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

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Docket #:

*This is a draft proposal, that has not yet been submitted. To submit this proposal, click on "Actions" in the toolbar above, then "Edit with form". At the bottom of the form, check the box that says "Submit this proposal to the IEEE History Committee for review. Only check this when the proposal is finished" and save the page. *

**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 Finite-Element-Method for Electromagnetics

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

From 1979 to 1991, an electromagnetic-wave research group in the department of electrical engineering, National Taiwan University, laid the theoretical foundation for the Finite Element Method for general linear Electromagnetic field problems. They also found a straightforward derivation as a generalized Galerkin's method. It has been widely applied in academic researches 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一般線性電磁學問題的有限元素 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 the 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 electromagnetic fields, from radiation, scattering, waveguide, to high-frequency circuits. To get a matrix equation for FEM, we have to derive a variational equation first. The applications of finite element methods for static fields were understood, and the variational nature of the formulation was well documented. However, for electromagnetic wave radiations and scattering problems, the underlying physics is non-Hermitian, and the corresponding variational principle was lacking in the 1980s.

Consequently, the work's significance is: proposing a systematic method for derivation of variational equations, including exterior fields in the variational equations, and discovering a shortcut for deriving the FEM matrix equations directly. The inclusion of exterior field results in the popularity of the hybrid element method, and the shortcut method was a generalized Galerkin's method. Such a shortcut and the hybrid finite element method have been widely used for academic research and successful commercial software development, like the early versions of HFSS software (High-Frequency Structure Simulation).

**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.

**Unsystematic derivation of variational formulation**

For mechanics, a variational equation is often obtained from the least action principle. For electromagnetics, the application of the FEM for exterior field problems is scarce before 1984. The most related papers are: Silvester and Hsieh (1971) [1], McDonald and Wexler (1972) [2], the Unimoment method (Mei, 1974) [3], and Morishita and Kumagai (1977) [4]. The variational equations in these works are derived case by case after setting up systems of partial differential equations from the Maxwell equation, except for Morishita and Kumagai in 1977, in which the formulation started from the least action principle generalized for electromagnetic 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) [5] and 1984 (with extension to general exterior problems) [6] that the FEM equations for any linear Electromagnetic 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 Dealing with Exterior Fields**

Silvester and Hsieh [1] 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 [2] 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 [3] 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 with 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 [6] and propagation modes in general waveguide [7]) or an integral over the artificial interior-exterior boundary [8]. Such expressions are then included in the functional with careful treatment for keeping 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 [9]. The exterior fields are thus taken into considerations 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 [5] and 1984 [6] were with a physical interpretation of general reaction. Note also the stationarity of the FVP is 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 [6], the FVP can be reduced to a form as oscillatory power if the problem is self-adjoint, which is 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 Ritz procedure are pretty tedious, though systematic. The FEM@NTU team's 1985 paper introduced the Variational Reaction Theory (VRT) [10] 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 [6] is equivalent to the Maxwell equations and
should hold for all linear electromagnetic field problems. The VRT [10] 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)
[11]. 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), where one 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 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.3. First inclusion of exterior field in the variational equation, which leads to the now popular hybrid finite element method.
- 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 becomes 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 [12], [13]; Cendes and Lee, 1988 [14]; Lee, Sun, and Cendes, 1991 [15]), and those emphasizing hybrid finite element methods for the scattering of objects in free space (Jin and Liepa, 1988 [16]; Jin and Liepa, 1988 [17]). The transfinite element method was then applied for developing the commercial software HFSS for dealing with RF, microwave, and millimeter-wave circuits. The hybrid finite element method has also been proved efficient and effective in solving realistic scattering problems (for example, Jin and Volakis, 1991 [18] and Shen et al., 1998 [19]). However, the impact of the generalized Galerkin's method is not limited by its applications to solving electromagnetic wave problems with the finite element methods. 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 electromagnetic 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 published papers in IEEE journals (as early as 1970 [20]). 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 [21], [22] and a rectangular waveguide in 1981 [23]. Later the FEM@NTU team began to focus on finding a general variational equation for electromagnetic field problems. They published papers on the fundamental variational principles (FVP) in 1980 [5] and 1984 [6], respectively. A shortcut, the VRT, for deriving FEM matrix equations from the FVP was then proposed in their 1985 paper [10]. Finally, in 1988, the FEM@NTU team proposed the partial variational principle [11] 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 [6], general scattering from anisotropic inhomogeneous slabs [24], [25] and cylinders [26], scattering from a magnetostatic slab in parallel-plate waveguide [7], general analysis of dielectric waveguides [8], [10], birefringence analysis of anisotropic optical fibers [27], and analysis of discontinuities in a dielectric waveguide [11] and planar dielectric waveguides [28], and planar dielectric antennas [29], For weakly guiding dielectric waveguides [9], microstrip lines [30], [31], and coplanar waveguides [32], [33], the FEM@NTU team conceived a delicate idea, the Variational Conformal Mapping technique [9], 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.

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