finite difference time domain (fdtd) analyses applied to ndt & e inductive sensor modelling

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2008; 24:1227–1237Published online 25 July 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1028

Finite difference time domain (FDTD) analyses applied toNDT & E inductive sensor modelling

I. Silva1,∗,†, J. Beck1, E. Costa1 and P. Gaydecki2

1Departamento de Engenharia Mecanica e Mecatronica, Faculdade de Engenharia, Pontifıcia UniversidadeCatolica do Rio Grande do Sul, Av. Ipiranga, 6681 - Predio: 30 - Bloco: E - sala 167,

CEP: 90619-900 - Porto Alegre RS, Brazil2Department of Instrumentation and Analytical Science, UMIST, Faraday Building, B9, P.O. Box 88,

Manchester M60 1QD, U.K.

SUMMARY

In this work, a simple but effective algorithm was developed that can perform 3D simulations of magneticfields emanating from coils and simple geometry objects in the time-harmonic domain. The softwarewas intended to provide information that would help in an inductive sensor design, by simulating theinteraction of the excitation field with objects with and without defects positioned within the field spaceof an inductive sensor. The object field was disturbed in all its three components in the presence of a 3Ddefect. The change in magnetic field intensity caused by the defect was of the order of 104 times smallerthan the excitation field at a distance of 15 cm from the objects. This suggests that a large amplificationfactor should be used in the sensor design. The main contribution of this article lies in the fact that apassive inductive sensor could be modelled by finite difference time domain, with enough details on howit would respond to metal objects and its defects. Copyright q 2007 John Wiley & Sons, Ltd.

Received 1 June 2006; Revised 30 May 2007; Accepted 31 May 2007

KEY WORDS: finite differences; inductive sensor design; simulation

INTRODUCTION

It is difficult, if not impossible, to visualize field lines and interactions with different materials byjust looking at the Maxwell’s equations. For some situations, it is possible to find an exact solutionfor a given problem, as presented by Dodd and Deeds [1] for axially symmetric eddy-currentproblems. In this work, the eddy currents were assumed to be produced by a circular coil drivenby a constant amplitude alternating current. In [2], the interaction of an eddy-current coil with

∗Correspondence to: I. Silva, Departamento de Engenharia Mecanica e Mecatronica, Faculdade de Engenharia,Pontifıcia Universidade Catolica do Rio Grande do Sul, Av. Ipiranga, 6681 - Predio: 30 - Bloco: E - sala 167,CEP: 90619-900 - Porto Alegre RS, Brazil.

†E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

1228 I. SILVA ET AL.

a conductive wedge is calculated. That work also presented the 3D TREE method that allowed thedetermination of the coil impedance changes due to the currents in the conductor.

The aim of this work is to provide mathematical and visual or graphical descriptions of the fieldbehaviour for certain simple situations involving objects positioned within the field space of aninductive sensor, such as the ones used in [3].

With the advances in newer generations of personal computers regarding speed, memory andstorage capacity, it is possible to develop and use algorithms to perform numerical calculationsthat would have been previously very limited or time-consuming.

There are many computational methods that are utilized in electromagnetism, such as finitedifferences (FD), finite element, boundary element and tube and slices [4]. The chosen techniquewas the FD method for reasons described in the next section. A simple but effective algorithmwas developed that can perform 3D simulations involving coils and simple geometry objects inthe time-harmonic domain. The programming language utilized was Pascal 7.0, and the code ranon a 1.7GHz PC, with 512MB of RAM, for the results shown here.

THE FINITE DIFFERENCE (FD) IN THE TIME-HARMONIC DOMAIN METHOD

FD is a very popular electromagnetic modelling method, mainly because it is easy to understand andimplement. FD formed the bases for most of the later developed techniques [5] and it is relativelyrapid to develop general-purpose programs using the equations provided for electromagnetism.Once again, the objective of the study was not to devote time and effort in developing softwarefor very complicated problems, but to provide information that would help in the sensor design.Having said that, the FD method was found to be the most suitable choice.

FD is a very versatile and intuitive modelling technique. When applied in the time domain, itpermits the finding of the H field, for instance, everywhere in the computational grid, and enablesthe development of animated displays of the field movement in time. It is very straightforward tospecify materials in all cells in the computational mesh. In fact, it is possible to model nearly allmaterials using this method [6]. When used to solve electromagnetic problems, FD provides theH field directly.

