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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017 7000109 Computation of Lorentz Force and 3-D Eddy Current Distribution in Translatory Moving Conductors in the Field of a Permanent Magnet Bojana Petkovi´ c 1 , Konstantin Weise 2 , and Jens Haueisen 1 1 Institute of Biomedical Engineering and Informatics, Technische Universität Ilmenau, 98693 Ilmenau, Germany 2 Department of Advanced Electromagnetics, Technische Universität Ilmenau, 98693 Ilmenau, Germany Determination of the 3-D eddy current distribution inside a translatory moving conductor under a permanent magnet can accurately be done by using finite-element method (FEM). However, FEM calculations are very expensive, as they require discretization of the whole conductor volume. In this paper, we propose a new technique, to be called boundary element source method (BESM), where only boundary layers are discretized. The BESM is a modification of the hybrid boundary element method (HBEM). In the BESM, the concentrated point sources placed at the centers of boundary elements for the HBEM are replaced by distributed charge density over the area of the boundary element. This is especially useful in the regions, where neighboring boundary meshes significantly affect one another and when calculation point of eddy current is very close or belong to the surface of a boundary element. The method can handle arbitrary geometries of the specimen as well as the defect and arbitrary orientation of the magnetization vector. The accuracy of the proposed method is verified by comparing the results with the solutions obtained from a finite-element model. The proposed BESM approach is shown to be simple, robust, and computationally accurate. Index Terms— Boundary element source method (BESM), eddy currents, Lorentz force, permanent magnet. I. I NTRODUCTION A NEW approach for nondestructive testing of solid con- ductive materials, called Lorentz force eddy current testing, has been introduced recently [1]. Different from the conventional eddy current testing method, the magnetic field of a permanent magnet generates eddy currents in the electrically conducting specimen, because of its movement with respect to the magnet. The interaction between the induced eddy currents and the magnetic flux density causes Lorentz forces, which intend to break the motion. According to Newton’s third law, the Lorentz force exerts the permanent magnet in the opposite direction. Material anomalies, such as changes in conductivity, cracks, or inclusions, distort the eddy current distribution in the object under test and, consequently, the Lorentz forces measured at the magnet system. Thus, any defect in the conductive material produces perturbations in the Lorentz force signals. The relationship between the force perturbations and material anomalies can be used to detect and reconstruct defects. The reconstruction of defects requires solving the inverse problem and embedding the solution of a forward problem. One approximate forward solution of the Lorentz forces has been introduced in [2]. However, this solution is limited to a laminated specimen consisting of a package of conductive sheets, where the conductivity in the direction perpendicular to the sheets is assumed to be zero. This enforces an eddy current flow in the plane of the sheets, allowing the assumption Manuscript received May 12, 2016; revised September 1, 2016 and October 13, 2016; accepted October 15, 2016. Date of publication October 27, 2016; date of current version January 24, 2017. Corresponding author: B. Petkovi´ c (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2016.2622223 that the eddy current in the direction perpendicular to the sheets is equal to zero. We now propose calculation of Lorentz forces without assumptions on the induced eddy current flow, satisfying the boundary condition of their normal component in a conductor along all the conductor/air interfaces. This makes the method applicable to any conducting specimen. The method enables the determination of the eddy current and Lorentz force profiles inside the bar. Consequently, the total Lorentz force acting on the permanent magnet can be derived. Notwithstanding that the method can be applied to arbitrary shaped geometries of both the specimen and the defect, the focus in this paper is on a parallelepipedical bar with a parallelepipedical defect. One of the most frequently used techniques for solving elec- trostatic problems is the charge simulation method (CSM) [3]. This method simulates the actual electric field superimposing the fields coming from a number of fictitious charges placed outside the region, where the electric field has to be calculated (or inside any equipotential surface such as metal electrodes). The magnitudes of these charges are determined by satisfying the boundary condition at a selected number of points on the boundary. Once the charges are determined, one can calculate the electric scalar potential and the electric field at any point of the region. Simulating the real charges distributed on the electrode surface by equivalent sources placed on the surface instead in the inner electrode volume, the surface CSM is intro- duced [4], [5]. Introducing the equivalent charge distributions on the boundary surfaces between two dielectrics, the solution of the electrostatic field problem is found for electrodes placed in inhomogeneous media [6], [7]. This formulation is extended to the case of floating electrodes and static current fields, introducing the general source simulation method [8]. An integral-free calculation procedure, based on the substitution of boundary segments of multilayered media 0018-9464 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Page 1: Computation of Lorentz Force and 3-D Eddy Current ... · PETKOVIC´ et al.: COMPUTATION OF LORENTZ FORCE AND 3-D EDDY CURRENT DISTRIBUTION 7000109 Fig. 2. BESM model of a bar with

IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017 7000109

Computation of Lorentz Force and 3-D Eddy CurrentDistribution in Translatory Moving Conductors

in the Field of a Permanent MagnetBojana Petkovic1, Konstantin Weise2, and Jens Haueisen1

1Institute of Biomedical Engineering and Informatics, Technische Universität Ilmenau, 98693 Ilmenau, Germany2Department of Advanced Electromagnetics, Technische Universität Ilmenau, 98693 Ilmenau, Germany

Determination of the 3-D eddy current distribution inside a translatory moving conductor under a permanent magnet can accuratelybe done by using finite-element method (FEM). However, FEM calculations are very expensive, as they require discretization of thewhole conductor volume. In this paper, we propose a new technique, to be called boundary element source method (BESM), whereonly boundary layers are discretized. The BESM is a modification of the hybrid boundary element method (HBEM). In the BESM,the concentrated point sources placed at the centers of boundary elements for the HBEM are replaced by distributed charge densityover the area of the boundary element. This is especially useful in the regions, where neighboring boundary meshes significantlyaffect one another and when calculation point of eddy current is very close or belong to the surface of a boundary element. Themethod can handle arbitrary geometries of the specimen as well as the defect and arbitrary orientation of the magnetization vector.The accuracy of the proposed method is verified by comparing the results with the solutions obtained from a finite-element model.The proposed BESM approach is shown to be simple, robust, and computationally accurate.

Index Terms— Boundary element source method (BESM), eddy currents, Lorentz force, permanent magnet.

I. INTRODUCTION

ANEW approach for nondestructive testing of solid con-ductive materials, called Lorentz force eddy current

testing, has been introduced recently [1]. Different from theconventional eddy current testing method, the magnetic field ofa permanent magnet generates eddy currents in the electricallyconducting specimen, because of its movement with respectto the magnet. The interaction between the induced eddycurrents and the magnetic flux density causes Lorentz forces,which intend to break the motion. According to Newton’sthird law, the Lorentz force exerts the permanent magnet inthe opposite direction. Material anomalies, such as changesin conductivity, cracks, or inclusions, distort the eddy currentdistribution in the object under test and, consequently, theLorentz forces measured at the magnet system. Thus, anydefect in the conductive material produces perturbations inthe Lorentz force signals. The relationship between the forceperturbations and material anomalies can be used to detectand reconstruct defects. The reconstruction of defects requiressolving the inverse problem and embedding the solution of aforward problem.

One approximate forward solution of the Lorentz forces hasbeen introduced in [2]. However, this solution is limited toa laminated specimen consisting of a package of conductivesheets, where the conductivity in the direction perpendicularto the sheets is assumed to be zero. This enforces an eddycurrent flow in the plane of the sheets, allowing the assumption

Manuscript received May 12, 2016; revised September 1, 2016 and October13, 2016; accepted October 15, 2016. Date of publication October 27, 2016;date of current version January 24, 2017. Corresponding author: B. Petkovic(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2016.2622223

that the eddy current in the direction perpendicular to thesheets is equal to zero. We now propose calculation of Lorentzforces without assumptions on the induced eddy current flow,satisfying the boundary condition of their normal componentin a conductor along all the conductor/air interfaces. Thismakes the method applicable to any conducting specimen.The method enables the determination of the eddy currentand Lorentz force profiles inside the bar. Consequently, thetotal Lorentz force acting on the permanent magnet can bederived. Notwithstanding that the method can be applied toarbitrary shaped geometries of both the specimen and thedefect, the focus in this paper is on a parallelepipedical barwith a parallelepipedical defect.

One of the most frequently used techniques for solving elec-trostatic problems is the charge simulation method (CSM) [3].This method simulates the actual electric field superimposingthe fields coming from a number of fictitious charges placedoutside the region, where the electric field has to be calculated(or inside any equipotential surface such as metal electrodes).The magnitudes of these charges are determined by satisfyingthe boundary condition at a selected number of points on theboundary. Once the charges are determined, one can calculatethe electric scalar potential and the electric field at any pointof the region. Simulating the real charges distributed on theelectrode surface by equivalent sources placed on the surfaceinstead in the inner electrode volume, the surface CSM is intro-duced [4], [5]. Introducing the equivalent charge distributionson the boundary surfaces between two dielectrics, the solutionof the electrostatic field problem is found for electrodes placedin inhomogeneous media [6], [7]. This formulation is extendedto the case of floating electrodes and static current fields,introducing the general source simulation method [8].

