# 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:**

Theoretical Foundation of Applying Finite-Element-Method to Linear High-Frequency Electromagnetic Wave Problems

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

From 1979 to 1991, a research group in the department of electrical engineering, National Taiwan University, laid the theoretical foundation for applying the Finite Element Method to general linear high-frequency electromagnetic wave problems. They also found a straightforward derivation as a generalized Galerkin's method. It has been widely applied in academic research and commercial software development, like the early versions of HFSS.

1979至1991期間，台大電機系的電波研究團隊建構了應用於線性高頻電磁波問題的有限元素法理論基礎。他們也發現有限元素法的矩陣方程式，可以由推廣的Galerkin方法直接完成。這種推廣的 Galerkin 方法很快就應用到許多電波學術研究以及廣為應用的商業模擬軟體開發，如 HFSS。

**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, Ruey-Beei Wu, and 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 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. This 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)?**

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. Based on it, the software HFSS (High-Frequency Structure Simulation) was the first commercially available software for solving EM 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 also arguably currently the most successful and most widely utilized commercial software for mission-critical EM devices dealing with high-frequency EM wave problems.

Of course, there exist compelling alternatives for simulating high-frequency EM 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. The problems may be a static or quasistatic, electrical or magnetic field, scalar and vectorial potentials, linear or nonlinear, with closed or open regions, including or not the forces and torques of moving parts in magnetic fields. Researchers have significantly advanced the modeling of such complicated phenomena since the 1970s [1]-[16]. They made FEM an almost dominant methodology in that area. We believe that developing theory and implementing FEM for low-frequency EM fields also deserves one or more IEEE milestones.

The FEM is a general approximation process applicable to various engineering problems, and its mathematical formulations already existed in the early 70s. Much of them have been reviewed in Zienkiewicz et al., 1977 [3]. With the general FEM framework proposed in [3], discretization schemes like the Rayleigh-Ritz procedure are independent of the engineering domain, resulting in matrix equations for solving unknowns.

However, each engineering domain, such as the low-frequency quasistatic field problems or our high-frequency EM wave problems, has its domain knowledge like the governing equations, such as Poisson equations or time-harmonic Maxwell equations, with different kinds of boundary conditions, for theoretical formulation and discussion of results.

The resultant FEM matrix equation in an engineering domain is often obtained by the Rayleigh-Ritz procedure to discretize a variational equation, by weighted residues, or by Galerkin's method. Among them, the variational method is often regarded as fundamental. The method of weighted residue or Galerkin's method is equivalent to the variational equation approach if they generate the same matrix equation. They will result in a matrix equation with a symmetric matrix, which is efficient to solve using numerical methods. Besides, the variational equation and its equivalents require less continuity of the bases (shape) functions to expand the solution approximately and linearly. In addition, only essential boundary conditions have to be enforced, and the natural boundary conditions may be taken care of automatically.

Thus, we must first derive a variational equation to take full advantage of the FEM for high-frequency EM wave problems. By the 1980s, the applications of FEM for static and quasistatic fields had been pretty mature, and the variational nature of some formulation was documented.

However, for high-frequency EM wave radiations and scattering problems, the underlying physics is non-Hermitian and with radiation condition. The corresponding variational principle was lacking in the 1980s.

Consequently, the work's significance is for high-frequency EM wave problems: proposing a systematic method for deriving variational equations, including exterior fields in the variational equations, and discovering a shortcut for deriving the FEM matrix equations directly. The inclusion of the exterior field results in the popularity of the hybrid element method, namely, a shortcut method as a generalized Galerkin's method.

Such a shortcut and the hybrid FEM have been widely used for academic research and successful commercial software development, like the early versions of HFSS software.

**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: difficulty in treating non-self-adjoint fields, unsystematic methods for the variational formulation, difficulty in dealing with exterior fields, difficulty to find physical interpretation, tedious approaches for obtaining the FEM matrix equations, and challenge to prove that the proposed shortcut method is consistent with the proposed fundamental variational equation.

