Soil Ionisation

Finite difference time domain simulation of soil ionization ingrounding systems under lightning surge conditions

G. Ala ∗1, E. Francomano 2,3, E. Toscano 4, and F. Viola 1

1 Dipartimento di Ingegneria Elettrica, Universita’ degli Studi di Palermo, Viale delle Scienze, 90128 Palermo,Italia

2 Dipartimento di Ingegneria Informatica, Universita’ degli Studi di Palermo, Viale delle Scienze, 90128Palermo, Italia

3 ICAR, Istituto per il CAlcolo e Reti ad alte prestazioni, CNR, Viale delle Scienze, 90128 Palermo, Italia4 Dipartimento di Fisica e Tecnologie Relative, Universita’ degli Studi di Palermo, Viale delle Scienze, 90128

Palermo, Italia

Received 30 June 2003, revised 30 October 2003, accepted 2 December 2003Published online 15 March 2004

Key words Finite difference, electromagnetic transient, grounding systemsSubject classification 65C20, 65N06

This paper proposes a Maxwell’s equations finite difference time domain (FDTD) approach for electromagnetictransients in ground electrodes in order to take into account the non linear effects due to soil ionization. A timevariable soil resistivity method is used in order to simulate the soil breakdown, without the formulation ofan initial hypothesis about the geometrical shape of the ionized zone around the electrodes. The model hasbeen validated by comparing the computed results with available data found in technical literature referred toconcentrated earths. Some application examples referred to complex grounding systems are reported to showthe computational capability of the proposed model.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Exact evaluation of electromagnetic transients in complex grounding system has a fundamental importance forthe lightning protection design. In fact, the earth electrodes constitute a fundamental part of the electric appa-ratus in industrial and civil structures. Earthing system should have a suitable configuration in order to avoid aserious human hazard, and to preserve electrical insulation in electric and electronic equipment and installations.Moreover, in electric power installations, the shape and dimensions of the earth termination system, as a part ofa lightning protection system (LPS), are more important than a specific value of the earth resistance; this in orderto disperse the lightning current into the earth without causing dangerous overvoltages.In technical literature, a large amount of papers deal with the transient behavior of grounding systems or LPS.The problem can be analyzed by a circuit approach or by a full-wave approach, in frequency or in time domains,and by using different numerical methods [1]-[16]. Usually, the non linear phenomena due to the soil ionizationprocess around the electrodes surfaces are neglected. On the other hand, experimental and theoretical studies haveshown that, when the surge current leaking into the soil by the different parts of the earthing system increases,the electric field on the lateral surface of the electrodes can overcome the electrical strength; so the ionizationregion takes place. This phenomenon has a great influence on the performance of concentrated electrodes fed byhigh magnitude currents [17]-[20]. On the other hand, it has been shown that also in extended electrodes the soilionization can take place [21]: this non-linear phenomenon modifies the electromagnetic transient behavior ofthe earthing system with respect to the case of absence of soil breakdown.A deionization process follows the local soil breakdown; it starts when the surge current begin to decrease andstops when the stationary values of soil parameters are restored.

∗ Corresponding author: e-mail: [email protected], Phone: +00 39 091 6615288, Fax: +00 39 091 488452


Appl. Num. Anal. Comp. Math. 1, No. 1, 90 – 103 (2004) / DOI 10.1002/anac.200310008

Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 91

In order to correctly take into account the non linear behavior of the earth electrodes during the ionization and thesubsequent deionization processes, this paper proposes a finite difference time domain (FDTD) model based onthe numerical solution of the Maxwell’s equations. Very few references have been found in technical literatureapproaching the problem by this full-wave method [22]-[23]; some papers approach the soil breakdown problemby using circuit theory [24]-[27].A time variable soil resistivity approach [17] is used in order to simulate the non linear effects of soil breakdown.The method has been validated by comparing the computed results with available data found in technical litera-ture referred to concentrated earths. Application results referred to more complex earth electrodes are reported toshow the capability of the simulation model.

