# Milestone-Proposal:Theoretical Foundation of Finite-Element-Method for Electromagnetics

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Docket #:2021-12

*This proposal has been submitted for review.*

**To the proposer’s knowledge, is this achievement subject to litigation?**
No

**Is the achievement you are proposing more than 25 years old?**
Yes

**Is the achievement you are proposing within IEEE’s designated fields as defined by IEEE Bylaw I-104.11, namely: Engineering, Computer Sciences and Information Technology, Physical Sciences, Biological and Medical Sciences, Mathematics, Technical Communications, Education, Management, and Law and Policy. **
Yes

**Did the achievement provide a meaningful benefit for humanity?**
Yes

**Was it of at least regional importance?**
Yes

**Has an IEEE Organizational Unit agreed to pay for the milestone plaque(s)?**
Yes

**Has an IEEE Organizational Unit agreed to arrange the dedication ceremony?**
Yes

**Has the IEEE Section in which the milestone is located agreed to take responsibility for the plaque after it is dedicated?**
Yes

**Has the owner of the site agreed to have it designated as an IEEE Milestone?**
Yes

**Year or range of years in which the achievement occurred:**

1979 -1991

**Title of the proposed milestone:**

Variational Reaction Theory for Finite Element Formulation, 1979-1991

**Plaque citation summarizing the achievement and its significance:**

Researchers at National Taiwan University's Electrical Engineering Department proposed Variational Reaction Theory (VRT) for linear non-self-adjoint boundary value problems, creating the first variational formulation to study electromagnetic wave scattering and propagation. The variational nature of the formulation established stable and accurate finite element solutions for solving full-wave Maxwell equations. VRT also helped build edge slot waveguide arrays for air defense radars in the 1990s, strengthening Taiwan's security.

應用於推導有限元素方程式的變分作用理論，1979-1991

臺大電機系研究團隊針對線性非自伴邊界值問題提出變分互作用理論 (VRT)，首次推導出探討電磁波散射和傳播的變分公式；公式的變分性質建立起穩定與準確的Maxwell方程式全波有限元素解。VRT還在1990年代協助建立用於防空雷達的導波管邊緣開槽陣列天線強化臺灣安全。

**200-250 word abstract describing the significance of the technical achievement being proposed, the person(s) involved, historical context, humanitarian and social impact, as well as any possible controversies the advocate might need to review.**

**IEEE technical societies and technical councils within whose fields of interest the Milestone proposal resides.**

**In what IEEE section(s) does it reside?**

IEEE Taipei Section

### IEEE Organizational Unit(s) which have agreed to sponsor the Milestone:

**IEEE Organizational Unit(s) paying for milestone plaque(s):**

**Unit:** IEEE Taipei Section

**Senior Officer Name:** Kea-Tiong Tang, Chairman

**IEEE Organizational Unit(s) arranging the dedication ceremony:**

**Unit:** IEEE Taipei Section

**Senior Officer Name:** Kea-Tiong Tang, Chairman

**IEEE section(s) monitoring the plaque(s):**

**IEEE Section:** IEEE Taipei Section

**IEEE Section Chair name:** Kea-Tiong Tang, Chairman

### Milestone proposer(s):

**Proposer name:** Shyh-Kang Jeng

**Proposer email:** *Proposer's email masked to public*

**Proposer name:** Ruey-Beei Wu

**Proposer email:** *Proposer's email masked to public*

**Proposer name:** Jin-Fa Lee

**Proposer email:** *Proposer's email masked to public*

**Please note:** your email address and contact information will be masked on the website for privacy reasons. Only IEEE History Center Staff will be able to view the email address.

**Street address(es) and GPS coordinates in decimal form of the intended milestone plaque site(s):**

Department of Electrical Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Rd., Taipei, Taiwan 10617. GPS coordinates: x 25.01953, y 121.54410

**Describe briefly the intended site(s) of the milestone plaque(s). The intended site(s) must have a direct connection with the achievement (e.g. where developed, invented, tested, demonstrated, installed, or operated, etc.). A museum where a device or example of the technology is displayed, or the university where the inventor studied, are not, in themselves, sufficient connection for a milestone plaque**.

