TWINS IIANNA ŠUŠNJARA, VICKO DORIĆ & DRAGAN POLJAK

Training School on Ground Penetrating Radar for civil engineering and cultural heritage managementRoma, Italy, May 14-18, 2018

Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture

University of Split, Croatia

Rome, Italy,

May 14-18,

2018

Training School on GPR for civil engineering and cultural

heritage management

2TWINS I = Thin WIre Numerical SolverSuzana = Sustav za analizu antena (Croatian)

Frequency domain solver (FD) Time domain solver (TD)

What is solved?

- Governing equations for a current distribution along the wire radiating above a half space

Numerical method?

- Galerkin-Bubnov scheme of Indirect Boundary Element Method (GB-IBEM)

Programming language:

- C++

- Used for educational purpose and scientific research

Rome, Italy,

May 14-18,

2018

Training School on GPR for civil engineering and cultural

heritage management

3TD vs. FD techniques

Time domain (TD) techniques:

Applied for a large frequency spectrum, but for single source

More complex formulation requiring more computational effort

Deeper physical insight

Frequency domain (FD) techniques:

Applied for more sources, but at a singlefrequency

Formulation is substantially less demanding

In general, regulatory standards are specified in FD form; convenient for analyzing EMC properties

Rome, Italy,

May 14-18,

2018

Training School on GPR for civil engineering and cultural

heritage management

4TWINS I – Frequency Domain (FD)

Rome, Italy,

May 14-18,

2018

Training School on GPR for civil engineering and cultural

heritage management

5TWINS I – Time Domain (TD)

Rome, Italy,

May 14-18,

2018

Training School on GPR for civil engineering and cultural

heritage management

6

Transmitted

wave

ϑ

x

z

h

T (x,z)

d1

d2

L

εr1 σ1

εr2 σ2

εr3 σ3

(0,h) (L,h)TWINS II

- Frequency Domain (FD)

- Presented on final GPR conference in Warsaw

What is solved?

Integral formulas for the electric field transmitted into

the subsurface.

NEW!

The subsurface is non-homogenous.

Talk Layout

• THEORETICAL

BACKGROUND

• Model geometry

• INSTALLATION PROCEDURE

• USER GUIDE:

• Antenna parameters

• Ground parameters

• Frequency domain (FD)

• Time domain (TD)

• Run the simulation

• EXAMPLES

• REFERENCES

7

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Training School on GPR for civil engineering and cultural

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Model geometry

Transmitted wave

ϑ

x

z

h

T (x,z)

d1

d2

L

εr1 σ1

εr2 σ2

εr3 σ3

(0,h) (L,h)

Ground penetrating radar dipole antenna

above a lossy half space.

The half space is modeled as a three

layered medium.

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𝐸𝑥𝑒𝑥𝑐 = 𝑗 𝜔

𝜇

4𝜋න−𝐿2

𝐿2𝑰 𝒙′ 𝒈 𝒙, 𝒙′ 𝑑𝑥′ −

1

𝑗4𝜋𝜔𝜀0

𝜕

𝜕𝑥න−𝐿2

𝐿2 𝝏𝑰 𝒙′

𝜕𝑥′𝒈 𝒙, 𝒙′ 𝑑𝑥′

Pocklington’s integro-differential equation:

𝒈 𝒙, 𝒙′ = 𝑔0 𝑥, 𝑥′ − 𝑹𝑻𝑴 𝑔𝑖 𝑥, 𝑥′

𝑔𝑖 𝑥, 𝑥′ =

𝑒−𝑗𝑘0𝑹𝒊

𝑹𝒊

I(x’) ... the unknown current

g(x,x’) ... Green’s function

g0(x,x’) ... for free space

gi(x,x’) ... from image theory

x σ, 𝜀r 𝜇0

ε0, μ0ϑ

x

z

Wire above a lossy half-space:

x

𝐑𝐢

x

x

𝐑𝟎

h

-h

𝑔0 𝑥, 𝑥′ =𝑒−𝑗𝑘0𝑹𝟎

𝑹𝟎

9

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𝐸𝑥 =1

𝑗4𝜋𝜔𝜀0න0

𝐿 𝜕𝐼 𝑥′

𝜕𝑥′

𝜕𝒈 𝒙, 𝒚, 𝒛, 𝒙′

𝜕𝑥𝑑𝑥′ − 𝛾2න

0

𝐿

𝐼 𝑥′ 𝒈 𝒙, 𝒙′ 𝑑𝑥′

𝐸𝑦 =1

𝑗4𝜋𝜔𝜀0න0

𝐿 𝜕𝐼 𝑥′

𝜕𝑥′

𝜕𝒈 𝒙, 𝒚, 𝒛, 𝒙′

𝜕𝑦𝑑𝑥′

𝐸𝑧 =1

𝑗4𝜋𝜔𝜀0න0

𝐿 𝜕𝐼 𝑥′

𝜕𝑥′

𝜕𝒈 𝒙, 𝒚, 𝒛, 𝒙′

𝜕𝑧𝑑𝑥′

Electric field integral relations:

𝒈 𝒙, 𝒙′ = 𝑻𝑻𝑴 𝑔0 𝑥, 𝑥′

Numerical procedure:

Galerkin Bubnov scheme of

Indirect Boundary Element Method

(GB-IBEM)

10

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z

x

L

h

~

ε0, σ = 0

μ0,

ε0 εr1

σ1

μ0, d1

ε0 εr2

σ2

μ0

ϑ0

ϑ1

ϑ2

Reflection/transmission coefficients for three layered media configuration:

Air + Layer1 + Layer2

The plane wave approximation with oblique

incidence at interfaces.

The continuity conditions for the tangential

components of electric and magnetic fields at two

interfaces (01 and 12) and the Snell’s law for wave

propagation yield the following equations

𝐸0+ =

1

𝜏01𝐸1+ +

𝜌01𝜏01

𝐸1−

𝐸0− =

𝜌01𝜏01

𝐸1+ +

1

𝜏01𝐸1−

𝐸1+ =

1

𝜏12𝑒𝛾1 cos 𝜗1𝑑1𝐸2

+

𝐸1− =

𝜌12𝜏12

𝑒−𝛾1 cos 𝜗1𝑑1𝐸2+

01

12

𝐸0+, 𝐸1

+, 𝐸0− & 𝐸1

− ...the magnitude of electric field at the

layer interfaces for the wave propagating in the negative

and positive direction of z axis, respectively

11

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𝜌𝑚𝑛 =𝑍𝑛 cos 𝜗𝑛 − 𝑍𝑚 cos 𝜗𝑚𝑍𝑛 cos 𝜗𝑛 + 𝑍𝑚 cos 𝜗𝑚

𝜏𝑚𝑛 =2𝑍𝑛 cos 𝜗𝑚

𝑍𝑛 cos𝜗𝑛 + 𝑍𝑚 cos 𝜗𝑚

𝑚 = 0,1 𝑛 = 1,2

Fresnel’s reflection/transmission coefficient approximation at

interfaces

𝑍𝑘 =𝑗𝜔𝜇0

𝜎𝑘 + 𝑗𝜔𝜀0𝜀𝑟𝑘𝛾𝑘 = 𝑗𝜔𝜇0 𝜎𝑘 + 𝑗𝜔𝜀0𝜀𝑟𝑘 𝑘 = 0,1,2

z

x

L

h

~

ε0, σ = 0

μ0,

ε0 εr1

σ1

μ0, d1

ε0 εr2

σ2

μ0

ϑ0

ϑ1

ϑ2

mn = 01

mn = 12

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complex propagation constantImpedance of medium

𝐸0+

𝐸0− = 𝑀01

𝐸1+

𝐸1−

𝐸1+

𝐸1− = 𝑃12𝑀12

𝐸2+

𝐸2−

𝑀𝑚𝑛 =1

𝜏𝑚𝑛

1

𝜌𝑚𝑛

𝜌𝑚𝑛

1𝑚 = 0,1, 𝑛 = 1,2

𝑃12 =𝑒𝛾1 cos 𝜗1𝑑1

0

0

𝑒−𝛾1 cos 𝜗1𝑑1

... matching matrix

... propagation matrix

The layers k = 3, ... are readily included by adding the propagation and matching matrices :

... Matrix form for equations that solve for field values at

layer’s interfaces

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By setting 𝐸0+ = 1 𝑉/𝑚, all the other magnitudes 𝐸𝑚

+ and 𝐸𝑚−are easily determined.

Electric field can be calculated using the following expressions:

𝐸0 = 𝐸1+ cos𝜗0 Ԧ𝑒𝑥 + sin 𝜗0 Ԧ𝑒𝑧 𝑒−𝛾0 𝑥 sin 𝜗0−𝑧 cos 𝜗0 + 𝐸0

− cos 𝜗0 Ԧ𝑒𝑥 − sin 𝜗0 Ԧ𝑒𝑧 𝑒−𝛾0 𝑥 sin 𝜗0+𝑧 cos 𝜗0

𝐸1 = 𝐸1+ cos 𝜗1 Ԧ𝑒𝑥 + sin 𝜗1 Ԧ𝑒𝑧 𝑒−𝛾1 𝑥 sin 𝜗1−𝑧 cos 𝜗1 + 𝐸1

− cos𝜗1 Ԧ𝑒𝑥 − sin 𝜗1 Ԧ𝑒𝑧 𝑒−𝛾1 𝑥 sin 𝜗1+𝑧 cos 𝜗1

𝐸2 = 𝐸2+ cos 𝜗2 Ԧ𝑒𝑥 + sin 𝜗2 Ԧ𝑒𝑧 𝑒−𝛾2 𝑥 sin 𝜗2−𝑧 cos 𝜗2

The reflection and transmission coefficients RTM and TTM appearing in the Green’s function

are calculated as the ratio between the field value at the ground surface and the field

value obtained from the equations above at the given observation point.

