Surface Integral Equation Methods for Multi-Scale Electromagnetic Problems

Zhen Peng∗1 and Jin-Fa Lee2

1 Department of Electrical & Computer Engineering, University of New Mexico, Albuquerque, NM, USA, pengz@unm.edu

2ElectroScience Laboratory., The Ohio State University, Columbus, OH, USA, lee.1863@osu.edu

Abstract

This work investigates the efficient and robust integral equation based solution of large multi-scale electromagnetic prob-

lems. The major technical ingredients in the proposed work include: (i) a scalable domain decomposition method for surface

integral equations via a novel multi-trace formulation, (ii) a discontinuous Galerkin boundary element method, which employs

discontinuous trial and testing functions without continuity requirements across element boundaries, and (iii) an optimized mul-

tiplicative Schwarz algorithm using complete second order transmission condition. The results obtained through this research

greatly simplify the model preparation and mesh generation for complex electromagnetic simulation. Moreover, It provide an

effective preconditioning scheme for the integral equation based solution of multi-scale problems. The strength and flexibility

of the proposed method will be illustrated by means of several challenge real-world applications.

1 Introduction

In the past decades, surface integral equation (SIE) methods have enjoyed great success in solving electromagnetic (EM)

radiation and scattering problems. Since surface-based modeling and analysis are used, it is easier to prepare analysis-suitable

models and often requires fewer unknowns to solve compared to a differential equation. However, the complexity of modern

engineering applications increases at a fast pace. We consider a plane wave scattering from a high-definition composite un-

manned aerial vehicle (UAV) at X-band, as shown in Fig. 1. It is a complex heterogeneous multi-scale EM scattering problem.

Of particular concern is that this complex platform is partially coated with multi-layer EM absorbers, which include frequency

selective surfaces, multi-layer impedance sheets and magnetic radar absorbing materials. Needless to say, such a multi-scale

EM problem is extremely challenging and taxes heavily on existing surface integral equation methods in terms of the desired

accuracy and the stability.

Fuselage B-Skin Fwd

Foam A ( r = 1.09 – j0.17)

Foam B ( r = 1.25 – j1.46)

Foam C ( r = 2.79 – j2.8)

Bulk Materials and Honeycomb

NWA 2 NWB 2

NWC 2

NWA 4 NWB 4

NWC 4

NWA 3

NWB 3

NWC 3 Wing Sleeve

Left

Wing Sleeve Right

Flap 4

Flap 3

NWA 1 NWB 1

NWC 1

Flap1

Flap 2

Tail Right

Rudder

Tail Left

Blade 1

Blade 3

Blade 2

Fiberglass ( r = 2.00 – j1.5)

Wood ( r = 5.00 – j0.0003)

Honeycomb, t=10mm (~ r = 1.24 – j0.006)

(a) A composite object

Wing T-Skin 4

Wing T-Skin 3

Wing T-Skin 2

Wing T-Skin 1

Wing B-Skin 1

Wing B-Skin 2

Wing B-Skin 4

Wing B-Skin 3

Fuselage-Skin Mid

Fuselage-Skin Fwd

Composite Skin Composite Skin Design Example

Absorber stackup unit cell.

εε

(b) Multi-layer absorber

Figure 1: Electromagnetic scattering from a mockup high-definition composite system

To overcome the difficulties, we have developed so-called multi-trace SIE methods [1–3]. These formulations have re-

ceived considerable attention recently as a promising domain decomposition approach to integral equation methods. The entire

computational domain is decomposed into a number of non-overlapping sub-regions. Each local sub-region is homogeneous

with constant material properties and described by a closed surface. Through this decomposition, we have introduced at least

two pairs of trace data as unknowns on interfaces between sub-regions. This multi-trace feature has two major benefits: the

localized SIE for the homogeneous sub-region problem is amenable to Calderon operator preconditioning, and the resulting

linear system of equations readily lend themselves to multiplicative Schwarz preconditioning. In this work, we will further

investigate a novel optimized second order transmission conditions to improve the convergence in the Schwarz iterations.