**Please give the address(es) of the plaque site(s) (GPS coordinates if you have them). Also please give the details of the mounting, i.e. on the outside of the building, in the ground floor entrance hall, on a plinth on the grounds, etc. If visitors to the plaque site will need to go through security, or make an appointment, please give the contact information visitors will need. **
The intended site is inside the main department building, which started its construction when Professor Chun Hsiung Chen, the team leader for the proposed milestone, served as the department head from 1982 to 1985. In this building, Professor Chen and his students in the team taught and developed further applications of the milestone. However, the major works of the proposed milestone were conducted earlier in the now civil engineering building, about 600 meters away. The Department of Electrical Engineering building now accommodates more than 2500 students, including 800 undergraduates, 1200 master, and 500 Ph.D. students, reside to study various areas in electrical engineering. It is the "powerhouse" in Taiwan to promote the innovation, education, and promotion of electrical engineering for humanity.

**Are the original buildings extant?**

The original buildings where the development took place now serve a different goal, are no longer publicly accessible, and will probably be demolished within a few years.

**Details of the plaque mounting:**

The intended location is near the entrance of the auditorium in the building, where seminars, conferences, and invited lectures are frequently held. A picture giving an overview of the entrance of the auditorium is shown below. The IEEE can consult the department of Electrical Engineering about the exact place and mounting within the building

**How is the site protected/secured, and in what ways is it accessible to the public?**

The building is opened on weekdays. The site is protected/secured by a management team and security cameras.

**Who is the present owner of the site(s)?**

Chairman Chung-Chih Wu, Department of Electrical Engineering, National Taiwan University

**What is the historical significance of the work (its technological, scientific, or social importance)? If personal names are included in citation, include justification here. (see section 6 of Milestone Guidelines)**

The Finite Element Method (FEM) has been popular in modeling linear high-frequency electromagnetic (EM) waves, from radiation, scattering, and waveguide, to high-frequency circuits.

By the 1980s, the applications of FEM for static and quasistatic fields had been pretty mature, and the variational formulation was widely available for the Computational Electromagnetics community.

However, the underlying physics for high-frequency EM wave radiations and scattering problems is non-Hermitian and with radiation conditions. The corresponding variational principle was lacking in the 1980s. Not until the Variational Reaction Theory (VRT) series of research publications did the corresponding variational formulation for 3D full-wave Maxwell equations become primarily elusive.

Consequently, the work's significance is for high-frequency EM wave problems: 1) proposing a systematic method for deriving variational equations; 2) including exterior fields in the variational equations; 3) discovering the VRT as a quick way for deriving the FEM matrix equations directly. The inclusion of the exterior field results in the popularity of the hybrid element method, while VRT is also widely used and re-discovered independently as a generalized Galerkin method.

The VRT and the hybrid finite element method have been widely used for academic research and helped explain the reliability and accuracy of early commercial software, which started appearing in 1990. It also helped Taiwan build edge slot waveguide arrays for air defense radars in the 1990s.

**What obstacles (technical, political, geographic) needed to be overcome?**

The major obstacles needed to be overcome by the FEM team led by Professor Chun Hsiung Chen (FEM@NTU) are: 1) difficulty in treating non-self-adjoint fields, 2) unsystematic methods for the variational formulation, 3) difficulty in dealing with exterior fields, 4) difficulty to find physical interpretation, 5) tedious approaches for obtaining the FEM matrix equations, and 6) challenge to prove that the proposed VRT is consistent with the more general but less straightforward fundamental variational equations also published by FEM@NTU earlier.

1) **Difficulty in Treating Non-self-adjoint fields**

Before 1980, most papers about variational formulation assume that the problem as an operator equation together with boundary conditions is self-adjoint. The inner product of a field with an operator operated on another field with associated boundary conditions is the same if we switch positions of the field and the operator. This reciprocity does not hold for a non-self-adjoint problem. For an example of the non-self-adjoint problem, consider the situation with anisotropic materials whose characteristics are described by non-symmetric and/or non-Hermitian matrices.

