IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 8, AUGUST 2003
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Loop Star Basis Functions and a Robust Preconditioner for EFIE Scattering Problems Jin-Fa Lee, Senior Member, IEEE, Robert Lee, Senior Member, IEEE, and Robert J. Burkholder, Senior Member, IEEE
Abstract—An electric field integral equation (EFIE) formulation using the loop-star basis functions has been developed for modeling plane wave scattering from perfect conducting objects. A stability analysis at the dc limit shows that the use of the Rao–Wilton–Glisson (RWG) basis functions will result in singular matrix operator. However, the use of the loop-star basis functions results in a well-conditioned matrix. Moreover, a preconditioner constructed from a two-step process, based on near interactions and an incomplete factorization with a heuristic drop strategy, has been proposed in conjunction with the Conjugate Gradient method to solve the resulting matrix equation. The approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the EFIE method. The computational complexity of the proposed approach is shown to be ( ) .
Fig. 1. Triangle pair and the geometrical parameters for RWG basis function for edge fi; jg.
Index Terms—Electromagnetic scattering, integral equations, iterative methods, moment methods.
I. INTRODUCTION
M
ETHOD of moment (MoM), or integral equation method, has been a very popular choice for solving three dimensional (3-D) scattering problems. However, as noted by many authors in the past, in its simple form, it suffers a few major problems: the storage and computational complexity are too prohibitive to do large problems; the electric field integral equation (EFIE) formulation, which is the first kind integral equation, is ill-posed and numerical solutions do not necessarily get better when finer discretizations are utilized; the use of the Rao–Wilton–Glisson (RWG) [1] basis functions exhibit low frequency instability, namely, the matrix equation becomes nearly singular at low frequencies; and the occurrences of the internal resonances when the scatter is a closed surface object. Some of these problems have already been answered with different degrees of success. Most advanced among them is the development of fast methods such as fast multipole method (FMM) and singular value decomposition (SVD) methods [2]–[4]. Both methods significantly cut down the storage requirement and computational complexity for matrix vector multiplication in the MOM computation. The low frequency instability can be resolved by employing the loop-star basis functions as pointed out by previous authors [5]–[7]. The Manuscript received October 23, 2001; revised March 4, 2002. This work was supported by the Dayton Area Graduate Studies Institute (DAGSI) under Grant SN-AFIT-00-06. The authors are with the ElectroScience Laboratory, Electrical Engineering Department, The Ohio State University, Columbus, OH 43212 USA (e-mail:
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/TAP.2003.814736
internal resonance problems can be most effectively corrected by using the combine field integral equation (CFIE) [8] formulation. However, for open surface objects, the only approach is still the EFIE. It is well known that the matrix equations resulted from EFIE are very bad conditioned, and when iterative matrix solutions such as conjugate gradient (CG) methods are used to solve the matrix equations, the convergences are in general less than satisfactory. We propose herein a simple two-step construction of a preconditioner, which results in superior convergence in CG methods for EFIE. The rest of the paper is organized as follows: Section II gives a detail description of the loop-star basis functions; Section III analyses the dc limit of the RWG and loop-star basis functions; Some useful implementation details are given in Section IV; Section V then details the construction of the proposed preconditioner; Various numerical examples are shown in Section VI to validate and demonstrate the performance of the current approach; and finally, in Section VII, we provide a brief summary and look ahead what need to be done next. II. LOOP STAR BASIS FUNCTIONS It is interesting to note that the commonly used RWG basis functions in the MoM community can be directly related to the edge elements in the FEM community [13]. Shown in Fig. 1 is a . usual description of the RWG basis function for an edge The RWG vector basis function associated with the edge is commonly written as
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(1) elsewhere.
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Fig. 2.
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Edge elements for triangular FEM. Fig. 3.
