12 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

Element-Free Galerkin Method for Static andQuasi-Static Electromagnetic Field Computation

Liang Xuan, Member, IEEE, Zhiwei Zeng, Member, IEEE, Balasubramaniam Shanker, Senior Member, IEEE, andLalita Udpa, Senior Member, IEEE

Abstract—Conventional finite-element methods (FEMs) rely onan underlying tessellation to describe the geometry and the basisfunctions that are used to represent the unknown quantity. Al-ternatively, however, it is possible to represent both the geometryand basis as a set of points. This alternative scheme has been usedextensively in solid mechanics to compute stress and strain distri-butions. This paper presents an adaptation of the scheme to theanalysis of electromagnetic problems in both the static and quasi-static regimes. It validates the proposed model against both analyt-ical solutions and benchmarked FEMs. The paper demonstratesthe efficacy of the proposed method by applying it to a range ofproblems.

Index Terms—Electromagnetic analysis, finite-element method,meshless method, nondestructive evaluation.

I. INTRODUCTION

THE finite-element method (FEM) is an establishedanalysis technique that has been successfully applied

in solving various problems in science and engineering. Thismethod is based on the fundamental idea that a continuousfunction over the entire solution domain can be replacedby a piecewise continuous approximation, usually based onpolynomials, over a set of subdomains called finite elements.The interconnecting structure of the elements via nodes iscalled a finite-element (FE) mesh. The need for an FE meshleads to some distinct disadvantages of the FEM. Generationof a mesh for a complex geometry is in itself a difficult andtime-consuming task [1]. For example, in electromagnetic com-putation, problems that involve geometrical deformation suchas inverse shape optimization, or large dynamic geometricalchanges such as propagating cracks, the use of an underlyingmesh creates difficulties in the treatment of discontinuities thatmight not necessarily coincide with the element boundaries.

Manuscript received November 18, 2002; revised October 13, 2003. Thismaterial is based on the work supported by the Federal Aviation Administra-tion under Contract DTFA03-98-D-00008, and performed at Michigan StateUniversity’s Center for Aviation Systems Reliability program through the Air-worthiness Assurance Center Excellence. The work of B. Shanker was sup-ported by the National Science Foundation under CCR: 9988347, and by theDefense Advanced Research Projects Agency VET Program under ContractF49620-01-1-0228.

L. Xuan was with the Department of Electrical and Computer Engineering,Michigan State University, East Lansing, MI 48824 USA. He is now with theDepartment of Electrical and Computer Engineering, University of Kentucky,Lexington, KY 40506 USA (e-mail: lxuan@uky.edu).

Z. Zeng, B. Shanker, and L. Udpa are with the Department of Electricaland Computer Engineering, Michigan State University, East Lansing, MI48824 USA (e-mail: zengzhiw@egr.msu.edu; bshanker@egr.msu.edu; udpal@egr.msu.edu).

Digital Object Identifier 10.1109/TMAG.2003.821131

Usually remeshing is required to handle the discontinuitiesat every step of the shape reconstruction [2]–[4]. For suchproblems, developing a numerical method that does not rely onan underlying mesh is desirable.

Recently, a new technique known as the “meshless method”has been developed where the unknown function is approxi-mated entirely in terms of “local” functions defined at a setof nodes. Elements, or usual relationships between nodes andelements, are not necessary to construct a discrete set of equa-tions. With the implementation of moving least squares (MLS)approximation in Galerkin formulation, meshless methods arenow widely applied to problems in fracture mechanics andstatic electromagnetics [5]–[9]. These methods are promising,although more expensive than the FEM in terms of constructionof the mass matrices.

This disadvantage notwithstanding, the principal attractivefeature of meshless methods is the possibility of: 1) workingwith a cloud of points that describes the underlying structure,instead of relying on a tessellation and 2) using this method forinverse problems. The principal contributions of this paper aretwofold: a) the development and validation of the element-freeGalerkin (EFG) method as applied to static and quasi-staticelectromagnetics problems via comparison against analyticalmodels and the standard FEM and b) formulation and applica-tion of this technique to static and quasi-static electromagneticproblems in both two and three dimensions.

The organization of this paper is as follows. Section II definesthe problems that will be analyzed. Section III outlines the EFGmethod in general, while Section IV presents details of numer-ical implementation. Section V presents a series of results thatboth validate this model as well as demonstrate its applicability.Finally, Section VI summarizes the contribution of this paper.