An integral-free calculation procedure, based on thesubstitution of boundary segments of multilayered media

0018-9464 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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7000109 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017

by equivalent charges placed in free space at the centersof boundary segments, is proposed in [9], and is calledhybrid boundary element method (HBEM). Source points onboundary layers coincide with collocation points, except inthe case of calculation of interaction of the element withitself. In that case, an equivalent radius of the boundarysource has to be determined and a collocation point isplaced on the surface of the equivalent source. The HBEMhas already been successfully applied to solve the problemsof cable terminations [10], grounding systems [11], [12],transmission lines [13], [14], as well as the calculationof magnetic fields and interaction forces of permanentmagnets [15].

In this paper, we introduce the boundary element sourcemethod (BESM), where the concentrated point source of theHBEM is replaced by distributed charge density over the areaof the boundary element. The distributed charge density overan element area allows for 3-D calculations in the cases ofboundary surfaces very close to one another and boundarylayers of irregular geometry involving sharp edges. When thecharge density is considered uniform, an analytical formulaof the electric scalar potential and the field of a uniformlycharged triangle [16] or a uniformly charged square [17]are used. The charge density of boundary elements could berepresented not only as a constant but also as a linear func-tion, or different special functions, such as the Fourier-typeexpansion, Tchebycheff-type expansion, and so on [4]. Thisenables overcoming an unrealistic discontinuity of the sourcestrength on the edges of neighboring mesh elements. Usingthe area distributed sources, the advantage of the diagonaldominance of the system of linear equations as in the HBEMis preserved. The method has no problems with singularityvalues also.

We now calculate, applying the BESM, the Lorentz forcesexerting on a permanent magnet because of the eddy cur-rents generated in an electrically conducting specimen movingunder the magnet. We assume a constant charge density oneach boundary element. Interaction between elements suf-ficiently far away from one another (e.g., at the distancetwo times the radius of the superscribed circle around themesh element) is considered as point–point interactions asproposed in [11]. For elements that are at a small distancefrom each other, an analytical formula [17] is used. Sat-isfying the boundary condition for the normal componentof the induced eddy current density at all conductor/airinterfaces, we obtain a system of linear equations with thecharges of the corresponding boundary elements as unknowns.Solving this system, the electric scalar potential can becalculated by superimposing the potentials from all equiv-alent sources. Calculation points very close to the surfaceof mesh elements or even belonging to them require anapplication of the analytical formula [17]. Together with thedriving term of motion, this allows to determine 3-D eddycurrent flow inside the conductor as well as the Lorentzforces acting on the magnet system. Notwithstanding that themethod could be applied to any kind of motion, we willconsider in this work translatory motion of a solid conductivespecimen.

TABLE I

GEOMETRY AND ELECTRICAL PARAMETERS OF THE PROBLEM

Fig. 1. Conducting bar with a single defect moving with constant velocitybelow a spherical permanent magnet.

II. THEORY AND METHODS

A. Model Definition

We consider a parallelepipedical conducting bar made ofaluminum with an isotropic electrical conductivity of σ . Thebar moves with a constant velocity v = 0.01 m/s in posi-tive x-direction. The bar contains a parallelepipedical defect.Geometry parameters and conductivities of the specimen andthe defect are given in Table I, and are used as input forthe BESM and finite-element method (FEM) computations.

A spherical permanent magnet of radius 7.5 mm with ahomogeneous magnetization along the z-axis, �M = Mz andM = 9.28 ·105 A/m, is placed at a liftoff distance δz = 1 mmabove the top surface of the conductor.

Two different depths of the defect are considered:2 and 10 mm below the top surface of the bar. The depthis defined as a shortest distance from the top surface of thebar to the top surface of the defect. The defect is positionedlongitudinally parallel to the direction of the bar motion(Fig. 1).