2 Some physical review

When a current is injected into the soil by a ground electrode, spatial electric field distribution will be generatedin accordance with the following equation:

E = ρ J, (1)

where ρ is the soil stationary resistivity and J is the current density vector at the observation point. Generally,the previous equation is time dependent. Wherever E exceeds the electrical strength, soil breakdown will occur.The soil ionization region starts at the electrode surface where the current density has its highest value. Thisregion extends up to a distance where the current density decreases to a value that makes the electric field lowerthan the critical breakdown value. Otherwise, when the surge current injected into the soil by the electrodes beginto decrease, the electric field in the ionized region begin to decrease also, and the deionization process takes placeso restoring the stationary steady soil characteristics.At the micro-structural level, most soils consist of basically non-conducting particles coated with water in whichsome salt is dissolved, with air filling the voids between the soil particles. The water coating provides intercon-nected water paths which determine the low stationary conductivity of the soil. Such conductivity will depend onboth the amount of water and the amount of salt present in the soil. The size of particles of a soil sample usuallyvaries within a wide range. The average size of the air voids within the soil will depend on the frequency distribu-tion of the size of particles. For example, a soil consisting of a very fine dust-like particles will have smaller sizevoids, while a sandy soil with coarse particles will have larger size voids. The shape of the voids is usually highlyirregular especially if the surrounding particles have sharp edges. This makes the maximum electric field withinthe voids of a soil gap significantly higher than the maximum electric field within an air gap having the sameconfiguration and dimensions. So when electric field in the void between the soil grains becomes large enough toionize the air in the void, ionization process starts and the current, that until this moment flows through the waterpaths of the soil, is now mainly conducted by ionized vacuum of air. Due to the irregular shape of the voids andthe effect of the relatively large dielectric constant of the soil, the critical value of the electric field can be muchsmaller than the breakdown field of an equivalent air gap. In this paper the critical value of the soil breakdown isassumed to be 1.1 kV/cm, or 3.0 kV/cm [17].

3 Soil ionization model

In technical literature, different models are employed to describe the soil ionization and deionization processes.The time variable soil resistivity approach [17], [19], [20] and the time variable electrodes geometry approach[24]-[26] are the most commonly used. The first one considers a time variable soil resistivity in the regionsurrounding the electrode. This time variable resistivity is a non-linear function of the electric field.The variable electrodes geometry approach models a given electrode embedded in an ionized soil as an electrodeof modified transversal dimensions into a non ionized soil. Therefore, this approach considers the soil resistivityunchanged, and the non linear behavior is given by the dependence of the equivalent electrode geometry onthe current flowing into the soil. For each value of the current, the effective radius of the electrode is obtainedby assuming that the electric field may not exceed the critical value. With this approach, the ionized region isassimilated to the conductor and the electric field in this region is assumed to be roughly null, as if the ionized


92 G. Ala, E. Francomano, E. Toscano, and F. Viola: FDTD simulation of soil ionization

region was short-circuited with the electrode. Obviously, this approach is far from the physics of the phenomenonand it needs a priori hypothesis of the shape of the ionized zones of the soil.In Fig. 1 the most used assumptions of the shape of the ionised zones of the soil are reported, for typicalconcentrated earths. Hemispherical surfaces and coaxial cylindrical shells are assumed in [17], shells of variableradius are considered in [24]-[25].

Air Air Air

Soil Soil Soil

Fig. 1 Shapes of the ionized zones for typical concentrated earths assumed in computations by other authors. From left toright cross sections of the ionized shapes: hemispherical shells, coaxial cylindrical shells and hemispherical surface assumedin [17], shells of variable diameter considered in [24]-[25].

In order to correctly take into account the non linear behavior of the earth electrodes during the ionization andthe deionization processes, the authors consider that a time variable resistivity approach can be usefully employedtogether with a FDTD method based on the numerical solution of the Maxwell’s equations. The half-spaceproblem is exactly solved by simulating the two media with different electrical parameters. The time variableresistivity approach proposed in [17] is used to simulate the non-linear effects of the ionization and deionizationprocesses. In particular, for the first one the time variable resistivity law is expressed by the following:

ρ = ρ0e− t

τ1 , (2)

where ρ0 is the stationary resistivity value of the soil, τ1 is the ionization time constant of the soil. Theionization process is driven by the electric field E: at the instants when E ≥ Ec, Ec is the electrical strength inthe soil, the resistivity behavior is driven by (2); for the deionization process the time variable resistivity law isexpressed by the following:

ρ = ρi + (ρ0 − ρi) · (1 − e−t

τ2 ) · (1 − E

Ec)2, (3)

where ρi is the minimal values reached by the soil resistivity during the ionization process and obtained by (2),τ2 is the deionization time constant of the soil, E is the actual amplitude of the electric field. In the formulationproposed in this paper, the deionization process is directly driven by the electric field, as proposed in [20] ratherthan by the current density as proposed in [17].As a great advantage, the proposed model enables to study the electromagnetic transient behavior of groundingsystems, directly in time domain, and without the formulation of an initial hypothesis about the geometrical shapeof the ionized zones. On the contrary, this hypothesis is necessary in the variable electrodes geometry approach,and it is also used in the papers related to ionization phenomena study of concentrated earths [17], [20] with atime variable resistivity approach.

4 Numerical model

The FDTD formulation of an electromagnetic field problem is a convenient scheme to numerically solve scat-tering problems [28]-[32]. By using Maxwell’s curl equations, together with the constitutive relations of themedium, the following formal equations can be written:



curl E(r, t) = −µ(r, t)∂ H(r, t)

∂t, curl H(r, t) = ε(r, t)

∂ E(r, t)∂t

+ σ(r, t) E(r, t), (4)

where E is the electric field vector, H is the magnetic field vector, ε(r, t) is the dielectric permittivity, µ(r, t)is the magnetic permeability and σ(r, t) is the conductivity of the medium. In general, as it is shown, all thesequantities are functions of space r, and time t.In a Cartesian coordinates system (x, y, z), the previous vector equations can be split into six scalar equations asfollows:

∂Hx

∂t=

1µ

(∂Ey

∂z− ∂Ez

∂y),

∂Hy

∂t=

1µ

(∂Ez

∂x− ∂Ex

∂z),

∂Hz

∂t=

1µ

(∂Ex

∂y− ∂Ey

∂x), (5)

∂Ex

∂t=

1ε(∂Hz

∂y− ∂Hy

∂z−σEx),

∂Ey

∂t=

1ε(∂Hx

∂z− ∂Hz

∂x−σEy),

∂Ez

∂t=

1ε(∂Hy

∂x− ∂Hx

∂y−σEz). (6)

In (5) and (6) the electric and magnetic fields components with respect to the reference geometric system, havebeen considered.In applying the FDTD method, the first step is to divide the region of interest into a grid of nodes; then, the givendifferential equations have to be approximated by finite difference equivalent equations, that relate the dependentvariables at a node in the solution region to their values at the neighbouring points. The obtained differenceequations are then solved subjected to the prescribed boundary and initial conditions.By using central finite difference approximation for space and time derivatives, and by considering the identities:i = i∆x, j = j∆y, k = k∆z, n = n∆t, the following relations valid for a generic scalar regular function,F = F (x, y, z, t), hold:

∂Fn(i, j, k)∂x

=Fn(i + 1/2, j, k) − Fn(i − 1/2, j, k)

∆x+ O(∆x2); (7)

∂Fn(i, j, k)∂t

=Fn+1/2(i, j, k) − Fn−1/2(i, j, k)

∆t+ O(∆t2). (8)

So operating, for example the first of relations (6) becomes:

En+1x (i + 1/2, j, k) = 1 − σ(i + 1/2, j, k)

ε(i + 1/2, j, k)En

x (i + 1/2, j, k)+

+∆t

ε(i + 1/2, j, k)Hn+1/2

z (i + 1/2, j + 1/2, k) − Hn+1/2z (i + 1/2, j − 1/2, k)+ (9)

+Hn+1/2y (i + 1/2, j, k − 1/2) − Hn+1/2

y (i + 1/2, j, k + 1/2).

The right side of (9) consists of five known values at the previous time (n + 1/2), while the left side has theunknown value of Ex at the actual time (n + 1).A graphic way of describing the difference formula (9) is throughout the so called computational molecule [28].In Fig. 2 the circle is used to represent the electric field value at the grid point and the square represents themagnetic field value. Electric and magnetic field components are evaluated at alternate half time steps.



t

y

n+1

n+1/2

n

j j+1/2 j+1

Ex (new)

Ex (old)

Hz (right)Hz

(left)

Electric field

Magnetic field

=new - old

t

right - left

y

Fig. 2 Computational molecule of equation (9), Hy is not showed.