II. MATHEMATICAL PRELIMINARIES

The interaction of static and low-frequency electromagneticfields with materials is the basis of nondestructive evaluation(NDE) of conducting samples. In order to model this interaction,the fields are typically expressed in terms of potentials and thegoverning equations for either the Poisson or the diffusion equa-tion are solved together with appropriate boundary and initialconditions. In what follows, we analyze the application of theEFG method to the numerical solution of both these equations.

Consider a domain of interest denoted by that is boundedby , where is the Dirichlet boundary and

0018-9464/04$20.00 © 2004 IEEE

XUAN et al.: EFG METHOD FOR STATIC AND QUASI-STATIC ELECTROMAGNETIC FIELD COMPUTATION 13

is the Neumann boundary. The outward pointing normal to theboundary is denoted by .

A. Poisson Equation

The Poisson equation with inhomogeneous boundary condi-tions in Cartesian coordinates can be expressed as

for (1)

for (2)

for (3)

where is the electric scalar potential, is electriccharge density, and is the material permittivity. Note thatrepresents the position vector in either two or three dimensions.

B. Diffusion Equation

The diffusion equation for eddy current problem in Cartesiancoordinates can be written as [10]

(4)

(5)

where is the permeability, is the conductivity, is thevector potential, and is the current density.

III. EFG METHOD

In the EFG method, a set of nodes is used to construct thediscrete equations. However, to implement the Galerkin proce-dure, it is necessary to compute the integrals over the solutiondomain; this is done by defining the support of the basis func-tions using either a set of quadrature points or a backgroundmesh. The manner in which this is numerically implemented iselucidated in Section IV.

Consider a function that is to be approximated. The EFGmethod utilizes a MLS approach, which relies on three compo-nents: 1) a weight function; 2) a polynomial basis; and 3) a set ofposition-dependent coefficients. The weight function is nonzeroonly over a small subdomain around a particular node, which istermed the domain of influence of that node.

A. MLS Approximation

In MLS approximation, the interpolant is given by[5]

(6)

where is the approximation point, is a particular node,are monomial basis functions, is the number of terms in thebasis function, and are coefficients that depend on the po-sition . The coefficients are determined by minimizing

the difference between the local approximation and the nodalparameters , i.e., by minimizing the following quadratic form:

(7)

Here, is a weight function with compact support, andis the number of nodes in the neighborhood of where the

weight function does not vanish. In matrix notation, (7) can berewritten as

(8)

where are the unknowns

(9)

(10)

The minimization of with respect to leads to

(11)

where

(12)

(13)

It follows that

(14)

Substituting (14) into (6), and letting , the MLS approxi-mation can be written as

(15)

where the shape functions are given by

(16)

Note that the shape function does not satisfy the Kronecker deltacriterion: ; therefore, , which makesit difficult to impose essential boundary conditions. Techniquesthat can be applied to address this issue include using either aLagrange multiplier or coupling with standard finite elementsat the boundary. Note, the construction of the shape function isidentical in both two and three dimensions.

B. Weight Function

In two dimensions, the solution domain is covered by the do-mains of influence of all nodes; while the choice of shape of thisinfluence domain is arbitrary, a circular or rectangular domain istypically used. In our implementation, a rectangular domain is

14 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

used and its corresponding weight function is derived as a tensorproduct weight, which at any given point can be expressed as [5]

(17)

where , for , are either Gaussians, exponentials,or cubic splines. These functions take the form

for

forGaussian (18)

for

forExponential (19)

for

for

forCubic spline (20)

where yields best convergence for the exponentialweight, and

(21)

(22)

Here, is a scaling factor, and ( , ) is the differencebetween node and its nearest neighbor. Note, is chosensuch that matrix is nonsingular. In three dimensions, theweight is a natural extension of those presented earlier for twodimensions.

C. Discontinuities Approximation

NDE problems typically involve multiply connected regionswith each region characterized by different conductivities andpermeabilities. This results in discontinuities of the normal com-ponent of current density when passing from one material toanother, which in turn implies that the derivatives of the shapefunction or the shape function itself should be discontinuous atthe interface. Since continuity of shape functions is inheritedfrom continuity of the weight function, it is necessary to intro-duce discontinuity into the weight function. This is realized byusing the visibility criterion [1].

As shown in Fig. 1, if there is no material discontinuity, thedomain of influence is the total area of the square. However, inthe presence of a discontinuity, the domain of influence of node

shrinks to the area covered by the dashed horizontal line, andthe weight function vanishes outside of that area. This proceduredirectly results in the discontinuity of weight function, whichin turn introduces the discontinuity of shape function and itsderivatives.