Where , tively, and have
are the area of triangles is the length of edge
and , respec. Furthermore, we
(2) Note, that in general situations, an edge is shared by at most two triangles. Special care needs to be taken (additional unknowns need to be assigned) when an edge is shared by more than two triangles. Similarly, we show in Fig. 2 the usual description of edge elements in FEM community. The edge element basis functions is then defined by associated with edge
Construction of Loop-Star basis functions.
shall start our discussion by first looking at the construction of the star-loop basis functions. being the usual RWG basis function for edge , the Let current can then be expressed by (5) The construction of the Loop-Star basis functions starts by using the Helmholtz decomposition theorem, namely
(3) where is the bary-centric (or simplex) coordinates of vertex . It can be shown that the RWG basis function, , and the , are related by edge element basis function, (4) , are the unit normals to triangles , respecwhere tively. Recognizing this fact has two main benefits. 1) The higher order basis functions for MoM computations are readily available through the work done in FEM community. Particularly, the hierarchical vector basis functions [14]. 2) The recently proposed loop-star basis functions, which will be described in more details later, are very similar to the tree-cotree decomposition in the vector finite element methods [15]. The reason for doing the tree-cotree splitting is to address the low frequency instability encountered in FEM computation. It is also true here; we shall need the loop-star splitting for obtaining reliable MoM solutions in the low frequency limit. This low frequency instability also results in deteriorating condition number (hence slow convergence in iterative matrix solvers) when smaller elements are used in MoM computations.
(6) The basis functions used to expand the solenoidal current, , is the loop basis function, whereas the basis functions for the irrotational component, , is the star basis functions. We note continuity does also that, for a general discretization where not exist, we can only enforce the Helmholtz splitting approximately. Therefore, in the loop-star splitting, we choose
(7) Namely, the loop basis functions are solenoidal vector functions, whereas, the star basis functions are not necessarily irrotational. Moreover, if the lowest order basis functions are used, then the , should be equal to the original total number of unknowns, formulation using RWG basis functions. B. Loop Basis Functions For structures without torus, referring to Fig. 3, we have one loop basis function associated with each interior vertex. Explicitly, they are
A. Inexact Helmholtz Splitting Our new formulation is tied very closely with the loop-star basis functions [6] that were proposed recently. Therefore, we
(8)
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where is the usual RWG basis function for edge divided by the edge length. From (2) it is easy to see that . A compact description of the loop basis functions can be written as
(9) Therefore, the loop basis functions are associated with vertices; one free vertex will result in one loop basis function. Fig. 4.
Sample triangular mesh.
C. Star Basis Functions On the other hand, the star basis functions are associated with triangles, one star basis function for one triangle. For example, again referring to Fig. 3, for triangles 1 and 5, the star basis functions are
we have and . Therefore, the number of loop basis functions for Fig. 4 is 2. 2) Number of Star Basis Functions: Taking the divergence , results in of (7), using the fact,
(10)
(14)
and do resemble the Although, irrotational type of vector fields, particularly along the loops defined by the loop basis functions.
When the unknown surface current, , is expressed in terms of the usual RWG basis functions, (14) is rewritten as
D. Degree of Freedom for Loop-Star Formulation
(15)
Let , , denote the number of vertices, edges, and faces (in our application here, triangles) of a 3-D surface (withour be the number of separated boundary contorus), and also tours, then the Euler formula states
, will be piecewise constant It is then obvious from (2), over the triangulation. Together with the fact that the total charge equals 0, viz.
