Lorentz Force Evaluation (LFE) is a technique to reconstruct defects in ...... of the Research Training Group âLorentz force velocimetry and Lorentz force eddy ...
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Permanent Magnet Modeling for Lorentz Force Evaluation
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Judith Mengelkamp1, Marek Ziolkowski2,3, Konstantin Weise2, Matthias Carlstedt2, Hartmut Brauer2,
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and Jens Haueisen1
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solution
Institute for Biomedical Engineering and Informatics, Technische Universität Ilmenau, DE-98693 Ilmenau, Germany Institute for Information Technology, Technische Universität Ilmenau, DE-98693 Ilmenau, Germany 3 Dept. of Applied Informatics, West Pomeranian University of Technology, Sikorskiego 37, PL-70313 Szczecin, Poland 2
Keywords: nondestructive testing, analytic modeling, force measurements, forward solution, inverse
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Abstract
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Lorentz Force Evaluation (LFE) is a technique to reconstruct defects in electrically conductive materials.
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The accuracy of the forward and inverse solution highly depends on the applied model of the permanent
13
magnet. The resolution of the technique relies upon the shape and size of the permanent magnet. Further,
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the application of an existing forward solution requires an analytic integral of the magnetic flux density.
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Motivated by these aspects we propose a magnetic dipoles model, in which the permanent magnet is
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substituted with an assembly of magnetic dipoles. This approach allows modeling of magnets of
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arbitrary shape by appropriate positioning of the dipoles and the integral can be expressed by elementary
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mathematical functions. We apply the magnetic dipoles model to cuboidal and cylindrically shaped
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magnets and evaluate the obtained magnetic flux density by comparing it to reference solutions. We
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consider distances of 2-6 mm to the permanent magnet. The representation of a cuboidal magnet with
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832 dipoles yields a maximum error of 0.02 % between the computed magnetic field of the magnetic
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dipoles model and the reference solution. Comparable accuracy for the cylindrical magnet is achieved
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with 1890 dipoles. Additionally, we embed the magnetic dipoles model of the cuboidal magnet into an
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existing forward solution for LFE and find that the errors of the magnetic flux density are partly
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compensated by the forward calculations. We conclude that our modeling approach can be used to
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determine the most efficient magnetic dipole models for LFE.
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1
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The aim of this paper is to investigate and compare different modeling approaches for permanent
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magnets applied in Lorentz Force Evaluation (LFE). LFE is a newly developed nondestructive testing
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technique for detection and reconstruction of defects in laminated electrically conductive materials
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[1, 2]. A conductive specimen moves with constant velocity v with respect to a permanent magnet
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(Fig. 1). The depth of a defect can be modified in the experimental setup easily if the specimen consists
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of stacked sheets. The permanent magnet has a homogenous magnetization along the z-axis M = Mez
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and is located at a liftoff distance δz above the top surface of the conductor. The specimen is assumed
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to have the conductivity σ0. Due to the relative movement eddy currents are induced in the conductor.
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The interaction of the eddy currents with the magnetic field of the permanent magnet results in three-
Introduction
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component Lorentz forces exerted on the conductor. Instead of determining the Lorentz forces on the
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moving conductor we measure the forces exerted on the fixed permanent magnet. Due to Newton’s third
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axiom they have equal magnitude but opposite direction as the Lorentz forces on the conductor. The
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presence of a material defect of dimensions L×W×H and depth d in the conductor yields perturbations
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in the induced eddy currents and, consequently, in the force signals. Based on the force perturbations
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the geometry of the defect can be reconstructed by solving an inverse problem. Inverse calculations
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require knowledge of a forward solution to calculate force signals based on a model of the permanent
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magnet, the conductive specimen and the relative movement. An existing approximate analytic forward
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solution is described in [1]. In this forward solution the permanent magnet is represented by one
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magnetic dipole.
47 48 49
Fig. 1: Experimental setup for Lorentz Force Evaluation. A laminated conductive specimen moves with constant velocity v with respect to a permanent magnet. A defect in the specimen is indicated by the dotted lines.
