unsolved problems in em and cem

270 IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

Open Problems in CEM

Özgür ErgülDepartment of Mathematics and StatisticsUniversity of StrathclydeLivingstone Tower, 26 Richmond StreetGlasgow G1 1XH UKE-mail: [email protected]

Unsolved Problems in EM and CEM: A Personal Perspective

Weng Cho Chew

Department of Electrical and Computer EngineeringUniversity of Illinois, Urbana-Champaign

1406 West Green St., Urbana IL 61801 USAE-mail: [email protected]

1. Introduction

A while ago, I was asked by Eric Michielssen and Balasubramanium Shanker to present a paper on unsolved

problems in computational electromagnetics [1]. (When I fi rst thought about it, it was actually a very diffi cult paper to present.) As the saying goes, “Beauty is in the eyes of the beholder:” so what remains unsolved is also in the eyes of the beholder.

Our community has researchers with broad interests, rang-ing from developing technologies that are immediately relevant to the users, to those who dwell on futuristic pursuits. There are some researchers who are interested in the mathe matics, some in the physics, some in both, while for others, in technologies. To this diverse range of researchers, what remains unsolved can vary greatly. I personally will always look for new, interesting, and important problems to solve. Of course, what is new, interesting, and important is also a very personal view.

2. Unsolved Problems: A First Cut

If one asks a fundamental question on electromagnetics (EM), there are still many unanswered questions, for instance, in Yang-Mills theory. Electromagnetic theory emerges as a special case of Yang-Mills theory, a gauge theory deeply related

to geometry. Why electromagnetic theory is deeply related to geometry is still a mystery. Pursuers of the Yang-Mills theory, a generalized electromagnetic theory, are inter ested in understanding the mass gap [2], for instance.

For a mathematician, there could be issues of uniqueness and existence of solutions [3], and more recently, convergence [4]. Convergence is highly relevant in light of the fact that many equations are numerically and iteratively solved, these days [5, 6]. To achieve rapid convergence, the operator has to be free from the null space, having a compactness property, and yielding well-conditioned formulations. Hence, formulat-ing well-conditioned equations is a prerogative in the area of computational electromagnetics (CEM) [8-13].

For engineers or technologists, the goal is the develop-ment of new technologies. Electromagnetics is a basic-science subject that has given birth to many new technologies. Some recent technologies are in the areas of metamaterials, small antennas, new simulation tools, and many more. Advances in computational electromagnetics (CEM), in particular, have stimulated advances in simulation tools. CEM is attractive to theorists because of the mathematics and physical insight needed to solve many of its ensuing problems.

Artifi cial materials emerged many years ago as artifi cial dielectrics [14]. Later, frequency-selective surfaces became in

vogue, which was pursued in earnest [15, 16]. Recently, the quest for artifi cial materials has emerged as metamaterials. The renewed interest was precipitated by two key papers by Pendry on superresolution [17] and cloaking [18], with experimental evidence in [19, 20]. The fl urry of activities in this area has defi nitely generated new knowledge in artifi cial dielectrics. (A representative work is [22]. I apologize for not providing the reference list for these activities, as it would fi ll volumes.) The area is further buoyed by the advent of nano-fabrication technology, involving nanometer sizes. Metamate rials have hence emerged under a new light that portends applications in the optical and quasi-optical regimes.

Small antennas are instrumental in the mobile-phone industry, which demands inconspicuously small antennas for aesthetics and convenience in addition to effi ciency and low power. Consequently, there are a plethora of current activities in this area [23-31].

As technological demand looms, so does the demand for modeling and simulation tools to replace expensive laboratory experiments. Many user-friendly commercial software prod-ucts have emerged. They have altered the modus operandi of scientists and engineers. In the past, a scientifi c and engineer-ing team consisted of both experimentalists and theorists. The role of the theorists was to provide theoretical models that could validate (or be validated by) the experimental fi ndings. However, these days, due to the user-friendliness of commer-cial software, many experimental cut-and-try methods are being replaced by simulation via commercial software. More-over, many experimentalists can run this software themselves, obviating the need to collaborate with theorists. The popularity of simulation has also been a boon to CEM. Hence, the quest for effi cient and multi-scale methods in CEM has been an important topic of research in recent years [32-41]. Some researchers have also taken CEM to new heights by solving problems of unprecedented sizes on existing computers [42-44, 50]. Others have tackled highly complex structures [46-52].