(11) (16) , , , and For instance, in Fig. 4, we have . 1) Number of Loop Basis Functions: Mathematically, the loop basis function can be defined through a scalar magnetic potential function, , by
we conclude that the number of independent charge density, , which will also be the number of independent star basis functions is
(12)
(17)
will be the For the lowest order loop-star basis functions, usual Lagrange interpolation polynomial associates with vertex . Note, that the essential boundary condition dictates that no current flow through the boundary contours. This can be accomplished by holding the potentials of the vertices on the same contour constant. Namely, the boundary contours also correspond to equal potential lines for . However, for separated contours, there could exist potential difference and hence allow current loop to form between them. Vecchi has addressed this issue in detail in [6]. If there is more than one contour in the problem domain, we can simply choose any of them and assign it the reference potential value, or just simply 0. Lastly, for closed sur, we still need to arbifaces and without holes, namely trarily assign a reference potential to a vertex. Consequently, the number of loop basis functions is
: Since the loop and the star basis func3) tions are the linear combinations of the RWG basis functions, they should span exactly the same function spaces. Therefore, it is necessary that the total degree of freedom in the loop-star formulation is the same as that of the usual RWG approach. Assuming regular topology for the triangulation, namely no edges are shared by more than two triangles, the degrees of freedom in RWG approach is the same as the number of internal edges, where is the number of edges on boundary contours. Summing up (13) and (17), we have (18) is the number of vertices on the boundary conwhere tours. With the help of (19)
(13) we have and are the number of interior vertices and sepwhere arated contours, respectively. Again, taking Fig. 4 for example,
(20)
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III. DC LIMIT OF THE EFIE USING RWG AND LOOP-STAR BASIS FUNCTIONS Application of the Galerkin (weighted residual) method to the EFIE results in
(24) (21) where where
is the problem domain or the scatterer. In the dc limit , (21) reduces to
Again, examining (23) and (24) at the low frequency limits reveals that (25)
(22) It is then evident that any solenoidal vector field, divergence free, will be a solution to (22). We turn our attention now to the situation where the unknown surface current is expressed in terms of the RWG basis functions. As described in the previous section, associated with each internal node, there would be a loop current, which is divergence free and a linear combination of RWG basis functions. That is to say using the RWG basis functions, there will be independent loop currents for a given mesh that are the solutions to (22). Consequently, the EFIE with RWG basis functions is ill posed at the dc limit and exhibits the low frequency instability, namely the condition number of the matrix as . Practically even for “normal” freapproaches quencies, this has a significant impact when iterative matrix solution techniques, such as CG methods, are employed to solve the matrix equations. Specifically, when dealing with problems whose geometric features range several orders of magnitude or h-type adaptive mesh refinements where very small sized elements (compared to wavelength) are used around current singularities for better accuracy. In these situations, in which small sized elements are present in parts of the computational domain, the convergence in the iterative process could be painfully slow or even fail to converge. The rationale behind decomposing the RWG basis functions into loop and star vector functions is exactly to eliminate this low frequency instability. By employing the property (for every loop basis function), the Galerkin testing of the EFIE can be splitted into two parts
Since the span of the star basis functions does not include any solenoidal vector field, we conclude from (25) (26) Furthermore, at low frequencies, substituting (26) into (23) results in
(27) For any small but nonzero values, the matrix equation in (27) can be transformed, simply through diagonal scaling, into
(28) Subsequently, a very well conditioned matrix equation to solve. IV. PRACTICAL IMPLEMENTATION CONSIDERATIONS In this section, we detail some of the implementation issues. Some of the tricks that we employed significantly speed up the matrix assembly process. To begin, we first introduce the transforming matrix, , which relates the new loop star basis functions to the conventional RWG basis functions. Note, our assembly process resembles the one commonly used in the FEM community. That is, our matrix is written as a summation of local element matrices. Namely (29) where
(23) and
(30)
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Fig. 6.
Surface triangulation of a conducting cone sphere.
The numerical integration is needed to evaluate the 3 by 3 local matrix. For example, we have
Fig. 5. Local unknown numbering for the loop-star basis functions.
and , are the pth and qth basis functions, respectively. Shown in Fig. 5 is a triangular element with local loop star basis functions numbering. As indicated in the figure, there are at most seven nonzero basis functions existing within the element; three loop and four star basis functions. In the local simplex coordinates, they can be written as
(31) and . These seven where basis functions can be simply expressed in terms of three basis through a mapping functions, , , and
(32)
Also, note that , and are the same as the common RWG basis functions within a scaling factor that is the length of the corresponding edge. Applying some geometric identities reveal that
(33) Then the element matrix following:
can be obtained through the (34)
where
(35)
(36)
, ; ), is the numerical inwhere ( , , whereas tegration rule for the outer/observation triangle , , ; ), is the integration rule ( . Furthermore, in (36), the used for the inner/source triangle denotes notation (37)
V. INCOMPLETE FACTORIZATION WITH HEURISTIC DROP STRATEGY AS A PRECONDITIONER In this section, we consider the efficient solution of dense by preconditioned iterative methods, linear system particularly GMRES method. We are concerned with those systems arising from the discretization of EFIE, where preconditioning is essential for convergence of GMRES. Most boundary integral equations possess singularities. In the special case, when these singularities are due to geometrical nonsmoothness (e.g., corners), the operator can be noncompact but the singularities occur at fixed points. Preconditioners based on separating these fixed points have been studied and iterative methods for the preconditioned systems have shown to be effective, see [10]. An insightful discussion of three types of preconditioners, the operator splitting preconditioner (OSP), the least squares approximate inverse preconditioner (LSAI), and the diagonal block approximate inverse preconditioner (DBAI), for dense matrices arising from the application of BIEs is provided in [11]. Interested readers are recommended to consult it for detail analyses of these preconditioning techniques. The preconditioner that is used in the current work is based upon a two-step process. In the first step, we extract from a sparse version, , which the full impedance matrix includes the near range interactions as well as a heuristic bias toward geometrical singularities. In this regard, our approach belongs to the mesh neighbor (MN) preconditioner in [12] is and is a special case of DBAI. Once the sparse matrix will be formed through obtained, the final preconditioner an incomplete factorization with a heuristic dropping strategy [17].
