IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018 6200105

Assessment of Two Forward Solution Approachesin Lorentz Force Evaluation

E.-M. Dölker 1, R. Schmidt 2, K. Weise2, B. Petkovic 1, M. Ziolkowski 2, H. Brauer2, and J. Haueisen1

1Institute of Biomedical Engineering and Informatics, Technische Universität Ilmenau, 98693 Ilmenau, Germany2Advanced Electromagnetics Group, Technische Universität Ilmenau, 98693 Ilmenau, Germany

Two different forward models for Lorentz force evaluation, the approximate forward solution (AFS) and the novel extended areaapproach (EAA), are evaluated using a goal function scan. A setup that contains a spherical permanent magnet and a specimen ofstacked aluminum sheets with a cuboidal defect of size 12 mm ×2 mm×2 mm at a depth of 2 mm is simulated by the finite-elementmethod (FEM). Deviations between force perturbation of FEM and AFS or EAA are compared in terms of normalized root meansquare errors (NRMSEs). The goal function scan yields minimal NRMSE for the correct depth with values of 6.1% for AFS and1.7% for EAA, corresponding to the determined defect dimensions of 7 mm × 10 mm × 2 mm for AFS and 11 mm × 2 mm × 2 mmfor EAA.

Index Terms— Eddy current, inverse problem, Lorentz force evaluation, nondestructive evaluation.

I. INTRODUCTION

THE increasing standards in quality and safety of materi-als, e.g., in aircraft, and the development of new materials

show the importance of high-resolution nondestructive evalu-ation methods during manufacturing and maintenance. In thefield of conducting materials, ultrasonics [1], radiography [2],thermography [3], magnetic induction tomography [4], mag-netic flux leakage [5], and eddy current testing [6] are used fornondestructive testing nowadays. Each method comes alongwith positive and negative properties, e.g., the eddy currenttesting is strongly limited to the detection of surface and sub-surface defects due to the skin effect. The recently introducedLorentz force evaluation is a promising evaluation techniquefor the identification of deep-lying defects in conducting mate-rials [7]. It enables the reconstruction of defects from perturba-tions in Lorentz forces that act on a permanent magnet, whichmoves relative to a conductive specimen. Previous defectreconstruction in this field used conductivity reconstructionbased on truncated singular value decomposition [7] and dif-ferential evolution [8], which are based on the approximate for-ward solution (AFS) as forward model [7]. Recently, the moreaccurate extended area approach (EAA) has been introducedas the forward model for Lorentz force evaluation [9]. Bothmodels, AFS and EAA show a reduced calculation time incomparison to finite-element method (FEM). The AFS is afast and simple method, which uses only the defect area inorder to calculate the force perturbation signal. In contrast,the EAA as an extended version of the AFS takes also thesurrounding area of the defect into account, what makes itmore expensive in calculation time. It is the aim of this paperto assess the defect reconstruction performance of these twodifferent forward models. For that purpose, a goal function

Manuscript received July 3, 2017; revised September 11, 2017; acceptedOctober 17, 2017. Date of publication November 15, 2017; date of currentversion February 21, 2018. Corresponding author: E.-M. Dölker (e-mail:eva-maria.doelker@tu-ilmenau.de).

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.2017.2765347

Fig. 1. Problem setup: a package of aluminum sheets with a cuboidal defectis moved relative to the spherical permanent magnet, where the interactionof the induced eddy currents (orange lines) with the magnetic field leads toLorentz forces. The figure is scaled for better visualization.

scan [10] is applied to a simulated data set. The referencedata, obtained by FEM, are used to model the setup shownin Fig. 1. It includes the permanent magnet and a speci-men consisting of stacked aluminum sheets and a cuboidaldefect.

