Two Far-field Approximation Based Fast IPO Methods for Scattering Analysis of Large Objects

Jeong-Un Yoo; Il-Suek Koh

doi:10.26866/jees.2026.4.r.373

J. Electromagn. Eng. Sci > Volume 26(4); 2026 > Article

Yoo and Koh: Two Far-field Approximation Based Fast IPO Methods for Scattering Analysis of Large Objects

Regular Paper

Journal of Electromagnetic Engineering and Science 2026;26(4):369-376.

Published online: July 31, 2026

DOI: https://doi.org/10.26866/jees.2026.4.r.373

Two Far-field Approximation Based Fast IPO Methods for Scattering Analysis of Large Objects

Jeong-Un Yoo

, Il-Suek Koh^*

Department of Electrical and Computer Engineering, Inha University, Incheon, Korea

^*Corresponding Author: Il-Suek Koh (e-mail: ikoh@inha.ac.kr)

Received August 28, 2025 Revised November 7, 2025 Accepted December 3, 2025

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

To reduce the numerical complexity of the conventional iterative physical optics algorithm to compute the scattering by a large scatterer, we propose two fast algorithms to accelerate the far-field interaction between the facets located within the far-field region for each facet. The interaction is approximated using the far-field approximation based on the octree grouping of the facets. The first algorithm considers the exact lit and shadow check between two facets, which corresponds to the physical optics approximation. The second algorithm mitigates the burden of the exact shadow check by checking at the leaf box, which can reduce the numerical complexity further. The accuracy and computation time of the two algorithms are examined for several objects and compared to those of the conventional iterative physical optics algorithm scheme and the multilevel fast multipole method.

Keywords: Electromagnetic Scattering, Iterative Physical Optics, Radar Cross Section

I. Introduction

The radar cross section (RCS) analysis for large-scale objects such as aircraft, ships, etc. is essential in electromagnetic scattering problems. Accurate and efficient computation of the RCS of electrically large targets poses significant challenges due to the substantial computational burden required to fully calculate the electromagnetic interactions over a widely distributed object.

Full-wave methods, such as the method of moments (MoM) and its accelerated variants like multilevel fast multipole method (MLFMM), can offer highly accurate solutions for scattering problems, but demand significant computational resources. Consequently, this can limit their application to electrically large scatterers [1].

Iterative physical optics (IPO) is a high-frequency approximation (HFA) technique that predicts the surface current based on the physical optics (PO) approximation [2]. Initially developed for analyzing open-ended cavities made of perfect electric conductor (PEC) materials, the IPO method was later extended to impedance media satisfying the impedance boundary conditions (IBC) [2, 3]. Beyond cavities, the IPO scheme has been applied to RCS prediction for realistic large objects like tanks, aircraft, and rough ocean surfaces [4–6]. This method offers significantly faster computation than full-wave solvers while maintaining sufficient accuracy.

To efficiently analyze electromagnetic scattering from large structures, the fast far-field approximation (FaFFA) technique has been introduced [7]. FaFFA, similar in principle to the fast multipole method (FMM), reduces computational complexity down to O(N^1.33) by approximating the interactions between spatially distinct sub-scatterers. In this approach, well-separated surface facets are grouped, and the interactions are computed between these groups rather than individual elements. The method was later extended to the IPO as demonstrated in [8], where it significantly accelerates the iterative process by approximating group interactions. Furthermore, in [9], a hybrid acceleration technique IPO-MLFMM was proposed, employing multilevel grouping and the addition theorem, leading to even greater computational efficiency for large-scale problems. Subsequently, various acceleration techniques based on domain decomposition were proposed in [10, 11], enabling more efficient scattering analysis of electrically large structures.

While the FaFFA scheme significantly accelerates IPO computation, it cannot exactly account for the lit/shadow check among facets, which is crucial for the PO approximation. This limitation can reduce the accuracy of the fast IPO scheme. To partially mitigate this negative effect, a simple shadow check scheme, widely used in various applications, is applied to the fast IPO scheme in [12]. Furthermore, acceleration through hardware advancements has been actively researched. Some approaches parallelize the iterative current update process on CPU-GPU heterogeneous platforms using frameworks like compute unified device architecture (CUDA) [13]. Others have focused on accelerating the primary bottleneck of determining shadow and lit regions by using dedicated ray-tracing engines like NVIDIA OptiX, which reduces the preprocessing time required for visibility checks [14].