**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. 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) [17].

**Unsystematic procedures for variational formulation**

A variational equation is often obtained from the least action principle for mechanics. For high-frequency electromagnetics, the application of the FEM for exterior field problems is scarce before 1984. The most related papers are: Silvester and Hsieh (1971) [18], McDonald and Wexler (1972) [19], the Unimoment method (Mei, 1974) [20], and Morishita and Kumagai (1977) [21]. 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) [22] and 1984 (with extension to general exterior problems) [23] 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.

**Difficulty in Including Exterior Fields**

By 1984 the exterior field has been dealt with for mechanics (Zienkiewicz, 1977 [3]) and quasistatic electric or magnetic field or simple wave problems using various approaches [2][4]-[10]. However, their methods cannot be applied directly to derive a variational equation for high-frequency EM wave problems.

Silvester and Hsieh [18] 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 [19] 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 [20] proposed the unimoment method. He imposed an artificial boundary and expressed the exterior and interior fields as sums of eigenmodes and pseudo modes, respectively. A pseudo mode for the internal problem was obtained by the FEM 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 [23] and propagation modes in general waveguide [24]) or an integral over the artificial interior-exterior boundary [25]. 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 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 a conformal mapping [26]. The exterior fields are thus taken into consideration automatically.

**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 [22] and 1984 [23] 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 that the principle of virtual work is equivalent to Newton's laws in statics and dynamics. Also, in the 1984 paper [23], the FVP can be reduced to oscillatory power if the problem is self-adjoint, similar to the least action principle for mechanics.

**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 Variational Reaction Theory (VRT) [27] to derive the required matrix equation and found it a generalization of the conventional Galerkin's method.

**Difficulty in Proving the Shortcut Being Consistent with the Fundamental Variational Equation**

The stationarity of the FVP proposed in 1984 [23] is equivalent to the Maxwell equations and should hold for all linear high-frequency EM wave problems. The VRT [27] 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's method work? Why does the way to deal with the exterior field in the generalized Galerkin's 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) [28]. By PVP, the variation on a functional of trial field f and its adjoint fa, like the partial differentiation, equals the sum of two functionals (PVPs), where one is obtained by taking partial variation with respect to f while the other is taken with respect to fa. 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 just 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 this series of work are given below:

- First variational formulation equivalent to the Maxwell equations and associated boundary conditions. Also proposed are their physical interpretations.
- A systematic approach to derive the FEM matrix equations.
- First inclusion of exterior field in the variational equation, which leads to the now popular hybrid FEM.
- First shortcut to derive the FEM matrix equations directly, which is equivalent to a generalized Galerkin's method and widely used in academic and commercial software development circles.
- A conformal mapping technique for dielectric 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 papers, the generalized Galerkin's 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 [29], [30]; Cendes and Lee, 1988 [31]; Lee, Sun, and Cendes, 1991 [32]), and those emphasizing hybrid FEM for the scattering of objects in free space (Jin and Liepa, 1988 [33]; Jin and Liepa, 1988 [34]). The transfinite element method was then applied for 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 [35] and Shen et al., 1998 [36]). However, the impact of the generalized Galerkin's 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's method with the reciprocity theorem in EM fields and the popular Bi-CG methods for solving complex symmetric but non-Hermitian matrices equations.

**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 [37]). 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 [38], [39] and a rectangular waveguide in 1981 [40]. 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 [27]. Finally, in 1988, the FEM@NTU team proposed the partial variational principle [28] 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: the scattering of the perfect electric conductor (PEC) cylinder with inhomogeneous dielectric coating [23], general scattering from anisotropic inhomogeneous slabs [41], [42] and cylinders [43], scattering from a magnetostatic slab in parallel-plate waveguide [24], general analysis of dielectric waveguides [45], [27], birefringence analysis of anisotropic optical fibers [44], and analysis of discontinuities in a dielectric waveguide [28] and planar dielectric waveguides [45], and planar dielectric antennas [46]. For weakly guiding dielectric waveguides [46], microstrip lines [47], [48], and coplanar waveguides [49], [50], the FEM@NTU team conceived a delicate idea, the Variational Conformal Mapping technique [26], to transform the problem onto a complex plane, and wherein solve a scalar field.