3. A New Niche for Theorists

The challenge to theorists in the CEM area is to come up with new techniques that make commercial software even better. The poignant truth is that this has made the traditional role of a theorist obsolete: a theorist now needs to provide solutions beyond those from commercial software.

However, as technologists, we seek out new technologies in all realms, be they in antennas, materials, energy sources and conversion devices, wireless power transfer, computer integrated circuits, communication systems, imaging, sensing and detection systems, medical electromagnetics, nonlinear phenomena, nano-optics and plasmonics, terahertz science and technology, as well as in simulations. Many of these research directions are being fervently discussed in ongoing confer-ences, for example, at [53]. Electromagnetics is indispensable in impacting these technologies, and so are the simulation tools developed. Moreover, in this changing landscape, a theo-

rist may not just be a solution provider: he or she needs to be an application-specifi c solution provider for those not met by commercial software. Increasingly, a theorist has to be one who can help formulate and identify new problems and issues to be solved that can impact new technologies.

4. Some Unsolved Problems

If one looks at unsolved problems in electromagnetics from a technologist’s viewpoint, there are many looming unsolved problems in electromagnetic technologies. Hence, in order to grow this area, and to identify new problems to be solved, it is imperative that CEM experts collaborate with other technologists to chart new frontiers. These technologies can be in the area of (I apologize for the incompleteness of this list):

• Antenna designs;

• Materials or metamaterials;

• Solar and thermo electricity;

• Energy-saving light sources such as LED, lasers;

• New sensing and imaging technology;

• Quantum technologies as emerging in the area of nano-electronics, communication, encryption, N/MEMS, and computing;

• New simulation software.

Just as in yesteryears, when a theorist often collaborated with an experimentalist, a way to grow EM technologies is for all CEM experts or theorists to collaborate with other tech-nologists and scientists. This will greatly expand the scope of their research, and enliven the fi eld. The future of CEM is then boundless.

5. Some of My Personal Favorites

Even though there are many unsolved problems in this world that I cannot personally tackle, here are a few of my personal favorites.

1. A fast algorithm for nonlinear inverse scattering and imaging: Many imaging algorithms that have

caught on and become popular have ( )logO N N complexity, where N represents the number of pix els in the image. This is the case for synthetic-aperture radar, MRI, ultrasound, X-ray computed tomography. Multiple scattering and multipath effects are neglected in these imaging modalities.

AP_Mag_Dec_2012_Final.indd 270 12/9/2012 3:52:29 PM

IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012 271

Open Problems in CEM

Özgür ErgülDepartment of Mathematics and StatisticsUniversity of StrathclydeLivingstone Tower, 26 Richmond StreetGlasgow G1 1XH UKE-mail: [email protected]

Unsolved Problems in EM and CEM: A Personal Perspective

Weng Cho Chew

Department of Electrical and Computer EngineeringUniversity of Illinois, Urbana-Champaign

1406 West Green St., Urbana IL 61801 USAE-mail: [email protected]

1. Introduction

A while ago, I was asked by Eric Michielssen and Balasubramanium Shanker to present a paper on unsolved

problems in computational electromagnetics [1]. (When I fi rst thought about it, it was actually a very diffi cult paper to present.) As the saying goes, “Beauty is in the eyes of the beholder:” so what remains unsolved is also in the eyes of the beholder.

Our community has researchers with broad interests, rang-ing from developing technologies that are immediately relevant to the users, to those who dwell on futuristic pursuits. There are some researchers who are interested in the mathe matics, some in the physics, some in both, while for others, in technologies. To this diverse range of researchers, what remains unsolved can vary greatly. I personally will always look for new, interesting, and important problems to solve. Of course, what is new, interesting, and important is also a very personal view.