First, we propose a fast IPO scheme that incorporates a full shadow check based on the far-field approximation and the MLFMM acceleration algorithm. To further enhance its speed, we introduce a second method that allows a small, controlled error in the shadow check, which further accelerates the proposed fast IPO scheme. Section II details the proposed scheme, while Section III presents several numerical examples. The convention of e^jwt is assumed and suppressed throughout the paper.

II. Fast Iterative Physical Optics

1. Conventional IPO Update Equation

The surface of the scatterer is discretized into many small flat triangular facets. The surface is assumed to consist of an inhomogeneous impedance material, whose normalized surface impedance is η(r→). Two surface currents on the scatterer are related to each other due to the IBC as J→m(r→)=η(r→)Z0J→e(r→)×n^, where n^=n^(r→) is the normal vector of the facet, and Z₀ is the free-space wave impedance. j→e(r→) and j→m(r→) are the electric and magnetic currents, respectively. The first-order PO current on the facets in the lit region of the incident wave is assigned by J→ePO(r→)=2n^×H→i(r→), where H→i(r→) is the incident magnetic field. On facets in the shadow region, the PO currents are set to zero [3]. From the first-order current, the IPO current is updated based on the magnetic field integral equation (MFIE).

The MFIE for the impedance object is written as

(1)

J→e(r→)=2n^×H→i(r→)+2n^×∇→×∬s′J→e(r→′)G0(R)ds′-2jη(r→)k0n^×∇→×∇→×∬s′[J→e(r→′)×n^′]G0(R)ds′

where n^′=n^(r→′), and k₀ is the free-space wavenumber. G₀(R) is the free-space Green’s function given by e⁻^jk^₀^R/4πR, where R=∣R→∣. |·| is the vector norm and R→=r→-r→′ is the vector between the source and observation points as seen in Fig. 1. The conventional IPO update equation is explicitly expressed [4] as

(2)

J→en(r→)=J→en-1(r→)-12πn^×∬s′G1(R)[R→×J→en-1(r→′)]ds′-η(r→)j2πk0n^×∬s′G2(R)[n^′×J→en-1(r→′)]ds′-η(r→)j2πk0∬s′G3(R)[n^′×J→en-1(r→′)·R→][n^×R→]ds′

where J→en(r→) is the surface current at the n-th iteration. G_1,2,3(R) is given by

(3)

G1(R)=1+jk0RR3e-jk0RG2(R)=-1-jk0R+k02R2R3e-jk0RG3(R)=3+3jk0R-k02R2R5e-jk0R

To accelerate the computation of Eq. (1), the facets are grouped and formed into the octree, and the interactions among the facets in cubes are approximated by the far-field approximation. The near-field interaction between the facets is directly calculated using Eq. (2) like the conventional IPO method.

2. Far-field Approximation between Cubes

Fig. 1 shows the coordinates of two cubes located in the far-field region relative to each other, whose length is L_c and L_c’, respectively. First, the far-field criteria for the cubes are derived.

Using the Taylor expansion, R= ∣Δr→+r→c-Δr→′-r→c′∣ can be approximated as

(4)

R=(Δr→+r→c-Δr→′-r→c′)·(Δr→+r→c-Δr→′-r→c′)=Rcc′1+2Rcc′R^cc′·ΔR→cc′+ΔRcc′2Rcc′2≈Rcc′+R^cc′·ΔR→+12Rcc′[ΔR^2-(R^cc′·ΔR→)2]

where ΔR→=Δr→-Δr→′. The subscript of R→cc′ indicates that it is the vector from the center of the box C’ to that of C. R^cc′ is the unit vector of R→cc′ defined as R^cc′=R→cc′/Rcc′. Due to the Rayleigh criterion, the maximum of k0 [ΔR^2-(R^cc′·ΔR→)2]/2R^cc′ should be less than or equal to π8 as k02Rcc′ΔR2=max {k02Rcc′∣Δr→-Δr→′∣2}≤π8. Therefore, the far-field criterion can be obtained as

(5)

6(Lc+Lc′)2λ0≤Rcc′

where λ₀ is the free-space wavelength. Interactions that do not meet this far-field criterion are treated as near-field and are calculated rigorously using Eq. (2).

