mom-based matrix operators in...

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MoM-Based Matrix Operators in AToM

Miloslav [email protected]

August 14, 2018

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Miloslav Capek MoM-Based Matrix Operators in AToM 1 / 29

Outl

ine 1. Preliminary Definitions

2. Impedance Matrix and Stored Energy Matrix

3. Dissipation Matrix

4. Impedance Matrix with Losses and Reactive Elements

5. Far-Field and Near-Field

6. Electric and Magnetic Dipole Moments

7. Useful Identities For RWGs

8. Practical Evaluation In AToM

9. Appendices

P+n

P−n

ρ+nρ−n

A+n

A−n

lnT+n

T−n

O

r

y

z

xψn (r) =

ln

2A±nρ±n

Definition of RWG function.




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Basi

cE

quati

ons Quadratic form of linear operator L reads

p =1

2

∫Ω

J∗ · L J dr, J = J (r) , r ∈ Ω. (1)

Complex power is defined as

Prad + 2jω (Wm −We) = −1

2

∫V

J∗ ·E dV (2)

Electric Field Integral Equation (EFIE) reads1

E (r) = −jZ0k

∫Ω

(1 +∇∇·k2

)J (r′)

e−jkR

4πRdS, (3)

with 1 being the identity matrix, 1 = [δpq].

1J. L. Volakis and K. Sertel, Integral equation methods for electromagnetics. Scitech PublishingInc., 2012.




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RW

GB

asi

sFunct

ions Usage of (finite) discretization of the source region Ω leads to

J (r) ≈N∑n=1

Inψn (r) , (4)

where I = [In] ∈ CN×1 are expansion coefficients, ψn are the basis functions.

Formula (4) transforms (1) into algebraic relation

p ≈ 1

2IHLI with Lmn ≡

∫Ω

ψm · L ψn dr. (5)

Complex power (2) is approximated as

Prad + 2jω (Wm −We) ≈ 1

2IHZI, (6)

where Z ∈ CN×N is the impedance matrix defined as follows.




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Imp

edance

Matr

ix The impedance matrix is constructed (from normalized data) as

Z = R + jX = jZ0a2

(ka(ZM,k + ZM,0

)− 1

ka

(ZE,k + ZE,0

)), (7)

where individual elements of the defining matrices are

ZM,kmn =

1

a3

∫Ω

∫Ω

ψm (r) ·ψn (r′)e−jkR − 1

4πRdS dS′, (8)

ZM,0mn =

1

a3

∫Ω

∫Ω

ψm (r) ·ψn (r′)1

4πRdS dS′, (9)

ZE,kmn =1

a

∫Ω

∫Ω

∇ ·ψm (r)∇′ ·ψn (r′)e−jkR − 1

4πRdS dS′, (10)

ZE,0mn =1

a

∫Ω

∫Ω

∇ ·ψm (r)∇′ ·ψn (r′)1

4πRdS dS′. (11)




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Exam

ple

:N

orm

aliza

tion

of

Ch

ara

cteri

stic

Modes Normalization of characteristic modes2 to unitary radiated power reads

1

2

∫Ω

J∗pZ Jq dr = (1 + jλp) δpq. (12)

The same normalization based on impedance matrix is3

1

2IHp ZIq = (1 + jλp) δpq. (13)

Physical units of impedance matrix Z are [Ω m2].

Matrices ZM,k, ZM,0, ZE,k, ZE,0 are made dimensionless.

2M. Capek, P. Hazdra, M. Masek, et al., “Analytical representation of characteristic modesdecomposition,” IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 713–720, 2017. doi:10.1109/TAP.2016.2632725.

3Complex and Hermitian conjugate can be removed in case as the characteristic modes for PECbodies are by definition real-valued.



http://dx.doi.org/10.1109/TAP.2016.2632725


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Sto

red

En

erg

yM

atr

ix The analytical differentiation of Z with respect to ω (normalized by ω) is

ω∂Z

∂ω= jZ0a

2

(ka(ZM,k + ZM,0 − jkaTM

)+

1

ka

(ZE,k + ZE,0 + jkaTE

)),

(14)where individual elements of the defining matrices are

TMmn =1

a4

∫Ω

∫Ω

ψm (r) ·ψn (r′)e−jkR

4πdS dS′, (15)

TEmn =1

a2

∫Ω

∫Ω

∇ ·ψm (r)∇′ ·ψn (r′)e−jkR

4πdS dS′, (16)

and

R = |r − r′| , a = maxr,r′∈Ω

R

2

, Z0 =

√µ0

ε0. (17)