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Fig. 7.
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Backscatter pattern for the cone sphere example, polarization.
where denotes the set of triangles that have “strong” cou. Let us first explain a few plings with respect to triangle, . is notations that we used in (38) in defining and , from center to center; is the distance between , in this project, we simple subthe characteristic length for ’s circumcircle for ; is the avstitute the radius of erage characteristic length of the mesh; and is the number of in the right-hand unknowns. The factor side of the inequality allows us to include more terms for geometric singularities since much smaller triangles are usually around these points. Moreover, we would like to include more layers of couplings when problem size becomes large; however, to grow in doing so we also do not want the sparse matrix increases. Thus, we have chosen to be drastically as a compromise between the two objectives that we strive to accomplish. The effectiveness of the chosen parameters will be demonstrated through numerical results in the next section. B. Incomplete Factorization With A Heuristic Dropping Strategy
Fig. 8. Perfectly conducting cone plate.
A. Sparse Matrix As mentioned earlier, the approach proposed here is a two-step process for constructing the preconditioner. As the first step, we borrow from the concepts of DBAI, or specifically, mesh neighbor (MN) preconditioner. Namely, we shall construct from the triangular mesh a sparse version of the , which includes only “strong” couplings impedance matrix of triangle pairs, viz.
(38)
is formed, we can perform a Once the sparse matrix complete factorization on it and use the factorization to be the preconditioner. However, the cost of constructing the preconditioner can be further reduced by employing an incomplete factorization with a heuristic dropping strategy [17]. Furthermore, in order to reduce the number of fill-ins significantly during the factorization process, a robust matrix-reordering algorithm must be performed. We recommend the METIS package from University of Minnesota [16]. The basic strategy in constructing the incomplete LU type of preconditioner is to let the factorization algorithm proceed as in the case when complete factors are computed, but simply drop certain nonzero entries in the factors as the algorithm progresses. Because nonzero entries are dropped, the algorithm results in less floating-point operations and subsequently requires less CPU time than the complete factorization.
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Fig. 9. Backscatter pattern of the open cone example at 3 GHz, polarization.