(i,j,k)

Ez Ez

Ez

Hz

Ex

Ex

Ex

Hx

Hy

Ey

Ey

Eyy

z

x

Fig. 3 Position of the field components in a unit cell of the Yee’s lattice.

This FDTD paradigm, was introduced by Yee [33] in order to numerically solve the two linked curl equations(4). In Fig. 3 the positions of the field components in a unit cell of the Yee’s lattice are shown.

As a further specification, in translating the hyperbolic system (5)-(6) into a computer code, one have to besure that, within the same time loop, one type of field components is calculated first, and then the obtained resultsare used in calculating the other type of components. So operating, the discretization of the partial differentialequations (5) and (6) can be completely carried out.In order to use the FDTD scheme for electromagnetic surge analysis, some particular features must be introduced.It is known that central finite difference approximation, for space and times derivatives, are second order accurate[28]-[31]. In order to ensure the accuracy and the stability of the computational algorithm, the FDTD methodrequires that the spatial increment must be small compared to the minimum dimension of the scattered object.Thus, the time increment ∆t must satisfy the so called Courant stability condition [34]:

∆t ≤ 1

umax

√1

∆x2 + 1∆y2 + 1

∆z2

, (10)

where umax is the maximum wave phase velocity within the model.As another fundamental consideration, it is to be underlined that the simulation of transient performance of earthelectrode constitutes an open boundary electromagnetic problem. One of the most important features of theFDTD method is the efficient solution of electromagnetic wave interaction problems in unbounded regions. Forsuch problems, in order to limit the extent of solution region, an absorbing boundary condition (ABC) must beintroduced at the outer lattice boundary, to create the illusions of the extension of the lattice to infinity. This must



be done without generate unpredictable reflections of propagating electromagnetic waves, because there is noway to determine which is the real wave and which is the reflected junk. An alternate point of view to realize anABC is to terminate the outer boundary of the space lattice in an absorbing material medium. This is analogous tothe physical treatment of the walls of an electromagnetic anechoic chamber. The perfectly matched layer (PML)firstly introduced in [35], [36] and revised in [31] is used in order to enforce the appropriate ABC in the openregion. The basic concept behind the PML method is that when a wave is propagating in a medium A (region ofstudy) and it impinges upon medium B (absorbing medium), the amount of reflection is dictated by the intrinsicimpedances of two media; so if two media have the same impedance no reflection would occur and if the secondmedium is also a lossy one, the wave will die out before it hits the boundary. The PML formulation adopted inthe paper, prevents spurious reflections from the artificial boundary of the problem, in such a way that waves ofarbitrary incidence, polarization, and frequency are matched at the boundary as reported in [30], [31].In order to introduce in the FDTD scheme current and voltage probe and source, useful for transient groundingsystems simulation, the following further considerations hold. Current source and probe can easily be modelledby using Ampere’s law:

I =∮

C

H · dl, (11)

that is, the current density flux through a surface is equal to the contour integral of the magnetic field.Voltage source and probe are more difficult to be modelled. In fact, unlike the static electric field, the timevariable electric field is rotational and the concept of voltage has no more a unique definition. However, byconsidering an electric field component of a cell, the voltage difference across a side of a cell can be defined asV = E∆s, because waves of which the wave length is shorter than 2∆s are not in FDTD calculation due tobandwidth limitation equal to ∆s.

5 Validation of the model

The model has been tested by using two different way of validation. Figure 4 shows a horizontal conductorsystem, where a straight conductor is placed above a copper plate; this scenario is reported in references [22]where experimental test are also reported.

Voltage generator

current lead wire

l= 4 m

h= 0.5 m

copper plate

current probe

Fig. 4 Conductor arrangement.

The horizontal conductor is excited by a pulse voltage generator through another straight vertical conductor.The conductors are supposed to be thin, so some further considerations have to be made in order to introduce thethin-wire hypothesis in the FDTD scheme. In a FDTD simulation a thin-wire has a radius that is smaller thanthe size of a cell, and it is simulated by forcing the appropriate resistivity value along the axis of the wire. If



the conductor is supposed to be perfect, the electric field along the axis is null. The so called ”intrinsic radius”concept of a FDTD scheme is used to correctly simulate a thin-wire arrangement [22], [23]. According to thismodel the intrinsic radius is set to r0 = 0.2298 ∆s, where ∆s is the step of the related spatial grid.In Fig. 5 the computed time profile of the current leading through the vertical wire of Fig. 4 is shown. The readeris invited to refer to the same profile reported in [22] where the second order Liao’s method is used to enforceABCs. By comparing the two profiles, a good agreement has been reached.