B. Numerical BESM Solution

The permanent magnet is modeled by a single equivalentmagnetic dipole with the magnetic moment �m = mz = MVm z,placed at the center of gravity (COG) of the magnet �r0 =x0�x + y0 �y + z0�z, where Vm is the volume of the magnet,and x, y, z represent the unit vectors in a Cartesian coordinatesystem (Fig. 2). The magnetic flux density vector at the point�r is calculated as [18]

�Bp = μ0

(3

⇀m · (�r − �r0)

|�r − �r0|5 · (�r − �r0) −⇀m

|�r − �r0|3)

. (1)

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PETKOVIC et al.: COMPUTATION OF LORENTZ FORCE AND 3-D EDDY CURRENT DISTRIBUTION 7000109

Fig. 2. BESM model of a bar with defect. The charge of a surface element isq, n is the outgoing unit normal vector, �r is the position vector to the centroidof mesh element and �r0 is the position vector of the magnetic dipole. Thesubscripts i and j represent the field and source point, respectively.

When the bar crosses the magnetic field lines with avelocity �v , eddy currents are induced. In the application ofthe BESM, we use the weak reaction approach (WRA) [19].According to the WRA, the magnetic field produced by thepermanent magnet �Bp is not influenced by the magneticfield produced by the induced eddy currents. The necessarycondition for applying the WRA is that magnetic Reynoldsnumber defined as Rm = μ0σvd is small, Rm � 1. Theparameter d denotes the characteristic length of the problem,which is defined as half of the width of the conductor,d = W/2. In the present case, Rm = 0.0064, confirming theassumptions of the model approaches. It is mentioned in [19]that for aluminum bar moving even up to velocities of 1 m/s,when Rm = 0.6, the conditions are met and the WRA could beapplied. However, one decisive drawback of the WRA is thatit does not account for the correct estimation of the lift force inthe unperturbed case, which intends to push the magnet awayfrom the conductor. This is reported in [19], and is a resultof the imposed symmetry of the eddy current distribution as adirect consequence of the neglect of secondary magnetic fieldproduced by induced eddy currents. Nevertheless, the defectsuppresses the symmetric eddy current distribution, and thelift-force perturbation in the case of a defect can be determinedcorrectly in terms of shape and peak-to-peak amplitude.

Modeling of the relative motion between the permanentmagnet and a conductor under test has been done using differ-ent positions of the magnetic dipole in the Cartesian coordinatesystem (Fig. 2), keeping the position of the conductive bar inthis coordinate system the same (center of the upper surfaceof the bar is placed at the coordinate origin).

The induced eddy currents are calculated using the Ohm’slaw for moving conductors

�J = σ(−∇ϕ + �v × �Bp). (2)

According to the BESM approach, we consider only theconductor/air interfaces, that is, in the case of a solid bar witha parallelepipedical defect, the six rectangular faces of theparallelepipedical bar and six faces of the parallelepipedical

defect. The electric scalar potential because of all surfacecharges can be written as [8]

ϕ(�r) = 1

4πε0

∫S

η(�r ′)|�r − �r ′|d S (3)

where η(�r ′) is the total surface charge density at the sourcepoint �r ′, S is the surface of the whole interface conductor/air,ε0is the permittivity of free space, �r is the position vector ofthe potential calculation point, and �r ′ is the position vector ofthe charge point.

After discretizing the bar and the defect faces into N and Msquare elements, respectively, assuming constant surfacecharge density on each square element, the electric scalarpotential (3) becomes

ϕ(�r) = 1

4πε0

N+M∑j=1

q j

S j

∫S j

d S

|�r − �r ′| (4)

where �r ′ ∈ Sj , Sj is the area of the j th surface element,N + M is the total number of the surface elements, and q j isthe charge assumed to be constant on the surface element.

Because of the boundary condition on the conductor/airinterface, �J · n = 0, where n represents the unit normal vectoron this interface, the condition

−∇ϕ · n + (�v × �Bp) · n = 0 (5)

has to be satisfied on all interfaces since σ �= 0. Assuming thatthe charge q j is concentrated at the centroid �r j and calculatingthe potential at the element i as a potential at the central pointof that element �ri (Fig. 2), we approximate the electric scalarpotential at the element i produced by the uniformly chargedelement j as a point source potential

ϕ(�ri ) = 1

4πε0

q j

|�ri − �r j | . (6)

The electric field at the centroid of the element i , �ri ,produced by the charge q j and projected onto the unit normalvector ni is equal to the scalar product of the gradient ofelectric scalar potential (6) and ni

�E(�ri ) · ni = −∇ϕ(�ri ) · ni = q j (�ri − �r j ) · ni

4πε0|�ri − �r j |3 , i �= j. (7)

However, the normal component of the electric field vectorat the centroid �ri at the conductor surface on its inner sideproduced by the charge qi of the same mesh element is equal to

�E(�ri ) · ni = − qi

2Siε0. (8)

It is important to emphasize that the normal derivative isdifferent at the opposite sides of the boundary carrying surfacecharges. The expression (8) is given for the inner side of theconductor and unit normal vector defined as in Fig. 2.