Fig. 1. Domain of influence of node adjacent to material discontinuity.

IV. NUMERICAL IMPLEMENTATION

A. Static Problem

The weak form solution to the Poisson equations (1)–(3), ,is computed by minimizing the functional [12]

(23)where vector space that is spanned by the shapefunctions and satisfies the Dirichlet boundary conditions,and is the Lagrange multiplier used to impose the boundaryconditions.

Let the EFG shape functions be the basis in , then, where is the total number of nodes in

domain . The functional is minimized with respect tovalues of function at all nodes in the domain , that is, withrespect to vector . The minimization offunctional results in the following equations:

(24)

which can be expressed as

for (25)

Combining the above set of equations with Dirichlet boundaryconditions

for (26)

yields a set of equations that can be written in matrix form. Inthe above equations, is a Lagrange multiplier, is the shapefunction associated with the Lagrange multiplier, and isthe number of nodes on where a Dirichlet boundary condi-tion is imposed. As mentioned before, a background mesh isrequired for integration calculations that appear in (25) and (26)

XUAN et al.: EFG METHOD FOR STATIC AND QUASI-STATIC ELECTROMAGNETIC FIELD COMPUTATION 15

since evaluation of or corresponds to computation of theJacobian of a background integration cell.

B. Quasi-Static Problem

In this section, Galerkin formulation and Lagrange multipliertechniques are applied to obtain the numerical solution of theeddy-current equations (4) and (5). The first task is to expresspotentials as a linear combination of shape functions, such as

, and , where ,or . Next, substituting the above expressions into (4)

and (5), performing the inner product on each of the resultingequations with a test function (shape function will be used), andapplying the Lagrange multiplier method

(27)

(28)

(29)

(30)

where is one of the three components of the magnetic vectorpotential and is the electric scalar potential, , are La-grange multipliers, is the shape function associated with theLagrange multipliers, and is the number of nodes wherethe Dirichlet boundary condition is imposed. In matrix notation,(27)–(30) may succinctly be written as

(31)

As in the standard FEMs, the matrix is sparse, and (31)is solved using a standard nonstationary iterative solver suchas transpose-free quasi-minimal residual (TFQMR) [11]. Thechoice of TFQMR is in some sense arbitrary. Although anyiterative solver can be used, TFQMR as opposed to biconjugategradient (BCG) and QMR, has excellent convergence proper-ties.

V. APPLICATIONS AND RESULTS

In this section, we first present the validation of the EFGmethod and compare solutions using this method to those ob-

Fig. 2. Surface plot of numerical solution to (32).

Fig. 3. Error estimation in L (dashed) and H (solid) norm for the linearEFG, as a function of mesh density.

tained using traditional FEM for two-dimensional (2-D) prob-lems. This technique is then applied to analyze 2-D quasi-static(eddy-current) NDE problems. Finally, both static and quasi-static problems are studied in three dimensions.

A. Convergence Study and Comparison With TraditionalFEMs

1) Effect of Mesh Density: Consider the Poisson equation

for (32)

with boundary condition

(33)

Uniform meshes of 9 9, 17 17, 33 33, and 65 65nodes are used with 4 4 Gaussian quadrature in each cell.The EFG method with linear bases is used in the study, andthe results are compared with those obtained by linear FEM. Atypical surface plot of a numerical solution is shown in Fig. 2.The error estimations in and norm are shown in Figs. 3and 4. The error functions and convergence rate are computedas

(34)

16 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

Fig. 4. Error estimation in L (dashed) and H (solid) norm for differentweight functions, as a function of mesh density.

(35)

(36)

where and are the derivatives of with respect to and, respectively, and is the space between nodes.It is observed that: 1) the convergence rate of EFG depends

on the scaling factor defined in (22) and 2) the convergence rateof the EFG method is higher than that of linear FEM in both

and norm, which varies from 2.1 to 3.1 in norm,and varies from 1.0 to 2.0 in norm. We should mention thatthe EFG method is inherently a high-order scheme that requiresmore computational overhead than the linear FEM that is usedfor comparison.