2. Unsolved Problems: A First Cut

If one asks a fundamental question on electromagnetics (EM), there are still many unanswered questions, for instance, in Yang-Mills theory. Electromagnetic theory emerges as a special case of Yang-Mills theory, a gauge theory deeply related

to geometry. Why electromagnetic theory is deeply related to geometry is still a mystery. Pursuers of the Yang-Mills theory, a generalized electromagnetic theory, are inter ested in understanding the mass gap [2], for instance.

For a mathematician, there could be issues of uniqueness and existence of solutions [3], and more recently, convergence [4]. Convergence is highly relevant in light of the fact that many equations are numerically and iteratively solved, these days [5, 6]. To achieve rapid convergence, the operator has to be free from the null space, having a compactness property, and yielding well-conditioned formulations. Hence, formulat-ing well-conditioned equations is a prerogative in the area of computational electromagnetics (CEM) [8-13].

For engineers or technologists, the goal is the develop-ment of new technologies. Electromagnetics is a basic-science subject that has given birth to many new technologies. Some recent technologies are in the areas of metamaterials, small antennas, new simulation tools, and many more. Advances in computational electromagnetics (CEM), in particular, have stimulated advances in simulation tools. CEM is attractive to theorists because of the mathematics and physical insight needed to solve many of its ensuing problems.

Artifi cial materials emerged many years ago as artifi cial dielectrics [14]. Later, frequency-selective surfaces became in

vogue, which was pursued in earnest [15, 16]. Recently, the quest for artifi cial materials has emerged as metamaterials. The renewed interest was precipitated by two key papers by Pendry on superresolution [17] and cloaking [18], with experimental evidence in [19, 20]. The fl urry of activities in this area has defi nitely generated new knowledge in artifi cial dielectrics. (A representative work is [22]. I apologize for not providing the reference list for these activities, as it would fi ll volumes.) The area is further buoyed by the advent of nano-fabrication technology, involving nanometer sizes. Metamate rials have hence emerged under a new light that portends applications in the optical and quasi-optical regimes.

Small antennas are instrumental in the mobile-phone industry, which demands inconspicuously small antennas for aesthetics and convenience in addition to effi ciency and low power. Consequently, there are a plethora of current activities in this area [23-31].

As technological demand looms, so does the demand for modeling and simulation tools to replace expensive laboratory experiments. Many user-friendly commercial software prod-ucts have emerged. They have altered the modus operandi of scientists and engineers. In the past, a scientifi c and engineer-ing team consisted of both experimentalists and theorists. The role of the theorists was to provide theoretical models that could validate (or be validated by) the experimental fi ndings. However, these days, due to the user-friendliness of commer-cial software, many experimental cut-and-try methods are being replaced by simulation via commercial software. More-over, many experimentalists can run this software themselves, obviating the need to collaborate with theorists. The popularity of simulation has also been a boon to CEM. Hence, the quest for effi cient and multi-scale methods in CEM has been an important topic of research in recent years [32-41]. Some researchers have also taken CEM to new heights by solving problems of unprecedented sizes on existing computers [42-44, 50]. Others have tackled highly complex structures [46-52].

3. A New Niche for Theorists

The challenge to theorists in the CEM area is to come up with new techniques that make commercial software even better. The poignant truth is that this has made the traditional role of a theorist obsolete: a theorist now needs to provide solutions beyond those from commercial software.

However, as technologists, we seek out new technologies in all realms, be they in antennas, materials, energy sources and conversion devices, wireless power transfer, computer integrated circuits, communication systems, imaging, sensing and detection systems, medical electromagnetics, nonlinear phenomena, nano-optics and plasmonics, terahertz science and technology, as well as in simulations. Many of these research directions are being fervently discussed in ongoing confer-ences, for example, at [53]. Electromagnetics is indispensable in impacting these technologies, and so are the simulation tools developed. Moreover, in this changing landscape, a theo-

rist may not just be a solution provider: he or she needs to be an application-specifi c solution provider for those not met by commercial software. Increasingly, a theorist has to be one who can help formulate and identify new problems and issues to be solved that can impact new technologies.

