Electro.oft, 1991, Vol. I, No. '-/3 combined

@1991 EI.evier Science Publi.her. Lid

Microwave NDT

N. IdaDepartment of Electrical Engineering, The University of Akron, Akron, DE /9£5-990./, USA

INTRODUCTION

Nondestructive testing at microwave frequencies is not a new practice, but, the use of nonmetallicmaterials and composites has made this method more common and applicable to a wider range ofproducts. Modeling at microwave frequencies demands special attention to both formulations usedto describe the physical phenomena and to the computer software implementing the formulations.

Electromagnetic fields at microwave frequencies are described by a general hyperbolic equation.From a formulation point of view, there are three domains that can be considered:

1. Time harmonic, externally driven problems. This includes geometries such as antennas overcomposites, or a coil (more often a loop) over a dielectric.

2. Time dependent, externally driven problems. Typical testing situations include pulsed andnon-periodic sources.

3. Time harmonic, sourceless problems. The most common of these is that of a microwavecavity where, although the coupling to the cavity implies a source, the solution consists ofmodal solutions and the source is not explicitly taken into account.

The first of these is an extension of the standard eddy current problem where both propagationeffects and losses may exist. In special cases, where losses can be neglected, the problem reducesto a wave equation in real variables, simplifying the analysis considerably. The second implies amore general approach and requires the solution of the transient wave equation. The third problemis of special interest in testing. It is used to solve for modes and fields in loaded waveguides andcavities. This approach allows characterization of materials by evaluating their properties basedon the shift in resonant frequencies and changes in Q-factors of the cavity.

Within each of these there are variations that affect the type of analysis and appropriate for-mulation. For example, a simple Helmholtz equation is sufficient to describe an ideal cavity (nolosses in the cavity or in materials within it). A modified equation, that includes the ,losSes isneeded if, either the cavity walls or materials ".ithin the cavity have low conductivities. This, ineffect requires solution of Helmholtz's equation in complex variables. If both sources and lossydielectrics must be modeled, the problem is really a general propagation problem with significanteddy currents.

From an algorithmic point of view, modeling of microwave NDT is different than other types ofelectromagnetic modeling. Because the displacement currents are always included in the formula.-tion, the propagation effects are present. This requires special treatment of boundary conditions.

Manuscript recei~ September 1990j and in fmal fonD April 1991.

116 N. Ida

Radiation or absorption boundary conditions are required for open boundary problems rather thana simple truncation of the mesh, to ensure that there are no reflections from artificial boundaries.For resonant cavities, the eigenmodes are often of interest requiring the solution of an eigenvalueproblem. This might be a simple algebraic eigenvalue problem in the case of lossless cavities but,in lossy cavities, a complex eigenvalue problem needs to be treated. This is particularly difficultbecause there are no simple, well established methods for solving complex eigenvalue problems.

The results presented here relate to the testing of materials at high frequencies. These includecharacterization of materials in microwave cavities and scattering applications in an open domain.The specifics of formulations are kept to a minimum for brevity.

FORMULATIONS

As might be expected there are a number of formulations that are especially amenable to compu-tation at high frequencies. In general, high frequency formulations have been limited to scatteringproblems and, in most cases to far field applications. Few formulations for near field calculationsand even fewer that are applicable to NDT exist. Some of these formulations are described nextin the context of high frequency NDT applications.

MODIFIED EDDY CURRENT FORMULATION

An eddy current formulation, based on the magnetic vector potentiall, can be easily modified toallow the computation of electromagnetic fields at high frequencies such as waveguides and cavitiesby adding the coupling between the electric and magnetic fields. This type of formulation is partic-ularly attractive for NDT applications since it allows calculation of conduction as well as dielectriclosses and is applicable to a ,,'ide range of practical geometries. The coupling between the electricand magnetic fields is normally taken out of eddy current formulations by assuming that thereare no displacement currents, The effect of displacement currents can be added by including theelectric field directly in the formulation. One particularly simple way is to use the magnetic vectorpotential as a variable while neglecting the electric scalar potential and adding an electric energyterm into the energy related functionaJ2.3

F(A) 1 B2 d 1 J.A d 1 jl.lD'A..A.d 1 eE2d= -v- -v+v+ -v,,211 ,,2 ,,2 " 2 (1)

where B = VXA and E = - 8A/Ot.The last term in the functional is the stored electric energy. .In addition to allowing calculation

at high frequencies, the losses due to eddy currents are also included in the third term in (1).A similar, but more general formulation takes into account the scalar potential as well

- .!.V2A = J, + IW2'A - uUIWA + VtjI) (2)JJ

If significant eddy currents exist, (as in the case of low conductivity materials), the eddy cur-rents must also be constrained in the solution domain to ensure uniqueness of the solution. Thecontinuity equation is used for this purpose.

