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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.
4 / 28
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.
5 / 28
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
9 / 28
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
12 / 28
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
18 / 28
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