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The magnetic field provided by the permanent magnet has a large influence on the eddy currents and
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Lorentz forces. Even more, the form and the magnitude of the observed perturbations and, therefore, the
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resolution of LFE depend on the size and shape of the applied permanent magnet [3]. The interest to
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achieve a high resolution has motivated us to develop appropriate permanent magnet models that can be
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applied to magnets of arbitrary shape. Moreover, we aim to maintain fully analytic forward calculations.
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Thus, for embedding the model of the permanent magnet into an existing analytic forward solution for
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LFE analytical expressions for induced eddy currents in the plate are required. Furthermore, due to the
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proximity of the magnet and the specimen in LFE the near magnetic field of the permanent magnet has
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to be considered. In this study, we focus on shapes of permanent magnets currently applied in our LFE
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experimental setup, i.e., cuboidal and cylindrical forms.
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An approximation of a complex-shaped permanent magnet with one magnetic dipole as in [1] provides
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a sufficient solution of the magnetic field only at large distances [4]. Analytical descriptions of the
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magnetic field are of great interest [5–16]. Unfortunately, such descriptions are not always applicable to
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inverse LFE calculations, especially not for magnets of complex shapes. In order to overcome these
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limitations we introduce the magnetic dipoles model (MDM) of a permanent magnet, in which the
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permanent magnet is substituted with an assembly of magnetic dipoles. The MDM allows modeling 2
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permanent magnets of arbitrary shape by appropriate placing of magnetic dipoles in the volume of the
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magnet. The integral of the magnetic flux density provided by the MDM is the linear sum of the integrals
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of the single magnetic dipoles.
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In the near field the accuracy of the approximation with one magnetic dipole depends on the shape of
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the modeled magnet [4]. Based on this aspect, we expect the position of the dipoles to have an impact
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as well. Therefore, we develop an optimization procedure to determine optimal dipole positions, instead
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of defining the positions of the magnetic dipoles arbitrarily. An MDM with optimal dipole positions has
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among all MDMs using equal number of dipoles the minimum error in the magnetic flux density
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compared to a reference solution. Since the accuracy of the MDM depends on the number of magnetic
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dipoles, we evaluate MDMs with varying number of dipoles. Further, we evaluate the improvement in
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the MDMs due to the use of optimal dipole positions by comparing the magnetic flux densities of MDMs
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with optimal and non-optimal dipole positions.
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For the investigated cuboidal and cylindrical permanent magnets analytic solutions of the magnetic field
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at any point outside the permanent magnet exist and serve as a reference solution for the MDMs. The
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charge model, also referred to as the coulombian model, provides an analytic solution of the magnetic
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field of the cuboidal magnet in terms of elementary functions [15, 17]. Alternatively to the charge model,
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we apply a surface current model (amperian model) for the cylindrically shaped magnet. Using this
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model the magnetic flux density of an axially magnetized cylindrical permanent magnet can be described
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with the help of generalized complete elliptic integrals [16].
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Additionally, we demonstrate how the accuracy of the model of the permanent magnet influences the
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exactness of the forward solution. Therefore, we embed selected MDMs of the cuboidal permanent
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magnet with optimal dipole positions into the existing forward solution for LFE and evaluate the
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resulting Lorentz forces.
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The paper is structured as follows. In Section 2 we outline the applied methods. We describe the MDMs
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followed by the charge and the surface current model of the cuboidal and cylindrical permanent magnets,
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respectively. Then, we explain our optimization procedure. In Section 3, we show the results, i.e., the
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optimal MDMs of the cuboidal and cylindrical magnets. Further, we evaluate the performance of the
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optimization procedure and the accuracy of the forward computed Lorentz forces. Finally in Section 4,
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we discuss the results and draw conclusions.