V.(jO'{lJA+uVtjI) = 0 (3)

These two equations represent a general formulation equivalent to equation (1). The onlyassumption implicit here is the use of Lorentz's gauge. Unlike equation (1) where the electricscalar potential is assumed to be insignificant, this formulation is general.

Microwave NDT 217

After the magnetic vector potential is calculated, either B or E can be calculated. The magneticflux density is calculated from B = VXA and the electric field intensity is:

(4)E = - (jIolA + V1/1)

The solution to equation (1) or (2)-(3) leads to the correct field quantities based on the general fieldrepresentation at any frequency. This is a deterministic approach where the source is assumed tobe known. For this reason the modes of a cavity cannot be obtained directly. However, the methodis not restricted to cavities or to high frequency problems. For any given frequency, a solution isobtained. By scanning a frequency range (i.e. solving a number of eddy current problems, each ata different frequency), the resonant frequencies can be detected.In addition, the Q-factor of a cavity can be calculated as:

(5)

where W. is the stored energy in the cavity, Pd are the dielectric losses and Pm are the losses inthe metallic walls of the cavity.Expressions for IV., P d and Pm in a cavity are:

W. = ~lEE.E-dV + ~lJlH.H-dV (6)

11Pd = '2 v D'E.E-dV (7)

11Pm = '2. R.H.H- dV (8)

where R, is the surface resistivity (R, = 1/0-6), 0- is the conductivity of the cavity walls and 6 theskin depth. In geometries where the penetration might be deep, surface resistivity is not definedand W" P d and Pm are calculated directly from the various terms of equation (1).

For the purpose of detecting resonance or shift in resonant frequency, the total stored energy inthe cavity is calculated from equation (6). A peak in the stored energy indicates resonance. Thischoice is convenient in that the actual energy in the system is not important, only the relativevalues. For a cavity, the actual fields are difficult to calculate because, either equation (1) or (2)requires the actual source distribution. This means that the sources and their coupling to thecavity must be modeled accurately. If only relative terms are required, an arbitrary source canbe used as long as the source supports the mode (or modes) required. From these relative energyterms, the resonant frequency and the Q.factor can be calculated. If the fields themselves areneeded, the source, and the coupling to the cavity must be modeled.

ResultsA simple cavity with dimensions a = b = d =11.4" (28.96 cm) is used to demonstrate the for-

mulations above. The cavity walls are conducting and are lined with a lossy dielectric, 1.7" (4.318cm) thick. This leaves an inner space 8" x 8" x 8" in size. The geometry is shown in Fig. l(a).Material properties for the dielectric are 'r = 6 or 'r = 9 (depending on the type of material) anda conductivity of 10-4 S/m. Samples made of dielectric or lossy dielectric materials (i.e. compositematerials) are introduced as shown in Fig. l(b) for testing purposes.

The empty cavity, without the dielectric lining, resonates at 732 mHz in a TM11O mode. Withthe lining ('r = 6) the measured resonant frequency is 554 mllz. A numerical solution for thecavity in Fig. 1, is shown in Fig. 2 where the frequency was scanned from 350 mHz to 580 mllz.

1.18 N./ill

~ 11."-..11... ..

I8-

t

18-

1

1.1... 11.4.

alumina---alumina. a- ..."4 8.

(b)(a)

Fig. 1. Dielectric lined cavity u$edfor modeling (a) Empty cavity, (b) cavity with a test sample

>-0~

~U)I:J>

gcJ~

Fig. I. Frequency $can for the empty cavity with tr = 6 for the dielectric lining. Re$onancei$ at 5.16 mHz

-1-Microwave NDT 119

The resonant frequency at 564 mHz is obvioua.'This is about 1.8% higher than the measured reso-nant frequency. For this calculation and the rest of the results presented here a simple cubic finiteelement mesh, with 512 hexahedral (8 node) elements and 729 nodes was used. Symmetry wasignored primarily because the original study looked at effects of spatially nonsymmetric loading.

To examine the shift in resonant frequencies due to presence of composite materials, the res-onant frequency with and without composite samples was calculated. This time the lining wasassumed to have a relative permittivity of 9. Figure 3 shows the frequency scan (400 mHz to 465mHz) for the empty cavity. The resonant frequency is at 440 mHz. A small sample, made of a di-electric material with £,. = 9, and dimensions 2.1" x 2.1" x 1.9" (5.334 x5.334 x4.826cm) is modeledinside the cavity as shown in Fig. l(b). The resonant frequency in the scan shown in Fig. 4 is now423 mHz. The same sample, with a conductivity of D' = 4 X 10-4 S/m, and permittivity of £,.= 9(graphite) was modeled and the results shown in Fig. 5. The resonant frequency has now shiftedto 442 mHz. These results are summarized in Fig. 6 where the amplitudes have been normalized.The shift in resonant frequency is consistent with perturbation theory in that an increase in thepermittivity of a material lowers the resonant frequency while an increase in conductivity (as isthe case with the graphite sample) increases the resonant frequency.