The FEM@NTU team in 1980 first proposed a general variational formulation with an adjoint problem to handle a non-self-adjoint problem [1]. The solution of the adjoint problem is related to problems with the material characteristics being the transpose or Hermitian of those of the original problem. Although the adjoint field is introduced, we still need to solve only the field distribution of the original problem with the FEM. The 1980 paper also provided physical interpretation through a concept of generalized reaction. However, it did not reveal how to extend the formulation for general exterior problems.

Consider the case of a lossless medium enclosed by a surface on which Dirichlet or Neumann boundary conditions are satisfied. For such a self-adjoint problem, a real stationary functional may be derived. On the other hand, a complex stationary functional can be deduced using the general variational equation proposed in the FEM@NTU 1980 paper. Both stationary functionals have been proved equivalent by Webb, et al. (1983) [2].

2) **Unsystematic procedures for variational formulation**

A variational equation is often obtained from the least action principle for mechanics. The application of the FEM for exterior field problems is scarce for high-frequency electromagnetics before 1984. The most related papers are: Silvester and Hsieh (1971) [3], McDonald and Wexler (1972) [4], the Unimoment method (Mei, 1974) [5], and Morishita and Kumagai (1977) [6]. The variational equations in these works are derived case by case after setting up systems of partial differential equations from the Maxwell equations, except for Morishita and Kumagai in 1977, in which the formulation started from the least action principle generalized for EM fields. However, their approach is inconvenient and indirect for use since the formulation involves vector and scalar potentials.

The FEM@NTU team first proposed in 1980 (primarily for interior problems and simple one-dimensional problems) [1] and 1984 (with extension to general exterior problems) [7] that the FEM equations for any linear EM field problems can be deduced from setting the Fundamental Variational Principle (FVP) to be stationary, equivalent to the Maxwell equations along with the associated boundary conditions. Variational equations can be deduced simply by applying problem-dependent constraints to reduce the FVP. This methodology is named Variational Electromagnetics (VEM) in [7].

3) **Difficulty in Including Exterior Fields**

By 1984 the exterior field had been dealt with for mechanics (Zienkiewicz, 1977 [8) and quasistatic electric or magnetic field or simple wave problems using various approaches [9]-[16]. However, their methods cannot be applied directly to derive a variational equation for high-frequency EM wave problems.

Silvester and Hsieh [3] applied Green's theorem to obtain a variational equation by treating the outer region as a single exterior element. However, they dealt with only 2D Laplace equations, i.e., static fields only.

McDonald and Wexler [4] treated an Integral equation as a constraint to replace the exterior element. Their paper also shows that they can handle only 2-D Poisson equations, again, static fields only.

Mei [5] proposed the unimoment method. He imposed an artificial boundary and expressed the exterior and interior fields as sums of eigenmodes and pseudo modes, respectively. The FEM obtained a pseudo mode for the internal problem by enforcing boundary conditions on the artificial boundary like an exterior eigenmode. The coefficients of both series were then acquired by matching the continuity conditions on the artificial boundary. The exterior field was not included in the variational equations directly.

The FEM@NTU team expressed the exterior field in the following ways: a sum of eigenmodes (including scattering wave modes [7] and propagation modes in general waveguide [17]) or an integral over the artificial interior-exterior boundary [18]. Such expressions are then included in the functional with careful treatment for their stationarity. In addition, the FEM@NTU team also proposed an attractive approach for dielectric waveguide and planar waveguide problems. For such problems, 2-D transverse trial fields extending to infinity in the stationary functional are transformed into the inside of a closed region in the complex plane by conformal mapping [19]. The exterior fields are thus taken into consideration automatically.

4) **Difficulty to Find Physical interpretation**

Many early publications about variational formulation describe their results with mathematic manipulation only. The variational principles derived by the FEM@NTU team in 1980 [1] and 1984 [7] were with a physical interpretation of general reaction. The stationarity of the FVP is also equivalent to the Maxwell equations along with the boundary conditions, just like the principle of virtual work is equivalent to Newton's laws in statics and dynamics. Also, in the 1984 paper [7], the FVP can be reduced to oscillatory power if the problem is constrained to be self-adjoint, similar to the least action principle for mechanics.

