Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)

Balanced Electric-Magnetic Absorber Green's Function Method for MoM Matrix Thinning and Conditioning

Naor R. Shay,1 Raphael Kastner1 and Daniel S. Weile2

1School of Electrical Engineering, Tel Aviv University 2Department of Electrical and Computer Engineering, University of Delaware

Abstract: A new numerical approach to solving the classic problem of electromagnetic (EM) scattering off a perfect electric object is studied with the objective of substantially reducing computation times. The method considered here is the in the frequency domain Method of Moments (MoM) formulation involving the use of a dyadic Green’s function (GF). Traditionally, this GF is formulated in free space, as afforded by the equivalence principle. However, since the resultant equivalent sources generate a null filed inside the scatterer volume, the door is open for the inclusion of arbitrary fillers therein.

We suggest the usage of balanced absorbers as fillers and using their Green’s function instead of the free space one. To this end, the solution of the essentially volumetric problem of the absorber is required as a preprocessing stage. Balanced absorbers have both electric and magnetic Weston-like or Perfect Matched Layers (PML) loss mechanisms. Many interactions between pairs of basic functions are then virtually eliminated. As a result, the MoM matrix, representing the GF, is significantly thinned.

The cost of calibrating this modified GF using the volumetric representation of the absorber is investigated. The effort incurred in the pre-processing stage can be alleviated by choosing absorber configurations that apply to many problems with high degrees of symmetry, thin absorbing shells rather than volumetric scatterers or homogeneous absorbers that lend themselves to surface formulations.

It is shown that this form of thinning has little effect on the accuracy. Moreover, most of the thinned elements need not be computed at all.

Keywords: Method of Moments, matrix thinning, spurious resonances.

References:

[1] W. W. Salisbury, "Absorbent Body for Electromagnetic Waves", US Patent 2599944 (A), June 10, 1952.

[2] V. H. Weston, ``Theory of Absorbers in Scattering," IEEE Trans. Antennas Propagat., Vol. 11, No. 5, 578-594, 1963.

[3] J. H. Richmond, "Scattering by a Dielectric Cylinder of Arbitrary Cross Section Shape", IEEE Trans. Antennas Propagat., Vol. 13, No. 4, 460 – 464, August 1966

[4] R. G. Rojas, "Scattering by an Inhomogeneous Dielectric/Ferrite Cylinder of Arbitrary Cross-Section Shape - Oblique Incidence Case", IEEE Trans. Antennas Propagat., vol. 36, No. 2, 238 – 246, Feb. 1988.

[5] A. Boag and V. Lomakin, `Generalized equivalence integral equations," IEEE Antennas Wireless Propag. Lett., Vol. 11, 1568-1571, 2012.

Naor R. Shay was born in Rehovot, Israel, in 1987. He received the B.Sc. degree in electrical engineering from Tel-Aviv University, Israel, in 2010. He is currently pursuing his M.Sc degree in Electrical Engineering, also at Tel-Aviv University. Since 2010 he has been employed with the Israel Defense Forces (IDF) as an RF and mm waves engineer, where he is engaged in research of advanced communication circuits for various applications.

Raphael Kastner was born in Haifa, Israel, in 1948. He received the B.Sc. (summa cum laude) and the M.Sc. degrees in electrical engineering, engineering from the Technion, Israel Institute of Technology in 1973 and 1976, respectively, and his Ph.D. degree from the University of Illinois, Urbana, in 1982.

From 1976 to 1988 he was with RAFAEL, Israel Armament Development Authority, where from 1982 to 1986 he headed the antenna section. He was a Visiting Assistant Professor at Syracuse University from 1986 to 1987, and a Visiting Scholar at the University of Illinois in 1987 and 1989. Since 1988 he has been with School of Electrical Engineering, Tel Aviv University, where is now a Professor. In 2000 he co-founded XellAnt Inc. and acted as its CEO until 2004. Since January 2016 he with the University of Pennsylvania as a visiting professor. He is a Life Fellow, a recipient of the IEEE Third Millenium medal and several excellence in teaching awards, and a member of Tau Beta Pi and Eta Kappa Nu. His research interests are in computational electromagnetics and antennas.