4. Some Unsolved Problems

If one looks at unsolved problems in electromagnetics from a technologist’s viewpoint, there are many looming unsolved problems in electromagnetic technologies. Hence, in order to grow this area, and to identify new problems to be solved, it is imperative that CEM experts collaborate with other technologists to chart new frontiers. These technologies can be in the area of (I apologize for the incompleteness of this list):

• Antenna designs;

• Materials or metamaterials;

• Solar and thermo electricity;

• Energy-saving light sources such as LED, lasers;

• New sensing and imaging technology;

• Quantum technologies as emerging in the area of nano-electronics, communication, encryption, N/MEMS, and computing;

• New simulation software.

Just as in yesteryears, when a theorist often collaborated with an experimentalist, a way to grow EM technologies is for all CEM experts or theorists to collaborate with other tech-nologists and scientists. This will greatly expand the scope of their research, and enliven the fi eld. The future of CEM is then boundless.

5. Some of My Personal Favorites

Even though there are many unsolved problems in this world that I cannot personally tackle, here are a few of my personal favorites.

1. A fast algorithm for nonlinear inverse scattering and imaging: Many imaging algorithms that have

caught on and become popular have ( )logO N N complexity, where N represents the number of pix els in the image. This is the case for synthetic-aperture radar, MRI, ultrasound, X-ray computed tomography. Multiple scattering and multipath effects are neglected in these imaging modalities.



However, when multiple scattering is accounted for, the problem quickly becomes nonlinear, and defi es

an ( )logO N N algorithm for its solution [54].

2. High-frequency computational electromagnetics solutions: In computational electromagnetics, the workload grows with the number of unknowns, N. Work on fast algorithms has enabled such work-

loads to grow close to ( )logO N N in complexity. However, N still grows as the electrical size grows, making the workload exorbitant. A different solu-tion approach has to be sought for high frequencies to pre-empt this growth in the workload [55-58].

3. Combining computational electromagnetics with modern physics: While CEM researchers have imbibed knowledge from mathematics, less so is the incorporation of knowledge from modern physics. Quantum mechanics is an intellectual achievement of the 20th century [59, 60], while electromagnetic theory, completed by Maxwell in 1864, will soon be 150 years old. There is a press-ing need to bring modern physics into electromag-netics and computational electromagnetics. This will help enliven the fi eld of both electromagnetics and computational electromagnetics [61-64] .

6. Conclusions

We often hear our colleagues say that electromagnetics is a dead fi eld. However, if we can imbibe new ideas into the fi eld, by using our mathematical skills and physics knowledge, it can be enlivened. We can use our knowledge and skills to identify and formulate new problems and research directions. If we can work together to grow this fi eld, the future for elec tromagnetics in its modern form has infi nite potential.

7. References

1. E. Michielssen and B. Shanker, “Ten Open Problems in CEM,” IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Spo-kane, Washington, July 2011.

2. http://www.claymath.org/millennium/Yang-Mills\_Theory/

3. R. Courant and D. Hilbert, Methods of Mathematical Phys-ics, New York, John Wiley and Sons, 1989.

4. G. H. Golub and C. F. Van Loan, Matrix Computation, Baltimore, MD, John Hopkins University Press, 1996.

5. G. Strang, Introduction to Applied Mathematics, Wellesley, MA, Wellesley-Cambridge Press, 1986.

6. R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems, Philadelphia, PA, SIAM, 1987.

7. D. R. Wilton and A. W. Glisson, “On Improving the Stabil ity of the Electric Field Integral Equation at Low Frequen cies,” URSI Radio Science Meeting, Los Angeles, CA, June 1981, p. 24.

8. G. C. Hsiao and R. E. Kleinman, “Mathematical Founda-tions for Error Estimation in Numerical Solutions of Integral Equations in Electromagnetics,” IEEE Transactions on Anten-nas and Propagation, AP-45, 3, March 1997, pp. 316-328.

9. R. J. Adams, “Physical and Analytical Properties of a stabi-lized electric fi eld integral equation,” IEEE Transactions on Antennas and Propagation, AP-52, 2, February 2004, pp. 362-372.

10. F. P. Andriulli, K. Cools, H. Baci, F. Olyslager, A. Buffa, S. H. Christiansen, and E. Michielssen, “A Multiplicative Calderon Preconditioner for the Electric Field Integral Equa-tion,” IEEE Transactions on Antennas and Propagation, AP-56, 8, August 2008, pp. 2398-2412.