A weakness with the FD method, however, which appears in other numerical methods, e.g.finite element method, boundary element method, MOM, when solving high-frequency Maxwell’sequations, is the fact that the cells in the computational mesh must be small compared withthe wavelengths and the smallest geometry in the model. Depending on the problem, very largecomputational domains or very fine meshes have to be used, which result in very long processingtimes.

In the FD time domain (FDTD) methods, Maxwell’s equations in their differential form areimplemented in software and solved interactively such that the electric field E is calculated for agiven instant of time. Next the magnetic field H, since it is dependent on the Curl of E, is solvedin the next instant of time. The process is repeated for as long as required. Of course, this is asimplified description of the complete algorithm, which is far more detailed. The method, however,is true for multiple dimensions regarding specific considerations.

A variation of the method is the implementation of the equations in the time-harmonic domain.In many applications, the source of electromagnetic field is sinusoidal. The advantage in theformulation and the solution of the field equations is that, by using the phasor notation, it ispossible to make the time-dependent field equations look like static equations [6].

Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1227–1237DOI: 10.1002/cnm

FINITE DIFFERENCE TIME DOMAIN ANALYSES 1229

For example, consider the time-varying H field given by

H = H0 sin(�t) (1)

which in the phasor notation gives

H = H0ei�t (2)

where H is a phasor. Note that, for t = 0, H = H0. The derivative of (2) with respect to time is

�H�t

= i�H0ei�t = i�H (3)

Similarly, the time-harmonic form for all Maxwell’s equations can be obtained by simply substi-tuting �/�t by i�. The second-order derivative �2/�t2 is substituted by −�2. No time appears inthe new set of equations, although a respective solution in the time domain can be obtained bytransforming Equation (2) into:

H(t) =Re(H0ei�t ) (4)

For low frequencies, i.e. frequencies below a few tens of kHz (where �D/�t can be neglected),electric and magnetic fields can be studied separately as independent quantities [7].

IMPLEMENTATION

This section describes the program ‘MField’ developed for magnetic field calculations. In order touse it, a computational domain was established. The computational domain is the volume of workwhere the simulation will be performed. The domain is created by a mesh that gives a volumecomposed of subvolumes in 3D. The mesh can assume different shapes, but in this work a regularsquare mesh, or cubic cells, also known as Yee cells, were used. For most of the calculationspresented here, a 51× 51× 51 grid was used.

The nodal points for the calculation were specified as the centre of each cell in the mesh. Theorigin of the domain was chosen to be the middle point of cell (26, 26, 26). The user can enterthe scale to the volume.

One number specifies the same dimensions for all three sides in the computational cubic domain.The simulation program treats coils as mesh-independent elements by allowing the user to enter

information such as diameter, length, number of turns and position in the grid, as well as excitationcurrent, frequency and winding direction, i.e. clockwise or counterclockwise. Since the sensorsunder design are coils, the software allows for coil-shape design, as cylindrical and conical coilscan be modelled. The program can take as many coils as it requires, and a coil can be placedoutside the computational volume if required by the user problem. Equation (5) is utilized tocalculate the H field at every point within the domain.

The program does not take into account mutual inductance effects, as only one coil was simulated,and, if more than one coil is modelled, the resulting field will be the summation of all contributionsfrom the coils, which can lead to significant errors to the resulting field

H= I

4�

∫C

dl× rr2

(5)



Figure 1. Effect of coil geometry and arrangement in the excitation field: y–z plot on plane x = 0.

The coils must be sectioned into small segments and two vectors determined, dl, along the wirewith the direction of the current flow, and r, the distance between the conductor section centralpoint and the point in the grid where the field is required. See Figure 1.

Each turn in the coil is subdivided by the program into 36 equally spaced segments, havingtherefore 10◦ of central angle. Equation (5) is applied for every field component Hx, Hy and Hzseparately in each cell of the volume. The total magnitude is also available from the program.Examples of the excitation field generated by different coil arrangements are shown in the followingplates and figures.

Plate 1 shows three cases of coil geometry and their effects on the magnetic field intensity: (a)cylindrical coil with 10 turns, length= 10mm, diameter= 10mm, placed in the origin (0, 0, 0); (b)conical coil, with 10 turns, length= 10mm, greater diameter= 10mm, smaller diameter= 5mm,placed in the origin (0, 0, 0); (c) two cylindrical coils, the lower with 10 turns and the current flowingclockwise, the upper with five turns and current flowing counter clockwise, placed symmetricallyspaced from the origin (0, 0, 0) by a distance of 2mm from each other. The field cancellation isachieved in the middle area between coils, as shown in the graph of magnetic field intensity versusthe z co-ordinate, at y = 0 cm. The current is 1A and the field intensity is given in Am−1. Plate 2shows the field components Hx, Hy and Hz, of the H field created by the coils in Plate 1(c) on az-plane 38mm high from the origin (0, 0, 0).