2) Effect of Weight Function: As mentioned before, choiceof weight function is crucial to the convergence of the numericalsolution. Cubic splines, exponentials, and Gaussians are someof the commonly used weight functions. It is found that cubicspline has the best convergence property, while the exponen-tial and Gaussian are better when the scaling factor is small asshown in Fig. 4. This is largely due to the fact that the latter twofunctions reduce the domain of influence, which in turn reducesthe number of nonzero entries in the stiffness matrix. This re-duction saves both computational time and memory.

B. Model Validation—2-D

1) Circular Disk: We consider a 2-D eddy-current problem.The governing equation in formulation is derived fromMaxwell equations [10]

(37)

Fig. 5. Nodes, quadrature points, and eddy current in the solution domain.

Fig. 6. Error estimation in L andH norm for eddy currents.

in a unit disk , with Dirichlet boundary conditions

(38)

on a unit circle , where . The analyticalsolution for this problem is

(39)

where is the modified Bessel function of the first kind. Thenumerical solution is shown in Fig. 5, where , , and

. The domain was uniformly discretized using a 6 6 gridof nodes, 4 4 cells for integration, and 4 4 Gaussian quadra-ture in each quadrature cell. As is expected, the eddy currentsflow in circular paths (see Fig. 5). The convergence of the EFGmethod is analyzed by studying increasingly denser discretiza-tion; more specifically, 9 9, 17 17, 33 33, and 65 65nodal points are used in this numerical experiment. The EFGmethod with linear bases was used in this study, and the resultsobtained were compared against those using linear FEM for the

XUAN et al.: EFG METHOD FOR STATIC AND QUASI-STATIC ELECTROMAGNETIC FIELD COMPUTATION 17

Fig. 7. Eddy current in presence of a tight crack by EFG.

Fig. 8. EFG discretization of solution region with a tight crack.

same discretization. The error estimations in and normare shown in Fig. 6, as a function of .

The convergence rate [defined in (36)] of the EFG methodis 2.39 in the norm, which is higher than that of the linearFEM, whose convergence rate is 2.0. The convergence rate ofthe EFG method is 1.35 in norm, which is also higher thanthat of linear FEM.

2) Plate With Tight Crack: Assume an infinite-size con-ducting plate with a tiny crack along the direction (width ,length cm) placed under a time-varying harmonic currentsource in the direction. The induced current will be along the

direction in the absence of crack, and will be perturbed inthe presence of a crack. This geometry was modeled using theEFG method, and results showing the induced current distribu-tion are presented in Fig. 7. The induced current flows aroundthe crack due to the electrical discontinuity confirming the va-lidity of EFG methods to model tight cracks. As shown in Figs. 8and 9, a tight crack is modeled without introducing any addi-tional nodes (total 144 nodes) in the region in contrast to con-ventional FEM, which requires a larger number of nodes (total180 nodes) and elements since at least two layers of elementsare needed to model the crack. Fig. 10 shows the difference inthe eddy-current distributions, indicating that the two solutionsare very close to each other.

3) Conducting Plate in Time-Varying Field: The thirdgeometry considered for the purpose of model validation

Fig. 9. FEM mesh of solution region with a tight crack.

Fig. 10. Difference of eddy current near the crack between EFG and FEM.

is a conducting plate, which is immersed in a time-varyingharmonic magnetic field as shown in Fig. 11. Using ,

, and , the induced eddy currents are calculatedwith both EFG and FE methods. The induced current calculatedby EFG and the difference between EFG and FE methodsare shown also in Fig. 11. Both methods correctly predictthe current continuity conditions and the eddy current flowaround the sharp corners in the geometry. Both models use adiscretization of 23 23 nodes uniformly distributed in region

cm cm. The relative total energydifference between them is 6.82%, which mainly occursat the corners

(40)

C. Model Validation—Three-Dimensional (3-D) Static Field

Consider the Poisson equation

for (41)

18 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

Fig. 11. (a) Numerical induced currents inside an “H” plate calculated by EFG.(b) Difference between FEM and EFG solutions.

with homogeneous boundary condition

(42)

The 3-D numerical model consists of integrationcells with 4 4 Gaussian quadrature in each cell. Linear basisis applied to construct the shape function. The error estimationsin and norm are shown in Fig. 12.

Again, the convergence rate of EFG depends on the scalingfactor. The convergence rate varies from 2.16 to 3.10 in normfor EFG, in contrast to 1.89 for linear FEM. The convergencerate in norm varies from 1.10 to 2.16 for EFG, also higherthan that of linear FEM. The results for computation cost aresummarized in Fig. 13. For a fixed size of the mesh, FEM ismuch less expensive than EFG; however, for a given precisionEFG uses less computation time than FEM. In this paper, theEFG has a high setup cost to assemble the matrix, which is abottleneck of meshless methods.