11. S. Yan, J. M. Jin, and Z. P. Nie, “EFIE Analysis of Low-Frequency Problems with Loop-Star Decomposition and Calderón Multiplicative Preconditioner,” IEEE Transactions on Antennas and Propagation, AP-58, 3, March 2010, pp. 857-867.

12. M. B. Stephanson and J.-F. Lee, “Preconditioned Electric Field Integral Equation Using Calderon Identities and Dual Loop/Star Basis Functions,” IEEE Transactions on Antennas and Propagation, AP-57, 4, April 2009, pp. 1274-1279.

13. S. Sun, Y. G. Liu, W. C. Chew, Z. H. Ma, “Calderon Multiplicative Preconditioned EFIE with Perturbation Method,” IEEE Transactions on Antennas and Propagation, AP-61, 1, January 2013, pp. 1-10.

14. R. E. Collin, Field Theory of Guided Waves, New York, McGraw-Hill, 1960; Second Edition, New York, IEEE Press, 1991.

15. C.-H. Tsao and R. Mittra, “Spectral-Domain Analysis of Frequency Selective Surfaces Comprised of Periodic Arrays of Cross Dipoles and Jerusalem Crosses,” IEEE Transactions on Antennas and Propagation, AP-32, 5, 1984, pp. 478-486.

16. B. A. Munk, Frequency Selective Surfaces, Theory and Design, New York, John Wiley and Sons, 2000.

17. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett., 85, 18, 2000, pp. 3966-3969.

18. J. B. Pendry. D. Schurig, D. R. Smith, “Controlling Elec-tromagnetic Fields,” Science, 312, 5781, 2006, pp. 1780-1782.

19. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, D. Schurig, “Calculation and Measurement of Bianisotropy in a Split Ring Resonator Metamaterial,” J. Appl. Phys., 100, 2, 024507, 2006.

20. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial Electromag-netic Cloak at Microwave Frequencies,” Science Express, 113362, 2006.

21. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, D. R. Smith, “Broadband Ground-Plane Cloak,” Science, 323, 366, 2009.

22. A. Alù and N. Engheta, “Achieving Transparency with Plasmonic and Metamaterial Coatings,” Phys. Rev., E72, 016623, 2005.

23. R. Hansen and R. E. Collin, Small Antenna Handbook, New York, John Wiley and Sons, 2011.

24. S. R. Best, “A Discussion on Power Factor, Quality Factor and Effi ciency of Small Antennas,” IEEE International Sym-posium on Antennas and Propagation Digest, 2007, pp. 2269-72.

25. A. D. Yaghjian, T. H. O’Donnell, E. E. Altshuler, S. R. Best, “Electrically Small Supergain End-Fire Arrays,” Radio Science, 43, 3, May 2008, p. RS3002.

26. H. L. Thal, “New Radiation Q Limits for Spherical Wire Antennas,” IEEE Transactions on Antennas and Propagation, AP-54, 10, October 2006, pp. 2757-2763.

27. J. J. Adams and J. T. Bernhard, “Tuning Method for a New Electrically Small Antenna with Low Q,” IEEE Antennas and Wireless Propagation Letters, 8, 2009, pp. 303-306.

28. H. S. Choo, R. L. Rogers, H. Ling, “Design of Electrically Small Wire Antennas Using a Pareto Genetic Algorithm,” IEEE Transactions on Antennas and Propagation, AP-53, 3, March 2005, pp. 1038-1046.

29. H. Wong, K.-M. Luk, C. H. Chan, Q. Xue, K. K. So, and H. W. Lai, “Small Antennas in Wireless Communications,” Proceedings of the IEEE, 100, 7, July 2012.

30. Z. N. Chen, T. S. P. See, and X. M. Qing, “Small Printed Ultrawideband Antenna with Reduced Ground Plane Effect,” IEEE Transactions on Antennas and Propagation, AP-55, 2, February 2007, p. 383.