II. MATERIALS AND METHODS

A. Problem Setup

A specimen with a conductivity of σ0 = 30.61 MS/m and asize of L×W ×H = 400 mm×400 mm×100 mm is moved inx-direction relative to the permanent magnet with the velocityv = 0.01 m/s (see Fig. 1). Due to the relative movement, eddycurrents are induced in the conducting aluminum layers. Theinteraction of the induced eddy currents with the magneticfield leads to Lorentz forces. In the presence of a defect,eddy currents are perturbed and so are the Lorentz forcecomponents (see Fig 1). The specimen consists of stackedaluminum sheets, each of thickness �z = 2 mm. The sphericalpermanent magnet of diameter Dm = 15 mm is characterizedby a homogenous magnetization M = Br/μ0ez (Br = 1.17 T),where Br describes the remanence. The permanent magnet is

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6200105 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018

Fig. 2. DRS �FFEM obtained from FEM simulations with components(a) �Fx , (b) �Fy , and (c) �Fz .

located at the lift-off distance δz = 1 mm above the top surfaceof the specimen. A cuboidal defect with the conductivity σd =1S/m and of the size Ld ×Wd × Hd = 12 mm×2 mm×2 mmis located at the depth d = 2 mm, which corresponds tothe second sheet.

B. Simulation of the Problem Setup

In order to model the setup described above, the weakreaction approach is applied [11]. In this approach, it isassumed that the magnetic flux density B in the specimen iscaused only by the permanent magnet since due to the smallvelocity v, the secondary magnetic field from the induced eddycurrent can be neglected. Thus the eddy current density insidethe conductor can be calculated directly by Ohm’s law formoving conductors

j = [σ ](−∇ϕ + v × B) (1)

where ϕ denotes the electric scalar potential. The tensor[σ ] = diag(σT) with σ T = [σx x σyy σzz]T = [σ0 σ0 σd]T

reflects the anisotropic character of the setup. The magneticflux density B can be calculated either numerically or analyti-cally depending on the shape of the magnet. In order to find ϕ,the following equation with appropriate boundary conditionsneeds to be solved:

∇ · ([σ ]∇ϕ) = 0 with n · j = 0 (2)

where n is the unit vector normal to the conductor surfaces.The problem mentioned in (2) is solved by FEM andimplemented in COMSOL Multiphysics (COMSOL Inc.,Burlington, MA, USA). The defect is discretized intotetrahedral elements with an edge length of 0.5 mm, whereasthe region around the defect is discretized into tetrahedralelements with an edge length of 1 mm. The remainingspecimen is discretized by tetrahedral elements whose sizes

Fig. 3. Parameter studies of appropriate extension in EAA: a stable saturationof NRMSE without further improvement is reached at ε ≈ 5 (arrow).

increase gradually to the outside. The resulting Lorentz forcesare calculated for 61×61 permanent magnet positions aroundthe x − y-center of the defect at x = [−30,−29, . . .30] mm,y = [−30,−29, . . .30] mm, and z = 8.5 mm, whichcorrespond to the center of the spherical permanent magnetpositions.

C. Defect Response Signal

The defect response signal (DRS) forms the basis forLorentz force evaluation. It is defined by

�F = F − F0 =∫

V −Vd

−j × BdV −∫

V−j0 × BdV (3)

where F describes the force acting on the permanent magnetwith the specimen containing a defect and F0 is the forcefor the same specimen without a defect. Similarly, the eddycurrent densities j and j0 describe the defect and the defect-freecase, respectively. The volumes of the specimen and the defectare denoted by V and Vd, respectively. The DRS �FFEM,obtained from FEM simulations (Fig. 2), provides the inputsignal used in the defect reconstruction procedure. Because thereconstruction algorithms require the intensive use of forwardcomputations, it is necessary to substitute the slow FEM bya fast approximating solver. Therefore, we present the twosolvers AFS and EAA. Both assume an ideal defect withconductivity of zero.

D. Approximate Forward Solution

Equation (3) can be reformulated to

�F = −∫

V −Vd

(j−j0) × BdV −∫

Vd

−j0 × BdV

︸ ︷︷ ︸AFS︸ ︷︷ ︸

EAA

(4)

where the forward solution AFS neglects the first term of (4).Only the defect, as a fictitious conducting region, is discretizedinto K volume elements (voxels) of volume VE = 1 mm ×1 mm × 2 mm. According to [7] the DRS is approximated by

�FAFS =⎡⎣�FAFS

x�FAFS

y�FAFS

z

⎤⎦ = −VE

K∑k=1

(�jk × Bk). (5)

Using the weak reaction approach, the distortion eddy currentdensity �jk = −j0 is directly calculated from (1) as �jk =−σ0(−∇ϕk + v × Bk) [7]. The spherical permanent magnet