**Further information**

Jin-Fa Lee is the major developer of the booming commercial software HFSS. He took Professor Chen's course on mathematical physics, which introduced variational formulation coupled with the FEM in 1981 when he was an undergraduate student in the National Taiwan University's electric engineering department. Afterward, he pursued his Ph.D. studies at Carnegie Mellon University under the supervision of Professor Zoltan J. Cendes, father of HFSS and the founder of Ansoft Corp. Note Ansoft Corp. was later acquired by Ansys Corp.

**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. General variational principle for interior field problems and simple one-dimensional **
transmission-reflection problems

The work [22] is included as a reference in Chapter 6, Variational Principles for Electromagnetics, The Finite Element Method in Electromagnetics, by Jianming Jin, New York, USA: John Wiley & Sons, Inc., 1993, a popular textbook for learning the FEM for electromagnetics. In that chapter, Professor Jin 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 FEM … However, even these methods remain unknown to many researchers and generally are not taught to graduate students. This situation contrasts Galerkin's 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's method. However, the FEM@NTU team also found this shortcut [27] and provided it with a more rigorous theoretical foundation [28].

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 [23], especially in including the exterior field into the variational equation. In Jin's book, the functional given is derived from the generalized Galerkin's method while fine-tuning some boundary terms to ensure that the functional is stationary.

Another book, Finite Elements for Wave Electromagnetics: Method and Techniques, edited by Peter P. Silvester and Giuseppe Pelosi, New York, USA: IEEE Press, 1994, collected papers important to applications of the FEM to electromagnetics. The FEM@NTU contributions [22] about the general variational formulation and [43] about the VRT were included.

**2. Fundamental Variational Principle (FVP) and inclusion of exterior field as hybrid finite **
element method

The FVP was proposed in 1984 [23]. It has been applied to model the scattering matrix coefficients for microwave circuit elements, such as [24]. The transfinite element method in [31] used a similar approach, though the FEM matrix equations are obtained from the generalized Galerkin's method.

A similar approach with another hybrid FEM for dielectric waveguide analysis was utilized by Su [25], also Professor Chun Hsiung Chen's student.

Such a hybrid FEM for scattering problems was a contribution of Jin [33].

**3. Variational Reaction Theory (VRT) as a generalized Glerkin's method**

As mentioned above, the VRT [27] became a popular shortcut as a generalized Galerkin's method in deriving equations for FEM. A lot of published FEM works were based on this approach. One typical example is applying the VRT to arbitrary two-dimensional EM problems with general anisotropic material [43]. Based on a unified (E_z, H_z) formulation, the scattering problem occurs when a plane wave is obliquely incident upon an inhomogeneous and anisotropic dielectric cylinder. The variational equation is then derived with all the exterior problems absorbed into the boundary operator and solved by the FEM. The scattering cross sections of radiation problems can be obtained. In addition, the guiding problems of dielectric waveguides are also solved by considering obliquely incident inhomogeneous waves.

**4. Partial Variational Principle to connect the VRT and the FVP and its applications to the **
analysis of discontinuities in dielectric waveguides

This is the topic of the FEM@NTU paper in 1988 [28]. Its two further similar applications are published in 1989 [45] and 1990 [46].

**5. Variational Conformal Mapping Techniques**

The 1986 paper of the FEM@NTU proposed such an idea in [26] for dielectric waveguide problems. Because it applied a clever idea to map the scalar potentials in the transverse plane into the complex plane. The original exterior field distribution is now in a closed region and can be solved easily by the FEM. With such a technique, problems with edge singularities like the microstrip and coplanar waveguide propagation problems can also be taken care of easily. References [47]-[50] are good examples.

**References**

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[17] J. P. Webb, G. L. Maile, and R. L. Ferrari, "Finite Element Solution of Three-Dimensional Electromagnetic Problems," IEE Proc. H, vol. 130, no. 2. pp. 153-159, 1983. doi:10.1049/ip-h-1.1983.0025.

