Lorentz force evaluation: A new approximation method for defect reconstruction

Author's Accepted Manuscript

Lorentz force evaluation: a new approximation method for defect reconstruction B. Petković, J. Haueisen, M. Zec, R.P. Uhlig, H. Brauer, M. Ziolkowski

www.elsevier.com/locate/ndteint

PII: DOI: Reference:

S0963-8695(13)00083-2 http://dx.doi.org/10.1016/j.ndteint.2013.05.005 JNDT1520

To appear in:

NDT&E International

Received date: 28 November 2012 Revised date: 17 May 2013 Accepted date: 20 May 2013 Cite this article as: B. Petković, J. Haueisen, M. Zec, R.P. Uhlig, H. Brauer, M. Ziolkowski, Lorentz force evaluation: a new approximation method for defect reconstruction, NDT&E International, http://dx.doi.org/10.1016/j.ndteint.2013.05.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Lorentz force evaluation: a new approximation method for defect reconstruction

B. Petković1, J. Haueisen1, M. Zec2, R. P. Uhlig2, H. Brauer2, M. Ziolkowski2 1

Institute of Biomedical Engineering and Informatics, Ilmenau University of Technology, PF 100565, 98684 Ilmenau, Germany

[email protected], [email protected] 2

Faculty of Electrical Engineering and Information Technology, Ilmenau University of Technology PF 100565, 98684 Ilmenau, Germany [email protected], [email protected], [email protected], [email protected]

Corresponding author: Prof. Dr. Jens Haueisen, [email protected], Institute of Biomedical Engineering and Informatics, Ilmenau University of Technology, PF 100565, 98684 Ilmenau, Germany, Tel.: +49 3677 69-2860, Fax: +49 3677 69-1311

Abstract We propose a new method for contactless, nondestructive evaluation of moving laminated conductors, the so-called Lorentz Force Evaluation (LFE). The Lorentz force (LF) exerting on a permanent magnet moving relative to the specimen is measured. We propose a novel fast forward calculation of the LF based on a three-dimensional finite volume discretization of the specimen and an approximation of defects using local

1

current distributions in the defect region. The approximate solution is compared with solutions from detailed finite element models developed for parallelepipedic subsurface defects. We obtain differences in LF that range between 1.7% and 6.7%, indicating that our approximation method yields sufficient performance. Furthermore, a linear inverse solution based on the novel forward method is presented. We invert the experimental data measured from a subsurface flaw with the dimensions of 2mm×2mm×12mm located within a laminated conductive bar. The reconstruction method yields the correct position of the flaw with an accuracy of 1mm in each direction. The reconstruction results are compared with high-resolution finite element analysis of the same crack configuration. We obtain correct lateral positions of the cracks, although the depth estimation shows a slight bias.

Keywords eddy currents, inspection systems, inversion, inverse problem, conductivity

1 Introduction Nondestructive material testing and evaluation is a vast interdisciplinary field as well as a challenge due to the wide variety of applications. Whereas the focus of nondestructive testing (NDT) is the detection and localization of anomalies within a specimen, the identification and reconstruction of defect properties (dimensions, shape, structure, composition) and their influence on the material´s usability is the focus of nondestructive evaluation (NDE). Defect identification and assessment are very important aspects of quality assurance.

2

Nondestructive material testing is understood as the non-invasive examination of any type of specimen without changing or altering the properties of the body under test to check whether the specimen contains anomalies. Anomalies are any type of defect or changes in the material properties which can be of natural or artificial origin, affecting the usefulness or serviceability of that object [1]. Nondestructive testing has turned from a rather empirical procedure dependent on the experience of the examiner into a more quantitative measurement technique that serves to determine the influence of material anomalies on the reliability of the object [2]. If the existing nondestructive testing techniques should be classified according to their limitations and not only according to the employed physical phenomenon, a separation in visual, surface and volumetric methods can be made [3]. However, usually these methods are classified as either electromagnetic or acoustic techniques [4-7]. Descriptions of most electromagnetic methods can be found in a number of the review papers [8-15]. Recently, a new approach for nondestructive testing of conductive materials has been introduced, called Lorentz force eddy current testing (LET) [16]. In contrast to the conventional eddy current testing technique, the magnetic field of a permanent magnet generates eddy currents in the electrically conductive specimen, as soon as the specimen is moved with respect to the magnet. The magnetic field caused by the eddy currents yields Lorentz forces, which try to break this motion. According to Newton’s third axiom, the Lorentz force is exerting on 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 magnetic system. Thus, any defect in the conductive material produces

3

perturbations in the Lorentz force signals. The direct relationship between the force changes and material anomalies can be used to detect defects. Due to the fact that the computation of transient field problems including the conductor movement is still a complicated and time consuming task, reconstruction of defects in laminated conductive materials based on Lorentz force measurements remains a challenge. The effects on the Lorentz force profile caused by a defect can be described by a subtraction of the forces acting on the permanent magnet in the defect-free system and the system containing a defect. The received difference of force components is referred to the defect response signal. For the calculation of the defect response signal, we propose a new approximation method (forward solution). First, we introduce a grid of point-like current dipoles placed in the defect region. Then, we calculate the sum of Lorentz forces acting on the current dipoles. This calculated sum directly represents the defect response signal. Based on the approximation method, we are able to formulate a reconstruction algorithm (inverse problem) of a single defect in a laminated bar, i.e. we are able to establish a kernel matrix and to apply a linear inverse scheme to estimate the unknown conductivity distribution in the region of interest (similarly to the approach given in [17]). Thus, for the first time, a method for the reconstruction of the defect geometry based on Lorentz force measurements is developed. This approach has been called Lorentz Force Evaluation (LFE). In this paper, we introduce the forward computation and the inverse solution, compare the new approach to standard finite element computations and apply finally the new technique to both simulated and measured data.