DÖLKER et al.: ASSESSMENT OF TWO FORWARD SOLUTION APPROACHES 6200105

Fig. 4. Detailed view on goal function scan based on AFS for metallayers 1-4. (a)-(d) NRMSE values for different length-width-combinationsof the possible defect are color coded. The limits of the color bar indicateminimal and maximal NRMSE of each layer.

is modeled as a magnetic dipole located at its center r0. Themagnetic flux density Bk in the center of gravity rk of kthvoxel is calculated as

Bk = μ0

4π

(3

m · (rk − r0)

|rk − r0|5 (rk − r0) − m|rk − r0|3

)(6)

where m = (BrVm)/μ0ez denotes the equivalent magneticmoment of the permanent magnet. The laminated structureof the specimen (Fig. 1) allows the assumption that theeddy currents flowing in z-direction, are negligible. Thus, thez-component of the distortion current density equals zero,i.e., � jk,z = 0. Furthermore, the electric scalar potential ϕk

has to fulfill the equation ∂ϕk/∂z = v Bk,y . The electric scalarpotential ϕk can be found easily for an extended conductor as

ϕk = −vmμ0

4π

(yk − y0)

|rk − r0|3 (7)

which is further used in the AFS.

E. Extended Area Approach

The EAA is an extension of the AFS. In addition to thedefect, an extended area around the defect is discretized intoE voxels of volume VE, whereas the extension is performedin x- and y-directions. The DRS in EAA [9] is approximatedby

�FEAA =⎡⎢⎣

�FEAAx

�FEAAy

�FEAAz

⎤⎥⎦ = −VE

E∑e=1

(�je × Be)+�FAFS (8)

where Be denotes the magnetic flux density at the centerof gravity of the eth extended voxel calculated accordingto (6).

Fig. 5. Detailed view on goal function scan based on EAA for metallayers 1-4. (a)-(d) NRMSE values for different length-width-combinationsof the possible defect are color coded. The limits of the color bar indicateminimal and maximal NRMSE of each layer.

The distortion current density �je [9] in the center ofgravity of the eth external voxel can be determined by

�je ∼= CdVE

2π�z

K∑k=1

[2�jk · (re − rk)

|re − rk |4 (re − rk) − �jk

|re − rk |2]

(9)

where the dipolar correction factor Cd = 1 + (π/4)(Ld/Wd)holds for cuboidal defects [9]. The position vectors of thevoxels’ centroids in the defect and the extended region aredenoted by rk and re, respectively.

The selection of an appropriate extension is an importantaspect of EAA. In order to find a sufficient size of theextended region for the shown setup, the DRS �FEAA is calcu-lated with the cuboidal defect for different extension factorsε = [0, 1, 2, . . . , 7], where 0 means that no extension, i.e.,AFS is applied. If ε ≥ 1 the area used for calculation equalsAext = ((1 + ε) · max(Ld, Wd))

2, which means that e.g., withε = 2 an area of Aext = 36 mm×36 mm = 1296 mm2 is takeninto account for EAA. For every extension, the normalized rootmean square error (NRMSE) between �FFEM and �FEAA iscalculated. The NRMSE is here defined by

NRMSE

= 1

2

∑i=x,z

⎡⎣

√√√√ 1

N

N∑n=1

(�FAFS/EAA

n,i − �FFEMn,i

)2

/ min[(

max(�FAFS/EAA

i

)−min(�FAFS/EAA

i

))

(max

(�FFEM

i

) − min(�FFEM

i

))]⎤⎦ (10)

where n indicates the current position of the magnetic dipole.

6200105 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018

Only the x- and z-components are used since including they-component yielded partly instable reconstruction results.

The expansion is chosen to ε = 5 as the NRMSE stopsimproving (Fig. 3, arrow). The DRS approximated byAFS (5) and EAA (8) is calculated for the magnetic dipolepositions according to the permanent magnet positions ofFEM simulations.

F. Goal Function Scan

In order to reduce the computational effort and focus on thecomparison of the defect reconstruction performance betweenthe forward models AFS and EAA, it is assumed that thedefect height Hd = 2 mm is known and equal to the aluminumsheet thickness �z. Further, it is assumed that the shape ofthe defect is cuboidal. Thus, the DRS is calculated by AFSaccording to (5) and by EAA according to (8), respectively.The different length-width-combinations of a possible defectLd = [1, 2, . . . , 50] mm and Wd = [1, 2, . . . , 50] mm areused for these calculations. These calculations have beenapplied from the first to the 11th metal layer. For every singlecalculation, the NRMSE (10) is determined as a goal functionvalue. The lowest NRMSE gives the results for depth andsize of the defect, produced by the goal function scan forAFS and EAA.