D. Experimental Geometry—3-D Quasi-Static Field

To illustrate the application of the EFG method to 3-D eddycurrent problems, we used a test geometry consisting of a single-layer aluminum plate 3 mm thick and of infinite width, withdefects, placed under a time-varying harmonic current sheet.

Fig. 12. Error estimation in L andH norm for the linear 3-D EFG.

Fig. 13. Product of CPU time and numerical error versus number of nodes for3-D Poisson problems.

Fig. 14. Geometry of conducting plate under current foil.

1) An infinite sinusoidal ac current foil 1 mm thick and12 mm wide is placed above the conducting plate. Boththe foil and plate are of infinite extent along the currentdirection, as shown in Fig. 14. Current density is 1 A/m .The solution region is mm mm

mm, and uniformly distributed nodeswere used for discretization. Figs. 15 and 16 show thecomponent of the magnetic vector potential at various ex-citation frequencies. It is observed that numerical resultsagree very well with the analytical solutions.

2) Next, we introduce a defect, a square notch of 5 mm widthand 1.5 mm depth in the conducting plate, and set the ex-citation frequency at 1 kHz. The solution region is chosenas mm mm mm with 11

XUAN et al.: EFG METHOD FOR STATIC AND QUASI-STATIC ELECTROMAGNETIC FIELD COMPUTATION 19

Fig. 15. Magnetic vector potential at frequency of 3 KHz.

Fig. 16. Magnetic vector potential at frequency of 10 kHz.

nodes in the and directions, and eight nodes in thedirection. This produces a discretization with a total of

700 integration cells and 968 nodes (3872 unknowns).

Fig. 17 shows the induced eddy currents in the conductingplate. It is observed that the induced currents flow around thedefect without passing across, which agrees with the underlyingphysics and experiments. Fig. 18 shows the induced currentsnearby a tight crack of 5 mm length and 3 mm depth (cuttingthrough the sample plate). The EFG method does not requirefiner meshing around the crack, and predicts the characteristicsof the induction phenomenon accurately.

VI. CONCLUSION

This paper presented a detailed description of the formula-tion, imposition of essential and interface boundary conditions,and implementation of the EFG method. A number of examplesin 2-D and 3-D were studied. Through these examples, we seethat the EFG method can be successfully applied to the elec-tromagnetic field problems in either static or quasi-static fields.Comparison of conventional FE and EFG methods with respectto accuracy, computation time, and data storage is summarized

Fig. 17. Side view of the real part of currents around notch (planey = 0).

Fig. 18. Side view of the real part of currents around tight crack (plane y = 0).

TABLE ICOMPARISON OF PERFORMANCES OF FE AND EFG METHODS

FOR THE 2-D POISSON PROBLEM (32 � 32 NODES)

TABLE IICOMPARISON OF PERFORMANCES OF FE AND EFG METHODS

FOR THE 2-D POISSON PROBLEM (48 � 48 NODES)

in Tables I–IV for different mesh discretization. The major ad-vantage of EFG methods is that they avoid the difficulties of

20 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 1, JANUARY 2004

TABLE IIICOMPARISON OF PERFORMANCES OF FE AND EFG METHODS

FOR THE 2-D POISSON PROBLEM (64 � 64 NODES)

TABLE IVCOMPARISON OF PERFORMANCES OF FE AND EFG METHODS

FOR THE 2-D POISSON PROBLEM (90 � 90 NODES)

large mesh changes in problems involving tight cracks, com-monly encountered in NDE applications.

REFERENCES

[1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Mesh-less methods: an overview and recent developments,” Comput. MethodsAppl. Mech. Eng., vol. 139, pp. 3–47, 1996.

[2] K. Miyata and K. Maki, “Air-gap remeshing technique for rotating ma-chines in 3D finite element modeling,” IEEE Trans. Magn., vol. 36, pp.1492–1495, July 2000.

[3] C. B. Rajanathan and G. Hu, “Electromechanical transient characteris-tics of an induction actuator by finite element analysis,” IEEE Trans.Magn., vol. 29, pp. 2001–2005, Mar. 1992.

[4] B. Bendjima, K. Srairi, and M. Feliachi, “A coupling model foranalysing dynamical behaviors of an electromagnetic forming system,”IEEE Trans. Magn., vol. 33, pp. 1638–1641, Mar. 1997.