Inserting (7) and (8) into (5) and moving the second addendin (5) to the right hand side, one obtains the system of linearequations

[A](N+M)×(N+M) · [q]N+M = [C](N+M) (9)

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7000109 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017

Fig. 3. Representation of the bar with a set of voxels with current elementsat their centroids.

where the elements of the system matrix [A] are ai, j =((q j (�ri − �r j ) · ni )/(4πε0|�ri − �r j |3)), i �= j , ai,i =−((qi )/(2Siε0)), ci = −(�v × �Bp)i · ni , and q j , j = 1, . . . , N +M are unknown surface charges. The sign − in ai,i is dueto the chosen orientation of the unit normal in Fig. 2. Theelements ai,i are calculated considering that the charge isuniformly distributed over the boundary element. The columnvector on the right-hand side is determined by the source of theprimary magnetic field (i.e., magnetic dipole) and its elementsci are obtained using (1) for �r = �ri . A unit outgoing normal atthe point �ri is denoted by ni , i = 1, . . . , N + M . Si representsthe surface area of the square, and the boundary conditionis satisfied at the centroid. The kernel matrix [A] is squareand diagonally dominant, leading usually to well-conditionedlinear systems with a stable solution [20].

After calculating the distribution of charges of the bound-aries, one can calculate the electric scalar potential with (4)and eddy current density distribution �J using (2) at any pointinside the bar. The integration of (4) is obtained by meansof a closed-form expression presented in [17]. The electricscalar potential further away than, say approximately, twotimes the equivalent radius of a mesh element (e.g., radius ofthe superscribed circle around the square), can be calculatedbased on the electric scalar potential of a point source placedat the centroid of a square element (6).

The interaction of the eddy currents with the magneticfield of the permanent magnet results in three-componentLorentz forces exerted on the conductor. To determine the netforce, the force density inside the solid bar is calculated in apost-processing step. Therefore, a uniform grid of volumetricelements (voxels) of volume V0 = x0y0z0 is definedinside the conductor as shown in Fig. 3. In each voxel, aninduced eddy current �Jk flows. The continuous distribution ofeddy currents inside the bar is replaced by a set of point currentelements located at the COGs of the corresponding voxels. TheLorentz force acting on the solid bar is then calculated by [2]

�FB =∫

V

�J × �BpdV ≈ V0

K∑k=1

�Jk × �Bk (10)

where V is the bar volume, V0 is the voxel volume, and Kis the total number of voxels located in the conductive region(Fig. 3). Finally, according to Newton’s third law, the Lorentzforce exerting the permanent magnet is equal to �F = − �FB .

A summary flowchart of the BESM approach is shown inFig. 4.

Fig. 4. Flowchart of BESM.

C. Reference Forward Solution Using Comsol Multiphysics

To obtain a reference forward solution, the FEM in combi-nation with the WRA [19] is used. According to the WRA, thesecondary magnetic field originating from the induced eddycurrents is neglected.

The computations reduce to the determination of the electricscalar potential ϕ inside the conducting domain. The governingequation for ϕ is given by the following elliptic second-orderdifferential equation with piecewise homogeneous materialproperties:

∇ · (σ∇ϕ) = 0. (11)

The primary magnetic field �Bp is calculated analytically,and imposed on every node in the mesh. Finally, definingthe boundary conditions �J · n = 0 at the boundaries of theconducting domain allows the computation of ϕ

∇ϕ · n = (�v × �Bp) · n. (12)

In this manner, the current density is enforced to flow insidethe conducting domain. Recently, the WRA has been extendedto the numerical calculation of the primary magnetic field inthe case of more complex magnet geometries as it is neededin the framework of optimization studies [21]. However, thegeneral procedure to determine the electric scalar potential isunaltered.