When two cubes are in the far-field region, the scalar Green’s function, e⁻^jk^₀^R/4πR can be approximated using the following approximations: for the phase term, R≈Rcc′+R^cc′·ΔR→=Rcc′+R^cc′·Δr→-R^cc′·Δr→′ and for the magnitude term, R ≈ R_cc’. Therefore, G₀(R) can be asymptotically written as

(6)

G0(R)~e-jk0Rcc′4πRcc′·e-jk0R^cc′·Δr→·ejk0R^cc′·Δr→′

Eq. (6) can be expressed in terms of the multiplication of three terms: the translation term, e⁻^jk^₀^R^_cc’/4πR_cc’, the disaggregation term, e-jk0R^cc′·Δr→, and the aggregation term, ejk0R^cc′·Δr→′. The first, second and third terms are dependent on only the displacement of the box center, observation point and source, respectively. Consequently, three terms can be calculated individually, which drastically improves the interaction computation efficiency. The del operators in Eq. (1) can be approximated in the far-field region as

(7)

∇→~-jk0R^cc′ and ∇→′~-jk0R^cc′

For the PO method, only the radiated field from facets located within the lit region of a facet is considered. Thus, two acceleration algorithms can be formulated, denoted as fast far-field IPO algorithm 1 (FF-IPO1) and FF-IPO2. The FF-IPO1 can exactly consider the lit and shadow condition between the facets like the conventional IPO scheme. On the other hand, the FF-IPO2 consider the lit and shadow condition between the leaf boxes, which is the smallest box in the octree. It can allow small errors but can improve numerical efficiency.

3. FF-IPO1

When the far-field condition is satisfied, Eq. (1) can be simplified by applying Eqs. (6) and (7) to Eq. (1). First, the required spatial derivatives of the scalar Green’s function in Eq. (1) can be asymptotically approximated as

(8)

∇→×[J→e(r→′)G0(R)]~-jk0e-jk0Rcc′4πRcc′R^cc′×J→e(r→′)e-jk0R^cc′·Δr→ejk0R^cc′·Δr→′∇→×∇→×[J→e(r→′)G0(R)]~-k02e-jk0Rcc′4πRcc′R^cc′×R^cc′×J→e(r→′)e-jk0R^cc′·Δr→ejkk0R^cc′·Δr→′

Fig. 2 shows the overall procedure of the accelerated computation for the FF-IPO1, which is very similar to MLFMM, except for the inexact computation of the Green’s function. The leaf box containing the observation facets is denoted as c_q, where the subscript q is the box index. First, determine the largest box, denoted as cq′, that satisfies Eq, (5) for the observation box c_q. All facets located within the largest box level ( cq′) are in the far-field region with respect to c_q. Therefore, the radiated field from all facets within cq′ can be efficiently calculated using Eq. (8). For one facet in c_q, first, J→en(r→′)ejk0R^cqc′·Δr→′ of each facet in cq′ is calculated. The terms of J→en(r→′)ejk0R^cc′·Δr→′ is summed over all facets in cq′, which lie within lit region of the observation facet. Then, the total summation for cq′ is translated and disaggregated into the observation facet in c_q with multiplied by e-jk0Rcqc′4πRcc′·ejk0R^cqc′·Δr→′, as depicted in Fig. 2. Therefore, the surface current at the n-th iteration can be updated in more simplified form than Eq. (2) as

(9)

J→en=J→en-1-jk02π∑q=0Nl∑q1=0Nf(q){e-jk0R^cqc′·Δr→qq1n^q1(r→)×e-jk0Rcqc′Rcqc′R^cqc′×∑p=0lit onlyNf′(q1)∫sfp[J→en-1(r→′)+ηR^cqc′×J→en-1(r→′)×n^′] ejk0R^cqc′·Δr→p′dsfp}

where s_f_{_p} is the surface of the p-th facet. N_l is the total number of leaf boxes. N_f₍_q₎ is the number of facets contained within the q-th leaf box (c_q). For the q₁-th facet in c_q, Nf′(q1) is the number of facets in cq′. Since only the facets located within the lit region are considered, this condition is emphasized by the “lit only” subscript in the summation symbols. Eq. (9) is the update algorithm for the FF-IPO1 scheme for the far-field interaction. As previously explained, the near-field interaction is directly calculated using Eq. (2). Fig. 2(a) and 2(b) illustrate the far-field interactions for two distinct observation facets, q₁ and q₂, respectively. Since q₁ and q₂ each possess a distinct set of lit facets, the radiation is calculated only from the corresponding lit facets for each case.