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[19] B. H. McDonald and A. Wexler, "Finite-element solution of unbounded field problems," IEEE Transactions on Microwave Theory and Techniques, vol. 20, no. 12, pp. 841-847, Dec. 1972. doi: 10.1109/TMTT.1972.1127895.

[20] K. Mei, "Unimoment method of solving antenna and scattering problems," IEEE Transactions on Antennas and Propagation, vol. 22, no. 6, pp. 760-766, Nov. 1974. doi: 10.1109/TAP.1974.1140894.

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[22] C. H. Chen and C.-D. Lien, "The variational principle for non-self-adjoint electromagnetic problems," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-28, no. 8, pp. 878-886, Aug. 1980. doi: 10.1109/TMTT.1980.1130186.

[23] S.-K. Jeng and C. H. Chen, "On variational electromagnetics: Theory and application," IEEE Transactions on Antennas and Propagation, vol. AP-32, no. 9, pp. 902-907, Sept. 1984. doi: 10.1109/TAP.1984.1143439

[24] H.-C. Chang, S.-K. Jeng, R.-B. Wu, and C. H. Chen," Propagation of waves through magnetoplasma slab within a parallel-plate guide," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-34, no. 1, pp. 50-54. Jan. 1986, doi: 10.1109/8.1074

[25] C.-C. Su, "A combined method for dielectric waveguides using the finite-element technique and surface integral equations method," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-34, no. 11, pp. 1140-1146, Nov. 1986. doi: 10.1109/ TMTT.1986.1133511

[26] R.-B. Wu and C. H. Chen, "A scalar variational conformal mapping technique for weakly guiding dielectric waveguides," IEEE Journal of Quantum Electronics, vol. QE-22, no. 3, 1986, pp. 603-609. doi: 10.1109/JQE.1986.1073014

[27] R.-B. Wu and C. H. Chen, "On the variational reaction theory for dielectric waveguides," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-33, no. 6, pp. 477-483, June 1985. doi: 10.1109/TMTT.1985.1133102

[28] S.-J. Chung and C. H. Chen, "Partial variational principle for electromagnetic field problems: Theory and applications," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-36, no. 3, pp. 473-479, Mar. 1988. doi: 10.1109/22.3537

[29] J.‐F. Lee and Z. J. Cendes, "Transfinite elements: A highly efficient procedure for modeling open field problems," Journal of Applied Physics 61, 3913 (1987). https:// doi.org/10.1063/1.338582

The variational formulation in this paper is similar to the one in a FEM@NTU 1982 paper: 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

[30] J.-F. Lee and Z. J. Cendes, "The transfinite element method for computing electromagnetic scattering from arbitrary lossy cylinders," in 1987 International Symposium on Antennas and Propagation, pp. 99-102, Jan. 1987, doi: 10.1109/ APS.1987.1150085.

The variational formulation in this paper is similar to the one derived in a FEM@NTU 1984 paper [6].

[31] Z. J. Cendes and J.-F. Lee, "The transfinite element method for modeling MMIC devices," IEEE Transactions on Microwave Theory and Techniques, vol. 36, no. 12, pp. 1639-1649, Dec. 1988. doi: 10.1109/22.17395. , [32] J. F. Lee, D. K. Sun and Z. J. Cendes, "Tangential vector finite elements for electromagnetic field computation," IEEE Transactions on Magnetics, vol. 27, no. 5, pp. 4032-4035, Sept. 1991. doi: 10.1109/20.104986.

[33] 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

[34] J.-M. Jin and V. V. Liepa, "A note on hybrid finite element method for solving scattering problems," IEEE Transactions on Antennas and Propagation, vol. 36, no. 10, pp. 1486-1490, Oct. 1988. doi: 10.1109/8.8638.

[35] J.-M. Jin and J. L. Volakis, "Scattering and radiation from microstrip patch antennas and arrays residing in a cavity," in 1991 International Symposium on Antennas and Propagation, pp. 657-660 vol.2. 1991, doi: 10.1109/APS.1991.174925.

[36] X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, "On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering," IEEE Transactions on Antennas and Propagation, vol. 46, no. 3, pp. 303-311, March 1998. doi: 10.1109/8.662648.