SoftwareA computer program based on the formulation in equations (2) and (3), specifically designed

for microwave NDT applications has been written. It is based on an existing three dimensionaleddy current programl. The computer code uses 3-D finite elements (8 node hexahedral elements).Solution of the system of equations is handled by a bi-conjugate gradient algorithm or I for smallproblems, by direct Gaussian elimination. The program handles general 3-D geometries but, is

>-0

~~>

~~

Fig. j. Fref.enc, scan for the empt, camt, with t',. = 9 for the Jielectric lining.i$ at ././0 mHz

Re80nance

220 N. Jda

>-0

IAJ>

~~

1~).Fig. ./. Frequency scan for a dielectric sample in a microu'ave cavity a$ in Fig.nance i$ at ./£3 mHz.

Reso.

>-0~aJ

?JaJ>

~aJ~

Fig. 5. Frequency scan for a lossy dielectric sample in a microwave cavity as in Fig.Resonance is at ././£ mHz.

1(6).

Microwave NDT !!I

>-

~~

~

~z

Fig. 6. Comparison of resonant frequencies for the cavity with and without test materials. Ampli-tudes have been normalized.

particularly efficient in handling cavities. Either magnetic or electric boundary conditions areincorporated directly into the computation. Surface impedance can also be incorporated but amore general method of treatment, extensively used in the code, is to model the material on theboundary as a lossy dielectric. This is possible because the formulation includes the eddy currenteffects. Typical finite element meshes are relatively small (a few hundred nodal points) but, if thegeometry is complex, or if the source must be modeled accurately, the number of nodes can bevery large. Most problems can be solved on a workstation while larger problems are solved with aCray version of the code. Typical solution time for a problem of the size described above is of theorder of 20 minutes on a workstation (per frequency point) or about 10 seconds on a Cray Y-MP.

FINITE DIFFERENCE TIME DOMAIN FORMULATION

Another approach to modeling of high frequency NDT is the use of Finite Difference Time Domain(FDTD) techniques. Finite difference methods in the time domain have been applied to solvetransient electromagnetic wave propagation problems over the atmosphere and the ground5, andtime-dependent eddy current problems6. The method presented here is a generalization of thiswork and is designed for nu~rical modeling of high-frequency electromagnetic wave propagationarising from nondestructive testing applications.

Geometries to which this method applies include examination of the scattering effects by cracksinside a piece of material (especially dielectrics) or due to surface variations when the materialis illuminated by a (TM) plane wave. The treatment here is in two dimensions (including ax-isymmetric geometries) and in terms of potential functions. The extension of this formulation tothree dimensions is possible13, but has not been attempted yet because of inherent difficulties in

222 N. Ida

designing meshes that satisfy the Courant stability criterion within a reasonable number of timesteps and, at the same time, models the types of geometries required for nondestructive testing.

A typical situation is depicted in Fig. 7, where the region is composed of two parts. The first

Fig. 7. Geometry of a general scattering problem in NDT

.8H~ .8H,a- + J-Ot 8t

8E= D'pE + £p~ forp = (a,m,B)

part is assumed to be free-space (£:G' O"G = 0). The source is located in this part of the solutiondomain. The medium below it is the test sample, typically a lossy dielectric, with parameters(£:m, O"m). For high frequency applications O"m is assumed to be very smaJl compared with theconductivity of a good conductor. The material under test may include flaws such as surface vari-ations, inclusions (buried or surface breaking), etc. These fla,,'s, shown in Fig. 7 as two cylinders,behave like scatterers. Their parameters are denoted by (',,0",). All materials have the perme-ability of free-space lJo.

For TM field incidence, the relevant equations are

~ + ~ = 0 (9)

8y

.8E- J- = -IJ I8z P

8H&--

(10)

(11)

K2 = (6I2L2poe",K2 = (.J2 L2/lo £'0 I0 [2 = c.1L2Jlo8m:m 2

[2 = c.1L JlO8j.

K2 = ",2L2POtm.m

8%

.8EJ-

8y

~--8% 8y . ,

By introducing a scalar function A such that pH = VX(ICA), (i.e. the single component vectorpotential, where IC denotes the direction of A) one obtains three scalar equations

Microwave NDT

The equations have been scaled by introducing z = z/ L, 1/ = 1// L, and t = wt. Interfaceconditions are obtained from the continuity of the tangential components of electric and magneticfields,

Absorbing boundary conditionsIn general, the so]ution domain is infinite. For a practical so]ution, either within the finite

element or the finite difference formalism, one needs to operate within a truncated domain. Anartificia] boundary condition must be specified on the truncated boundary to simu]ate the infinitedomain. A free space boundary condition can be written as8,9;

0

Ora 1

+ kaa"i + 2; A. = 0

1~ '+- 2km.