The next step is to specify the material of each cell, so that the H field can interact withit according to the geometry modelled. The material is typically specified by its permeability,permittivity, and conductivity. Figure 2 represents a computational domain composed of twomaterials, labelled 0 and 1.

In Figure 3, the nodes N represent the centre of the cells in the mesh. The third dimensionis not shown. As the mesh utilized was a regular square mesh, the modelling of objects withcross-sections other than rectangular e.g. circular was limited by the mesh resolution. In this study,


Plate 1. Effect of coil geometry and arrangement in the excitation field: y–z plot on plane x = 0.

Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24(11)DOI: 10.1002/cnm

Plate 2. Hx, top left Hy, top right and Hz, bottom, components of the H field, on a z-plane38mm high from the origin (0, 0, 0).



Figure 2. Computational domain composed of two materials, labelled 0 and 1. SurfacesS1–4 represent the boundaries.

Figure 3. Notation used for FD method implementation, 2D representation.

only rectangular cross-section objects were simulated. They were placed somewhere within themesh, which had free space as its initial condition.

As the field encounters an object, it interacts with it, according to the equation applicable tolow frequencies:

∇ × H = J (6)

The implementation of Equation (6) in the FD model is given by the following Pascal code:

deix1 := (((hz2 − hz0) − (hy2 − hy0))/(2 ∗ scale))

deiy1 := (((hx2 − hx0) − (hz2 − hz0))/(2 ∗ scale)) (7)

deiz1 := (((hy2 − hy0) − (hx2 − hx0))/(2 ∗ scale))

where dei is the curl of H in a given direction, h is the magnetic field, letters refer to the fieldcomponent direction and numbers refer to the nodes in the mesh. ‘Scale’ represents the distancebetween two consecutive cells, and is identical in all directions.



OBJECT MODEL

Two models were created for this investigation, a model of a perfect square bar and a model of asquare bar with a cut. Geometric details of those bars are shown in Figure 4. The computationalvolume was initially set to contain a 27× 106 mm3 volume of air. The x , y and z coordinatesranged from −150 to 150mm. For the simulations, the bars’ centres were located at (0, 0, 70)from the origin. The excitation coil with characteristics similar to that shown in Plate 1(a) andexcited at a frequency of 1 kHz was placed in the domain at coordinates (0, 0,−150).

The current density J is responsible for the creation of a field H0 opposite in direction to theoriginal field H. Ida [7] shows how the tangential components of fields decrease exponentially inmagnitude with depth of penetration in a conducting material. The phenomenon is known as the‘skin effect’. According to Ida, the skin depth is defined as the depth at which the amplitude ofthe field is reduced to 1/e (or 37%) of the surface field value. The expression for that is given by

� =√

2

��(8)

where � is the skin depth. It is a function of the conductivity, permeability and frequency ofexcitation. The higher the frequency, the shallower the penetration. This phenomenon can beillustrated by running the simulation at different frequencies. It can also be compared with thetheoretical value for the penetrating field, as can be seen in Figure 5, for 1Hz and 1 kHz frequencies.

In Plate 3, the skin effect of the magnetic property current density with frequency is studied. Atzero frequency, i.e. under magnetostatic conditions, it is observed that a deeper field penetrationresults in a strong object field, as expected and is supported by Equation (8). As the frequencyincreases, the field intensity decreases with the inverse of the square root of the frequency, andthe larger currents are concentrated in at shallower depths, closer to the surface.

Simmonds [8] found that, at low excitation frequencies, where Biot–Savart can be applied,and for single-bar models, no difference in the magnitude of flux density is seen with change infrequency. The same was found here, for the magnitude of the object magnetic field for frequenciesbelow 20 kHz.

Figure 4. Object geometries utilized in the simulations.



Figure 5. Skin effect of Jz penetrating a bar; calculated by both software and theory.