31. K. L. Wong, Planar Antennas for Wireless Communica-tions, New York, Wiley, 2003.

32. L. Greengard and V. Rokhlin, “A Fast Algorithm for Par ticle Simulations,” J. Comp. Phys., 73, 2, 1987, pp. 325-348.

33. R. Coifman, V. Rokhlin, and S. Wandzura, “The Fast Multipole Method for the Wave Equation: A Pedestrian Pre-scription,” IEEE Antennas and Propagation Magazine, 35, 3, June 1993, pp. 7-12.

34. C. C. Lu and W. C. Chew, “A Multilevel Algorithm for Solving Boundary Integral Equations of Wave Scattering,” Micro. Opt. Tech. Lett., 7, 1994, pp. 466-470.

35. J. M. Song and W. C. Chew, “Multilevel Fast-Multipole Algorithm for Solving Combined Field Integral Equations of Electromagnetic Scattering,” Micro. Opt. Tech. Lett., 10, 1, September 1995, pp. 14-19.

36. W. C. Chew, J. Jin, E. Michielssen, and J. M. Song (eds.), Fast and Effi cient Algorithms in Computational Electromag-netics, Norwood, MA, Artech House, July 2001.

37. J. M. Rius, J. Parrón, E. Úbeda, J. R. Mosig, “Multilevel Matrix Decomposition Algorithm for Analysis of Electrically Large Electromagnetic Problems in 3-D,” Micro. Opt. Tech. Lett., 22, 3, August 1999, pp. 177-182.

38. R. Mittra, “Characteristic Basis Function Method (CBFM) – An Iterative-Free Domain Decomposition Approach in Computational Electromagnetics,” ACES Journal, 24, 2, April 2009, pp. 204-223.

39. P. Ylä-Oijala and M. Taskinen, “Electromagnetic Scatter-ing by Large and Complex Structures with Surface Equiva lence Principle Algorithm,” Waves in Random and Complex Media, 19, 1, February 2009, pp. 105-125.

40. T. F. Eibert, “A Diagonalized Multilevel Fast Multipole Method with Spherical Harmonics Expansion of the k-Space Integrals,” IEEE Transactions on Antennas and Propagation, AP-53, 2, February 2005, pp. 814-817.

41. E. Garcia, C. Delgado, L. Lozano, I. Gonzalez-Diego, and M. F. Catedra, “An Effi cient Hybrid-Scheme Combining the Characteristic Basis Function Method and the Multilevel Fast Multipole Algorithm for Solving Bistatic RCS and Radiation Problems,” PIER B, 34, 2010, pp. 327-343.

42. J. M. Taboada, M. G. Araujo, J. M. Bertolo, L. Landesa, F. Obelleiro, and J. L. Rodriguez, “MLFMA-FFT Parallel Algo-rithm for the Solution of Large-Scale Problems in Electro-magnetics,” PIER, 105, 2010, pp. 15-30.

43. M. L. Yang and X. Q. Sheng, “Parallel FE-BI-MLFMA for Scattering by Extremely Large Targets with Cavities,” ICEAA, Sydney, Australia, September 20-24, 2010.

44. O. Ergul and L. Gurel, “Rigorous Solutions of Electro-magnetics Problems Involving Hundreds of Millions of Unknowns,” IEEE Antennas and Propagation Magazine, 53, 1, February 2011, pp. 18-26.

45. Y. Shao, Z. Peng, K. H. Lim, and J. F. Lee “Non-Confor mal Domain Decomposition Methods for Time-Harmonic Maxwell Equations,” Proc. R. Soc. A, 468, 2145, September 8, 2012, pp. 2433-2460.


IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012 273

However, when multiple scattering is accounted for, the problem quickly becomes nonlinear, and defi es

an ( )logO N N algorithm for its solution [54].

2. High-frequency computational electromagnetics solutions: In computational electromagnetics, the workload grows with the number of unknowns, N. Work on fast algorithms has enabled such work-

loads to grow close to ( )logO N N in complexity. However, N still grows as the electrical size grows, making the workload exorbitant. A different solu-tion approach has to be sought for high frequencies to pre-empt this growth in the workload [55-58].

3. Combining computational electromagnetics with modern physics: While CEM researchers have imbibed knowledge from mathematics, less so is the incorporation of knowledge from modern physics. Quantum mechanics is an intellectual achievement of the 20th century [59, 60], while electromagnetic theory, completed by Maxwell in 1864, will soon be 150 years old. There is a press-ing need to bring modern physics into electromag-netics and computational electromagnetics. This will help enliven the fi eld of both electromagnetics and computational electromagnetics [61-64] .