5) **Tedious Approaches for Obtaining the FEM matrix equations**

The process of obtaining a FEM matrix equation from the FVP by applying problem-related constraints and the Rayleigh-Ritz procedure are pretty tedious, though systematic. The FEM@NTU team's 1985 paper introduced the VRT [20] to derive the required matrix equation and found it a generalization of the conventional Galerkin method.

6) **Difficulty in Proving the VRT Being Consistent with the Fundamental Variational Equation**

The stationarity of the FVP proposed in 1984 [7] is equivalent to the Maxwell equations and should hold for all linear high-frequency EM wave problems. The VRT [20] is more efficient in deriving the required FEM matrix equations. However, its consistency with the variational equation of FVP was not clear. In other words, why does the generalized Galerkin method work? Why does the way to deal with the exterior field in the generalized Galerkin method lead to the correct solution? Such essential theoretical questions were answered in a FEM@NTU paper in 1988, in which the team proposed the concept of the Partial Variational Principle (PVP) [21]. By PVP, the variation on a functional of trial field f and its adjoint f^a, like the partial differentiation, equals the sum of two functionals (PVPs). One functional is obtained by taking partial variation with respect to f while the other is taken with respect to f^a. If only field f is to be solved, the functional by taking partial variation to f can be ignored. The resultant partial variational principle is the same as the starting equation of VRT. The consistency of VRT and the variational equation on the FVP is thus proved.

**What features set this work apart from similar achievements?**

Features of VRT and this series of work are given below:

1. First variational formulation equivalent to the Maxwell equations and associated boundary conditions. Also proposed are their physical interpretations.

2. A systematic approach to derive the FEM matrix equations.

3. First inclusion of exterior field in the variational equation, which leads to the now popular hybrid FEM.

4. The VRT for deriving the FEM matrix equations directly, which is equivalent to a generalized Galerkin method and widely used in academic and commercial software development circles.

5. A conformal mapping technique for dielectric waveguide or planar waveguide problems to transform the two-dimensional transverse field into the inside of a closed region. Solving the fields in this closed region by the FEM handles the exterior fields automatically.

**Later development**

Since the FEM@NTU team published their 1985 VRT paper [20], the generalized Galerkin method has become very popular in solving the scattering of objects in free space and waveguide structures. The primarily related papers are those introducing the transfinite element method (Lee and Cendes, 1987, 1987 [22], [23]; Cendes and Lee, 1988 [24]; Lee, Sun, and Cendes, 1991 [25]), and those emphasizing hybrid FEM for the scattering of objects in free space (Jin and Liepa, 1988 [26]; Jin and Liepa, 1988 [27]). The transfinite element method was then applied to developing the commercial software HFSS for dealing with RF, microwave, and millimeter-wave circuits. The hybrid FEM has also been proved efficient and effective in solving realistic scattering problems (for example, Jin and Volakis, 1991 [28] and Shen et al., 1998 [29]). However, the impact of the generalized Galerkin method is not limited by its applications to solving EM wave problems with the FEM. Professor Jin-Fa Lee at the Ohio State University also recognized the strong correlation of the generalized Galerkin method with the reciprocity theorem in EM fields and the popular Bi-CG methods for solving complex symmetric but non-Hermitian matrices equations.

HFSS (High-Frequency Structure Simulation), which appeared in 1990 [60], is the first commercially available finite-element software for solving electromagnetic radiation and scattering problems. With HFSS, researchers and designers can conduct "virtual prototyping" for the first time by running accurate S-parameter simulations of fully 3D microwave devices and high-frequency circuits.

The HFSS is arguably the most successful and most widely utilized commercial software for mission-critical EM devices dealing with high-frequency electromagnetic wave problems.