Fig. 5 Time profile of the current leading through the vertical wire of Fig. 4.

Secondly, the proposed model has been validated by comparing transient grounding systems simulations withexisting data reported in technical literature. A single rod, 0.61 m length, embedded in homogeneous soil (ρ0 =50 Ωm; εr = 8; µ = µ0) has been firstly considered. This in order to compute different quantities comparablewith the same data found in technical literature.The earth electrode is schematized in the FDTD grid as sketched in Fig. 6. Four lead wires and a copper plateare used in order that the fields distribution become symmetrical. The rod is driven by an ideal current sourcegenerator with a defined wave form.

copper plate

Air

current lead wire

PML

Copper

soill= 0.61 m

gap = 0.30 m

1.52 mcurrent source

Fig. 6 Arrangement for ground electrode simulation. A portion of the FDTD grid is reported also.

In Fig. 7 the cross-section of the physical scenario sketched in Fig. 6 is reported; current lead wires are placedin different media, far from the rod.

The rod is simulated with a thin-wire conductor. According to the intrinsic radius concepts already pointedout, since ∆s = 0.061 m in this simulation, the radius of the thin-wire is set to r0

∼= 0.014 m. The rod isfirstly fed by a 10 A pulse waveform current source, in order to simulate the achievement of the steady state;a comparison between the computed resistance value reached in stationary steady condition, and the theoreticalvalue can be carried out. In Fig. 8 the transient time profile of the earth resistance is reported. In order to obtainthis resistance profile, the potential to remote ground of the input section of the rod has to be calculated. For the



current lead wires

soil

surge current

0.30 m

0.19 m

copper plate

air

PML

pec boundary

pec

bo

un

daryrod

1.52 m

0.61m

0.30 m

0.30 m

Fig. 7 Cross-section of the physical scenario sketched in Fig. 6. PML and perfect electric boundary are shown.

single vertical rod configuration of Fig. 6, in each transversal section of the rod it is possible to rigorously definethe potential to remote ground function; in fact, in this case, this voltage is the line integral of the electric fieldbetween the surface of the electrode and the remote ground, and this integral can be calculated along any linebelonging to the transversal plane. In fact, this line integral is not affected by the contribution of the rotationalcomponent of the electric field related to the time varying magnetic field. The voltage is computed by handlingthe nodal values of the y-component of the electric field with a trapezoidal rule. The resistance profile is thenobtained by dividing the voltage to remote ground of the input section along with the injected current.

Fig. 8 Transient time profile of the earth resistance of the rod of Fig. 1.

The stationary steady condition value reached during the simulation and shown in Fig. 8 (49.58 Ω), can bematched with the analytical one (50.10 Ω). A very good agreement has been reached.The rod of Fig. 7 has been then fed by a surge current. The double exponential waveform used in [17] has beenselected, in order to compare the transient resistance time profile computed with the proposed model and thatreported in the same reference [17] (Fig. 3, p.127). In particular, the 5-16.5 µs waveform with a peak valueof 3.5 kA is used for the current source time profile. The electrical strength is set to 1.1 kV/cm, and the timevarying resistivity laws reported in equations (2) and (3) are assumed. In Fig. 9 the time profile of the computedearth resistance of the rod is shown. In the first time steps of the simulation, when the injected current increase,the ionization of the soil in the close proximity of the electrode, occurs: this is accomplished by the decreasing



profile of the earth resistance. Then, when the current decrease, the earth resistance values increase since thedeionization process of the soil takes place, so restoring the stationary steady condition. This profile can bematched with that reported in [17]: a satisfactory agreement is reached, also taking into account that the model in[17] is based on the assumption that the soil around the rod is ionized by assuming as geometrical shape, adjacentcylindrical shells.

Fig. 9 Transient time profile of earth resistance for the rod of Fig. 6 fed by a surge current generator as reported in [17].

Fig. 10 Soil conductivity map for a 2 m vertical rod fed by a surge current.