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2
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2.1
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The idea of the magnetic dipoles models consists in representing the permanent magnet by a regular grid
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of ND volume elements of identical volume, i.e., voxels. The shape of the voxels depends on the shape
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of the permanent magnet. For cuboidal magnets the voxels are cuboids whereas for cylindrical magnets
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the central voxels are cylinders and the others are hollow cylinder sectors (Fig. 2). One magnetic dipole 3
Methodology Magnetic Dipoles Models
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is positioned in each voxel. The voxels have the same volume. Consequently, the magnetic moments of
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the inserted magnetic dipoles are equal. The dipole positions in the MDM depend on one parameter α
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for the cuboidal permanent magnet and two parameters (α,β) for the cylindrical magnet.
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Fig. 2: Magnetic dipoles model of a) a cuboidal and b) a cylindrical permanent magnet.
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α -MDM of Cuboidal Permanent Magnet
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The cuboidal permanent magnet has the base edge length a, height h, volume V0 = a2h, and is located at
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point P0 = [x0, y0, z0]T corresponding to the center of gravity of the lower face of the magnet (Fig. 2a).
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The edges of the magnet are parallel to the axes of the global Cartesian coordinate system.
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According to the idea of the MDM, the permanent magnet is composed of a set of ND = Na2Nh voxels,
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with Na being the number of voxels along the base edges and Nh the number of voxels along the height
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edge. The volume of each voxel equals VE = Δa2Δh = (a/Na)2(h/Nh) = V0/ND. The magnetic dipoles
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positioned in the voxels have an equal magnetic moment m = mez = MV0/ND = MVE, with V0 and VE
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denoting the volume of the permanent magnet and an elementary voxel, respectively. The magnetic flux
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density B = [Bx, By, Bz]T at any point P = [x, y, z]T outside the permanent magnet can be calculated as a
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linear superposition of the magnetic flux densities of all magnetic dipoles of the α-MDM ND
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B( x, y, z ) = ∑ b m ( x, y, z | xm , ym , zm ) , m =1
4
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with bm being the magnetic flux density of the m-th dipole located at Qm = [xm, ym, zm]T.
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Due to the symmetry of the cuboidal permanent magnet and the identity of all voxels it is not expected
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that any (xm, ym) position of the magnetic dipoles other than the center of gravity (COG) of the bottom
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and top face of the voxels results in an improvement of the α-MDM. Thus, the coordinates (xm, ym) are
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fixed to this position. However, we expect the z-coordinates of the magnetic dipoles to have an impact
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on the model accuracy in the near magnetic field below the permanent magnet. Exploiting this aspect
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the z-coordinate of the magnetic dipoles depends on a parameter zα = α∆h, which defines a local z-
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position of the magnetic dipole in the corresponding voxel. The magnetic flux density of the α-MDM
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depends on a proper selection of the parameter α. Then, the position Qm of the m-th magnetic dipole is
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defined as
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xm Q m = P0 + qijk + q m = ym = zm
x0 − a2 + (i − 12 )∆a i = 1,..., N a a 1 y0 − 2 + ( j − 2 )∆a , j = 1,..., N a , z0 + (k − 1)∆h + zα k = 1,..., N h
(2)
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with m = i+(j-1)Na+(k-1)Na2. The variable qijk denotes the position (COG) of the corresponding voxel
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with respect to the COG of the permanent magnet, and qm describes the position of the magnetic dipole
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with respect to a local coordinate system having its origin at the center of the lower face of the
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corresponding voxel. The parameter α is constraint on the interval [0, 1] with 0 and 1 corresponding to
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the bottom and top face of the voxels. All magnetic dipoles in the α-MDM have an equal local position
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inside their respective voxel.