Since for the Galerkin EFIE formulation, both the original full and the sparse version matrices are symmetric, our incomplete factorization generates a preconditioner (39) and is a lower triangular matrix. The where problem then is to decide which nonzeros to keep and which set to drop in order to strike a balance between the cost of constructing the preconditioner, the cost of using the preconditioner, and the speed-up in convergence rate. Our implementation based on the incomplete factorization by value process, where the sparsity of the final preconditioner is dynamically determined, see [17]. In particular, we have chosen at r-th it(we use ), the entry eration, if will simply be deleted. The incomplete factorization by value procedure has been well documented in literature; therefore, we shall not describe its implementation details except by directing the interested readers to [17]. VI. NUMERICAL RESULTS A. Cone Sphere Example The first example in using the loop star MoM procedure is a conducting cone sphere, shown in Fig. 6. This example is taken from [20] to validate the current approach. The backscatter patpolarization is plotted in. Fig. 7. tern of this example for the Moreover, in Fig. 7, we also compare our numerical results to data obtained from [20] which was computed using a 2-D body-of-revolution (BOR) MoM code. B. Open Cone Plate Example The previous example is a closed surface scatterer. The second example that we tried here is an open cone plate as
shown in Fig. 8. The cone has a height of 20 cm and a base whose diameter is also 20 cm. The backscatter pattern for the polarization is shown in Fig. 9 and compared to results obtained from a BOR code. The slight disagreement around is mainly due to the fact that the geometry of the cone may not be sufficiently sampled and approximated by only 10 segments along the circumcircle. C. Performance of the Preconditioned Loop-StarEFIE Formulation Table I summarizes the performances of the current preconditioned GMRES procedure for solving the matrix equations arising from the application of the loop-star EFIE formulation for PEC scattering problems. In Table I, N is the number of unknowns; sparsity denotes the sparsity of the final preconditioner matrix; CPU(MoM) is the CPU time used to assemble the matrix equation; CPU(PC) indicates the total CPU times for building the two-step process preconditioners; CPU(CG) is the CPU times in applying the GMRES(10) to solve the precon; and ditioned matrix systems with a relative residual of finally NI denotes the number of iterations for GMRES(10) to converge. Moreover, next to every number of unknowns, there are labels C for cone plate example, CS for conesphere example, and AP for an air plane example as in [21]. The current proposed preconditioner works very well together with the GMRES procedure as evidenced through the table. In generating the data in Table I, every triangular mesh is generated with the average discretization 0.2 , of course, near singularities or geometrically small feature, the discretization tends to be relatively small in those regions. Regardless of the matrix sizes and the PEC scatterers, the GMRES(10) can converge in tens of iterations. Furthermore, we note that in Table I the preconditioner grows sparser as the matrix size increases.
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TABLE I PERFORMANCES OF THE PRECONDITIONED LOOP-STAR EFIE FORMULATION FOR PEC SCATTERING PROBLEMS
VII. CONCLUSIONS AND LOOK AHEAD This paper presents a method of moment formulation employing the loop star basis functions. The employment of the loop star basis functions removes the low frequency instability that is inherent in the conventional RWG basis functions. Moreover, this paper also proposes a two-step construction of a robust preconditioner for the EFIE: a sparsification of the impedance matrix based on near field interactions and incomplete factorization using a heuristic drop strategy. It is shown through numerical examples that the proposed preconditioner provides fast and smooth convergence in conjunction with the GMRES(10) algorithm. In summary, the loop-star method of moment together with the novel preconditioner has been demonstrated to be reliable and stable through two examples: a cone sphere (a closed surface object) and a cone plate (an open surface object). The overall performance of the proposed methodology is . To further improve the performance of EFIE formulation, an efficient algorithm needs to be in place to calculate the matrix vector multiplication in the MoM formulation. This has been the main focus of the vast literature existed today in method of moment research. Techniques like FMMs [2], [3] and SVD [4] have been demonstrated to be highly efficient. It is the authors’ intention to adopt these fast methods to further reduce the computational complexity of current approach.
REFERENCES [1] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409–418, May 1982. [2] V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Computat. Phys., vol. 60, no. 2, pp. 187–207, Sep. 1985. [3] W. C. Chew, J.-M. Jin, C.-C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propagat., vol. 45, pp. 533–543, Mar. 1997. [4] S. Kapur and D. E. Long, “IES : a fast integral equation solver for efficient 3-dimensional extraction,” in Proc. 37th Int. Conf. Computer Aided Design, Nov. 1997.