978-1-4673-5225-3/14/$31.00 ©2014 IEEE

Another numerical ingredient is the use of discontinuous Galerkin boundary element methods for the numerical solution

of sub-regions. The main objective of this work is to allow the implementation of the generalized combined field integral

equation using square-integrable, L2, trial and test functions without any considerations of continuity requirements across

element boundaries. The proposed work has advantages over the classical approaches in that different basis functions can be

seamlessly integrated to best approximate the unknown currents locally. Another significant advantage of the method is that it

can be applied conveniently to non-conformal discretizations and different types of elements. In this way, the mesh generation

task of complex, multi-scale targets can be facilitated considerably.

2 Multi-Trace Surface Integral Formulation

The aim of this section is to summarize the key ideas in multi-trace surface integral equation formulation. For simplicity,

we consider the solution of EM scattering from a bounded composite penetrable object Ω, which composes of 2 sub-regions

Ω = Ω1∪Ω2, as illustrated in Fig. 2. The exterior region is denoted by Ω0(= R3\Ω), which is unbounded free space. Each sub-

region Ωm, m = 0, 1, 2, is a homogeneous medium. The field Em inside Ωm satisfies Maxwell’s equations with wave-number

km. Two adjacent sub-regions Ωm and Ωn are separated by their common interfaces Γmn.

(a) A composite object (b) Notations for the multiple traces

Figure 2: Electromagnetic scattering from a composite object

The starting point of deriving the multi-trace surface integral equations is the characterization of the traces of a local

solution as a range of surface integral operator. We introduce two set of traces on each sub-region surface, ∂Ωm. These

traces are the Neumann trace jm and Dirichlet trace em, defined by: jm = 1ık0

γτ

(1

μrm∇×Em

)and em = πτ (Em). They

represent the (scaled) electric current and the tangential components of the electric field. The magnetic currents can be written

as em×n̂m. Subsequently, by using these traces as input arguments, the generalized combined field integral equation (G-CFIE)

formulation [1] is applied to individual sub-regions. It can be stated formally, on each the sub-region surface, ∂Ωm, as

αem + (1− α) η̄mjm − η̄mCkmα (jm; ∂Ωm)− n̂m × Ckm

(1−α) (em × n̂; ∂Ωm)=

{αeinc + (1− α) jinc m = 00 m = 1, 2

Similar to the non-penetrable PEC case, the G-CFIE is comprised of tangential components of both electric and magneticfields, which guarantees the removal of the resonance solution and yields accurate and stable numerical solutions. We remark

that we have assigned separate Dirichlet and Neumann traces for each sub-region Ωm. It results in multi-trace spaces at the

interfaces. In turn, with these traces treated as unknowns, it yields two pairs of unknowns on each interface, twice the number

used in most other boundary integral formulations. This accounts for the attribute multi-trace formulation [2].

3 Discontinuous Galerkin Discretization

Surface integral equation methods are usually solved via the Galerkin method, which is based on a variational formu-

lation in suitable trial and testing function spaces. Therefore, conforming boundary element spaces defined on a conformal

discretization of the target’s surface are commonly required. Subsequently, mixing different types of basis functions, em-

ploying non-conformal discretizations, and/or incorporating the underlying physics to construct special basis functions within

local regions are quite complicated. The goal of this research is to develop a discontinuous Galerkin (DG) boundary element

method for surface integral equations. It supports various types and shapes of elements, non-conformal discretizations and

non-uniform orders of approximation. Initial work has been done to demonstrate the potential of the method in solving the

EM wave scattering problem from non-penetrable objects [4]. This research will further investigate the DG formulation for

penetrable composite objects composed of piecewise homogeneous materials.

Based on the trace theorem [5, 6], the function spaces for the Neumann and Dirichlet trace are jm ∈ H(divτ , ∂Ωm)and em ∈ H(curlτ , ∂Ωm), the space of surface div-conforming and curl-conforming, respectively. In the proposed work, we

will uplift both trial function spaces to be square-integrable vector functions, jm ∈ L2 (∂Ωm), and em ∈ L2 (∂Ωm). More

precisely, individual discontinuous Galerkin discretizations are employed for the solution of G-CFIE at each sub-region surface

∂Ωm. As a numerical example, we consider a plane wave scattering from a homogeneous dielectric sphere in free space, for

which analytic solution is available in the form Mie series. Three methods are included in this study: (a) classical single-

trace formulation, PMCHWT [7, 8], (b) multi-trace formulation, G-CFIE using a conforming Galerkin discretization [1], and

(c) multi-trace formulation, G-CFIE using a discontinuous Galerkin discretization. As evident from the Fig. 3, the numerical

solution of G-CFIE with DG discretization is stable and accurate.