One of the proposers of this proposal, Prof. Jin-Fa Lee, was the original developer of the HFSS under Professor Zoltan J. Cendes, the father of HFSS and the founder of Ansoft Corp. The success of HFSS can be attributed to three major technical accomplishments: the automatic mesh generation, the vector finite elements, and the transfinite element method. Automatic mesh generation resulted from Dr. David Shenton's Ph.D. thesis, the vector finite elements were due to Prof. Nedelec's seminal papers [30], and the transfinite element method [22]-[25] was invented and documented fully in Prof. Lee's Ph.D. thesis [31] at Carnegie Mellon University. With the transfinite element method, the S parameters of the 3D microwave components are not only directly embedded in the solution vector but are also stationary. Consequently, HFSS offers better accuracy in S parameters, even to this day, than other alternatives. As indicated in Prof. Lee's thesis, the transfinite element formulation was in no small part inspired by VRT. It should not be surprising since Prof. Lee obtained his BSEE from NTUEE in 1982. Specifically, he was first introduced to FVP and VRT in his senior year when Prof. Lee took the Mathematical Physics course that Prof. Chun Hsiung Chen taught. The variational nature of the transfinite element formulation should not be overlooked.

Of course, there exist compelling alternatives for simulating high-frequency electromagnetic waves, such as the method of moment (MoM), the finite-difference time-domain (FDTD) method, and the finite integral technique (FIT). Each method has its niche and has been implemented as commercial/non-commercial software.

Aside from the high-frequency EM wave problems, another application of the FEM is in low-frequency electromagnetics, where the displacement current is neglected. Researchers have significantly advanced the modeling of such complicated phenomena since the 1970s [8]-[16], [32]-[38]. They made FEM an almost dominant methodology in that area.

**The FEM@NTU team**

The FEM@NTU leader, Professor Chen, is the first faculty in Taiwan who publish papers in IEEE journals (as early as 1970 [39]). He and his students published their first FEM paper on applying the FEM to a one-dimensional problem with inhomogeneous dielectric in free space in 1979 [40], [41] and a rectangular waveguide in 1981 [42]. Later the FEM@NTU team focused on finding a general variational equation for EM field problems. They published papers on the fundamental variational principles (FVP) in 1980 [22] and 1984 [23], respectively. A shortcut, the VRT, for deriving FEM matrix equations from the FVP was then proposed in their 1985 paper [20]. Finally, in 1988, the FEM@NTU team proposed the partial variational principle [21] and proved that the VRT is consistent with the variational equation on the FVP.

Not restricted to pure and abstract mathematic formulation, based on the FVP, VRT, and PVP, Professor Chen and his students published various applications. The applications include: 1) the scattering of the perfect electric conductor (PEC) cylinder with inhomogeneous dielectric coating [7], 2) general scattering from anisotropic inhomogeneous slabs [43], [44] and cylinders [45], 3) scattering from a magnetostatic slab in parallel-plate waveguide [17], 4) general analysis of dielectric waveguides [20], 5) birefringence analysis of anisotropic optical fibers [46], and 6) analysis of discontinuities in a dielectric waveguide [21], planar dielectric waveguides [47], and planar dielectric antennas [48], 7) the Variational Conformal Mapping technique [19] for weakly guiding dielectric waveguides [49], microstrip lines [50], [51], and coplanar waveguides [52], [53], where the problem is transformed onto a complex plane, and wherein a scalar field is solved.

**Industry-Oriented 3-D Applications in the 1990s**

One of the proposers, Professor Ruey-Beei Wu, lead a team to apply the VRT to some real industry-oriented 3-D problems in the 1990s, for example, [56]-[59]. Due to industry considerations, not all of those works have been published.

In [56]-[57], Professor Wu's team applied VRT to calculate the admittance of an inclined slot in a rectangular waveguide, which is a very challenging problem. Paper [57] results from years' corporation of National Taiwan University and the antenna division in the National Chung Shan Institute of Science and Technology (former Chung Shan Institute of Science and Technology, CSIST). One of the coauthors of [57], Professor Dau-Chyrh Chang, is an IEEE Life Fellow and the director (1982-1998) of the antenna division, CSIST, which is the most extensive antenna research and development center in Taiwan. Techniques developed in [56]-[57] helped CSIST build many slot antennas for military applications. One of the slot array antennas was used for S-band PODAR (point defense array radar) and another for X-band PODAR. In addition to the ground base radar applications, many fuse slot antennas on various missiles were developed using the FEM technique.