4

2

Methodology

2.1

Problem description

We assume that the permanent magnet described by the magnetization M = M1z can be replaced by a single equivalent magnetic dipole with the magnetic moment m = m1z = MV1z. The dipole is located at the center of gravity of the magnet r0 = x01x+ y01y+ z01z, where V is the volume of the magnet and 1x, 1y, and 1z represent the unit vectors in a Cartesian coordinate system. A conductive specimen located below the magnetic dipole is moving with a constant velocity v = v1x. The conductive specimen is approximated by a set of thin Aluminum sheets containing a single parallelepipedic defect (Fig. 1). Two types of parallelepipedic defects with respect to the direction of movement of the laminated conductive bar are investigated, a long defect and a wide defect as presented in Fig. 1.

Fig. 1 – The laminated conductive bar - a package of thin Aluminum sheets moving with constant velocity below the permanent magnet, together with a long and a wide defect used in the analysis.

Simulation I: The first simulation is performed for a conductive laminated bar with a long parallelepipedic subsurface defect of dimensions w×h×l = 2mm×2mm×12mm. The defect is located at a depth of d = 2mm below the top surface of the bar and is positioned longitudinally parallel to the direction of the bar motion (Fig. 1). The bar is

5

50mm wide (W), 50mm high (H), and 250mm long (L), made of Aluminum with an electrical conductivity of σ0 = 20.4 MS/m, and is placed 10mm (h0 = δz) below a magnetic dipole (magnetic moment m = 3.5 Am2). The magnetic moment is equal to the equivalent magnetic moment of the permanent magnet used in the experiment (Section 3). The bar moves with the velocity v = 0.16 m/s. Simulation II: The second simulation is carried out for a wide subsurface defect of same dimensions as the long defect (w×h×l = 2mm×2mm×12mm) but located perpendicularly to the direction of the bar movement. Further setup parameters are the same as in the first simulation.

2.2

Forward problem - approximation method

Lorentz forces exerted on the magnetic dipole due to induced eddy currents in the moving bar are equal to F0 and F for the defect-free system and the system with defect, respectively. Then, the influence of a defect on the Lorentz force profile, the defect response signal (DRS), can be calculated as ΔF = F − F0. The DRS calculation in this paper is restricted to the use of the weak reaction approach (WRA), i.e. the influence of the magnetic field produced by induced eddy currents on the primary field (the field produced by the magnetic dipole) can be neglected [18]. In this case, the velocity of the moving specimen has to be rather small. We additionally assume that the electrical conductivity of the defect equals σd = 0, i.e. we consider only ideal defects without eddy currents flowing inside the defect region. The eddy currents distribution in the system with defect can be modeled by superimposing the eddy currents in the system without defect and the distribution of eddy currents in the defect region with changed conductivity σd :=σ0 flowing exactly in

6

opposite direction. Thus, we can conclude that the eddy currents located in the defect region are responsible for the defect response signal ΔF. In the defect region we define a uniform grid of conductive volumetric elements (voxels) with conductivity σ0 (Fig. 2).

Fig. 2 – Modeling of DRS with a set of voxels with current dipoles (approximation method).

Since in this study are only considered rectangular defects whose walls are parallel to the walls of the bar, the voxels are small cuboids of volume V0 = Δx0Δy0Δz0. In each voxel of volume V0 and conductivity σ0 flows an induced eddy current described by a current density jk. Thus, the continuous distribution of eddy currents is replaced by a set of point current dipoles pk = jkV0 located at the centers of gravity (COG) of the corresponding voxels. Taking the WRA into account, the eddy current density jk in the k-th voxel of the defect region can be calculated with the help of the Ohm’s law for moving conductors, i.e. jk = σk (−∇ϕk + v×Bk), where ϕk is an electric potential, Bk is the magnetic flux density produced by the magnetic dipole located at r0: Bk =

μ0 4π

⎧⎪ [m ⋅ (rk − r0 ) ] m ⎫⎪ , r r − − ( ) ⎨3 k 0 5 3⎬ rk − r0 rk − r0 ⎭⎪ ⎩⎪

(1)

and rk = xk 1x + yk 1 y + zk 1z denotes the COG of the k-th voxel.

7

To determine the electric potential ϕk in (1), we assume that the z-component of induced eddy currents in the conductive bar vanishes. This assumption is motivated by the fact that in the laminated bar, which is a package of conductive sheets (Fig. 1), the vertical conductivity σzz of the whole package is usually much lower than the conductivity of the single sheets. This enforces a dominant eddy currents flow in xyplanes. Using jzk = 0 and

∂ϕk

ϕk = −vm

∂z

= vByk , we find ϕk at the COG of the k-th voxel as:

μ0 ( yk − y0 ) . 4π ⎡( x − x ) 2 + ( y − y ) 2 + ( z − z ) 2 ⎤ 3 2 0 k 0 k 0 ⎣ k ⎦

(2)

The Lorentz force connected with eddy currents flowing in the defect region is equal to Fd = ∫ j × BdV ≈ V0 ∑ k =1 jk × B k = ∑ k =1 p k × B k , where Vd is the defect volume, V0 is K

K

Vd

the volume of voxel, pk is a point-like current dipole located in the COG of k-th voxel, and K is the number of voxels located in the defect area. This force is exactly equal to that exerted on the magnetic dipole but in opposite direction, i.e. the defect response signal equals ΔF = -Fd. The components of the DRS are given by the following expressions: ⎛ μ ⎞ ΔFx = vV ⎜ m 0 ⎟ ⎝ 4π ⎠

⎡ ( y − y )2 + ( z − z )2 0 0 k k

2 K

∑ σ k ⎢3 ⎣⎢

k =1

⎛ μ ⎞ ΔFy = −3vV ⎜ m 0 ⎟ ⎝ 4π ⎠ ⎛ μ ⎞ ΔFz = −3vV ⎜ m 0 ⎟ ⎝ 4π ⎠

2 K

∑σ k k =1

2 K

∑σ k k =1

rk − r0

5

−

⎤ ⎡ ( z − z )2 1 ⎤ 0 k (3) − 3 ⎥ ⎢ 3 5 3⎥ rk − r0 ⎦⎥ ⎣⎢ rk − r0 rk − r0 ⎦⎥

2

( xk − x0 )( yk − y0 ) ⎡ ( zk − z0 ) 2 1 ⎤ − ⎢3 5 5 3⎥ rk − r0 rk − r0 ⎥⎦ ⎢⎣ rk − r0 ( xk − x0 )( zk − z0 ) ⎡ ( zk − z0 ) 2 2 ⎤ . − ⎢3 5 5 3⎥ rk − r0 rk − r0 ⎥⎦ ⎢⎣ rk − r0

,

(4)

(5)

8

Changing the position of the magnetic dipole r0={x0 y0 h0}T according to the measurement grid points ( P: {xi yi δz}T, i=1…N ) and taking x0 = xi as well as y0 = yi into account, we can calculate a set of approximated Lorentz force profiles describing the forward defect response signals.