Daniel S. Weile obtained his B.S.E.E and his B.S. (in Mathematics) at the University of Maryland at College Park and 1994, and M.S. and Ph.D. in Electrical Engineering at the University of Illinois at Urbana-Champaign in 1995 and 1999, respectively. Currently, he is an Associate Professor of Electrical Engineering at the University of Delaware. In 1994, he worked at the Institute for Plasma Research developing interactive software for the design of depressed collectors for gyrotron beams. As a research assistant and Visiting Assistant Professor at the University of Illinois, Dr. Weile worked on the efficient design of electromagnetic devices using

stochastic optimization techniques, and fast time-domain integral equation methods for the solution of scattering problems. His current research interests include computational electromagnetics (especially time-domain integral equations), periodic structures, and the use of evolutionary optimization in electromagnetic design. Dr. Weile is the recipient of an NSF CAREER Award and and an ONR Young Investigator Award. He is a member of Eta Kappa Nu, Tau Beta Pi, Phi Beta Kappa, and URSI Commission B.

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Balanced Electric-Magnetic Absorber

Green’s Function Method for MoM

Matrix Thinning and Conditioning

Naor R. Shay,1 Raphael Kastner1 and Daniel S. Weile2

1School of Electrical Engineering, Tel Aviv University

2Department of Electrical and Computer Engineering, University of Delaware

PIERS 2016 ShanghaiSession 2P 14, , organizer: Dan Jiao

SC1: Fast Methods in Computational ElectromagneticsTuesday, August 9, 2016, 15:00–15:20.

1 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

2 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

3 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Introduction

Conventional surface integral equation formulations employequivalent sources with null field in the internal region of thescatterer. Therefore, this region can be filled, at leastconceptually, with a balanced absorber with both electric andmagnetic conductances. The Green’s function of the balancedabsorbers is then employed instead of the conventional free spacefunction.By reducing interactions between pairs of opposing basisfunctions, this method enables an MoM matrix with fewer than25% non-zero elements with little effect on the result. Moreover,the annulled elements need not be computed.

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Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

What is a balanced absorber?

Let us require that a double-lossy medium has the same intrinsicimpedance as the homogeneous medium with (✏0, µ0):

Z =

rµ0

✏0

s1� | �?

!µ0

1� | �!✏0

=

rµ0

✏0(1)

The balanced condition is then

�/✏0 = �?/µ0. (2)

Eq. (2) is sufficient for a 100% absorption of a normally incidentplane wave at a planar interface between the homogeneous andabsorbing medium.

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Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

The idea of balanced EM absorbers is not entirely new

The concept of the balanced absorber dates back to Salisbury inthe 1940’s1 and to Weston in the 1960’s2. It was laterelaborated into the seminal PML by Bérenger.Balanced absorbers can work with very high or very lowpermittivities and permeabilities, even with little inherent losses3.

1W. W. Salisbury, "Absorbent Body for Electromagnetic Waves", USPatent 2599944 (A), June 10, 1952.

2V. H. Weston, “Theory of Absorbers in Scattering," IEEE Trans.

Antennas Propagat., Vol. 11, No. 5, 578–594, 1963.3R. Kastner, “High Electromagnetic Conductance Media”, IEEE Trans.

on Antennas and Propagation, vol. 61, no. 2, pp. 775 –778, Feb. 2013.6 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

7 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Equivalence Principle with Balanced Absorber

8 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Balanced EM absorber method (BEMA)

The unknown source J(r0) is supported by the surface C0

.In view of the null total field inside C

0

, we embed abalanced absorber in it.The Green’s function of the absorber, denoted G

mod

(r, r0),and a corresponding modified incident field are to be used.

Weston-type absorber

scatterer

Jinc

Modified incident field

C1

C0 J

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Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Rationale

The rationale of using Gmod

is in the isolation it providesbetween basis functions located across the absorber. As a result,a typical row in the MoM matrix will decay faster away from thediagonal, relative to the conventional free space matrix:

Norm

alize

dGreen’sm

atrix Red:Freespacematrix

Blue:Modifiedmatrix

i

PECs and impedance surfaces have been used as fillers before4.4Boag, A. and V. Lomakin, ‘Generalized equivalence integral equations,"

IEEE Antennas Wireless Propag. Lett, Vol. 11, 1568–1571, 2012.10 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Formulating an integral equation with the Balanced EM

Absorber

Reall: The balanced absorber is characterized by (2):

�/✏ = �?/µ.