[37] C. H. Chen, "Some remarks on exterior electromagnetic boundary value problems for spheres," IEEE Transactions on Antennas and Propagation, vol. 18, no. 5, pp. 705-707, Sept. 1970. doi: 10.1109/TAP.1970.1139747.

[38] C. H. Chen and C.-D. Lien, "A finite element solution of the wave propagation problem for an inhomogeneous dielectric slab," IEEE Transactions on Antennas and Propagation, vol. 27, no. 6, pp. 877-880, Nov. 1979. doi: 10.1109/TAP.1979.1142199.

[39] C.-H. Chen and Y.-W. Kiang, "A variational theory for wave propagation in a one-dimensional inhomogeneous medium," IEEE Transactions on Antennas and Propagation, vol. 28, no. 6, pp. 762-769, Nov. 1980. doi: 10.1109/TAP.1980.1142435.

[40] C.-T. Liu and C. H. Chen, "A variational theory for wave propagation in inhomogeneous dielectric slab loaded waveguides," IEEE Transactions on Microwave Theory and Techniques, vol. 29, no. 8, pp. 805-812, Aug. 1981. doi: 10.1109/TMTT.1981.1130451.

[41] S.-K. Jeng and C. H. Chen, "Variational finite element solution of electromagnetic wave propagation in a one‐dimensional inhomogeneous anisotropic medium," Journal of Applied Physics 55, 630, 1984. https://doi.org/10.1063/1.333115

[42] S.-K. Jeng, R.-B. Wu and C. H. Chen, "Waves obliquely incident upon a stratified anisotropic slab: A variational reaction approach," Radio Science, vol. 21, no. 4, pp. 681-688, July-Aug. 1986, doi: 10.1029/RS021i004p00681.

[43] R.-B. Wu and C. H. Chen, "Variational reaction formulation of scattering problem for anisotropic dielectric cylinders," IEEE Transactions on Antennas and Propagation, vol. 34, no. 5, pp. 640-645, May 1986, doi: 10.1109/TAP.1986.1143874.

[44] R.-B. Wu and C. H. Chen, "Birefringence analysis of anisotropic optical fibers using variational reaction theory," IEEE Transactions on Microwave Theory and Techniques, vol. 34, no. 6, pp. 741-745, June 1986, doi: 10.1109/TMTT.1986.1133428.

[45] S.-J. Chung and C. H. Chen, "A partial variational approach for arbitrary discontinuities in planar dielectric waveguides," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-37, no. 1, pp. 208 - 214, Jan. 1989. doi: 10.1109/22.20040

[46] S.-J. Chung and C. H. Chen, "A partial variational analysis of planar dielectric antennas," IEEE Transactions on Antennas and Propagation, vol. 39, no. 6, pp. 713 - 718, Jun. 1991. doi: 10.1109/8.86867

[47] C. Shih, R. -B. Wu, S.-K. Jeng, and C. H. Chen, "A full-wave analysis of microstrip lines by variational conformal mapping technique," IEEE Transactions on Microwave Theory and Techniques, vol. 36, no. 3, pp. 576-581, Mar. 1988, doi: 10.1109/22.3551.

[48] C. Shih, R.-B. Wu, S.-K. Jeng, and C. H. Chen, "Frequency-dependent characteristics of open microstrip lines with finite strip thickness," IEEE Transactions on Microwave Theory and Techniques, vol. 37, no. 4, pp. 793-795, April 1989, doi: 10.1109/22.18856.

[49] C. –N. Chang, Y. –C. Wong, and C. H. Chen, "Full-wave analysis of coplanar waveguides by variational conformal mapping technique," IEEE Transactions on Microwave Theory and Techniques, vol. 38, no. 9, pp. 1339 - 1344, Sep. 1990, doi: 10.1109/22.58662.

[50] C. –N. Chang, W. –C. Chang, and C. H. Chen, "Full-wave analysis of multilayer coplanar lines," IEEE Transactions on Microwave Theory and Techniques, vol. 39, no. 4, pp. 747 - 750, April 1991, doi: 10.1109/22.76444.

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