) A. = 0

FINITE DIFFERENCE IMPLEMENTATION

.e.e... N. Ida

~t~z

At--l+ak~z

+(17)

where the constants a through h are evaluated from the parameters of the problem and are given inRef. 5. These equations are then solved (explicit schemes) directly to obtain the magnetic vectorpotential. Other quantities, if required are calculated from the magnetic vector potential.

As with any finite difference schemes for wave propagation in the time domain, the term(~t/ ~z)2 appears in the expressions. In order to obtain a stable solution, the discretizationin the time domain must be made fine enough to satisfy the stability criterion. (~t2 / ~z2 $ 0.5).This normally results in a very large number of time steps.

-,.;-

~

.,.~,'

::~.

~c'~ao F

,'--

---

v00 - F to

t . "-'-' ~ ~~~~ Or "'~-; ~ /.../ /U;: " ,/" /~ot ..- :/ L---,/ ~/./' .,-'

~ "-- ... t.,'- ~.

-' "

'..'{ ~~'

.'"

~ / .t ./. ;'

....'

/

/

,'",.~. .'.: .'.'.~.' '.'" .',I ,'.'" .'.'.- .'.,.,. .~ ,. . . .'

,I;'" ...,.;. -~'.'w '~.,

0.0

Fig. 8. Numerical solation (FDTD) without scatterers

,(--~..-..

Microwave NDT ttS

NUMERlCAL EXAMPLES AND DISCUSSION

The finite difference code based on the formulation above, has been applied to calculate the fielddistribution for a number of configurations under plane wave incidence. The general NDT modelin Fig. 7 is used to demonstrate this formulation. The medium below the interface is a lossydielectric (Po, £0' O'm = 0.05 S/m) under test. The foreign material inside the test material is agood conductor Po, £0' O'm = 107 S/m). The diameter of the cylinders is: L = 0.2m.

An oblique plane wave is incident on the test sample from the ambient medium. The incidentwave penetrates into and reflects from the material surface. Figure 8 is the field contour plottingfor 45° incidence with respect to the y-axis, with frequency f = 4GHz and 0' = 0.05 S/m, whenthere are no defects or scatterers in the material being tested. The field distribution is quite fa-miliar. Inclusions are now introduced into the material. Fipre 9 shows the wave surface plot forthis case, where the conductivity of the foreign material is 107 S/m, and the incidence is the sameas above. Since the foreign material behaves essentially like a scatterer, the penetrating wave isscattered by the cylinder. The scattered wave then propagates through free-space, after a refrac-tion at the interface. The field distortion, due to the presence of foreign materials is detectable

Fig. 9. N.merical sol.tion with two cylinders as scatterers

on the surface of the material. Figure 10 gives a comparison of the field distributions just abovethe material surface with and without scatterers. If one subtracts the two fields, the field due toscattering by the conducting foreign materials is obtained. The corresponding field surface is givenin Fig. 11. The solutions given here are typical of composite materials with fiber reinforcement.The scatterers modeled here were good conductors. This is not necessary in order to apply theformulation. Results for other materials are given in Ref. 5.

226 N. Ida

Fig. 10. Comparison of solutions with and without scatterers

Fig. 11. Scattered field

Microwave NDT tt7

Another example of a significant NDT problem at high frequencies is the interaction of elec-tromagnetic fields produced by a current excited small loop with lossy and lossless materials. Theproblem is solved in the time domain in order to show the generality of the approach even though,in most cases, the fields are time harmonic. The loop is considered to be very small so that thecurrent distribution in the loop is uniform12.

The problem considered here has a source in the solution domain. The source gives the incidentfield component and is modeled by the incident components of the potential at a group of gridpoints, which are near to and enclose the source. For time harmonic fields, A is a complex phasor.The incident magnetic vector potential produced by a small loop, carrying a spatially uniformcurrent I, under time harmonic excitation is13:

Ai - ~ei(k."'-"'C}(& + -.!.-) sin 0 (18)

4'8 r' (r')2

where r is the radius of the loop, r' is the distance of the observation point from the loop's center,'" is the angular velocity of the exciting current and 0 is the angle which the position vector r'makes with the z-axis. On the axis itself, 0 = 0 and therefore the condition A = 0 is imposed.