2.3

Inverse problem - reconstruction algorithm

We understand the defect reconstruction as the estimation of its size and the position of its center of gravity with respect to the coordinate system assigned to the conductive specimen. Using the idea of the relation between defect response signals and the distribution of current dipoles presented in Section 2.2, we define a region of interest (ROI) with unknown distribution of electrical conductivity where the reconstruction of the defect will be performed. In this study, the ROI is a part of the conductive bar and is defined arbitrary as a fixed cuboidal region in the vicinity of the region where the distortion of measured signals due to an artificial defect is observed. Then, a uniform mesh of volumetric elements (voxels) of constant volume (V = ΔxΔyΔz ) is generated in the ROI (Fig. 3).

Fig. 3 – The region of interest – a set of volumetric elements (voxels) where the defect reconstruction algorithm is applied.

9

There are no restrictions for the definition of the mesh of voxels in the ROI with respect to the position of defect walls, i.e. voxels can fit or not to the real defect. Unknown constant electrical conductivities σk are prescribed to each voxel. The set of all voxel conductivities in the ROI forms a vector {σ} = {σk: k=1…K}T, where K is the total number of voxels in the region. If we would like to reconstruct a defect, we first have to find the conductivity vector {σ} in the ROI. Once {σ} is known, we apply various conductivity thresholds Th to find the truncated conductivity distribution {σTh} and to visualize the corresponding cluster of voxels which we interpret as the reconstructed defect. Calculating the weighting average of centers of gravity of voxels in the cluster, we can estimate the position of the defect, with a conductivity σk as a weighting factor. Similarly, we can find the size of the reconstructed defect. The components of the defect response signal produced by the current dipoles located in the ROI are linearly dependent on the conductivity vector {σ}, as shown in equations (3)-(5). Thus, the Lorentz force exerted on the magnetic dipole at N different positions above the conductive bar and conductivities of the K voxels from the region of interest are linearly related through a kernel matrix [L]: ⎡ l11x ⎢l ⎢ 11 y ⎢ l11z ⎢ ⎢ ⎢ lN 1x ⎢ ⎢l N 1 y ⎢l ⎣ N 1z

l12 x l12 y l12 z lN 2 x lN 2 y lN 2 z

… l1Kx ⎤ … l1Ky ⎥⎥ … l1Kz ⎥ ⎥ ⎥ … lNKx ⎥ ⎥ … lNKy ⎥ … lNKz ⎥⎦ 3 N ×K [ L ]3 N ×K

⎧ σ1 ⎫ ⎪σ ⎪ ⎪ 2⎪ ⎨ ⎬ ⎪ ⎪ ⎩⎪σ K ⎭⎪ K {σ }K

⎧ ΔF1x ⎫ ⎪ ΔF ⎪ ⎪ 1y ⎪ ⎪ ΔF1z ⎪ ⎪ ⎪ =⎨ ⎬ , ⎪ ΔF ⎪ ⎪ Nx ⎪ ⎪ΔFNy ⎪ ⎪ ΔF ⎪ ⎩ Nz ⎭3 N

(6)

{ΔF }3 N

where

10

2 2 2 2 ⎤ ⎡ ( zk − zn ) 2 1 ⎤ ⎛ μ ⎞ ⎡ ( y − yn ) + ( z k − z n ) , (7) − − lnkx = vV ⎜ m 0 ⎟ ⎢3 k 3 5 3⎥⎢ 5 3⎥ ⎝ 4π ⎠ ⎢⎣ rk − rn rk − rn ⎥⎦ ⎢⎣ rk − rn rk − rn ⎥⎦

lnky

lnkz

2 2 1 ⎤ ⎛ μ0 ⎞ ( xk − xn )( yk − yn ) ⎡ ( zk − zn ) , = −3vV ⎜ m − ⎢3 ⎟ 5 5 3⎥ ⎝ 4π ⎠ rk − rn rk − rn ⎥⎦ ⎢⎣ rk − rn

(8)

2 2 2 ⎤ ⎛ μ0 ⎞ ( xk − xn )( zk − zn ) ⎡ ( zk − zn ) , 3 = −3vV ⎜ m − ⎢ ⎟ 5 5 3⎥ ⎝ 4π ⎠ rk − rn rk − rn ⎦⎥ ⎣⎢ rk − rn

(9)

and rk − rn = ( xk − xn ) 2 + ( yk − yn ) 2 + ( zk − zn ) 2 is the distance between the COG of

the k-th voxel and the n-th position of the magnetic dipole above the bar. Equation (6) can be written in a matrix form as:

[ L]{σ } = {ΔF } ,

(10)

where {σ} is a vector containing unknown conductivities associated with voxels from the ROI, {ΔF} is a vector of three-component DRS measured/calculated at N measurement points above the conductive bar, and [L] denotes a kernel matrix. To find the unknown voxel conductivities, we apply the singular value decomposition of the matrix [L]:

[ L ]3 N ×K = [U ]3 N ×3 N [ S ]3 N ×K [V ]K ×K T

(11)

where [U]T[U]=[I], [V]T[V]=[I], the columns of [U] are orthonormal eigenvectors of [L][L]T, the columns of [V] are orthonormal eigenvectors of [L]T[L], and [S] is a diagonal matrix containing the square roots of the non-zero eigenvalues of both [L][L]T and [L]T[L] in descending order. The unknown voxel conductivities {σ} can be found by minimizing the difference between the forward calculated data, which is obtained after applying our

11

approximation method, and the measured Lorentz forces at N points above the specimen. This solution is a minimum norm least squares solution and is obtained using the expression:

{σ } = [ L ] {ΔF } , +

(12)

where [L]+ = [V][S]+[U]T is the pseudoinverse of the kernel matrix [L] and [S]+ is the pseudoinverse of [S] found by replacing every nonzero diagonal entry of [S] by its reciprocal and transposing the resulting matrix. Like most inverse problems, this problem suffers from a lack of stability in the solution: the solution is very sensitive to both noise and a priori information used in the inverse analysis. To overcome the loss of stability, we use the truncated singular value decomposition (TSVD) of the kernel matrix [L], which is approximated with the matrix ⎡⎣ L ⎤⎦ under the constraint that rank( ⎡⎣ L ⎤⎦ ) = r , i.e. as many small singular values as

necessary are ignored. The truncation level r in this study is chosen using the value which corresponds to the minimum of the normalized root mean square deviation (NRMSD). This is a measure of the difference between the force signal calculated for the truncated distribution {σTh} approximating the defect and the measured signal. A variety of other methods for estimating r can be found in the literature [19-21]. They are mostly suitable for continuous, but not truncated distributions which we use in the interpretation of the reconstructed defect. By taking into account r singular values, we +

form the pseudoinverse matrix ⎡⎣ S ⎤⎦ and calculate unknown conductivities as: +

{σ } = [V ] ⎣⎡ S ⎦⎤ [U ] {ΔF } . T

(13)

If the conductivity values calculated from (13) are negative it means that the real current density vector in the voxel is in the opposite direction with the respect to the assumed

12

one in pk. Therefore, we take as a solution the absolute values of voxel conductivities obtained by (13). Additionally, we normalize them using the maximum conductivity found in the ROI yielding finally to a range of conductivities from 0 to 1. When only one scanning plane with measurement data is available, one can try to solve the inverse problem in a fully three-dimensional source space (ROI). However, this reconstruction approach might require the use of high computational resources, including large memory space and long computation time. To avoid the inversion of very large kernel matrices, we split the defect reconstruction into three steps. In Step I, we take the x- and z-components of the DRS along the symmetry line of the bar (the ycomponent of the Lorentz force equals zero if the defect is located symmetrically in the bar) to find the depth zi corresponding to the z-position of the intermediate plane of the defect. In this case, we use a rectangular grid of voxels distributed on X0Z-plane. The density of voxels grid in the z-direction is higher than the density in the x-direction to assure a good z-localization of the intermediate plane of the defect. In Step II, using the same set of DRS as in Step I and the found zi position of the intermediate plane, we look for the x-size of the defect. In this step, we use only voxels regularly distributed on the line z = zi, y = 0. In Step III, we use all three components of DRS measured/simulated at the scanning plane and perform the defect reconstruction on the XY-plane using rectangular grid of voxels uniformly distributed at z = zi.

2.4

Reference forward solution

To obtain a reference solution for the direct problem, we apply the finite element method (FEM). The reference solution is computed for two different defects (see Section 2.1) and serves as a benchmark for the new approximation method. The relative

13

motion between the magnet and the conductor can be efficiently modeled, if the logical expression approach (LEA) is applied. The LEA enables fast 3D simulations on fixed computational grids [22]. Using LEA, the spatial coordinates of the moving parts are modeled with time-dependent logical expressions (LE). The shapes of the moving parts are determined on the fly by calculating the constraints given by LE and selecting the finite elements within the domain in which the logical expressions are introduced. Because we are only interested in the Lorentz force perturbations caused by defects, we use LEA to model the defect motion. In the moving defect implementation of LEA, the global coordinate system is fixed to the magnet, and logical expressions are used to model the motion of the defect (Fig. 4). Implementation of the moving defect approach requires a definition of the moving domain within the conductor, where movement of the defect is realized. The moving domain is defined by the size of the defect (w×h×l), its depth d, and its initial position X start (Fig. 4).

Fig. 4 – Implementation of the moving defect approach. Δx is the mesh size in the moving direction, and Δt is the simulation time step.

To simulate the defect motion, the electrical conductivity assigned to the moving domain is modified as σ = σ (1 − LEC ) , where LEC is the logical expression used to model the parallelepipedic defect [22]:

14

X start + vt ≤ x ≤ X start + vt + l ⎧1, . LEC = ⎨ ⎩0, x < X start + vt ∨ x > X start + vt + l

(14)

Omitting displacement current and using the magnetic vector potential A ( B = ∇×A ), the following magnetic field equation applies in the reference frame associated with the permanent magnet ⎛ 1 ⎞ ⎛ ∂A ⎞ + ∇ϕ − v × ∇ × A ⎟ + ∇ × ⎜ ∇ × A − M ⎟ = 0 , [σ 0 ] ⎜ ⎝ ∂t ⎠ ⎝ μ0 ⎠

(15)

where ϕ is the electric scalar potential, M is the magnetization vector of the permanent magnet, [σ0] is a diagonal conductivity tensor of the conductor, and μ0 is the permeability of the vacuum. We use the diagonal tensor [σ0] with σ0xx = σ0, σ0yy = σ0,

σ0zz = 0 instead of scalar conductivity σ0, for modelling the laminated conductive bar and to force vanishing of the z-component of induced eddy currents in the conductive bar. Instead of using both potentials A and ϕ, it is more efficient to use a modified vector potential defined as A* = A − ∫∇ϕ dt. In this case, the electric scalar potential ϕ can be excluded from (15), see [23]. Additionally, for small velocities, the time derivative in (15) can be neglected. Thus, the transient problem is transformed into a quasi-static one leading to a reduced simulation time [22]. The lateral position of the permanent magnet δy is changing during the simulation to perform the scan of the whole area of the conductor in the XY-plane (Fig. 4).

3 Experiment For the validation of the calculations, a model experiment is considered. The permanent magnet is attached to a commercial multi-component force sensor (ME-Meßsysteme

15

GmbH, Hennigsdorf, Germany) [24] and stays in a fixed position. The movement of the set of thin Aluminum sheets mounted in a special holder is realized by a linear beltdriven drive. The experimental setup is presented in Fig. 5 (see [25] for details). 3D-force-sensor Al-sheets Permanent magnet

(a)

(b)

Fig. 5 – (a) The experimental setup, comprised of a linear belt-driven drive, a multi-component force sensor, a package of Al-sheets, a y-z-positioning stage and a data acquisition unit (not shown). (b) The package of Al-sheets in the vicinity of the permanent magnet (NdFeB, Ø15mm×25 mm) attached to the multi-component force sensor.