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Ele

ctri

can

dM

agn

eti

cS

tore

dE

nerg

ies The stored energy matrix4 for ka < 1 is defined as

W = Im

ω∂Z

∂ω

. (18)

and the stored electric and magnetic energies are

Xm =1

2(W + X) , (19)

Xe =1

2(W −X) . (20)

Therefore, the following identities holds

X = Xm −Xe, (21)

W = Xm + Xe. (22)

4R. F. Harrington and J. R. Mautz, “Control of radar scattering by reactive loading,” IEEETrans. Antennas Propag., vol. 20, no. 4, pp. 446–454, 1972. doi: 10.1109/TAP.1972.1140234.





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Basi

s-Funct

ion

Overl

ap

Matr

ixand

Loss

yM

atr

ix The basis-function overlap matrix Ψ = [Ψmn] is defined as5 sparse matrix

Ψmn =

∫Ω

ψm (r) ·ψn (r) dS. (23)

Properties:

all eigenvalues are strictly positive (non-negative),

it is a Gram matrix,

it is sparse; sparsity for a sphere of T triangles with N = 3T/2 RWG basisfunctions is equal6 to 1− 5/N .

5L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces,” IEEE Trans.Antennas Propag., vol. 65, no. 1, pp. 329–341, 2017. doi: 10.1109/TAP.2016.2624735.

6Topologically minimal S2 sphere is a 3-simplex (tetrahedron), then N = 6 and each basisfunction ψn sees all the others except of one. All opened and simple structures have higher sparsity.





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Loss

yM

atr

ix The dissipation (ohmic) losses matrix RΩ = [RΩ,mn] ∈ RN×N as

RΩ,mn =

∫Ω

Rs (r)ψm (r) ·ψn (r) dS, (24)

with Rs (r) ≥ 0 being spatially dependent surface resistivity. For Rs (r) = Rs

RΩ = RsΨ (25)

and inheriting all properties of matrix Ψ.




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Adja

cency

Matr

ixof

the

Radia

tor The topological adjacency matrix A = [Amn] can then be defined as

Amn =

1 ⇔ Ψmn 6= 0,0 ⇔ Ψmn = 0.

(26)




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Imp

edance

Matr

ixw

ith

Loss

es

and

React

ive

Ele

ments The dissipation loss matrix RΩ and matrix of lumped elements ZL = [ZL,mn]

can be added to the impedance matrix as

Z = R + jX = RΩ + ZL + Z. (27)

The lumped element matrix elements read

ZL,nn = l2n

(Rn + j

(ωLn −

1

ωCn

)). (28)




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Far-

Fie

ldQ

uanti

ties The far-field is defined as

F (r, e) = limr→∞

rejkre ·E (r)

. (29)

Explicitly,

F (r, e) = − jZ0k

4π

∫Ω

e · J (r′) ejkr·r′ dS, (30)

and

U (r, e) =1

2Z0|F (r, e)|2 , U (r) = U (r, ϕ) + U

(r, ϑ

), (31)

Prad =

∫S2

U (r) dS, D (r, e) =4πU (r, e)

Prad. (32)




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Far-

Fie

ldand

Far-

Fie

ldM

atr

ix Radiation intensity matrix U = [Umn] for direction r and polarization e isdefined7 as

Umn (r, e) =Z0k

2

32π2

∫Ω

∫Ω

(e ·ψm) (e ·ψn) e−jkr·(r−r′) dS dS′, (33)

and it is a rank-one matrix as the far-field vector F = [Fn] defined as

Fn (r, e) =−jZ0k

4π

∫Ω

e ·ψnejkr·r dS (34)

can be used

U (r, e) =1

2Z0FH (r, e) F (r, e) . (35)

Total radiation intensity can be calculated as

U (r) = U (r, ϕ) + U(r, ϑ

). (36)

7L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped surfaces,” IEEE Trans.Antennas Propag., vol. 65, no. 1, pp. 329–341, 2017. doi: 10.1109/TAP.2016.2624735.





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Spheri

cal

Waves

Pro

ject

ion

Matr

ix Spherical waves can be projected onto basis functions as8

Sαn = k√Z0

∫Ω

u(1)α (kr) · ψn (r) dS, (37)

α is multi-index σ,m, l, and u(1)α (kr) are the spherical modes9.