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( α , β )-MDM of Cylindrical Permanent Magnet
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The axially magnetized cylindrical permanent magnet has radius R, height h, and volume V0 = πR2h. It
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is positioned at the center of the cylinder bottom face (P0 = [x0, y0, z0]T) and the cylinder axis is parallel
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to the z-axis (Fig. 2b). The (α,β)-MDM is composed of Nh layers. Each layer contains one central
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cylindrical voxel (axial voxel) of radius r0 and height ∆h, and NR concentric rings consisting of voxels
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of the form of hollow cylinder segments. They are described with the inner radius ri, the outer radius
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ri+1 with i indexing the i-th ring of voxels, the segment angle ϕ0, and the height ∆h. Thus, the total
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number of magnetic dipoles ND is calculated as
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NR = N D N h 1 + ∑ N Si . i =1
(3)
The variable NSi denotes the number of voxels in the i-th concentric ring and is defined as
π i N= 4 (i − 12 ) ≥ 4, =i 1,..., N R . S 2 5
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(4)
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It is always a multiple of 4 to ensure the symmetry of the (α,β)-MDM. The operator ⋅ denotes the floor
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(greatest integer) function. The magnetic dipoles in the central voxels are positioned on the main axis
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of the cylinder. Further, the radius r0 of the central voxels is defined as
r0 =
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VE . π∆h
(5)
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The magnetic dipoles in the ring voxels are located on the symmetry plane of the corresponding voxel.
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Thus they are positioned at half of the segment angle spanning the ring voxel ϕ0/2. The inner radius ri
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and outer radius ri+1 of the voxels in the concentric rings are calculated by the recurrence
ri +1 =
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VE N Si + r 2 , r1 = r0 . π∆h i
(6)
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The radial and axial positions of the dipoles in the ring voxels depend on the parameters α and β,
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respectively. The parameters α and β are equal for all voxels and are constraint to the interval [0, 1]
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ensuring that the dipoles are located inside the corresponding voxels. With respect to a local coordinate
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system placed at the center of the bottom face of the respective layer of voxels (on the z-axis), the
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positions of the dipoles can be summarized as
0 (axial voxel) i= 0, riβ = 1,..., N R (ring voxels) . (1 − β )ri + β ri +1 , i = z = α∆h α
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(7)
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Then, the position Qm = [xm, ym, zm]T of the m-th magnetic dipole in the global Cartesian coordinate
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system is calculated as
xm x0 + riβ cos θ j j − 12 θ π , 2 , Q m =P0 + qijk = ym = y0 + riβ sin θ j = j N Si zm z0 + (k − 1)∆h + zα
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i = 1,..., N R i j =1,..., N S , k = 1,..., N H
(8)
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with qijk denoting the position of the dipole with respect to a local coordinate system placed at the center
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of the bottom face of the cylindrical magnet. The magnetic flux density B = [Bx, By, Bz]T at any point
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P = [x, y, z]T outside the permanent magnet is calculated according to (1).
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2.2
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The analytic models of the magnetic flux density of a cuboidal and a cylindrical permanent magnet are
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shown in Fig. 3. Applying the charge model the cuboidal permanent magnet is represented by fictitious
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magnetic charges situated on the bottom and top face of the magnet (Fig. 3a) [15, 17, 18]. The surface
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charge density equals σm = M⋅n = M⋅ez =M for the top face located at z = δz+h and σm = M⋅n = M⋅(-
Analytic Models of the Magnetic Flux Density
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ez) = -M for the bottom face positioned at z = δz. Using the magnetic charges as a source term the
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magnetic flux density at any point P = [x, y, z]T outside a cuboidal permanent magnet is calculated as
B(P )
0 m (P ')(P P ') | P P ' |3 ds ' 4 S
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a /2 a /2 a /2 a /2 0 (P P ') (P P ') , M dx ' dy ' dx ' dy ' 3 P P 4 a /2 a /2 | P P ' |3 | ' | a /2 a /2 z h z z z
(9)
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with P´ = [x´, y´, z´]T, and B = [Bx, By, Bz]T. An expression of the integrals in (9) in terms of elementary
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functions is given in [15] and Appendix A.
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Fig. 3: Analytic models of the magnetic flux density: (a) charge model for cuboidal and (b) surface current model for cylindrical permanent magnets.
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In the surface current model the cylindrical permanent magnet is represented by an equivalent infinite
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thin solenoid of radius R and height h. Thus, the magnet is reduced to a current flowing in azimuthal
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direction on the lateral surface of the cylinder. The surface current density is calculated as
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JS = M×n = M×er = Meϕ and the magnetic flux density at any point outside the cylindrical permanent
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magnet is expressed as
B(P )
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0 J S (P´) (P P´) P P´ 3 ds´ , 4 S
(10)
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with S denoting the lateral surface of the cylinder. An analytic expression of the integral in (10) is based
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on generalized complete elliptic integrals [6]. The mathematical formulations of the magnetic field
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components are given in Appendix B.