This formulation was applied to a testing geometry where a small loop carrying a current islocated over a flat composite material. The geometry, material properties, and loop parametersare given in Fig. 12. The field distribution in Fig. 13, clearly shows the effect of the slab on thefield, including the guiding effect of the composite.

(120.120) (120,0) (120,-4) (120.-120)

~~

(0.120) (0.0) (0,-4) (0..120)

Fig. 12. Geometry and parameters for a loop over a composite material. The numbers at cornersshow the finite difference grid size.

The method can also be applied to low frequency testing geometries6 as long as an appropriateboundary condition is used, but may be less efficient because of the large number of time stepsrequired. One problem remains unresolved. Since the artificial absorbing boundary condition is,more or less, frequency-dependent, the finite difference method in the time domain fails for wavescontaining low and high frequencies (e.g., a very short pulse input). One remedy is to take atransform, (Fourier or Laplace), solve a sequence of problems in the transformed domain, and theninvoke a numerical inversion scheme to get the transient solution.

I..

££8 N. Ida

Fig. 13. Contour plot for the geometry in Fig. 1./

SoftwareThe formulation presented here has been impJemented in a series of programs that form a

software package in two-dimensional and axisymmetric geometries for computation of transientand time-harmonic fields using the FDTD method. Radiation and absorption boundary conditionsfor two-dimensional and axisymmetric geometries for low and high frequency applications wereimplemented and the method was applied to a large number of NDT problems. These includelow frequency eddy current problems, transient NDT problems and time-harmonic applications.Typical problems may require on the order of 5000 nodes in the finite difference mesh and on theorder of 2,000 to 5,000 time steps, depending on frequency and material properties. A problemof this magnitude can be solved in 2 to 10 minutes on a workstation (SUN-3). In some cases,particularly at low frequency, transient problems, larger meshes are required (the geometry in Fig.12 uses over 16,000 nodes). A typical problem of this kind may require as many as 40,000 nodesand over 20,000 time steps. Typical solution times are about 5 hours on a workstation or about10 minutes on a Cray Y-MP.

Microwave NDT t£9

Although the method can be extended to three dimensional analysis, it does not seem tobe a viable option at this time because of inherent difficulties. One is the radiation boundarycondition. However, a more important concern is the stability criterion. If adequate modeling isto be maintained, the finite difference mesh must be relatively fine, requiring a large number oftime steps. While the finite difference schemes are explicit, the large number of time steps willmake the solution quite expensive. Until these problems can be resolved, there seems to be littleincentive to proceed with a 3-D formulation.

SOLUTION OF HELMHOLTZ'S EQUATION

Computation of steady state fieldsA number of important high frequency problems can be solved by the scalar Helmholtz equation.

These include fields in waveguides and resonant cavities where the scalar Helmholtz equation isparticularly simple to solve. The boundary conditions are simple and well defined on metallicinterfaces and, in most cases, the problem can be solved in real variables. This is certainly thecase in empty cavities and in cavities loaded with dielectrics. In cases where the loading includesconducting materjals (or lossy dielectrics) or the metallic walls are made of poor conductors, thecomplex permittivity must be used leading to a problem in complex variables.

The problem is therefore one of solving the following equation in two or three dimensions.

v2q, + k,2q, = 0

The simplest way of formulating this equation in terms of finite elements is to ,,'rite a functional1.

2'Bt/J

,By+

Introduction of a finite element approximation based on standard shape functions for any typeof element and minimization of the functional with respect to the unknowns ~i results in a systemof linear equations to be solved.

[51{~} = k2[M]{~} (21)

The necessary boundary conditions are the Dirichlet boundary conditions while the Neumannboundary conditions are implicit in the formulation. Equation (19) is, an eigenvalue problem.Solution proceeds by finding the eigenvalues k2 of the system, which give the cutoff (or resonantfrequencies) of the geometry. This method of formulation has been used for calculation of dielec-trically loaded waveguides in Ref. 14, using tetrahedral finite elements in three dimensions andtriangular finite elements in two dimensions.

A general method for dielectrically loaded waveguides is presented in Ref. 15 for 2-D geometriesThis formulation uses both field variables E and H. The formulation follows identical steps asabove when the variables E. and H. are viewed as scalars and where k2 = (.J2£Jl - .82.

It is also pointed in Ref. 15 that the solution to the eigenvalue problem in (21) yields spurioussolutions. These solutions corresponds to eigenvalues of equation (19) but do not correspond tophysical solutions. While this is not a major problem for two dimensional geometries (mostlysimple cross section waveguides), it becomes a major difficulty in 3-D calculations. The numberof modes is large and the modes are close together. Identification of correct modes becomes quitedifficult, especially if higher order modes are required. As an example of two-dimensional solutionsat microwave frequencies, T E and T M fields in rectangular waveguides were calculated using afinite element representation. However, since the waveguides in these examples were assumed tobe uniformly loaded, it is sufficient to solve either for E or H rather than both or

1.30 N.lda

V2E + };2£ = 0 V2H + k2H = 0or

where l' = w211E:.