The multi-component force sensor, with a measurement range of 3N in x- and ydirection and 10N in z-direction, works on the basis of strain gauges that indicate the deflection of the 3D deformation body caused by the acting force (Lorentz force in our case). The voltage change produced by the force sensor is recorded with a real-time PXI-system (National Instruments Corporation, Austin, Texas) at a sampling rate of 10kHz. The lift-off distance between the permanent magnet and the specimen is adjusted using a microscope table as a planar positioning stage with an accuracy of better than 50nm. A conductive specimen of dimensions W×H×L = 50mm×50mm×250mm consists of 25 aluminum sheets with a thickness of 2mm each. This allows us to introduce artificial defects in easy way in any of the sheet positions. The specimen moves with a constant velocity of 0.5 m/s below the permanent magnet. We place a defect of dimensions

16

w×h×l = 2mm×2mm×12mm (long defect) in the second sheet, parallel to the direction of motion of the sheet package and with the center in the XY-plane coinciding to the center of the bar surface area. The permanent magnet is positioned 1mm above the specimen. We scan the specimen for different lateral positions of the permanent magnet. The scanning area contains 191 lines in the y-interval -9.5mm ≤ y ≤ 9.5mm. The concentration of measurement points along y−axis is much higher in the area above the defect than in the rest region.

4 Results 4.1 Comparison of the forward computed defect response signals First, we have performed the comparison of the Lorentz forces calculated for the conductive laminated bar containing a long defect (Simulation I). The reference forward solution was obtained using FEM, as indicated in Section 2.4. The approximated defect response signals were calculated applying our approximation method on a grid of voxels, regularly distributed in the cuboidal defect region with Δx = 1mm, Δy = 0.1mm, and Δz = 0.5mm used as voxels densities. The LF components were calculated using a grid of non-uniformly distributed measurement points on the rectangular scanning plane (Fig. 2). The scanning plane was located at h0 = δz = 10mm above the conductive bar. The density of measurement points was chosen larger in the central part of the scanning plane than in the rest region, i.e. in the vicinity of the region where the defect was expected, to assure a good resolution of the defect response signals recording. Normalized reference defect response signals for the long subsurface defect (Simulation I) are calculated as

ΔFα =

Fα − Fα 0 , α ∈ {x, y, z} , max( Fα − Fα 0 )

(16)

17

where Fα0 and Fα are componnents of the LF L calculated with the FEM F for the conductive bar wiithout and with w the defecct, respectiveely. The resuults are preseented in Fig.. 6a-c. Thee distributioons of normaalized approoximated deefect responsse signals obtained o by our approximatio a on method are shownn in Fig. 6d d-f. The noormalized signals s are determ mined using the t followinng formula:

ΔFα =

ΔFα , α ∈ {x, y, z} , m ΔFα ) max(

(17)

where ΔFα are giv ven by (3)-(55). a)

ΔFx

d) ΔFx

b)

ΔFy

e) ΔFy

c)

ΔFz

f) ΔFz

Fig g. 6 – Simulatio on I - distributiions of reference (ΔF) and appproximated ( ΔF ) normalizeed defect response signals s evoked by the long suubsurface defecct.

The defect d respon nse signals obtained byy FEM and predicted byy our methood were in accepttable agreem ment: the normalized n rroot-mean-square deviaations acrosss all field calculaation points in the scannning area aree equal to 4.8%, 2.6% aand 2.6% forr the x-, y-, and z-componentss, respectivelly. The seecond comp putation of the t Lorentz forces was performed for f the case of a wide defect (Simulatio on II). A reeference soolution was again obttained by FEM. F The

18

approxximated defeect responsee signals of tthe wide deffect was sim mulated by ap pplying our approxximation meethod on a voxel v grid created c withh density Δxx = 0.1mm, Δy Δ = 1mm, and Δzz = 0.5mm. All three coomponents of o the LF foor the FEM and a the approximation methood were calculated at non-uniform n mly distributted points oof the scannning plane located d at h0 = δzz = 10mm above a the cconductive bar. b Normallized reference defect responnse signals of o the wide defect d calcullated accordiing (16) are presented inn Fig. 7a-c. The distributions of the DRS S predicted bby our approoximation m method and normalized n using (17) are pressented in Fig g. 7d-f. a)

ΔFx

d) ΔFx

b)

ΔFy

e) ΔFy

c)

ΔFz

f) ΔFz

Fig. 7 – Simulation n II - distributiions of referencce ( ΔF ) and approximated a ( ΔF ) normalizzed defect response signals evoked by the wide su ubsurface defecct.

The DRS D obtaineed by FEM M and our m method are in the acceeptable agreeement: the normaalized root-m mean-squaree deviationss across alll field calcculation poinnts in the scanniing area aree equal to 5.9%, 5 1.7% and 6.7% for the x-, yy-, and z-coomponents, respecctively.

19

4.2 Reeconstructioon of a simu ulated long ssubsurface defect d We usse the three steps reconsstruction alggorithm desccribed in Secction 2.3 to reconstruct r the lonng subsurfacce defect in the t conductivve laminated d bar. Stepp I: The x − and z − co omponents oof the Loren ntz forces aloong the sym mmetry line ( y = 0 ) obtained d from FE EM (Simulaation I) foor 65 non--uniformly distributed measu urement poin nts located in the inteerval −50mm m ≤ x ≤ 50m mm at h0 = δz δ = 10mm above the bar (Fiig. 8) are ussed to define the refereence forwardd solution (nnormalized defect response, NDR). N

Fig. 8 – Simulation I - normalized defect d responsee signals of thee long defect calculated c on th he symmetry line, 100mm above thee conductive laaminated bar.