The relationship between resistance part of impedance matrix and sphericalwaves matrix

R = STS. (38)

8D. Tayli, M. Capek, L. Akrou, et al., “Accurate and efficient evaluation of characteristicmodes,” , 2017, submitted, arXiv:1709.09976. [Online]. Available: https://arxiv.org/abs/1709.09976.

9G. Kristensson, Scattering of electromagnetic waves by obstacles. Edison, NJ: SciTechPublishing, an imprint of the IET, 2016.



https://arxiv.org/abs/1709.09976


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Pro

pert

ies

of

pro

ject

ion

matr

ixS Matrix S ∈ RNα×N is a projection matrix between RWG basis and spherical

waves basis. It is low-rank matrix in α, suitable for compression techniques.

For Ω being a spherical shell, the rows of matrix S contains directly theexpansion coefficients of corresponding spherical waves, Iα, since

IHαpS

TSIαq = Kαδpq, (39)

where Kα is arbitrary constant depending on scaling.Real part of the impedance matrix Z represented in basis of spherical waves is

Rα = SST. (40)




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Pro

ject

ion

of

Arb

itra

ryC

urr

ent

Into

Sph

eri

cal

Waves The radiated far-field can be expressed as

F (r) =1

k

∑α

jl−τ+2fαYα (r) , (41)

where Yα (r) are the real-valued spherical vector harmonics and

[fα] = SI, fα ∈ R1×1. (42)

Then we get a series of equalities

Prad ≈1

2IHRI =

1

2|SI|2 =

1

2

∑α

|fα|2 = IH

∫S2

(U (r, ϕ) + U(r, ϑ)

)dS I, (43)

with S2 being unit sphere.




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Ele

ctri

cand

Magn

eti

cN

ear-

Fie

lds Electric and magnetic fields are approximated as

N e,n (r) = −jZ0k

∫Ω

((r ×ψn)× r + (3r (r ·ψn)−ψn)

(1

(kR)2 +

j

kR

))e−jkR

4πRdS,

(44)

Nm,n (r) = −jk

∫Ω

(r ×ψn)

(1 +

1

jkR

)e−jkR

4πRdS. (45)

with Ne = [N e,n] ,Nm = [Nm,n] ∈ CN×3.

Notice then that

E (r) ≈ Ne (r) I, (46)

H (r) ≈ Nm (r) I. (47)




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Ele

ctri

can

dM

agn

eti

cP

ola

riza

bilit

yM

atr

ices Electric dipole moment matrix p = [pn] ∈ RN×3 is defined as

pn =j

ω

∫Ω

r∇ ·ψn dS =1

jω

∫Ω

ψn dS. (48)

Magnetic dipole moment matrix m = [mn] ∈ RN×3 is defined as

mn =1

2

∫Ω

r ×ψn dS (49)

Squared magnitude of the electric and magnetic dipole moments can becalculated as

|p|2 = IHPI, (50)

|m|2 = IHMI, (51)

where matrices P ∈ RN×N and M ∈ RN×N are

P = ppH, (52)

M = mmH. (53)




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Use

ful

Identi

ties

For

RW

Gs Useful relations10 are

∇ ·ψn (r) = ± ln

A±n, (54)

∫Ω

ψn (r) dS = ln(rc+n − rc−n

), (55)

∫Ω

r ×ψn dS =ln2

(P+n × rc+n −P−n × rc−n

), (56)

∫Ω

∇ ·ψm (r) dS =ln

A+n

∫A+n

dS − ln

A−n

∫A−n

dS = 0, (57)

where Ω = A+n ∪A−n .

10S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces ofarbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, 1982. doi:10.1109/TAP.1982.1142818.





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Tri

angula

rD

om

ain

Inte

gra

tion

Wit

hB

ary

centr

icSubst

ituti

on For r ≡ [x y z] the transformation from/to simplex coordinates (α, β, γ) reads

r =[α β γ

] PA

PB

P±

. (58)

Notice that γ = 1− α− β, and

∫T

f (r) dS = 2A

1∫0

1−α∫0

f(αPA + βPB + (1− α− β) P±

)dβ dα. (59)




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Evalu

ati

on

inA

ToM

,P

art

#1 Discretization grid and RWG basis functions:

Listing 1: atom code1 mesh.m

% Frequencyf = atom.selectedProject.physics.getFrequencyListValues;% or manually specified by user:f = 1e9; % 1 GHz

