interpolating the values between nodes, plots ofthe field distribution for various propagating modes can begenerated. The plots in Figure 5 are sorted by cutoff frequency.

The dispersion diagram of the modes can be calculatedfrom the solutions. This is shown in Figure 4 for the firstseven modes. Note that below cutoff, the wave is attenuatedand does not propagate. There are three degenerate pairs(TE01 & TE20, TM11 & TE11, TM21 & TE21).

C. Waveguide Solutions With Uncertainties

With the validity of the FEM program verified, we study thewaveguide with stochastic dimensions. We assume that eachdimension will vary around 10 percent, which correspond toa standard deviation of approximately 3 percent. The PDFof the dimensions are modeled using the random variablesdescribed in (26). In order to obtain the PDF of kc, we usethe sparse grid stochastic collocation method outlined in(20) and (23). We also verify the results using Monte Carlomethod with N = 5000 and analytical method (24).

50 100 150 200 250 300−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Frequency (MHz)

TE10

TE01, TE20

TM11,TE11

TM21,TE21

| βk0|

| αk0|

Fig. 4. Dispersion diagram for the first 7 modes in a 2m × 1m rectangularwaveguide calculated using FEM. There are 3 degenerate pairs.

We can see in Figure 6 that the results show good agree-ment. On a 2.3 GHz Intel Xeon system, the stochasticcollocation method takes about 70 seconds while the MCmethod took over 80 minutes to complete. The statistics ofmode dispersion can also be easily calculated from stochasticcollocation results. Figure 7 shows the dispersion diagram forthe first three modes of the waveguide, with the mean and±3σ values. These results gives us insight into the range ofpropagation constants we can expect when the geometry isuncertain.

5


TABLE ICUTOFF ANALYSIS OF A 2M × 1M RECTANGULAR WAVEGUIDE: ANALYTICAL SOLUTION VS. FEM SOLUTION FOR THE FIRST 10 MODES.

Mode k2c (Analytical) k2

c (FEM) fc (Analytical) fc (FEM) % Error[m−2] [m−2] [MHz] [MHz]

TE10 2.4674 2.4676 74.9481 74.9513 0.0043TE01 9.8696 9.8724 149.8962 149.9175 0.0142TE20 9.8696 9.8734 149.8962 149.9251 0.0193TM11 12.3370 12.3413 167.5890 167.6186 0.0176TE11 12.3370 12.3416 167.5890 167.6206 0.0188TM21 19.7392 19.7514 211.9852 212.0510 0.0310TE21 19.7392 19.7516 211.9852 212.0521 0.0315TE30 22.2066 22.2244 224.8443 224.9345 0.0401TM31 32.0762 32.1089 270.2292 270.3669 0.0509TE31 32.0762 32.1108 270.2292 270.3749 0.0539

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.40

0.5

1

1.5

2

2.5

3

k2c

ρ(k

2 c)

TE10

Monte CarloAnalyticalStochastic Collocation

7 8 9 10 11 12 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k2c

ρ(k

2 c)

TE01


7 8 9 10 11 12 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k2c

ρ(k

2 c)

TE20


10 11 12 13 14 15 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k2c

ρ(k

2 c)

TM11


10 11 12 13 14 15 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k2c

ρ(k

2 c)

TE11


16 17 18 19 20 21 22 23 240

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

k2c

ρ(k

2 c)

TM21


Fig. 6. Probability density functions (PDFs) of k2c for TE10, TE01, TE20, TM11, TE11, and TM21 modes in a rectangular waveguide with geometricuncertainty. Note that the degenerate pair TE01 and TE20 have different probability distributions.

IV. REMARKS

The Finite Element Method is a highly robust numericalscheme for solving boundary value problems. This chapterstudied an example with regular geometry, but any irregularand complex geometry can be computed using the sameformulation by generating a mesh for it. Due to FEM’s easilyadaptivity to changing geometries, it is the ideal method tostudy stochastic problems where geometry is random. Byusing affine transformation of the finite elements, we are ableto model the random variables into the Jacobian matrix, thuseliminating the need to re-mesh. This same technique can beeasily applied to other problems where the size is random.

The stochastic collocation method used for this chapterallows for extremely fast computation of random variableswithout the need for lengthy Monte Carlo type simulations.The benefits and accuracy of the method are demonstratedwith our example. We can also see that uncertaintiesin electromagnetic problems can significantly affect theresults. In many engineering applications, knowledge of thedistribution of results is more important than having themost accurate result for a deterministic problem. Numericalmethods such as stochastic collocation will give engineerspowerful tools in modeling electromagnetic problems withrandom inputs.

6


Fig. 5. Transverse field distribution of the first 30 modes in a 2:1 rectangularwaveguide.