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PETKOVIC et al.: COMPUTATION OF LORENTZ FORCE AND 3-D EDDY CURRENT DISTRIBUTION 7000109

We employ two models for the reference finite-elementsimulations: one for the current density calculations and theother one for calculation of the force over the whole bar, thatis, the force acting on a magnet system. Because we apply theWRA, only the conductor volume is densely discretized [19].In the current density model, we use 882 278 third-orderelements. In the region of interest right under the magnet,525 100 hexahedral elements are used with a volume of (0.5×0.5 × 0.5) mm3. The remaining elements are tetrahedral. Forthe force profile model, we use 378 108 tetrahedral elements.A fine mesh is generated along the line, where the magnetmoves along the conductor.

D. Comparison of Methods

At first, the eddy current profile obtained by the BESMis compared with the reference FEM solution, taking intoconsideration a bar without defect. The three components ofeddy current density vectors are calculated at 13 237 points dis-tributed in the volume −0.03 m ≤ x ≤ 0.03 m, −0.015 m ≤y ≤ 0.015 m, and 0 ≤ z ≤ −0.03 m. As a measure ofdifferences between amplitude values, L2 relative error is used

L2 relative errork =√√√√∑N0

i=1

(J BESM

i,k − J FEMi,k

)2

∑N0i=1

(J FEM

i,k

)2

k ∈ {x, y, z} (13)

where J BESMk and J FEM

k are the components of the eddycurrent density obtained by the BESM and the FEM, respec-tively. N0 is the number of current calculation points, and icorresponds to the point index.

The second verification of the BESM is done through thecomparison of forward computed x- and z-components ofthe Lorentz force with the values obtained by the FEM. Thesignals are calculated along the line −0.140 m ≤ x ≤ 0.140 m,for y = 0 and z = 0.0085 m above the upper surface ofthe bar. The lateral force Fy vanishes along the line y = 0(symmetry line). We compare two solutions using the L2

relative error (13), taking FFEMi,k instead of J FEM

i,k and FBESMi,k

instead of J BESMi,k , k ∈ {x, z}.

III. RESULTS

A. Eddy Current Profiles

We first investigate the influence of the mesh density on thesimulation accuracy using the defect-free solid bar. The L2

relative error is shown in Fig. 5. It is calculated with respectto the reference FEM solution for element sizes in the rangefrom 2 to 5 mm with the step of 1 mm. In the cases where theside lengths of the bar are not multiple integers of the elementsize , the sides of the parallelepiped are divided into as manyparts to ensure that the mesh length is closest to the length under consideration.

The induced eddy currents Jx and Jy already show sat-isfactory accuracy with mesh element sizes of = 5 mm,having L2 relative error equal to 1.9% and 2.1%, respectively,Differently, Jz has an error 106% for the same mesh elementsize = 5 mm. Therefore, a denser mesh is required. Usingthe element size of 2 mm, we obtained L2 relative error equal

Fig. 5. L2 relative error (in logarithmic scale) of the eddy current profilesobtained by the BESM with respect to the reference FEM solution for differentsizes of the BESM mesh elements at 13 237 points inside the volume of thedefect-free bar defined by −0.03 m ≤ x ≤ 0.03 m, −0.015 m ≤ y ≤ 0.015 m,and 0 ≤ z ≤ −0.03 m.

to 0.37%, 0.29%, and 2.4%, for Jx , Jy , and Jz , respectively.The following simulations are performed with a size of themesh elements equal to 2 mm, resulting in 13 750 boundaryelements.

Visualization of the eddy current flows in the solid bar at xyplane obtained by the BESM 5 mm below the upper surfaceof the bar is shown in Fig. 6(a)–(d). The errors are quantifiedas the difference between the FEM and the BESM solution,and presented in Fig. 6(e)–(h). The resulting difference showsthat the amplitudes of eddy currents obtained by the FEM aremainly larger than the BESM values. Although the differencebetween the two solutions of current components Jx and Jy

shows the same pattern as the currents themselves, the differ-ence of Jz has two hotspots in the area below the permanentmagnet [Fig. 6(g)]. However, the Jz component in the middleof the bar (far away from the edges) is much smaller thanJx and Jy , producing almost no influence on the total currentmagnitude J and on the Fx and Fy force density distributioninside the bar.

The presence of a defect in a solid conductive materialalters the eddy current profile. The eddy current distributionin the case of a specimen without defect and distorted bythe presence of a defect are presented in Fig. 7(a) and (b),respectively. The defect is placed 10 mm below the top surfaceof the bar. The induced eddy currents are obtained by theBESM and presented at xy plane for z = −11 mm, that is, themiddle plane of a defect.