The X-/S- bands PODARs are hybrid radar systems with E-plane in phase scanning by ferrite phase shifter and H-plane in both frequency and mechanical scanning over the whole hemisphere with a radius from several kilometers up to 300 km. The H-plane frequency scanning used edge slot waveguide arrays. The root mean square (RMS) side lobe level (SLL) of the antenna pattern for these two radars should be quite low to reduce the strong land clutter and reject the high-speed anti-radiation missiles. Without the VRT-FEM technique, the admittance of the edge slots can only be computed based on an infinitely thin waveguide wall. The measured side lobe level based on an infinitely thin waveguide wall cannot reach the design goal of low RMS SLL. Using the VRT-FEM technique, the admittance of each edge slot can be calculated accurately with finite waveguide thickness. Finally, the measured SLL of these two radar antennas is pretty low and agrees with the VRT-FEM simulation results. Due to the low RMS SLL and hybrid scan capability, these two radar systems were in production and deployed around many places in Taiwan.

In [58], the VRT has also been used to deal with arbitrary discontinuities in 3-D waveguides, which is again challenging and critical for practical industrial applications. This approach was used to design a rectangular to dielectric-filled circular waveguide transition with less than -20 dB return loss over 40% bandwidth using a suitable modified dielectric rod transformer.

Another interesting effort [59] is a pioneering work combining FDTD and FEM to solve transient electromagnetic problems associated with structures of curved surfaces. A tetrahedral edge-based finite element scheme is only used to model the region near the curved surface, while the FDTD method is employed for most of the regular regions. Without any interpolation of the field on the curved surface nor additional stability constraints due to finer divisions near the curved surface, the scheme was found to have second-order accuracy, unconditional stability, programming simplicity, and computational efficiency. The hybrid approach was applied to solve electromagnetic scattering from 3-D arbitrarily shaped dielectric objects.

**Supporting texts and citations to establish the dates, location, and importance of the achievement:** Minimum of five (5), but as many as needed to support the milestone, such as patents, contemporary newspaper articles, journal articles, or chapters in scholarly books. 'Scholarly' is defined as peer-reviewed, with references, and published. **You must supply the texts or excerpts themselves, not just the references**. At least one of the references must be from a scholarly book or journal article. All supporting materials must be in English, or accompanied by an English translation.

**1). E. W. Lucas and T. P. Fontana, "A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays," in IEEE Transactions on Antennas and Propagation, vol. 43, no. 2, pp. 145-153, Feb. 1995, doi: 10.1109/8.366376. https://ieeexplore.ieee.org/document/366376 **

In this seminal paper, Lucas and Fontana extended the Fundamental Variational Principle (FVP) [7][54] and the Partial Variational Principle (PVP) [21] (the theoretical foundation of the VRT) along with the hybrid finite element method to radiation and scattering of general infinite periodic arrays.

This paper introduces the three FEM@NTU works, "We use the reaction-based techniques of Jeng [18], [19] and Chung [20], to formulate a stationary functional in a systematic manner. The methodology is consistent with the inclusion of both periodic boundary conditions as well as vector-Floquet boundary elements into the variational functional."

The authors then clearly explain how to derive a new variational equation by including the periodic condition in the FVP and PVP. The resultant formulation is then applied for numerically simulating radiation and scattering of many essential engineering devices, including a multilayer Frequency Selective Surface (FSS), a circular-loaded waveguide phased array antenna, and a wideband stripline-fed "mecha-notch" phased array.