III. RESULTS AND DISCUSSION

The results of the goal function scan within the chosenLd − Wd-combinations for the AFS are shown in Fig. 4 foraluminum layers 1 to 4. It shows the NRMSE color-coded,where the minimal NRMSE of 6.1% at the correct depthcorresponds to the defect size Ld × Wd = 7 mm × 10 mm[asterisk, Fig. 4(b)]. The layer-wise minimum (asterisks,Fig. 4) further moves from the left bottom corner to thetop center for layers 5 to 11, which are not depicted here.That means that a larger defect is estimated for deeperaluminum layers. The results for the forward model EAAare shown in Fig. 5. The lowest NRMSE can be foundin layer 2, too [Fig. 5(b)]. Thus, the correct defect depthis detected. The NRMSE of 1.7% is smaller compared toAFS and corresponds to a length and width of the defectLd × Wd = 11 mm × 2 mm, which is closer to the correctextensions of Ld × Wd = 12 mm × 2 mm than achieved byAFS. A larger defect is estimated for deeper metal layers asthe layer-wise minimum (asterisks, Fig. 5) moves from thebottom left corner to the right center for layers 5 to 11.

In order to show the direct comparison of AFS and EAAregarding defect reconstruction performance, the minimalNRMSE values of each metal layer (asterisks, Figs. 4 and 5)are shown with the corresponding defect extensions Ld × Wd(indices, Fig. 6) in Fig. 6. It can be observed that the NRMSEvalues are larger for AFS compared to EAA. A strongerincrease in defect extensions occurs from layers 6 to 9 for AFS(Fig. 6, blue curve), where the estimated defect extensionsfor layers 10 and 11 are smaller again. This behavior occursbecause the goal functions based on AFS in the observedparameter range show a structure with two minima. It occursfor metal layers 5 to 11, whose goal functions are not

Fig. 6. Results of the goal function scan based on AFS and EAA fora cuboidal defect with x − y- extensions (12 × 2) mm2 at the depth2 mm: The minimal NRMSE and its corresponding estimated defect extension(Ld × Wd) mm2 are shown for each layer. The correct defect depth (layer 2)has been found for both forward solutions, whereas the EAA estimates thedefect shape more accurately. The NRMSE values are shown in table.

shown here. The goal function scan based on EAA showsa steady increase of the estimated defect extension for deepermetal layers. This can be explained by the fact that thegoal functions for the observed parameters show a structurewith only one minimum that is present for all 11 observedmetal layers.

The more accurate EAA is more expensive than AFSregarding the computational cost. The computation time ofone single DRS point ranges from [1.1, 232.9 ms] for AFSand [3.1 ms, 7.4 s] for EAA for the proposed length-width-combinations. In the case of the correct defect size AFS,EAA and FEM need 5.5 ms, 86.1 ms, and 6 min and 41 s,respectively, which shows the advantage of AFS and EAAas forward models compared to FEM. The computations areperformed with the software MATLAB R2017a and COMSOLMultiphysics with an Intel Xeon E5-2697v3 processor and384 GB of RAM, which is used for the expensive FEM calcu-lations. Parallelization and GPU programming could improvethe computation time of EAA.

IV. CONCLUSION

The assessment of the AFS and the EAA for Lorentzforce evaluation based on a goal function scan shows thatthe correct defect depth can be estimated with both methods.The EAA shows a more stable goal function behavior as onlyone minimum is present for the observed parameters of metallayers 1 to 11. The EAA gives the better shape reconstruction.Future work will focus on the application of the EAA tomeasurement data.

ACKNOWLEDGMENT

This work was supported by the Deutsche Forschungsge-meinschaft in the framework of the Research Training Group“Lorentz force velocimetry and Lorentz force eddy currenttesting” under Grant GRK 1567.

DÖLKER et al.: ASSESSMENT OF TWO FORWARD SOLUTION APPROACHES 6200105

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