6. Conclusions

We often hear our colleagues say that electromagnetics is a dead fi eld. However, if we can imbibe new ideas into the fi eld, by using our mathematical skills and physics knowledge, it can be enlivened. We can use our knowledge and skills to identify and formulate new problems and research directions. If we can work together to grow this fi eld, the future for elec tromagnetics in its modern form has infi nite potential.

7. References

1. E. Michielssen and B. Shanker, “Ten Open Problems in CEM,” IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Spo-kane, Washington, July 2011.

2. http://www.claymath.org/millennium/Yang-Mills\_Theory/

3. R. Courant and D. Hilbert, Methods of Mathematical Phys-ics, New York, John Wiley and Sons, 1989.

4. G. H. Golub and C. F. Van Loan, Matrix Computation, Baltimore, MD, John Hopkins University Press, 1996.

5. G. Strang, Introduction to Applied Mathematics, Wellesley, MA, Wellesley-Cambridge Press, 1986.

6. R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems, Philadelphia, PA, SIAM, 1987.

7. D. R. Wilton and A. W. Glisson, “On Improving the Stabil ity of the Electric Field Integral Equation at Low Frequen cies,” URSI Radio Science Meeting, Los Angeles, CA, June 1981, p. 24.

8. G. C. Hsiao and R. E. Kleinman, “Mathematical Founda-tions for Error Estimation in Numerical Solutions of Integral Equations in Electromagnetics,” IEEE Transactions on Anten-nas and Propagation, AP-45, 3, March 1997, pp. 316-328.

9. R. J. Adams, “Physical and Analytical Properties of a stabi-lized electric fi eld integral equation,” IEEE Transactions on Antennas and Propagation, AP-52, 2, February 2004, pp. 362-372.

10. F. P. Andriulli, K. Cools, H. Baci, F. Olyslager, A. Buffa, S. H. Christiansen, and E. Michielssen, “A Multiplicative Calderon Preconditioner for the Electric Field Integral Equa-tion,” IEEE Transactions on Antennas and Propagation, AP-56, 8, August 2008, pp. 2398-2412.

11. S. Yan, J. M. Jin, and Z. P. Nie, “EFIE Analysis of Low-Frequency Problems with Loop-Star Decomposition and Calderón Multiplicative Preconditioner,” IEEE Transactions on Antennas and Propagation, AP-58, 3, March 2010, pp. 857-867.

12. M. B. Stephanson and J.-F. Lee, “Preconditioned Electric Field Integral Equation Using Calderon Identities and Dual Loop/Star Basis Functions,” IEEE Transactions on Antennas and Propagation, AP-57, 4, April 2009, pp. 1274-1279.

13. S. Sun, Y. G. Liu, W. C. Chew, Z. H. Ma, “Calderon Multiplicative Preconditioned EFIE with Perturbation Method,” IEEE Transactions on Antennas and Propagation, AP-61, 1, January 2013, pp. 1-10.

14. R. E. Collin, Field Theory of Guided Waves, New York, McGraw-Hill, 1960; Second Edition, New York, IEEE Press, 1991.

15. C.-H. Tsao and R. Mittra, “Spectral-Domain Analysis of Frequency Selective Surfaces Comprised of Periodic Arrays of Cross Dipoles and Jerusalem Crosses,” IEEE Transactions on Antennas and Propagation, AP-32, 5, 1984, pp. 478-486.

16. B. A. Munk, Frequency Selective Surfaces, Theory and Design, New York, John Wiley and Sons, 2000.

17. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett., 85, 18, 2000, pp. 3966-3969.

18. J. B. Pendry. D. Schurig, D. R. Smith, “Controlling Elec-tromagnetic Fields,” Science, 312, 5781, 2006, pp. 1780-1782.

19. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, D. Schurig, “Calculation and Measurement of Bianisotropy in a Split Ring Resonator Metamaterial,” J. Appl. Phys., 100, 2, 024507, 2006.

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