4. FF-IPO2

In the computation of FF-IPO1, the filtering process for facets located within the shadow region can be somewhat cumbersome. Since the leaf box typically contains a very small number of facets (e.g. fewer than ten), performing the shadow check only at the leaf-box-to-leaf-box level introduces an approximation error. However, this error is often sufficiently small to be acceptable. The box-wise shadow check offers a significant advantage: the aggregation for each leaf box can be calculated once and then reused for all far-field interactions, which drastically reduces the numerical complexity. This constitutes the main difference between FF-IPO1 and FF-IPO2. Apart from this box-wise approach, the overall procedure of FF-IPO2 is otherwise identical to that of FF-IPO1. The updating equation, Eq. (9) can be expressed as

(10)

J→en=J→en-1-jk02π∑q=0Nl∑q1=0Nf(q)e-jk0R^cqc′·Δr→qq1n^q1(r→)×e-jk0Rcqc′Rcqc′R^cqc′×∑p=0lit boxNl′(q)ejk0R^cqc′·r→c′J→˜p

where Nl′(q) is the number of leaf boxes in cq′ and J→˜p is calculated once for every leaf box and defined as ∑o=0Nf(p)∫sfo[J→en-1(r→′)+ηR^cqc′×J→en-1(r→′)×n^′] ejk0R^cqc′·(Δr→o′+r→o)dsfo. r→p is the vector to the center of the p-th box. r→o and r→c′ are shown in Fig. 3, where the overall procedure and geometry of the FF-IPO2 scheme are illustrated.

III. Numerical Results

To validate the far-field approximation of the dyadic Green’s function, Eq. (8), the source cube, c’ with a side length of 4λ₀, is centered at the origin. A point source is located at one of the cube’s vertices, specifically at (2λ₀, 2λ₀, −2λ₀). When considering a point observation ( Δr→=0), the minimum distance for the far-field region can be calculated as 96λ₀ using Eq. (5). A total of 150 observation points are generated as (96λ₀sinθ_scosφ_s, 96λ₀sinθ_scosφ_s, 96λ₀cosθ_s), where θ_s and φ_s are simultaneously varied from 0° to 180° and from 0° to 359°, respectively. J→e=13(x^+y^+z^) is assumed. Fig. 4 shows the comparison between the exact and approximate computation of Eq. (8). The exact formulation of Eq. (8) can be given by

(11)

∇→×[J→eG0(R)]=-G1(R)4πR→×J→e∇→×∇→×[J→eG0(R)]=G2(R)4πJ→e+G3(R)4π(R→·J→e)R→

It is observed that the far-field approximation is very accurate, and its accuracy may increase for greater distances between the box center and the observation point. Consequently, the proposed far-field interaction calculation scheme is very accurate. The phase comparison for the double curl equation shows behavior very similar to that for the single curl case shown in Fig. 4(b), which is omitted.

To evaluate the performance of the two proposed FF-IPO algorithms, three large and complicated objects are considered, which have different dimensions and operating frequency. To assess computational efficiency and accuracy, the scattering from the objects is computed using the conventional IPO, the proposed FF-IPO schemes, and the commercial FEKO MLFMM. The iterative process for three IPO schemes is terminated by the same stopping criterion, detailed in [4]. The iterations stop when the residual error falls below a tolerance of 10⁻³ or when the error begins to diverge.

The accuracy is then quantified using the normalized root mean squared error (NRMSE), defined as

(12)

NRMSE=1σmaxMLFMM-σminMLFMM1Nσ∑i=1Nσ(σiMLFMM-σiIPO)2

where N_σ is the number of observation points. σ^MLFMM and σ^IPO are the RCS values, computed by the MLFMM and IPO, respectively. The subscripts, “max” and “min’ denote the maximum and minimum values of σ^MLFMM.