Finally, in this paper's Conclusion, the authors wrote: "The formulation itself applies the rigorous VEM methodologies from the perspective of the reaction reciprocity which exists between the direct and adjoint problem systems. An E-field partial variational principle was formulated to include all continuity, radiation and periodic boundary conditions in a systematic fashion through these reaction-based concepts. The vector Floquet BEM formulation was also developed and absorbed into the stationary E-field functional. . . The technique handles general periodic array structures, both of the open and aperture-types. It handles nonorthogonal grids, arbitrary oblique scan conditions, two or three dimensional geometries, lossy media, arbitrary polarizations, and both internal as well as external excitations."

**2). K. S. Chiang, "Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides," Opt Quant Electron 26, S113–S134 (1994). https://doi.org/10.1007/BF00384667 **

In this review paper, Chiang commented, 'Using the variational reaction theory, Wu and Chen, formulated an Ez-Hz method without spurious solutions. This method can handle general anisotropic media and also solve scattering problems, but does not yield a standard eigenvalue equation."

**3) J. -F. Lee, D. . -K. Sun and Z. J. Cendes, "Full-wave analysis of dielectric waveguides using tangential vector finite elements," in IEEE Transactions on Microwave Theory and Techniques, vol. 39, no. 8, pp. 1262-1271, Aug. 1991, doi: 10.1109/22.85399. https://ieeexplore.ieee.org/document/85399**

This paper cited the VRT paper [20] as one of three on applying the finite element method to dielectric waveguide analysis. The authors commented, "The finite element method is probably the waveguide analysis method that is the most generally applicable and most versatile."

**4). J.-M. Jin and V. V. Liepa, "Application of hybrid finite element method to electromagnetic scattering from coated cylinders," IEEE Transactions on Antennas and Propagation, vol. 36, no. 1, Jan. 1988, pp. 50-54. doi: 10.1109/8.1074 https://ieeexplore.ieee.org/document/1074**

This paper is the first on the hybrid finite element method. It mentioned the FEM@NTU 1984 paper [7] as one of three works on solving dynamic scattering problems by the finite element method. The other two are Mei's Unimoment method [5] and a master thesis of Guo at Nanjing University, China, 1985 [55]. However, the concept of the hybrid finite element method had been briefly described in [7] and used for dielectric waveguide analysis by Su [18], also Professor Chun Hsiung Chen's student.

**5) Matthew N.O. Sadiku, Computational Electromagnetics with MATLAB®, 4th ed., Boca Raton: CRC Press, 2018. https://www.routledge.com/Computational-Electromagnetics-with-MATLAB-Fourth-Edition/Sadiku/p/book/9781032339030**

This wide-adopted textbook on computational electromagnetics cited the FEM@NTU work [7][21] in Section 4.4, "Construction of Functionals from PDEs." It commented, "Other systematic approaches for the derivation of variational principles for EM problems include Hamilton's principles or the principle of least action [9, 10], Lagrange multipliers [10]-[14], and a technique described as variational electromagnetics [15][16]." References [15] and [16] are [7] and [21] in this proposal, respectively.

This book also mentioned the VRT in Subsection 4.7, "Eigenvalue Problems," with "Examples of eigenvalue problems for which variational methods have been applied include [28]-[37]," where its reference [33] is our VRT paper [20].

In Section 4.9, "Concluding Remarks," the author Sadiku said, ". . . our discussion on variational techniques in this chapter has been only introductory. An exhaustive treatment of the subject can be found in [1, 6, 10, 11],[41]-[43]. Various applications of variational methods to EM-related problems include

waveguides and resonators [28]-[37]

transmission lines [38, 39], [44]-[47]

acoustic radiation [48]

wave propagation [49]-[51]

transient problems [52]

scattering problems [53]-[59].

. . ..” References [33][49][50][57] here correspond to references [20][41][43][45] of this proposal.

**6) Ramesh Garg, Analytical and Computational Methods in Electromagnetics , Artech House, 2008. https://ieeexplore.ieee.org/document/9100322**

In Subsection 4.6.2, "Microstrip Line with a Cover Shield," this book said, "The popularity of the conformal mapping method to planar transmission lines arises from the fact that an open geometry is transformed into a closed polygonal geometry easily. ... Another advantage of conformal mapping is that metal strips are stretched, and ground planes are compressed during transformation to parallel plate capacitor geometry. Due to the stretching of strips, the charge and current singularities on the strips vanish in the transformed plane. The capacitance of the transformed geometry may be determined using computational methods, and this does not require finer discretization near the strip edges [16]." The reference [16] here is the FEM@NTU 1988 paper [50] analyzing microstrip lines with the VRT method and conformal mapping techniques.