For T M modes, with propagation in the z direction, the following equation it solved

~ ~ - _L2E (23)8%2 + 8r2 - ~ I

A similar form is obtained for T E modes. More general forms of the formulation can be foundin Refs 14 and 15.

This describes propagation in a lossless waveguide. Losses in the waveguide can also be ac-counted for by defining k~ = '-I211(E: - jD'/'-I) and replacing k2 in (20) through (23). The maindifference is in the need for eigen\-alue solution in complex variables.

The solution follows the standard procedure of finding the eigenvalues and then finding a solu-tion for any of the required eigen\'alues.

After discretization of the solution domain in terms of finite elements (isoparametric quadri-lateral elements ".ere used in this case) and application of boundary conditions an eigensystem isobtained &8 in (21).

The necessary boundary conditions are defined by the metal wall surfaces. At the surface of aperfect conductor 8H./8n = 0 in a TE mode and E. = 0 in TM mode.

RtsultsAs an example consider the waveguides shown in cross section in Fig. 14(a) and 15(a). Figure

14(b) shows a TM1.2 solution for a ridged wa\"eguide. Figure 15(b) shows a TEl.3 solution for arectangular waveguide. Although only simple waveguides have been considered here, the methodis quite general and can be similarly applied to other geometries with minor modifications. 3-Dproblems are solved in a similar manner4.

(a) (b)

Fig- 1./.aide

Crou section of two simple waveguides (a) rectangular waveg.ide, (6) ridged waveg-

A GENERAL 3-D FORMULATION FOR CHARACTERIZATION OF SPECIMENSIN A MICROWAVE CAVITY

Characterization of a specimen in a microwave cavity can be done by the resonant frequencies andQ-factors of the cavity. Since any change in the volume of the cavity or its material propertiesresults in a shift in frequency and its Q-factor, this method is a very simple way of detecting,identifying and monitoring of materials in a cavity. It is used extensively, especially with open

Microwave NDT 131

fa)

(b)

Fig. 15. Modes in waveguides: numerical solution (a) TE13 solution in the rectangular waveguide,(b) TMlt solution in the ridged waveguide

cavities for such application as detection of gases and monitoring of water content in materials.Modeling of a partia]]y loaded cavity is quite complex because the modes of the cavity can

change, in addition to some modes shifting in frequency. Analytical treatment of the problem isonly possible for sma]] insertions, using the perturbation method. In the method presented here,the insertion can be of any shape and size. The model uses the recently established edge finiteelements with divergence free shape functions to solve the vector eigenvalue problem. It has theadvantage of accurately computing the eigenmodes and eliminating the nonphysical modes thatplague all finite element methods based on nodal-based shape functions18.19.

N. IdaP.3!

In NDT, the problem is one of finding the resonant frequencies and the corresponding fielddistributions in a microwave cavity containing a test specimen. The cavity wall S is assumed tobe highly conducting (not perfectly conducting). The interior of the cavity may contain materialswith constant material properties. Since a sample is introduced in the cavity (Fig. 16), n is char-acterized by {poPrT, t:o(t:r - j£'~), O'} were a spatial dependence for £' and 0' is implied. Po and £'0

s

Fig. 16. A rectangular carity with a dielectric spherical specimen at its center

are the free space permeability and permittivity. Assuming time harmonic fields, the modes in acavity must be the solution of one of the follo,,"ing curlcurl equations

'24)v x

(25)

1-V x E - k2t:E = 0P1- V x H - k2 JlH = 0t:

v x

where,

t = t,.(1-jtan6)k = w,fjiOi; I

and L is a spatial scaling factor.The loss tangent is related to conductivity by,

'10 = JPJ£:tan 6 = 0"1)o/k£,.,

On material interfaces, E and H must be tangentially continuous. Two types of boundary condi-tions are possible. On a perfect electric wall nXE = 0; and on a perfect magnetic wall nXH = O.

The problem defined in (24) or (25) is a generalized eigenvalue problem. It is, in general, dif-ficult to solve. To do so, the so-called weak forms for the E and H formulations are considered.

For the E formulation:

1 (;V x E) .(V x wm)dv - k21 ~E.wmdV = -jk'1o 1 (n x H).wmdS (28)

-LMicrowave NDT

and for the H formulation:

where Wm is any set of real vector weighting functions.In the E formulation, a symmetry boundary condition nXE and a natural boundary condition

nXH exist. Similarly in the H formulation, nXH is the symmetry condition and nXE is thenatural boundary condition. At high frequencies it is common to assume that the tangentialelectric field vanishes on the cavity wall. When it is required to take into account the finite (butusually large) conductivity in the cavity wall, the following impedance boundary condition can beused IS.