It is a fact that the computational mesh density can contribute towards the accuracy of theresults. However, a finer mesh would not affect the qualitative result of the tests, and the samemesh was employed in the rest of the study. A conclusion that may be drawn from the above isthat the frequency of excitation would mainly be determined by the sensor circuit’s characteristicof operation, rather than the object’s response to it.

The flow of the currents in the material generates a magnetic field which opposes the excitationfield. The opposing field, also called the object field, is calculated in the program by applying avariation of Equation (5), see [9]

B = �

4�

∫∫∫J × rr2

dv (9)

where dv is the differential volume.The magnetic field H is obtained from Equation (9). The total field is given by

HT = HE + HO ⇒ all volume

∇2 J = ��J ⇒metal(10)

where the subscripts T stands for total, E for excitation and O for object. The second equation inEquation (10) is the homogeneous equation for the current density in the time-harmonic form, andis responsible for the skin effect as J penetrates the object [6]. The current density J was also takeninto account previously in Equation (9) as a source of object magnetic field. The algorithm usedrequires that Equations (6), (9) and (10) be revisited interactively until the convergence parameteris reached. The next section gives details on the boundary conditions applied and the convergencecriteria.



Figure 6. Boundary conditions for the FD method.

BOUNDARY CONDITIONS AND CONVERGENCE CRITERIA

In this study, two different materials were selected within the computational grid for the simulations.Free space was used as the initial medium in each cell in the grid and a conductor as the second ma-terial. For free space, the values for � and � were 4� × 10−7 (Hm−1) and 8.8542× 10−12 (Fm−1),respectively. For the conductor, the used values for �r, �r and � were 300, 300 and 1× 107 (�m)−1,respectively. The boundary conditions imposed for the computational domain are shown in Plate3.

In Figure 6, P means a property, and is valid for H , B and J . As for the second material,subscript 1, the same conditions were imposed, i.e. the tangential component of the propertyis continuous. Therefore, the same value is assumed immediately below the surface. This wasachieved by the code (Hsup = HE + HO), which is part 1 of Equation (10). With the boundarycondition implemented, calculation throughout the material is performed by Equation (10). Theconvergence conditions for the FDTD method is normally given by the expression

�x�� (11)

where �x is the distance between two consecutive nodes in the grid and � is the wavelength. Thewavelength is a function of the medium and frequency. For the frequency range in the order ofkHz, either in free space or in metal, the wavelength is of the order of metres. The criterion thenused to monitor convergence and hence when to stop the process was

� = (Pnew − Pold)/Pnew<0.001 (12)

Although simulations normally terminate when a 10% variation is reached, the code showed thatthe lower value chosen would only take from 7 to 12 iterations to satisfy the conditions forconvergence, for simulations of conducting bars or plates, with square cross-section, which is avery simple geometry. For more complex geometries, this lower limit would take too much timeto reach convergence, and may not be applicable.

PROGRAM FLOW CHART

Figure 7 shows a simplified diagram of the simulation program used in this investigation. Thefirst block start values represents the input options the program asks the user, such as domaindimensions, coils’ information and second material (bars) details. The program next calculates theexcitation field emanating from the coils, in He from coils. With the excitation field calculated


Plate 3. Current density component Jy and skin effect for different excitation frequencies. Plot oflongitudinal plane AA for a bar with a cut.

Plate 4. Magnetic field intensity magnitude in Am−1, at z = 70mm, for a perfect bar (left) and a bar witha cut (right), the excitation frequency fE = 1 kHz.



Figure 7. Flow chart for the FD method.

initially for the air, the metal is then taken into consideration. The current density penetrating themetal is calculated in J , which is responsible for the creation of an object field Ho. The originalfield is perturbed by the object’s presence in HT = HE + HO.

The process is repeated until convergence is achieved in conv?. The program then stops calcu-lating and saves the current values of the excitation and object fields and current density.

SIMULATION RESULTS

Some results obtained with the program ‘MField’ are shown next.Plates 4 and 5 show the H field and its component Hz, respectively, for a perfect bar and a bar

with a cut. The field perturbation caused by the defect is visible at around y = 0, where the cutwas located.

In Plate 6, the Hz component of the excitation field is calculated along the line z =−150 to150mm, at y =−50 to −100mm. Remembering that the excitation coil is located at coordinates(0, 0,−150) from the origin, the field lines have their peak values at z = − 150mm. This meansthat the sensing element is exposed to large field values unless that field can be cancelled in itsvicinity by a compensating coil. Also an important aspect is the magnitude of the signal undermeasurement, i.e. the object field, when compared with the excitation field. As the simulationresults show, in the next figure, the object field is 104 times smaller than the excitation field atz = − 120mm, for the case under study. The discontinuity found in the right graph of Figure 8was due to the change in direction, in the resulting z-component of the object field, as the graphshows the magnitude of the field.