The objects considered are a fighter aircraft, an unmanned aerial vehicle (UAV) [15], and a missile. Only the missile is positioned above two rough ocean surfaces. The fighter comprises three distinct materials: a dielectric radome, dielectric canopy, and an impedance body. The dielectric radome and canopy are modeled as an impedance surface with an appropriate surface impedance as η_r = 0.53 [16] and η_r = 0.71 + j0.1 [17] for the radome and canopy, respectively. The body’s normalized surface impedance (η) is 0.38 – j0.06 [17]. The UAV consists of a homogeneous material with a PEC substrate coated with a laminate radar-absorbing structure (RAS). The RAS is a nonmagnetic dielectric layer, whose thickness and relative permittivity are given by t = 0.02λ₀ and ɛ_r = 6.6 – j2.8, respectively. Consequently, the normalized impedance of the UAV can be approximated as η=j1/ɛrtan (k0tɛr)=0.38-j0.06 [18]. The final object includes two distinct surfaces: η_m = 0.38 – j0.06 for the missile surface and η₀ = 0.0926 – j0.0371 for the ocean surface. The impedance of the ocean surface is determined based on environmental parameters, including the temperature and salinity of 20°C and 32.54% [19]. The ocean surface is generated using the Pierson-Moskowitz (PM) spectrum with wind speed of 10 m/s. The root-mean-square (RMS) height of the generated rough surface is 10.6λ₀. To increase the simulation dimension, two sizes of the ocean surface are considered: 160λ₀ × 160λ₀ and 320λ₀ × 320λ₀. Table 1 summarizes the geometries, material parameters, operating frequency, number of facets, average number of facets in the leaf box, and other relevant details. The subscripts “i“ and “s“ in θ and φ, respectively indicate the incidence and observation point. The far-ratio is defined as the average, over all leaf boxes, of the ratio of the number of boxes in the far-field region (relative to that leaf box) to the total number of boxes. As previously mentioned, the average number of facets per leaf box is approximately 5, remaining well below 10.

Fig. 5 compares the RCS results for three different targets, which are computed by the conventional IPO, two proposed FF-IPO schemes and the MLFMM as a reference solution. The NRMSE for the conventional IPO, FF-IPO1, and FF-IPO2 are given by 17.12 m², 17.12 m², and 17.19 m², respectively. As expected, the error for the FF-IPO2 shows a slight increase. In Fig. 5(a) and 5(b), the error between FF-IPO1 and FF-IPO2 can be observed in the low RCS intervals, but the difference is negligible.

Fig. 6 illustrates the source of this error. It occurs because the group-wise visibility check in the FF-IPO2 scheme erroneously includes some shadowed facets (light gray) in the lit region interaction. Since the leaf box contains a small number of facets, the number of these misclassified facets is very small compared to the total number of facets. Fig. 7 compares the computational cost of the two proposed FF-IPO schemes. The cost is computed as the ratio of the FF-IPO scheme’s cost to that of the conventional IPO scheme. As expected, both computation time and the number of multiplications decrease with increasing the far-ratio in Table 1. The FF-IPO2 scheme demonstrates superior efficiency compared to the FF-IPO1 scheme. For the larger ocean case, for example, the simulation time of FF-IPO1 and FF-IPO2 is reduced to approximately 33% and 25%, respectively.

Finally, the asymptotic numerical complexity (number of multiplication) is estimated as the number of facets (N) increases. Fig. 8 shows this complexity: the conventional IPO exhibits a complexity of O(N²), while those of FF-IPO1 and FF-IPO2 are reduced to O(N^1.8) and O(N^1.33), respectively. Here, O(·) is the big O symbol. The numerical complexity of the (dis)aggregation and translation is summarized in Table 2 for the simulation geometry. The translation is the main bottleneck of the FF-IPO2. To compute the complexity in Table 2, only the far-field case is considered, so that the number off multiplications for the aggregation is less than O(N).

IV. Conclusion

This paper introduces two FF-IPO algorithms to accelerate the RCS analysis of large-scale targets. Both algorithms apply the far-field approximation to reduce the computational burden of the conventional IPO scheme. Specifically, the FF-IPO1 scheme precisely performs the lit/shadow check on a facet-wise, while the FF-IPO2 scheme conducts this check solely between leaf boxes. Although this approximation in FF-IPO2 may introduce a slight error, it significantly enhances numerical efficiency. For three complicated objects, the proposed FF-IPO schemes are compared against the MLFMM. As expected, the accuracy of the FF-IPO1 scheme is maintained accuracy without degradation. The FF-IPO2 scheme’s accuracy was only slightly reduced, compared to that of the conventional IPO scheme. However, the numerical burden is drastically lowered, exhibiting a numerical complexity of O(N^1.33). For instance, the maximum simulation time reduction for the larger ocean simulation is 33% for FF-IPO1 and 25% for FF-IPO2. Further reductions are attainable for larger scatterers, as the far-field interactions increase rapidly with size. Consequently, these fast IPO schemes prove highly effective for analyzing scattering from large objects.