Xu Chen received the B.S. and M.S. degrees in elec-trical engineering from the University of Illinois atUrbana-Champaign in 2005 and 2014, respectively.From 2005 to 2012, he was an engineer with IBMSystems and Technology Group in Poughkeepsie,NY. In summer of 2013 and 2015, he was a Hard-ware Intern with Apple Inc. in Cupertino, CA. He iscurrently pursuing his Ph.D. degree in electrical andcomputer engineering at the University of Illinois atUrbana-Champaign, conducting research in the areaof stochastic modeling techniques for computational

electromagnetics and signal integrity. He is a recipient of the Ernst A. ReidFellowship Award and IEEE CPMT Student Travel Award.

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Bet

a/ko

TE10

TE01

TE20

Fig. 7. Stochastic dispersion diagram for the first 3 propagating modes ofthe waveguide with geometric uncertainties. Ranges represent ±3σ.

Jose E. Schutt-Aine received the B.S. degree inelectrical engineering from the Massachusetts Insti-tute of Technology, Cambridge, in 1981, and theM.S. and Ph.D. degrees from the University ofIllinois at Urbana-Champaign (UIUC), Urbana, in1984 and 1988, respectively. He joined the Hewlett-Packard Technology Center, Santa Rosa, CA, asan Application Engineer, where he was involvedin research on microwave transistors and high-frequency circuits. In 1983, he joined UIUC, andthen joined the Electrical and Computer Engineering

Department as a member of the Electromagnetics and Coordinated ScienceLaboratories, where he is currently involved in research on signal integrityfor high-speed digital and high-frequency applications. He is a consultant forseveral corporations. His current research interests include the study of signalintegrity and the generation of computer-aided design tools for high-speeddigital systems. Dr. Schutt-Ain was a recipient of several research awards,including the 1991 National Science Foundation (NSF) MRI Award, theNational Aeronautics and Space Administration Faculty Award for Researchin 1992, the NSF MCAA Award in 1996, and the UIUC-National Center forSuperconducting Applications Faculty Fellow Award in 2000. He is an IEEEFellow and is currently serving as Co-Editor-in-Chief of the IEEE Transactionson Components, Packaging and Manufacturing Technology (T-CPMT).

7


Andreas C. Cangellaris received the Diploma de-gree in electrical engineering from the AristotleUniversity of Thessaloniki, Thessaloniki, Greece, in1981, and the M.S. and Ph.D. degrees in electricalengineering from the University of California atBerkeley, Berkeley, CA, USA, in 1983 and 1985,respectively. After a two-year tenure with the Elec-tronics Department, General Motors Research Lab-oratories, he joined the Electrical and Computer En-gineering (ECE) Department, University of Arizona,initially as an Assistant Professor (19871992) and

then as an Associate Professor (19921997). In 1997, he joined the ECEDepartment, University of Illinois at Urbana-Champaign, Urbana, IL, USA,as a Full Professor in 1997, where he is currently the M. E. Van Valken-burg Professor and Dean of the College of Engineering. He has authoredor coauthored over 250 papers in peer-reviewed journals and conferenceproceedings on the topics of finite methods for electromagnetic field modeling,multi-conductor transmission line modeling, and signal and power integrityof integrated electronic systems. His expertise and research interests arein applied and computational electromagnetics and its applications to theadvancement of modeling methodologies and computer-aided design (CAD)tools in support of electrical performance analysis and noise-aware design ofintegrated electronic systems. His research group has conducted pioneeringwork in the advancement of modeling methodologies and computer toolsfor predictive electromagnetic response performance of the signal and thepower distribution networks of integrated electronics at the board, package,and chip levels. Several of the prototype tools developed by his group havebeen transferred to the microelectronics industry and have contributed to thedevelopment of several EDA signal integrity tools. Prof. Cangellaris wasthe cofounder of the IEEE Topical Meeting on Electrical Performance ofElectronic Packaging in 1991. He is currently editor of the IEEE Press Serieson Electromagnetic Wave Theory. He is an active member of the IEEEMicrowave Theory and Techniques Society (IEEE MTT-S) and the IEEEComponents Packaging and Manufacturing Technology Society, contributingto numerous professional activities, conferences, and symposia organized andsponsored by them. From 2010 to 2012, he was a Distinguished Lecturer forthe IEEE MTT-S. He was the recipient of the 2000 University of Illinois, ECEDepartment Outstanding Faculty Teaching Award and the 2005 Alexander vonHumboldt Research Award for outstanding contributions to electromagnetictheory.

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Documents

Finite Element Analysis of Rectangular Waveguide with ......The Finite Element Method (FEM) is a versatile numerical procedure to ﬁnd approximate solutions to partial differential