Figure 9 shows the object magnetic field intensity components, along the z-coordinate, aty = 0mm and x =−100mm. These field values are for a perfect bar under excitation. A bar with



Figure 8. Excitation against object plots for the magnetic field intensity magnitude (left) and Hz component(right) plots along the z coordinate, at x = 0 and y = 0.

Figure 9. Object field intensity components, along the z-coordinate, at y = 0mm and x = −100mm.

cut would give similar responses, as can be seen in the next figure, having, however, differencesin the signal magnitude.

Although the difference in field magnitude between the two conditions under investigation issmall, it can still be visualized in the graph. The next figure therefore suggests that the objectsignal must be amplified in order to detect it and subsequently process it.

In Plate 7, the magnetic field intensity components, along the y-coordinate, are provided againstthe components for a perfect bar. As the bars run along the direction given by x = 0mm, the fieldcomponents in the graph are for lines along the bars’ direction, at 80mm of height. There is visiblereduction in the field magnitude for a bar with defect.


Plate 5. Hz component of the magnetic field intensity magnitude in Am−1, at z = 70mm, for a perfectbar (left) and a bar with a cut (right), the excitation frequency fE = 1 kHz.

Plate 6. Hz components of the magnetic excitation field intensity along the line z =−150 to150mm, at x = 0mm and i ranging from 50 to 100mm in 10mm steps. The excitation coil is

placed at coordinates (0, 0,−150).


Plate 7. Magnetic field intensity components, along the y-coordinate, at x = 0mm and z =−50mm, fora bar with a cut. The Hx, Hy and Hz components for a bar without a cut are provided for reference.



CONCLUSIONS

In this chapter, a series of models for two bars, one with a cut and another without, was discussed.The bars were excited by an AC magnetic field produced by a coil. Frequencies ranging from 0to 10 kHz were studied.

Three important conclusions could be drawn from the study:

(1) For some aspects of the simulations, e.g. skin effect investigation, the discretization usedwas not found to be fine enough. However, the qualitative results were not affected by thatfact. The frequency parameter did not show any significant influence on the object magneticfield.

(2) The object field was disturbed in all its three components by the presence of a 3D defect.The change in magnetic field intensity caused by the defect was of the order of 104 timessmaller than the excitation field at a distance of 150mm from the bars. This suggests thata large amplification factor should be used in the sensor design.

(3) In order to make it possible to amplify the sensor signal, the large excitation componentcan be cancelled by the use of a compensation coil carrying current flowing in the oppositedirection to that of the excitation. That could be the most important conclusion, as withoutthat cancellation, the object field would appear negligible, when compared with the originalfield.

No attempt was made in this study to simulate different materials or to consider the phaseinformation contained in the results. Future works will include the detection of different materialsand corrosion.

REFERENCES

1. Dodd CV, Deeds WE. Analytical solutions to eddy-current probe-coil problems. Journal of Applied Physics 1968;39(6):2829–2838.

2. Theodoulidis T, Bowler JR. Eddy current interaction with a right-angled conductive wedge. Proceedings of theRoyal Society of London 2005; 461(2062):3123–3139.

3. Silva INL. Portable inductive scanning system for imaging of steel bars in concrete structures. Ph.D. Thesis,UMIST, 1999.

4. Glossop KJ. Imaging steel in concrete: generation, modelling and restoration of inductive images. Ph.D. Thesis,UMIST, 1995.

5. Sykulski KJ. Computational Magnetics. Chapman & Hall: London, 1995; 373.6. Ida N. Numerical Modelling for Electromagnetic Non-Destructive Evaluation. Chapman & Hall: London, 1995;

511.7. Ida N, Bastos JPA. Electromagnetics and Calculation of Fields. Springer: Berlin, 1992; 458.8. Simmonds J. Detect visualization and location for reinforcing bars and cables in concrete using active

electromagnetic induction sensors. Ph.D. Thesis, UMIST, 1998.9. Kraus JD. Electromagnetics (4th edn). McGraw-Hill: New York, 1992; 847.


finite difference time domain (fdtd) analyses applied to ndt & e inductive sensor modelling

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