[5] D. R. Wilton and A. W. Glisson, “On improving the electric field integral equation at low frequencies,” in Proc. URSI Radio Science Meeting Dig., Los Angeles, CA, June 1981, p. 24. [6] G.Giuseppe Vecchi, “Loop-star decomposition of basis functions in the discretization of the EFIE,” IEEE Trans. Antennas Propagat., vol. 47, no. 2, pp. 339–346, Feb. 1999. [7] J. S. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1635–1645, Oct. 2000. [8] J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektron Übertragungstech., Electron. Commun., vol. 32, no. 4, pp. 159–164, 1978. [9] R. Kress, “Linear integral equations,” in Applied Mathematical Series 82. Berlin, Germany: Springer-Verlag, 1989. [10] K. E. Atkinson and I. G. Graham, “Iterative solution of the linear systems arising from the boundary integral method,” SIAM J. Sci. Statist. Comput., vol. 13, pp. 694–722, 1992. [11] K.Ke Chen, “An analysis of sparse approximate inverse preconditioners for boundary integral equations,” SIAM J. Matrix Anal. Appl., vol. 22, pp. 1058–1078, 2001. [12] S. Vavasis, “Preconditioning for boundary integral equations,” SIAM J. Matrix Anal. Appl., vol. 13, pp. 905–925, 1992. [13] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propagat., vol. 45, pp. 329–342, Mar. 1997. [14] S. V. Polstyanko and J. F. Lee, “Two-level hierarchical FEM method for modeling passive microwave devices,” J. Computat. Phys., vol. 140, no. 2, pp. 400–420, Mar. 1998. [15] D.-K. Sun, J. Manges, X. Yuan, and Z. J. Cendes, “Spurious modes in finite element methods,” IEEE Antennas Propagat. Mag., vol. 37, pp. 12–24, Oct. 1995. [16] G. Karypis, V. Kumar, and METIS. A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, Version 4.0. Univ. Minnesota, Dept. of Computer Science/Army HPC Research Center. [Online] http://www.cs.umn.edu/~karypis [17] I. Gustafson, “Modified incomplete Choleski (MIC) methods,” in Preconditioning Methods: Analysis and Applications, D. J. Evans, Ed. New York: Gordon and Breach, 1983, pp. 265–293. [18] O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: Cambridge Univ. Press, 1996. [19] O. Axelsson and N. Munksgaard, “Analysis of incomplete factorizations with fixed storage allocation,” in Preconditioning Methods: Analysis and Applications, D. J. Evans, Ed. New York: Gordon and Breach, 1983, pp. 219–241. [20] J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics. Piscataway, NJ: IEEE Press, 1998. [21] J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propagat, vol. 45, no. 10, pp. 1488–1493, 1997.
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Jin-Fa Lee (S’85–M’85–SM’88) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C. in 1982 and the M.S. and Ph.D. degrees from Carnegie-Mellon University, Pittsburgh, PA, in 1986 and 1989, respectively, all in electrical engineering. From 1988 to 1990, he was with ANSOFT Corporation, where he developed several CAD/CAE finite element programs for modeling 3-D microwave and millimeter-wave circuits. His Ph.D studies resulted in the first commercial 3-D FEM package for modeling RF/Microwave components, HFSS. From 1990 to 1991, he was a Postdoctoral Fellow at the University of Illinois at Urbana-Champaign. From 1991 to 2000, he was with Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA. Currently, he is an Associate Professor at the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University. His current research interests include analyses of numerical methods, fast finite element methods, integral equation methods, hybrid methods, 3-D mesh generation, and nonlinear optic fiber modeling.
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Robert Lee (S’82–M’83–SM’01) received the B.S.E.E. degree from Lehigh University, Bethlehem, PA, in 1983 and the M.S.E.E. and Ph.D. degrees from the University of Arizona, Tucson, in 1988 and 1990, respectively. From 1983 to 1984, he was a Microwave Engineer at Microwave Semiconductor Corporation, Somerset, NJ. From 1984 to 1986, he was a Member of the Technical Staff at Hughes Aircraft Company, Tucson. From 1986 to 1990, he was a Research Assistant at the University of Arizona. During the summer from 1987 through 1989, he worked at Sandia National Laboratories, Albuquerque, NM. Since 1990, he has been at The Ohio State University where he is currently a Professor of electrical engineering. His major research interests are in the analysis and application of finite methods to electromagnetics.
Robert J. Burkholder (S’85–M’89–SM’97) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1984, 1985, and 1989, respectively. Since 1989, he has been with the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, where he currently a Research Scientist and Adjunct Associate Professor. His research specialties are high-frequency asymptotic techniques and their hybrid combination with numerical techniques for solving large-scale electromagnetic radiation and scattering problems. He has contributed extensively to the EM analysis of large cavities, such as jet inlets/exhausts, and is currently working on the more general problem of EM radiation, propagation, and scattering in realistically complex environments. Dr. Burkholder is a Full Member of the International Scientific Radio Union (URSI) Commission B, and the Applied Computational Electromagnetics Society (ACES).