The

region

of

interest

contains

22420

voxels

in

the

rectangular

region

(−60m mm ≤ x ≤ 60m mm, −10mm m ≤ z ≤ 0). Settting threshoold Th for th he normalized conductivity in the ROI R to 50% %, we have found for the TS SVD the truuncation levvel equals r = 45 whicch corresponnds to the minim mum of the normalized n root r mean sqquare deviatiion betweenn DRS produuced by the

20

truncated conductivity distribution and the reference DRS. The corresponding truncated conductivity distribution {σ50} is shown in Fig. 9. a)

b)

Fig. 9 – Step I – plane X0Z, distribution {σ50} of color-coded reconstructed normalized conductivities for the long defect (conductivity threshold Th = 50%, red color denotes high conductivity) (a) spurious solution, (b) filtered solution.

We observe spurious solutions (artifacts) at the first line of grid of voxels (Fig. 8a), i.e. at the line closest to the top surface of the bar. The artifacts can be easily filtered if we use the a priori information that only one subsurface defect is sought. The rescaled filtered conductivity distribution is shown in Fig. 8b. The cluster of voxels with a color other than blue can be interpreted as a representation of the reconstructed defect in the X0Z-plane. Using conductivities from the found cluster in the weighting averaging of voxels COGs, we estimated the z−position of the intermediate z−plane of the long defect as zi = −2.8mm. This depth position is used in the next two steps. Step II: We take the same DRS as in the Step I (Fig. 8). At zi = −2.8mm, we generate a uniform grid of voxels in the range −50mm ≤ x ≤ 50mm. The ROI consists of 200 voxels. Similarly to Step I, we set the threshold Th for the normalized conductivity in

21

the ROI to 50%. The found truncation level in TSVD is equal to r = 8. Fig. 10 shows the truncated normalized conductivity distribution {σ50}.

Fig. 10 – Step II – distribution {σ50} of reconstructed normalized conductivities for the long defect along X−line at zi = −2.8mm. The real defect is indicated as a black line segment.

Taking into account only voxels with a color other than blue, the x−length of the defect is estimated as le = 11.5mm which is almost the length of the real defect. We did not observe any artifacts in the solution. Step III: All three components of the Lorentz force at 40×19 measurement points non-uniformly distributed on the plane δz = 10mm above the bar in the rectangular region −25mm ≤ x ≤ 25mm, −4mm ≤ y ≤ 4mm, are used as DRS. The grid of voxels used for the reconstruction contains 800 voxels uniformly generated in the rectangle −20mm ≤ x ≤ 20mm, −10mm ≤ y ≤ 10mm at the depth zi = −2.8mm. Applying the TSVD procedure with a truncation level r =38 found for the conductivity threshold Th = 50%, a good reconstruction of the defect region in the XY−plane at the depth of zi = −2.8mm is provided (Fig. 11).

22

Fig. 11 – Step III – distribution {σ50} of reconstructed normalized conductivities for the long defect in the plane XY at zi = −2.8mm. The real defect is marked with the black rectangle.

The defect reconstruction in the XY−plane is satisfactory, i.e. the length le and the width we of the defect have been found with an acceptable error: le = 9mm, we = 3mm. The lateral position corresponding to the COG of the defect was precisely determined.

4.3 Reconstruction of a simulated wide subsurface defect

The reconstruction of the wide subsurface defect was performed using the same procedure as in Section 4.2. Step I: The x − and z − components of the Lorentz forces along the symmetry line ( y = 0 ) obtained from FEM (Simulation II) for 65 non-uniformly distributed measurement points located in the interval −50mm ≤ x ≤ 50mm at h0 = δz = 10mm above the bar (Fig. 12) are used to define the reference forward solution (normalized defect response).

23

Fig. 12 – Simula ation II - normaalized defect reesponse signalss of the wide deefect calculated on the symmetry lin ne, 10mm abovve the laminateed conductive bbar.

The uniform u grrid of voxxels containns 2420 voxels v in the rectanggular ROI (−60m mm ≤ x ≤ 60m mm, −10mm m ≤ z ≤ 0), below the scanning line ( y = 0). Appplying the reconsstruction sch heme presennted in Section 2.3 with the TSVD truncation leevel r = 33 found for the conductivity threshold t Thh = 50%, wee obtain thee defect shape in the X0Z−p plane shown n in Fig. 13 after a filteringg surface artiifacts.

Fiig. 13 – Step I – distribution {σ50} of color--coded reconstrructed normaliized conductiviities for the widee defect in the pplane X0Z - filttered solution.

24

Although the reconstructed conductivity {σ50} smeared around the real position of the defect in the area twice larger than the real defect, the found weighted zi position of the intermediate plane equals zi = −2.1mm is in acceptable range of error. Step II: We use DRS shown in Fig. 12. We generate a uniform grid of 364 voxels in the interval −60mm ≤ x ≤ 60mm, y = 0, at the depth zi = −2.11mm. Applying the reconstruction scheme presented in Section 2.3 (with the TSVD truncation level r = 45 found for the conductivity threshold Th = 50%), a defect length of 1.98mm in the x−direction is successfully determined (Fig. 14).

Fig. 14 – Step II – distribution {σ50} of reconstructed normalized conductivities for the wide defect along X−line at zi = −2.1mm. The real defect is indicated as a black line segment.

Step III: All three components of the Lorentz force at 41×29 measurement points non-uniformly distributed on the plane δz = 10mm above the bar in the rectangular region (−21mm ≤ x ≤ 21mm, −8.5mm ≤ y ≤ 8.5mm) are used as DRS. The grid of voxels used in the reconstruction contains 1120 voxels uniformly generated in the rectangle (−21mm ≤ x ≤ 21mm, −14mm ≤ y ≤ 14mm) at the depth zi = −2.1mm. The best reconstruction was obtained using in the TSVD procedure a truncation level r = 40 corresponding to the conductivity threshold Th = 50%. The found reconstruction of the defect contour {σ50} in the XY-plane is presented in Fig. 15.

25

Fig. 15 – Step III – distribution {σ50} of reconstructed normalized conductivities for the wide defect in plane XY at zi = −2.1mm. The real defect is marked with the black rectangle.

The reconstruction of the defect in the XY−plane with we = 3mm and le = 11mm is satisfactory, with similar errors as for the long defect.