6 Application examples

In order to show the flexibility of the proposed model, some simulation results are reported in the following. Atfirst, in Fig. 10 the soil conductivity map is reported for a 2 m vertical rod in a homogeneous soil. The rod isdirectly injected at the top section by a lightning current simulated with a 1.2-50 µs double exponential waveformwith a peak value of 10 kA. The rod is modelled with 20 cells; the spatial grid dimensions are: ∆x = ∆y = 5cm, ∆z = 10 cm; the soil parameters are: ρ0 = 100 Ωm; εr = 8; µ = µ0. The map of Fig. 10 is referred to thetime step of 0.25 µs. The electrical strength value is set set to 3 kV/cm [17].



Fig. 11 Time profiles for a 2 m vertical rod directly injected at the top section - observation point 20 cm away from the rod,on the soil surface.

current lead wire

copper plate

current source

PML

Copper

soil

Air

0.42 m

0.61 m

Fig. 12 A two buried electrode scenario: a rod with a horizontal electrode.

0

5

10

15 0

10

20

30

40

50

012

no. of cells

σ [ 1

/Ω]

no. of cells

soil

surfa

ce

Fig. 13 Soil conductivity map for the arrangement of Fig. 12.

As it can be deducted from Fig. 10, the soil ionization takes place in the close proximity of the electrodesurface.In order to better underline the time variant behavior of the soil characteristics, in Fig. 11, the time profiles ofthe electric field E and of the resistivity of the soil ρ, 20 cm away from the rod, on the soil surface, are reported.These profiles are normalized with respect to the electrical strength Ec and to the stationary steady resistivityvalue of the soil ρ0, respectively. The ionization and deionization processes which occur in the soil can be clearlydeducted from the ρ/ρ0 time profile.



current lead wire

copper plate

current source

PML

Copper

soil

Air

0.42 m

1.22 m

Fig. 14 A complex mesh earth electrode fed by a insulated wire at the central point.

0

10

20

30

40

50

010

2030

4050

0.020.030.04

no. of cells

σ [1

/Ω]

no. of cells


As it is shown in Fig. 10, the ionized zone of the soil surrounding the electrode takes shape depending on thelocal value of the electric field. It is to be underlined that, as already pointed out, no assumption has been maderegarding the shape of the ionized zone. Moreover, the first parts of the rod are much more ionized than the otherones, and this is in accordance with experimental observations: the soil zones more closer to the input section ofthe earth electrode are more dry than the other ones after the drain of a surge current.In Fig. 12 an upside-down T electrode is sketched.

The spatial grid steps are: ∆x = ∆y = ∆z = 6.1 cm; the soil parameters are set equal to: ρ0 = 50 Ωm;εr = 8; µ = µ0. Ec = 1.1 kV/cm. The 5-16.5 µs waveform with a peak value of 3.5 kA is used as surgecurrent. In Fig. 13 the soil conductivity map is reported for the time step of 10.15 µs. The ionized zones of thesoil near the surface of the electrodes are emphasized in the figure.

In Fig. 14 a meshed earth electrode is shown. It is to be underlined that, in technical literature, in theknowledge of the authors there are not present results and data referred to this complex grounding systems fromthe soil ionization non-linear behavior point of view. The electrode is fed at the central point by an insulated



PML

Copper

0.42 m

1.22 m

current lead wire

copper plate

current source

soil

Air

Fig. 16 A complex mesh earth electrode fed by a insulated wire at one edge.

0

10

20

30

40

50

010

2030

4050

0.02

0.03

0.04

no. of cells

σ [1

/Ω]

no. of cells


conductor carrying the surge current. This current is modeled by a 1.2-50 µs double exponential waveform with12.5 kA as peak value.

The soil parameters and the grid dimension are the same of those adopted for the example reported in Fig. 13.In Fig. 15 the soil conductivity map is reported for the time step of 1 µs. The ionized zones of the soil near thesurface of the electrodes are emphasized in the figure.

In Fig. 16 the meshed earth electrode of Fig. 14 is fed by the insulated conductor at one edge. All the sameparameters of the previous example are considered.

In Fig. 17 the soil conductivity map is reported for the time step of 1 µs.

The reported examples show the flexibility and the capability of the proposed numerical scheme. However,the simulation results are referred to geometry of complex shapes but of limited extension. Electrodes of largerextension can be easily simulated requiring more computational resources.



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Soil Ionisation

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