% (It is expected that MoM requests are correctly set up.)Res = atom.selectedProject.solver.MoM2D.results;

Mesh = Res.mesh; % mesh gridBF = Res.basisFcns; % basis functions

% Or without running MoM solver directly from AToM:Mesh = atom.selectedProject.mesh.getMeshData2D();BF = models.solvers.MoM2D.basisFcns.getBasisFcns(Mesh);




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Evalu

ati

on

inA

ToM

,P

art

#2 Impedance matrix(-related) operators Z, W, ZM,k, ZM,0, ZE,k, ZE,0, TM , TE :

Listing 2: atom code1 MoM1.m

% Impedance matrixZ = Res.zMat.data;

omega = 2*pi*f;% Stored energy matrixW = omega*imag(Res.zMatD.data); % omega*DZ!

% Individual parts of impedance matrix:ZMk = Res.zMatMk.data;ZM0 = Res.zMatM0.data;ZEk = Res.zMatEk.data;ZE0 = Res.zMatE0.data;TE = Res.tMatE.data;TM = Res.tMatM.data;




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Evalu

ati

on

inA

ToM

,P

art

#3 Alternatively, Z and W can be calculated directly from normalized blocks:

Listing 3: atom code1 MoM2.m

% Circumscribing spherea = models.utilities.meshPublic.getCircumsphere(Mesh.nodes);% Wavenumber and electrical sizek = models.utilities.converter.f0tok0(f);ka = k*a;

% Normalization constantC = aˆ2*Z0;

% Impedance matrix and stored energy matrixZ = 1j*C*(ka*(ZMk + ZM0) − 1/ka*(ZEk + ZE0));W = imag(1j*C*(ka*(ZMk + ZM0 − 1j*ka*TM) + 1/ka*(ZEk + ZE0 + 1j*ka*TE)));




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Evalu

ati

on

inA

ToM

,P

art

#4 Basis-function overlap matrix Ψ and related matrices RΩ and A:

Listing 4: atom code1 LossyMatrix.m

% The basis−function overlap matrixPsi = models.utilities.matrixOperators.ohmicLosses.lossyMatrix(Mesh, BF);

% Dissipation losses matrixRomega = Rs*Psi;

% RWG adjacency matrixA = double(logical(Psi));




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Evalu

ati

on

inA

ToM

,P

art

#5 Far-field vectors F and matrix U:

Listing 5: atom code1 farfield.m

% Theta and phi used to specify $\widehat\boldsymbolr$theta = pi/2; % from 0 to piphi = 0; % from 0 to 2*pi% Component used to specify $\widehat\boldsymbole$component = 'theta'; % component = 'theta' | 'phi' | 'total'

% Far−field matrix (U) and vectors (F)[U, F phi, F theta] = ...

models.utilities.matrixOperators.farfield.computeU...(Mesh, BF, f, theta, phi, component)




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Evalu

ati

on

inA

ToM

,P

art

#6 Spherical wave projection matrix S:

Listing 6: atom code1 sphericalwaves.m

nQuad = 1; % order of quadrature ruleLmax = 15; % highest degree of used Legendre polynomial

% Spherical wave projection matrix of size NxM% N − number of spherical waves, M − number of RWG functionsS = models.utilities.matrixOperators.SMatrix.computeS...

(Mesh, BF, f, Lmax, nQuad);




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Evalu

ati

on

inA

ToM

,P

art

#7 Electric and magnetic dipole matrices P and M, and vectors p and m:

Listing 7: atom code1 elmagMoments.m

% Electric moment matrix (P) and vector (p)[P, p] = models.utilities.matrixOperators.electricMoment.computeP...

(Mesh, BF, f);

% Magnetic moment matrix (M) and vector (m)[M, m] = models.utilities.matrixOperators.magneticMoment.computeM...

(Mesh, BF, f);




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Use

dFar-

field

Ap

pro

xim

ati

on Far-field approximation

Efar (r) = −jωA (r) , A (r) =µ0

4π

∫Ω

J (r′)e−jk|r−r′|

|r − r′|dS, (60)

with amplitude R ≈ |r| = r and phase R ≈ r − r · r′ (61)

leads to

Efar (r) = − jZ0k

4π

e−jkr

r

∫Ω

J (r′) ejkr·r′ dS. (62)




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Questions?

For a complete PDF presentation seecapek.elmag.org

Miloslav [email protected]

August 14, 2018



www.capek.elmag.org

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