Notes

This work was supported by the Agency For Defense Development Grant Funded by the Korean Government (UD230016JD).

Fig. 1

Far-field approximation between two cubes.

Fig. 2

Proposed facet-wise far-field interaction for FF-IPO1: (a) for Δr→qq1 and (b) for Δr→qq1.

Fig. 3

Proposed group-wise far-field interaction for FF-IPO2.

Fig. 4

Comparison of exact and approximate computations of two curl equations in eq. (8). (a) Magnitude of ∇→×[J→eG0(R)], (b) phase of ∇→×[J→eG0(R)], and (c) magnitude of ∇→×∇→×[J→eG0(R)].

Fig. 5

RCS comparison for three targets: (a) fighter, (b) UAV, and (c) missile above rough ocean surface (160λ₀×160λ₀×76λ₀).

Fig. 6

Example of visibility misclassification in the FF-IPO2, showing the exact lit facets (dark gray) and erroneously classified facets (light gray).

Fig. 7

Computational efficiency of FF-IPO1 and FF-IPO2 compared to conventional IPO for four scatterers.

Fig. 8

Comparison of numerical complexity of conventional IPO, FF-IPO1, and FF-IPO2 as a function of N.

Table 1

Objects and simulation information

Object	Frequency dimension (x×y×z)	# of facets (N)	Incidence observation angle	Material	Average # of facets per leaf box	Far-ratio
Fighter	2 GHz 106λ₀×88λ₀×28λ₀	60,244	θ_i = 45°, φ_i = 0° θ_s = 0°–360° φ_s = 0°	radome: η_r = 0.53 [16] canopy: η_c = 0.71–j0.01 [17] body: η = 0.38–j0.06 [17]	4.8	63%
UAV [15]	10 GHz 125λ₀×125λ₀×10λ₀	84,232	θ_i = 45°, φ_i = 0° θ_s = 0°–360° φ_s = 0°	η = 0.38–j0.06	6.0	75%
Missile over ocean	4 GHz missile: 104λ₀×11λ₀×11λ₀ ocean: 160λ₀×160λ₀×76λ₀	112,068	thetas;_i = 45°, φ_i = 0° θ_s = 90° to −90° φ_s = 90°	missile: η_m = 0.38–j0.06 ocean: η_o = 0.09–j0.04 [19]	4.9	82%
Missile over ocean	320λ₀×320λ₀×76λ₀	243,380			4.8	90%

Table 2

Computational complexity of three processes for FF-IPO2

Object	Aggregation	Translation	Disaggregation
Fighter	40,403 (N)	2.3×10⁶ (N^1.32)	56,564 (1.4N)
UAV	55,495 (N)	4.6×10⁶ (N ^1.33)	80,467 (1.4N)
Missile (160λ₀)	75,339 (N)	6.3×10⁶ (N ^1.34)	97,940 (1.3N)
Missile (320λ₀)	126,807 (N)	1.4×10⁷ (N ^1.34)	152,168 (1.2N)

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Biography

Jeong-Un Yoo, https://orcid.org/0000-0003-2363-296X received the B.S. degree in electronic engineering from Gangneung-Wonju National University, Gangneung, Korea, in 2017. He is currently working toward a Ph.D. degree in electronic engineering at Inha University, Incheon, Korea. His research interests include electromagnetic numerical analysis.

Biography

Il-Suek Koh, https://orcid.org/0000-0003-0014-2466 received B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Korea, in 1992 and 1994, respectively, and a Ph.D. from the University of Michigan, Ann Arbor, MI, USA, in 2002. In 1994, he joined LG Electronics Ltd., Seoul, as a research engineer. He is currently a professor at Inha University, Incheon, Korea. His research interests include wireless communication channel modeling and numerical and analytical methods for electromagnetic fields.