E x n = Zmn X (n x H) (30)

where Zm = (1 + j)/u6, is the surface impedance. The generalized eigenvalue problem (28) or(29) possesses an infinite number of modes. In reality, only a few of these, especially the lowestmodes are of interest. Corresponding to these modes, there exists a set of Q-factors as defined inequations (5) through (8).

Finite element implemtrrtation for lossles$ caritiesIt is well kno\\'n that the standard (node based) finite element solution to the vector eigenvalue

problem (25) contains both physical and nonphysical solutions. One reason is that the curlcurlequation, is solved directly leaving the divergence free condition, V.H = 0 unspecified. Methodsfor overcoming this difficulty have been introduced18.19 to remove the nonphysical modes. Thesemethods are only partially successful. The penalty method18 is relatively simple to implement butrequires a penalty parameter that must be guessed. It does not remove spurious modes but rathershifts them to the domain of higher modes where, presumably one is not interested. The reductionmethod 19 can remove all spurious modes but is difficult to implement and expensive in terms of

resources needed. Here, a linear "edge" finite element built on a tetrahedral model is used. Thesix vector shape functions in a tetrahedral element are20

Wn = Sgn(n)f(P7-n'l x P7-n,2 + e7-n x r)

These functions possess zero divergence. The approximation is

MH = L Hnwn

n=1

where H" are the tangential components of H along edges. H is therefore divergence free withinthe cavity and tangentially continuous on material interfaces. In a loss free case (£r is real), withvector weighting functions chosen to be the same as the shape functions, (31), equation (29) reducesto the generalized algebraic eigen\'&lue problem

[A}{B} = i'[B]{H}

Lossy cavitiesIf the cavity has losses due to the presence of conductors, (nonzero imaginary part in e"r in the

insertion or possibly due to imperfect cavity walls), the formulation must be modified. Assumethat on S the impedance condition (30) is satisfied. By applying the boundary conditions andintroducing the vector shape functions into (6), the following eigenvalue problem is obtained

~N. Ida134

([A(~)] - ~2[B]) {H} = 0

[A(k)] denotes that the matrix [A] depends on k and it can be evaluated only when k is given.[B) is a constant matrix. The dependence of the loss tangent on frequency has made the solutionof (34) more difficult as it cannot be reduced to an algebraic eigenvalue problem considered inthe previous section. However, (34) has a nonzero solution only if the determinant of the stiffnessmatrix is zero

det(R) = det ([A(k)] - k2[BJ) = 0 (35)

The problem now becomes one of searching for the zeros of (35) along the real axis in thecomplex plane.

In a similar manner to the H formulation, the E formulation can also be derived but, unlikethe H formulation, it provides an explicit expression. Assuming that the dielectric constant doesnot depend on frequency, discretizing (28) and imposing all necessary boundary and symmetryconditions, yields a complex quadratic eigenvalue problem of k:

([A] - k2[B] + jk[C]) {E} = 0 (36)

Next, an important case where the imaginary part of dielectric constant is proportional tofrequency is considered

£~ = kK"

Assuming no wall losses, by decomposing [B] in (36) into two parts, one ends up with a third-order eigensystem:

([A] - k2[B] + jk[C] - jk3[D]) {E} = 0 (38)

Once the solutions to either eigenvalue problem in equations (33), (34), (36) or (38) are avail-able, a mode identification is performed if such an identification is needed and can be made. TheQ-factors can be computed using the eigenvectors through the definition in equation (5) through(8). The field expansion (32) is then inserted into the above expressions for accurate integration.

2.5fr = 1

e"-(/)

"0~'""

~uZtaJ::>0taJ~C&.

e-oz<z0C/)taJ~

2.01

l.~

1.0

£,. = 98o. c; I

I0.0' . . . . . . . . . ,0.0 0.2 0.4 0.6 0.8 1.0

aJbFig. 17. Mode chart lor 0 dielectric .,here in 0 nctong.lor cavit,. The .ize and dielectric constant01 the sphere ore varied. Resulu shown (or dominant mod~.

Microwave NDT £35

ResultsTo demonstrate the foregoing arguments, two cavities are analyzed. The first consists of a

rectangular cavity with a spherical inclusion at its center as shown in Fig. 16. The inclusion isassumed to be a lossless dielectric. In the analysis, both the dielectric constant and the size ofthe inclusion are varied. The results are shown in Fig. 17. Each curve corresponds to a differentdielectric constant. The curves shown are for dielectric constants of 1, 1.5,2,3,4,8, 16,36,54,72,and 98, shown from top to bottom in Fig. 17. Only the first (t,. = 1) and last (t,. =98) are marked.The variation in the resonant frequency is considerable with either parameter. The results shownare for the dominant mode but similar curves can be obtained for any required mode.