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ϑinc

T(x,z)

T(x’,z’)

The MIT approach assumes the normal incidence of

a propagating EM wave, therefore the incidence angles are zero: ϑm = 0º.

The expressions for the reflection and transmission

coefficients for the two adjacent media indexed

with m and n are given as:

𝜌𝑚𝑛𝑀𝐼𝑇 =

𝑍𝑚2 − 𝑍𝑛

2

𝑍𝑚2 + 𝑍𝑛

2

𝜏𝑚𝑛𝑀𝐼𝑇 =

2𝑍𝑚2

𝑍𝑚2 + 𝑍𝑛

2

𝑚 = 0,1 𝑛 = 1,2

Alternative simplified approach to calculate tranmission / reflection coeffiicients:

- Modified image theory based approach

- convenient for TD computations

- a comparison with the plane wave approximation in FD needed for benchmark

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ϑ=0

Installation procedure

Run transmitted_E_field_pkg.exe.

This action installs MATLAB Compiler Runtime (MCR) version 8.1. and

extracts the following files:

- transmitted_E_field.exe

- MCRInstaller.exe

- readme.txt file

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. . .

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User guide

▪ ANTENNA PARAMETERS

▪ GROUND PARAMETERS

▪ FREQUENCY DOMAIN (FD)

▪ TIME DOMAIN (TD)

▪ RUN THE SIMULATION

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1.

2.

3.

4.5.

Graphical user interface

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Antenna parameters

L

(0,h) (L,h)

Ground parameters

d1

d2

εr1 σ1

εr2 σ2

εr3 σ3

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Frequency domain (FD)

Figures generated:

- E vs. x (z) if only one z (x) point is defined

- contours if xz grid is defined

Figure generated:

- E vs. frequency

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Figure generated:

- None

- The results are saved in .txt file

Freq. (MHz) x(m) z(m) Real(Ex) Imag(Ex) Real(Ez) Imag(Ez)

Frequency domain (FD)

22

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Time domain (TD)

▪ The excitation is given as a Gaussian waveform

where the parameters tw and t0 represent the

half width and time delay of a pulse

▪ Figure generated upon the calculation:

- E vs. time for tangential component of electric

field

23

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Examples1. E vs. xz

2. E vs. z

3. E vs. freq

4. E vs. time

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1. E vs. xz25

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f = 10 MHz 26

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1. E vs. xz

f = 10 MHz 27

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1. E vs. xz

f = 300 MHz 28

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1. E vs. xz

f = 300 MHz 29

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1. E vs. xz

30

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1. E vs. xz

Freq. (MHz) x(m) z(m) Real(Ex) Imag(Ex) Real(Ez) Imag(Ez)

Freq. (MHz) x(m) Real(I) Imag(I)

31

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1. E vs. xz

2. E vs. z

32

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3. E vs. f

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3. E vs. f

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3. E vs. time

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36REFERENCES

[1] D. Poljak, “Advanced Modelling in Computational Electromagnetic Compatibility,” John Wiley andSons, New York 2007.

[2] D. Poljak, V. Dorić, “Transmitted field in the lossy ground from ground penetrating radar (GPR)dipole antenna”, Computational Methods and Experimental Measurments XVII 3, WIT Transactions onModelling and Simulation, Vol 59, pp. 3-11, 2015

[3] A. Šušnjara, D. Poljak, S. Šesnić, V. Dorić “Time Domain Integral Equation versus Frequency DomainBoundary Element model for calculation of Transmitted Electrical Field for Ground Penetrating Radar(GPR) Antenna

[4] A. Šušnjara, D. Poljak, V. Dorić, “Electric Field Radiated By a Dipole Antenna Above a Lossy HalfSpace: Comparison of Plane Wave Approximation with the Modified Image Theory Approach,”SoftCOM conference, Split, 2017

[5] A. Šušnjara, V. Dorić, D. Poljak, “ Electric Field Radiated By a Dipole Antenna and Transmitted Into aTwo-Layered Lossy Half Space,” submitted for SpliTECH conference, Split, 2018