(a) Accuracy study (b) Convergence of iterative solver

Figure 3: Electromagnetic scattering from a homogeneous sphere with different permittivities

4 Optimized Schwarz Preconditioning

An important motivation for the development of multi-trace SIE formulation was the desire to obtain linear systems of

equations that readily lend themselves to Schwarz preconditioning. This amounts to solving localized G-CFIE on Ωm using

Dirichlet and Neumann boundary data on the adjacent sub-domains from the previous iteration. A general Schwarz algorithm

for a three sub-domain problem, Ωm, m = 0, 1, 2, can be described as follows:

Solve iteratively for iteration p = 1, 2, · · · the following sub-domain problems

Gm (epm, jpm) = yincm on ∂Ωm

Bn̂m (epm, jpm) = Bn̂m

(ep−1n , jp−1

n

)on Γmn, ∀Γmn ∈ Γm

where Gm denotes the G-CFIE for sub-domain Ωm, Γmn is the interface between Ωm and Ωn, and Bn̂m , m = 0, 1, 2, denote the

transmission condition (TC) used to couple the Dirichlet and Neumann traces at the interfaces between adjacent sub-domains.

As pointed out in many previous works, the choice of TCs is of vital importance in determining the convergence behavior of the

Schwarz method. The use of first order Robin-type TCs in [1,3] makes the Schwarz iteration converge quickly for propagating

eigenmodes, though the evanescent modes fail to converge. The aim of the proposed research is to develop more effective TCs

for the best possible performance of the Schwarz algorithm. We investigate the complete second order TC:

Bn̂ (e, j) := (I + κ1∇τ ×∇τ ×+κ2∇τ∇τ ·) e− (I + κ3∇τ ×∇τ ×+κ4∇τ∇τ ·) η̄j (1)

where κ1, κ2, κ3, and κ4 are the parameters that can be chosen in order to optimize the performance. We remark that bysetting κ1 = β, κ2 = 0, κ3 = 0 and κ4 = γ, we recover the second order TC that was proposed in for finite element domain

decomposition method in [9], and a complete analysis is available in [10]. The convergence analysis of this new complete

SOTC and optimal choices of the parameters are examined in the proposed work and will be reported in the conference.

5 Numerical Results

We present a complex large-scale simulation to highlight the capability of the proposed method. As shown in Fig. 1, we

consider a plane wave scattering from a high-definition composite aircraft at 10GHz. Of particular concern is that this large

platform comprises a variety of real-world EM materials. As depicted in Fig. 1, we have colored different material regions with

different colors. Following the strategy of the proposed multi-trace SIE domain decomposition method (SIE-DDM), the entire

aircraft is decomposed into 70 sub-domains. Each sub-domain is a closed-surface homogeneous object and G-CFIE is used the

sub-region solver. For the simulation at X-band, the number of degree of freedom (DOFs) is about 46 millions. The SIE-DDM

requires 13 iterations to reach a relative residue ε = 10−2. The peak memory requirement is 68 GB and the total CPU time is 5

days. Moreover, we plot the electric current distributions on the exterior surfaces of the composite aircraft. Judging from these

two figures, we have witnessed smooth current distributions, without noticeable discontinuities, across sub-domain interfaces.

(a) Top view (b) Side view

Figure 4: Surface electric current distribution on a composite UAV at X-band

6 Conclusions

In the past decade, there have been increasing growing research activities aiming at developing a robust and efficient

integral equation solution strategy combating both large-scale and multi-scale electromagnetic problems. We present herein a

multi-trace surface integral equation formulation for full wave analysis of multi-scale composite systems. A novel discontinuous

Galerkin boundary element method is introduced and analyzed. It allows the solution of the combined field integral equation

using square-integrable, L2, trial and testing function spaces. Due to the local characteristics of L2 vector functions, it permits

us to employ non-conformal surface discretizations of the targets. Another significant advantage of the method is to mix

different types of elements and use different orders of basis functions within the same discretization. Therefore, the proposed

method is highly suitable for the application of adaptation techniques.

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