In the vicinity of a defect, the eddy currents are dis-torted and forced to flow around the defect. Some currenthas to flow along the defect surface. Furthermore, in thepresence of a defect, the eddy currents become weaker,as indicated by the maximum values of the two colorbars.

One of the criteria by which the efficiency of Lorentz forceeddy current testing can be judged is the depth of penetrationof the eddy currents into the material. A representation of thetotal eddy current flow in the central xz plane (y = 0) obtainedby the BESM for a defect-free system and a bar containinga defect is shown in Fig. 8(a) and (b), respectively. The eddy

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7000109 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017

Fig. 6. Eddy current profiles at the xy plane located 5 mm below the top surface of the bar, obtained by the BESM (a) Jx , (b) Jy , (c) Jz , and (d) J andthe difference of the flows obtained by the FEM and the BESM (e) Jx , (f) Jy , (g) Jz , and (h) J . Unit of the color bar is A/m2.

currents flow around the defect, clearly indicating the xz crosssection of the defect [Fig. 8(b)].

B. Lorentz Force on a Permanent Magnet

The BESM scheme is applied to compute the Lorentz forcesexerted on the permanent magnet for a bar containing a defectat a depth of d = 2 mm. We have performed a convergencestudy using a square mesh of length = 1 mm and

= 2 mm for the upper, right, and left face of the barand d = 0.25 mm, d = 0.5 mm, and d = 1 mmfor defect faces. Having a smaller influence on the magnetalong symmetry line, front and back faces of the bar werediscretized with = 2 mm and bottom face with =3 mm. The L2 relative error of Lorentz forces Fx and Fz

with respect to the reference FEM solution are presented inTables II and III, respectively. The voxel dimensions are set to

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Fig. 7. Induced eddy currents at an xy plane located at z = −11 mm (middleplane of a defect) obtained using the BESM for a specimen (a) without and (b)with defect. Black solid lines: contours of constant current density. Contourvalue increment is 500 A/m2. The unit of the color bar is A/m2.

Fig. 8. Induced eddy currents in the central xz plane obtained using theBESM in the case of (a) defect-free and (b) defective specimen. Black solidlines: contours of constant current density. Contour values are from 1000 to4500 A/m2 with increment 500 A/m2. The unit of the color bar is A/m2.

x0 = y0 = z0 = 1 mm up to the depth of 10 mm insidethe bar, and x0 = y0 = z0 = 2 mm for −50 mm < z <−10 mm to accelerate the solution process.

The length of the mesh element of the bar = 2 mm,irrespective of the size of mesh elements of the defect,provides insufficient accuracy of the Lorentz force exerting thepermanent magnet. The L2 relative error of Fx and Fz remainsalmost constant with decreasing mesh elements size at defectfaces. However, when = 1 mm is used, the L2 relative error

Fig. 9. Lorentz force exerting the permanent magnet along the line−0.140 m ≤ x ≤ 0.140 m at the height z = 8.5 mm above the barcontaining a defect at 2 mm depth. (a) Fx . (b) Fz . (c) |FFEM

x − FBESMx |

and |FFEMz − FBESM

z |.TABLE II

L2 RELATIVE ERROR OF Fx ALONG THE SYMMETRY LINE USING

SQUARE MESH ELEMENTS OF LENGTH OF THE BAR FACES AND

LENGTH δd OF THE DEFECT FACES

decreases with increasing the density of mesh elements of thedefect, providing satisfying accuracy when d = 0.25 mm isused. The x- and z- components of the Lorentz forces alongthe centerline −0.140 m ≤ x ≤ 0.140 m at z = 8.5 mmabove the bar, obtained by the FEM and the BESM, arefor this case presented in Fig. 9(a) and (b), respectively.Absolute errors |FFEM

x − FBESMx | and |FFEM

z − FBESMz | are

shown in Fig. 9(c).The maximal error of the x- component of the Lorentz force

obtained by the BESM with respect to the FEM solution occurs

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7000109 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 2, FEBRUARY 2017

TABLE III

L2 RELATIVE ERROR OF Fz ALONG THE SYMMETRY LINE USING

SQUARE MESH ELEMENTS OF LENGTH OF THE BAR FACES AND

LENGTH δd OF THE DEFECT FACES

exactly above the edge of the bar, while the z-component hasthe largest deviation before approaching the edge of the bar.However, the edges of the bar are not considered during thetypical Lorentz force eddy current evaluation process [2], [22].Local maxima of absolute errors of Fx and Fz appear in thevicinity of the defect, and for x = 0.85 m and x = 0.9 m,when the curves Fx and Fz start to decrease and increase,respectively [Fig. 9(c)].