[5] J. Dolbow and T. Belytschko, “An introduction to programming themeshless element free Galerkin method,” Arch. Comput. Mech., vol. 5,no. 3, pp. 207–241, 1998.

[6] T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,”Int. J. Numer. Eng., vol. 37, pp. 229–256, 1994.

[7] V. Cingoski, N. Miyamoto, and H. Yamashita, “Element-free Galerkinmethod for electromagnetic field computations,” IEEE Trans. Magn.,vol. 34, pp. 3236–3239, Sept. 1998.

[8] S. A. Viana and R. C. Mesquita, “Moving least square reproducingKernel method for electromagnetic field computation,” IEEE Trans.Magn., vol. 35, pp. 1372–1375, May 1999.

[9] S. L. Ho, S. Yang, J. M. Machado, and H. C. Wong, “Applications of ameshless method in electromagnetics,” IEEE Trans. Magn., vol. 37, pp.3198–3202, Sept. 2001.

[10] O. Biro and K. Preis, “On the use of the magnetic vector potential in thefinite element analysis of three-dimensional eddy currents,” IEEE TransMagn., vol. 25, pp. 3145–3159, July 1989.

[11] R. W. Freund, “Transpose-free quasiminimal residual methods for non-Hermitian linear systems,” in Advances in Computer Methods for PartialDifferential Equations, IMACS Series in Computational and AppliedMathematics, 1992, pp. 258–264.

[12] J. Jin, The Finite Element Method in Electromagnetics. New York:Wiley, 2002.

Liang Xuan (M’04) was born in China in 1972. He received the B.S. and M.S.degrees in mathematics from the University of Science and Technology ofChina, Hefei, in 1994 and 1997, respectively, and the Ph.D. degree in electricalengineering from Iowa State University, Ames, in 2002.

Since January 2003, he has been a Postdoctoral Scholar in the Electricaland Computer Engineering Department, University of Kentucky, Lexington. Hisresearch interests include electromagnetic computation and microwave circuitmodeling.

Zhiwei Zeng (M’04) was born in China in 1974. He received the B.S. degree inelectrical engineering from Nanjing University of Aeronautics and Astronautics,Nanjing, China, in 1996 and the Ph.D. degree in electrical engineering fromIowa State University, Ames, in 2003.

Currently, he is a Postdoctoral Research Associate in the Department of Elec-trical and Computer Engineering, Michigan State University, East Lansing. Hisresearch interests include aspects of computational electromagnetics, nonde-structive evaluation, and applied statistics.

Balasubramaniam Shanker (M’96–SM’01) received the B.Tech degree fromthe Indian Institute of Technology, Madras, India, in 1989 and the M.S. andPh.D. degrees from The Pennsylvania State University, State College, in 1992and 1993, respectively.

From 1993 to 1996, he was a Research Associate in the Department of Bio-chemistry and Biophysics, Iowa State University, Ames, where he worked onthe Molecular Theory of Optical Activity. From 1996 to 1999, he was with theCenter for Computational Electromagnetics at the University of Illinois at Ur-bana-Champaign, Urbana, as a Visiting Assistant Professor, and from 1999 to2002, he was with Iowa State University as an Assistant Professor in the De-partment of Electrical and Computer Engineering. Currently, he is an AssociateProfessor in the Electrical and Computer Engineering Department, MichiganState University, East Lansing. He has authored or coauthored over 130 articlesin archival journals and conference proceedings. His research interests includeall aspects of computational electromagnetics, and electromagnetic wave prop-agation in complex media.

Dr. Shanker is a full member of the USNC-URSI Commission B.

Lalita Udpa (M’85–SM’90) received the Ph.D. degree in electrical engineeringfrom Colorado State University, Fort Collins, in 1986.

She joined the Department of Electrical and Computer Engineering, IowaState University, Ames, as an Assistant Professor where she served from 1990to 2001. Since January 1, 2002, she has been with Michigan State University,East Lansing, where she is currently a Professor in the Department of Electricaland Computer Engineering. She works primarily in the broad areas of nonde-structive evaluation (NDE), signal processing, and data fusion. Her research in-terests include development of computational models for the forward problemin electromagnetic NDE, signal processing, and data fusion algorithms for NDEdata, and development of solution techniques for inverse problems.

Dr. Udpa is an Associate Technical Editor of the American Society of Non-destructive Testing Journals on Materials Evaluation and Research Techniquesin NDE.

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