The second example is shown in Fig. 18. A spheroidal inclusion is centered in a cylindrical

Fig. 18. A spheroidal, lossy dielectric sample in a cylindrical cavity

e-rn"t110)..I

-

~u:z.c.J::>0

~fs.

E-t

~0C/)~

%)

Fig. 19. Mode chart for a lossy dielectric spheroid in a cylindrical cavity. The dielectric con.stant varies linearlv with water content. Results shown for the first few dominant modes.

N. Ida1.96

cavity. The inclusion is a lossy dielectric. The loss is included in the model by assuming thatthe inclusion contains a certain percentage of water. The dielectric constant of the inclusion isdescribed as t'r = 9 + (70 - j25)~ where ~ is the percentage of water in the inclusion. The results

for this geometry are shown in Fig. 19 for the first few modes. The water content is varied fromzero to 20%.

While these results clearly show the variation in resonant frequency with changes in materialproperties, the modes shown are not identified. The formulation in (28) or (29) does not allowidentification of the modes in the cavity. If the modes need to be identified, this can be done byimposing various symmetry conditions in the cavity and identifying the modes that can exist underthe imposed conditions. A more general way is to evaluate the eigenvectors corresponding to eachmode and inspecting the field distributions.

SoftwareThe formulation presented above has been implemented in a computer program on a Cray Y-

MP computer. The need for eigenvalue solution all but precludes the solution of any realistic NDTgeometries on smaller computers. An eigenvalue solver based on Ericsson and Ruhe's method21 isused. The use of edge elements results in physical modes only. Pre- and post-processing is doneon a workstation using the KUBIK mesh generator23 and a simple version of a post-processor thatallows display of modes and fields.

CONCLUSIONS

The formulations and results presented here represent various aspects of modeling at high fre-quencies. The use of modeling for NDT purposes is relatively new but its benefits to testing ofcomposite and nonconducting materials are obvious. One ,,'ould expect this use and the need foraccurate modeling to grow in the future.

REFERENCES

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3. Ida, N. Computation of resonant frequencies for nonuniformly loaded cavities using a modified3D eddy current formulation, Proceedingl of the International Seminar in Electromagnetic FieldAnalYli.. Oxford, April 23-25, 1990, 267-274

4. Song, H. and Ida, N. An eddy current constra.int formulation for 3D electromagnetic field calculations,to appear

5. Hariharan, 5.1., Wang, J.S. and Henriksen, B.B. Transient electromagnetic wave propagation over ahalf-plane, to appear

6. Lee, M.E., Hariharan, S.I. and Ida, N. Solving time-dependent two-dimensional eddy current prob-lems, ICOMP Report, ICOMP-88-10, NASA Lewis Research Center, June 1988

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9. Lee, M.E., Hariharan, S.I. and Ida, N. Solving general time dependent two-dimensional eddy currentproblems, Journal of Computational Ph'Jlic.. August 1990, 89(2), 319-348

Microwave NDT !91

10. Wang, J.S., Ida., N. and llariharan, 5.1. Numerical modeling of transient wave propagation for highfrequency NDT, in Review of Progre.. in Quantitative Nonde.troctive Evaluation, Thompson, D.O.and Chimenti, D.E. (Eds), Plenum Press, New York, 1989, Vol. 8A,259-266

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15. Cendes, Z.J. and Silvester, P. Numerical solution of dielectric loaded waveguides: I - finite elementanalysis, IEEE Tran.action.. on Microwave Theory and Technique" December 1970 MTT-18(12)

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18. Webb, J.P. Efficient generation of divergence free fields for finite element analysis of 3-D cavities,IEEE Tran,action on Magnetic., Jan 1988, MAG-24(1), 162-165

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20. Barton, M.L. and Cendes, Z.J. New vector finite elements for three-dimensional magnetic fieldcomputation, JoumcJ of Applied Phy,ic., 1987, 61(8), 3919-3921

21. Ericsson, T. and Ruhe, A. The spectral transformation Lanczos method for the numerical solutionof large sparse generalized symmetric eigenvalue problems, Mathematic, of Computation, Oct. 1980,

35(152),1251-126822. Wang, J.S. and Ida, N. A numerical model for characterization of specimens in a microwave cavity,

Review of Progre., in Quantitative Nonde,troctwe Evaluation, La Jolla, July 15-20, 1990, to appear

23. Pissanetzky, S. KUBIK: an automatic three-dimensional finite element mesh generator, InternationcJJournal for NumeriCtJl Method.. in Engineering, 1981, 17, 255-269