Memory and time costs of meshing, matrix composition,matrix inversion, and determination of boundary sources usingN = 25178 and M = 1664 elements are 3 GB of RAMmemory and 30 min, respectively. The computations are per-formed on a computer with CPU (Intel Core i7 860 2.8 GHz)and 8 GB of RAM. The current implementation of the BESMcould be accelerated by additional parallelization, for exampleusing special coding techniques or a graphics processing unitimplementation of the BESM.

IV. CONCLUSION

The HBEM is adopted for the calculation of induced eddycurrents in the case of translatory moving conductors in thefield of a permanent magnet. The accuracy of the proposedmethod is evaluated by comparing the individual componentsof the eddy current density as well as the Lorentz force onthe permanent magnet with the results from finite-elementcomputations.

Besides the WRA, the BESM does not involve any assump-tions on the eddy current flow and can be applied on arbitrarilyshaped specimen and defect geometries. In this paper, a zero-conductivity defect has been studied. However, the BESM canbe used for the determination of the induced eddy currentdistribution and the electric scalar potential in solid conductivematerials with the defects of any conductivity. Multiple defectscan be considered as well, and each defect can have its ownconductivity. The position and orientation of a defect insidea specimen has no influence on the implementation of thismethod. In the case of a defect very close to one of the surfacesof the specimen, an analytical expression of the electric fieldproduced by a uniformly charged boundary element has tobe applied instead of the field of a point source. The BESMdoes not have the difficulties of numerical integration. It issimple to implement, accurate, and robust. The relatively highcomputation demands are one drawback of the BESM in itscurrent implementation.

Future study will be focused on the establishment of aninverse reconstruction scheme based on the proposed forward

formulation. Instead of a single permanent magnet, morecomplex magnet geometries will be considered, as proposedin [21]. These magnets will be modeled by an assembly ofmagnetic dipoles, appropriately positioned according to themagnetic dipoles model of the magnets [23].

ACKNOWLEDGMENT

B. Petkovic would like to thank Prof. S. Aleksic for fruit-ful scientific discussions. This work was supported by theDeutsche Forschungsgemeinschaft within the framework ofthe Research Training Group, Technische Universität Ilmenau,through the Project entitled Lorentz Force Velocimetry andLorentz Force Eddy Current Testing under Grant GK 1567and Grant PE 2389.

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Bojana Petkovic was born in Niš, Serbia, in 1977. She received the M.S.degree in telecommunications from the Faculty of Electronic Engineering,Niš, in 2002 and the Dr.Ing. degree from Technische Universität Ilmenau,Ilmenau, Germany, in 2013.

Since 2014, she has been a Post-Doctoral Researcher with the Instituteof Biomedical Engineering and Informatics, Technische Universität Ilmenau.Her current research interests include mathematical modeling in biomedicalengineering and nondestructive testing and evaluation of solid materials.

Konstantin Weise (S’12) was born in Leipzig, Germany, in 1986. He receivedthe B.Eng. degree from the University of Applied Science Leipzig, Leipzig,in 2009, and the M.Sc. and Dr.Ing. (equivalent to Ph.D.) degrees in electricalengineering, from Technische Universität Ilmenau, Ilmenau, Germany, in 2012and 2016, respectively.

He is currently with Technische Universität Ilmenau. His current researchinterests include biomedical engineering problems and nondestructive testingapplications.

Jens Haueisen received the M.S. and Ph.D. degrees in electrical engineeringfrom Technische Universität Ilmenau, Ilmenau, Germany, in 1992 and 1996,respectively.

From 1996 to 1998, he held a post-doctoral position and from 1998to 2005, he was the Head of the Biomagnetic Center, Friedrich-Schiller-University, Jena, Germany. Since 2005, he has been a Professor of BiomedicalEngineering and the Director of the Institute of Biomedical Engineering andInformatics, Technische Universität Ilmenau. His current research interestsinclude the investigation of active and passive bioelectric and biomagneticphenomena and medical technology for ophthalmology