The problem can then be formulated by the following IE

ˆ

C0

Gmod

(r, r0)J(r0)dl0 = �ˆ

C1

Gmod

(r, r0)J inc(r0)dl0. (3)

Here, J inc(r0) is an equivalent source distribution that hasgenerated the original incident field under free space conditions.It is now used to generate the modified incident field in thepresence of the absorber, seen in the right hand side of (3). Itmay pay to choose C

1

as a straight line for the sake of simplicity.

11 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Generating Gmod

(r, r0)

The ability to solve (3) hinges on the availability of Gmod

(r, r0).This function is evaluated in a potentially taxing volumetricpre-processing stage.

P1

P0

P2

LOS

LOS

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Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Economizing on the pre-processing stage

Several configurations can beused: the absorber can besmall (red) or large (blue).Results for bothconfigurations are similar.The absorber can be hollow,making the pre-processingproblem almost a surface one.

P1

P0

P2

LOS

LOS

A high degree of symmetry also helps.A given G

mod

can be used for many problems.

13 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

A hollow filler can do the job

14 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Rule of thumb for avoiding the computation of annulled

elements

Two lines-of-sight (LOSs) are shown for a basis function at thepoint P

0

. These lines of sight skim the circumference of the(flattened, red) absorber. It turns out that only the interactionsover the arc {P

1

: P0

: P2

} between the two LOSs are needed aselements in the column representing the basis function at P

0

.This results in substantial saving in matrix fill-up time.

P1

P0

P2

LOS

LOS

15 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Thinning according to the line-of-sight-criterion

TypicallineinGmod

16 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

17 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Current distributions on conditioned and ill-conditioned

circular cylinder problem

The following graphs show the currents, computed with theconventional Green’s function. The EFIE is conditioned for thetop graph and ill-conditioned for the bottom.

R=1.375l

R=1.275l

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Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Symptom: condition number

19 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

BEMA conditions the EFIE

The graph below shows the condition numbers for the MoMmatrices: free space Green’s function is red, BEMA is blue.Resonant frequencies, where the condition number blows up, areseen only for the free space case.

20 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

21 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Sanity check: using fully populated Gmod

Modifiedincidentfield(amp)

Modifiedincidentfield(phase)

RowsinGmod

Result:J (amp) Result:J (phase)

22 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Detail: The behavior of Gmod

(fully populated) for the case of

the flattened absorber

23 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Detail: Five typical rows in the MoM matrix representing

Gmod

.

The non-self terms in Gmod

appear to be significantly smallercompared with the free space function. This effect provides themeans for intense thinning of the matrix.

24 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Thinned using the line-of-sight rule - large circular filler

The number of non-zero elements in the matrix is then cut byabout half along each row and each column, therefore the totalnumber of remaining elements is about 25% of the originalmatrix. Resultant current distribution is shown in the figure (blueline) compared with the conventional MoM solution (red line).

25 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

1 Introduction

2 Thinning the EFIE Matrix with the Balanced EM AbsorberMethod (BEMA)

3 Conditioning of the EFIE with BEMA

4 Example: PEC Circular Cylinder with Diameter d = 3.1�Excited by a TM Plane Wave

5 Conclusions

26 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

Conclusions

A surface scattering problem can be solved using about 25% ofthe MoM matrix elements with little effect on the accuracy.Conditioning of the EFIE is also achieved.

By inspecting lines of sight between basis functions it maybe possible to avoid the computation of the matrix elementsthat represent severe attenuation.Solution is robust.The effort incurred in the pre-processing stage can bealleviated as described above.Most annulled elements need not be computed at all.

The thinned matrix then forms the basis for either a direct

method or pre-conditioned iterative method for

formulating and solving surface integral equations.

27 / 28

Balanced

Electric-

Magnetic

Absorber

Green’s

Function

Method for

Matrix

Thinning

Shay,

Kastner,

Weile

Introduction

BEMA

Conditioning

Example

Conclusions

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