4.4 Reconstruction of a long subsurface defect using the measurement data

The reconstruction algorithm presented in the previous sections has to be verified in the presence of noise. Therefore, we have performed reconstructions of the long subsurface defect using measurement data recorded by the system described in Section 3. Step I: The x − and z − components of the Lorentz forces along the symmetry line ( y = 0 ) at 69 uniformly distributed points in the interval −50mm ≤ x ≤ 50mm at δz = 1mm above the bar extracted from the measurement data are used to define the reference forward solution.

26

The normalized defect response signals are calculated according the following formulae:

ΔFα ( xi , y j ) =

Fα j 0 ( y j ) =

Fα ( xi , y j ) − Fα j 0 ( y j ) max( Fx ( xi , y j ) − Fxj 0 )

, i = 1… N x , j = 1… N y , α ∈ {x, y, z}

Fα ( x1 , y j ) + Fα ( xN x , y j ) 2

,

(18)

(19)

where Nx and Ny are the numbers of measurement positions in the x- and y-direction, respectively. Fig. 16 shows DRS on the symmetry line y = 0.

Fig. 16 – Experiment - normalized defect response signals of the long defect on the symmetry line, recorded 1mm above the laminated conductive bar.

The region of interest contains a grid of 2420 voxels defined in the rectangle (−60mm ≤ x ≤ 60mm, −10mm ≤ z ≤ 0). The filtered truncated conductivity distribution {σ85}, found for the truncated TSVD level r = 46, is shown in Fig. 17.

27

Fig. 17 – Step I – distribution {σ85} of color-coded normalized conductivities representing the long defect reconstructed from the measurement data in the plane X0Z - filtered solution.

The distribution {σ85} forms a highly concentrated cluster of voxels with high conductivities around the center of the defect. Although the x-length of the reconstructed defect is much too short, whereas the depth of the defect represented by the position of the intermediate plane (zi = −3.0mm) is found correctly. Step II: We use the DRS shown in Fig. 16. We generate a new grid of 261 regularly distributed voxels along the line: y = 0, −50mm ≤ x ≤ 50mm at the estimated depth zi = −3.0mm. Applying the reconstruction scheme presented in Section 2.3 (with a truncation value r = 5 for the conductivity threshold Th = 85%), we obtained a defect length of 12mm but with a center shifted about 0.5mm to the left compared to the original defect position (Fig. 18).

Fig. 18 – Step II – distribution {σ85} of reconstructed normalized conductivities for the wide defect along X−line at zi = −3.0mm (measurement data). The real defect is indicated as a black line segment.

Step III: The DRS calculated according (18)-(19) for all three components of the Lorentz force at 40×81 measurement points non-uniformly distributed on the plane δz = 1mm

above

the

bar

in

the

rectangular

region

(−25mm ≤ x ≤ 25mm,

28

−4mm m ≤ y ≤ 4mm) was used as a the refereence signal. Distributionns of normalized defect responnse signals used u in the reeconstructionn are shown in Fig. 19.

Fig. 19 – Stepp III – distribuutions of normaalized DRS at z = 1mm used in i the reconstru uctions.

Thee region of interest i is loocated at zi = −3.0mm annd consists oof a rectangu ular grid of 800

voxels,

uniformly

distributedd

in

thhe

area

(−20mm ≤ x ≤ 20mm,

−10mm m ≤ y ≤ 10m mm). The TS SVD proceduure (with a trruncation levvel r = 2 fou und for the threshold Th = 85% %) gives a defect d map prresented in Fig. F 20.

29

Fig. 20 – Step III – distribution {σ85} of reconstructed normalized conductivities for the long defect in the plane XY at zi = −3.0mm (measurement data). The real defect is marked with the black rectangle.

This step provides a truncated conductivity distribution which is regularly smeared around the defect (le = 13mm, we = 5mm). The found truncation level (r = 2) shows that the reference DRS is influenced by noise (Fig. 19). The use of only a very few singular values in the reconstruction algorithm would enable a proper interpretation of reconstructed conductivity distribution.

5 Discussion

We have demonstrated the application of a novel fast forward method for the computation of the eddy current and magnetic field distribution in a laminated conductor which has been moving with respect to a fixed permanent magnet. This new approach has further been used to introduce a new inversion scheme for the Lorentz force evaluation based on the analysis of simulated and measured data. The quality of defect reconstruction has been found to depend on parameters of the defect under consideration, in particular on the position of the defect with respect to the

30

moving direction of the specimen and the signal-to-noise ratio of the data. Using a priori information of the defect under investigation (i.e. assuming only a single, subsurface defect) it was possible to find a conductivity threshold which together with an artifacts filter gave an appropriate interpretation of the estimated conductivity distributions. The corresponding truncation level used in TSVD has been determined using the minimum normalized root mean square deviation criterion calculated for the truncated conductivity distribution. This approach has been found to work more effective in our problem than for example the L-curve technique which can be found in the literature [21]. Reconstructions in the X0Z-plane have been qualitatively correct, i.e. the subsurface defect has been found in the proper region. The reconstruction quality strongly depends on the localization of the defect on the moving direction of the bar. For long defects, we have received focal concentrations of voxels representing the defect which x-size was much smaller than the original defect length. However, the defect intermediate plane position calculated as a weighted average of voxels COGs has been estimated correctly, for simulated as well as for measurement data. In contrary, for the wide defect, the estimated conductivity distribution was smeared around the original defect covering much more area than the original defect. In this case, the position of a defect intermediate plane has been determined with less accuracy and was shifted towards the surface of the bar. Generally, it is significantly harder to determine the depth of a defect than to locate its lateral position when only measurement values in one plane above the specimen are available. The reason is the difficulty to dissect the effect of depth from that of source strength. In the future, scanning in more than one plane will be performed. The other problem which should be studied in details is the use of depth

31

weighting techniques which compensate a bias toward superficial sources [26-27]. They can be applied to those reconstructions where the defect is localized close to the bar surface (wide defect) to correct the positioning of the intermediate reconstruction plane. The length of the defect in the x − direction was successfully determined (Step II of the reconstruction algorithm) in all three cases although the used conductivity threshold strongly depends on the noise level (Th = 50% for simulated, 85% for measured data). Reconstructions in the XY-plane using simulated data (long and wide defect) were correct, fitting almost perfectly to the defect shape. For the measured data, the use of a high threshold conductivity (85%) combined with a small number of singular values in TSVD has produced a smeared XY-distribution of voxels representing the long defect. This is the result of a relatively high noise which can be observed in the measured signals. The noise yields a form of oscillations of DRS in the directions parallel and vertical to the bar movement (see Figs 16 and 19), with amplitudes of about 10% of the useful signals. The reason for these oscillations has not been found so far. The regions of interest (ROI) used in the reconstructions have been chosen using a priori information about the position of DRS in the recorded/simulated signals and their duration. The size of ROI and the density of voxels are set in arbitrary way. In the future, it is necessary to study the influence of these parameters on the quality of the reconstruction as well.