In Subsection 10.3.7, "FEM Analysis of Open Boundary Problems," Garg told us, "For the problems in electrostatics, one may use conformal mapping method (Chapter 4) to transform the open region problem into a closed boundary problem, which can be analyzed using FEM. Analysis of planar lines using this hybrid approach has been reported [11]." The citation [11] here is the last FEM@NTU paper [53] in 1991 and is for the analysis of multilayer coplanar lines with the VRT and conformal mapping.

**7) Finite Elements for Wave Electromagnetics: Method and Techniques, Peter P. Silvester and Giuseppe Pelosi, ed., New York, USA: IEEE Press, 1994. https://www.semanticscholar.org/paper/Finite-elements-for-wave-electromagnetics-%3A-methods-Silvester-Pelosi/b5a6b35469285212c4a3618e23f3a00755d4d26a **

This IEEE book collected papers important to applications of the FEM to electromagnetics. The FEM@NTU contributions [1] about the general variational formulation and [20] about the VRT were included.

**8). Jianming Jin, The Finite Element Method in Electromagnetics, New York, USA: John Wiley & Sons, Inc., 1993. https://www.wiley.com/en-us/The+Finite+Element+Method+in+Electromagnetics%2C+3rd+Edition-p-9781118571361**

This textbook is famous for learning the FEM for electromagnetics. At the beginning of chapter 6, "Variational Principles for Electromagnetics," Professor Jin cited [1]. It commented on the use of variational equations: "It is only in recent years that the variational formulation has been discussed more extensively, mainly to satisfy the need for finite element method … However, even these methods remain unknown to many researchers and generally are not taught to graduate students. This situation contrasts the Galerkin method, which is a popular choice both in research and instruction, possibly because of its simplicity."

We agree that starting the FEM with a general variational formulation requires much mathematical background, and its derivation is not as straightforward as the generalized Galerkin method. However, as far as we know, the FEM@NTU team's VRT [20] is the first publication about the generalized Galerkin method for FEM and provided it with a more rigorous theoretical foundation [21].

Jin also said: "If we indeed establish a general procedure to derive the variational formulation for any given problem, there will be no major obstacles, except for personal preference, to prevent us from employing the variational method for the finite element formulation." Here Jin skipped our general variational equation of FVP, which could be due to the abstract mathematics in the FEM@NTU work [7], especially in including the exterior field into the variational equation. In Jin's book, the functional given is derived from the generalized Galerkin method while fine-tuning some boundary terms to ensure that the functional is stationary.

**9) Weng Cho Chew, Waves and Fields in Inhomogeneous Media, New York, USA: Van Nostrand Reinhold, 1990. https://ieeexplore.ieee.org/book/5270998 **

This book is well-known for treating inhomogeneous media. It mentions the FEM@NTU contributions [1] in Section 5.3, "Variational Expressions for Non-Self-Adjoint Problem." The book says, "[I]f the medium is nonreciprocal, then the corresponding linear operator is neither self-adjoint nor symmetric. For such a class of problems, an auxiliary problem has to be defined to derive a variational expression. (see Chen and Lien 1980)." Major derivation of the variational equation followed those given in [1]. This Chew’s book also listed other FEM@NTU publications [7],[18]-[20],[43][44][49] as further reading for Chapter 5, “Variational Methods.”

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This paper is the first on the transfinite elements. The variational formulation in this paper is similar to the one in a FEM@NTU 1982 paper, though not cited explicitly: S.-K. Jeng and C. H. Chen, "A new variational theory for linear field problems and its application to electrostatics," Journal of the Chinese Institute of Engineers, vol. 5, no. 2, 1982, pp. 99-107. https://doi.org/10.1080/02533839.1982.9676696

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