6 Summary and conclusions

A novel approximation method for the calculation of the magnetic fields and Lorentz forces exerted on a permanent magnet where in its vicinity a laminated conducting specimen is moving has been developed. This development has been done taking into

32

account the main assumption that the weak field reaction approach can be applied. Compared with the finite element analysis for small subsurface defects, the method shows an error smaller than 7%. Based on the new approximation method for the forward problem, a new inverse procedure for the reconstruction of a single, ideal defect is proposed. Successful reconstructions using simulated data from finite elements analysis and measured for various (long, wide) defects have been obtained. Thus, a new contactless, nondestructive evaluation method for conductive materials, named Lorentz force evaluation, is presented in this paper.

Acknowledgement

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Research Training Group "Lorentz force velocimetry and Lorentz force eddy current testing" (GK 1567).

References

[1] Hellier CJ. Handbook of nondestructive evaluation. New York: McGraw-Hill; 2003. [2] Achenbach J. Quantitative non-destructive evaluation. Int J Solids Struct 2000;37:13-7. [3] Aastroem T. From fifteen to two hundred NDT-Methods in fifty years. Proceedings of the 17th World Conference on Nondestructive Testing; 2008 Oct 25-28; Shanghai, China.

33

[4] Ma X. Electromagnetic NDT and condition monitoring - A personal view. Proceedings of the 17th International Conference on Automation & Computing; 2011 Sept 10; Huddersfield, United Kingdom. p. 266-1. [5] Mook G, Michel F, Simonin J. Electromagnetic imaging using probe arrays. Strojniški vestnik – J Mech Eng 2011;3:227-6. [6] Lim MK, Cao H. Combining multiple NDT methods to improve testing effectiveness. Constr Build Mater 2011;1–6. [7] Heckel T, Thomas HM, Kreutzbruck M, Ruehe S. High speed non-destructive rail testing with advanced ultrasound and eddy current testing techniques. National Seminar & Exhibition on Non-Destructive Testing; 2009. p. 261-5. [8] Jiles DC. Review of magnetic methods for nondestructive evaluation. NDT Int 1988;21:311-9. [9] Jiles DC. Review of magnetic methods for nondestructive evaluation (Part 2). NDT Int 1990;23:83-2. [10] Auld BA, Moulder, JC. Review of Advances in Quantitative Eddy Current Nondestructive Evaluation, J Nondestruct Eval 1999;18(1):3-36. [11] Krause H-J, Kreutzbruck, M v. Recent developments in SQUID NDE. Physica C 2002;368:70-9. [12] Sposito G, Ward C, Cawley P, Nagy PB, Scruby C. A review of non-destructive techniques for the detection of creep damage in power plant steels; NDT & E Int 2010;43(7):555-567. [13] García-Martín J, Gómez-Gil J, Vázquez-Sánchez E. Non-destructive techniques based on eddy current testing; Sensors (Basel, Switzerland) 2011;11(3):2525–65.

34

[14] Yunze He, Feilu Luo and Mengchun Pan. Defect characterisation based on pulsed eddy current imaging technique. Sens Actuators, A 2010;164:1-7. [15] Grimberg R. Electromagnetic nondestructive evaluation: present and future. Strojniški vestnik – J Mech Eng 2011;57:204-7. [16] Brauer H, Ziolkowski M. Eddy current testing of metallic sheets with defects using force measurements. Serb J Elect Eng 2008;5:11-20. [17] Haueisen J, Unger R, Beuker T, Bellemann ME. Evaluation of inverse algorithms in the analysis of magnetic flux leakage data. IEEE T Magn 2002;38:1481-8. [18] Ziolkowski M, Brauer H. Fast computation technique of forces acting on moving permanent magnet. IEEE T Magn 2010;46:2927-30. [19] Hanson RJ. A numerical method for solving Fredholm integral equations of the first kind using singular values. SIAM J Numer Anal 1971;8:616-2. [20] Xu P. Truncated SVD methods for discrete linear ill-posed problems. Geophys J Int 1998;135:505-4. [21] Hansen PC, O’Leary DP. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comp 1993;14:1487-3. [22] Zec M, Uhlig RP, Ziolkowski M, Brauer H. Lorentz force eddy current testing: Modelling of permanent magnets in dynamic simulation using logical expressions. IET 8th International Conference on Computation in Electromagnetics; 2011 Apr 11-14; Wroclaw, Poland. p. 1-2. [23] Emson C, Simkin J. An optimal method for 3-D eddy currents. IEEE T Magn 1983;19:2450-2. [24] Mehrachsen-Kraftsensor K3D40Th.

35

http://www.me-systeme.de/de/datasheets/k3d40.pdf, downloaded 2012-07-10. [25] Uhlig RP, Zec M, Ziolkowski M, Brauer H, Thess A. Lorentz force sigmometry: a contactless method for electrical conductivity measurements. J Appl Phys 2012;111:094914-1 - 094914-7. [26] Phillips JW, Leahy RM, Mosher JC. MEG based imaging of focal neuronal current sources. IEEE T Med Imag 1997;16:338-8. [27] Köhler Th, Wagner M, Fuchs M, Wischmann HA, Drenckhahn R, Theißen A. Depth normalization in MEG/EEG current density imaging. 18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society; 1996 Oct 31-Nov 3; Amsterdam, Netherlands. p. 812-3.

Highlights

•

Lorentz Force Evaluation is introduced as a new technique for NDE.

•

A novel approximate force calculation method is introduced.

•

A new inverse procedure for the reconstruction of defects is successfully applied to simulated and measured data.

36