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2.3
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The aim of the following optimization procedure is to determine optimal parameters αo for the α-MDM
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(2) as well as optimal parameters αo and βo for the (α,β)-MDM (8). We perform the optimization for a
Optimization Procedure
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predefined number of magnetic dipoles. We define a test region G in the specimen, which is positioned
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in the region below the permanent magnet. Motivated by the laminated structure of the specimen
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explained in Section 1, the test region consists of Nz XY-layers: G = {Gk, k=1…Nz}. The XY-layers are
194
equidistantly distributed along the z-axis {zk = -d0-(k-1)∆z, k=1…Nz}, with d0 indexing the z-coordinate
195
of the uppermost layer and ∆z the distance between adjacent layers. Each layer is composed of a regular
196
grid of Nx and Ny points in the x- and y-coordinate directions, respectively. Thus, it holds Gk: {xi = (i-
197
1)∆x, yj = (j-1)∆y, i=1…Nx, j=1…Ny }, with ∆x and ∆y denoting the distance between the grid points.
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Due to the symmetry of the permanent magnets, we restrict the test region in the specimen to the first
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quadrant (x≥0, y≥0). In the test region we evaluate the global normalized root mean square error
200
NRMSEG between the magnetic flux density obtained from the α- or (α,β)-MDM and the flux density
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provided by the corresponding reference solution. The reference solutions are the charge model for the
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cuboidal and the current model for the cylindrical permanent magnet (Section 2.2). The objective
203
function to be minimized is defined as
205
1 Nz
NRMSEG =
204
Nz
∑ NRMSE k =1
k G
2
(⋅) .
(11)
The NRMSE in the k-th XY-layer NRMSEGk(⋅) is calculated as N
NRMSEGk (⋅)
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2 1 3 Nx y BnD ( xi , y j , zk ) − BnA ( xi , y j , zk ) ∑∑∑ N x N y =n 1 =i 1 =j 1
3 3 2 2 max ∑ ( BnA ) − min ∑ ( BnA ) n 1= n1 = z = zk
⋅100 % ,
(12)
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with BnD being the n-th component of the magnetic flux density obtained from the α- or (α,β)-MDM,
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and BnA being the n-th component of the reference solution. Further, the index n∈{1,2,3} corresponds
209
to the {x,y,z}-components of B.
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In order to minimize (11) for the α-MDM and (α,β)-MDM we apply the golden section algorithm [19]
211
and the simplex search method [20], respectively. The parameters are bounded to the interval
212
(α,β) ∈ [0, 1] and the initial values are set to α0 = β0 = 0.5.
213
3
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3.1
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α-MDM of Cuboidal Permanent Magnet
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The cuboidal permanent magnet under investigation has the parameters a = 15 mm, h = 25 mm,
217
δz=1 mm, and the remanence Br =1.17 T. The COG of the magnet is located at x = y = 0. These are
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characteristic values for benchmark problems in LFE, e.g., in [21]. We optimize the parameter α for
Results Magnetic Dipoles Models
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219
different α-MDMs specified by combinations of Na = {2:2:14} and Nh = {1:1:25} by applying the
220
optimization procedure explained in Section 2.3. The test region G consists of Nz = 5 XY-layers, with
221
d0 = -1 mm and ∆z = 1 mm, i.e., the closest layer is located at a distance of 2 mm below the permanent
222
magnet. Each layer is composed of a grid of Nx×Ny = 31×31 points, which are equidistantly distributed
223
in the range of 0 ≤ (x,y) ≤ 30 mm in the first quadrant of the XY-layer. Hereinafter, we refer to the MDMs
224
with optimal parameter αo resulting from the optimization procedure as αo-MDMs.
225
The results are depicted using semi-logarithmic scaling in Fig. 4. We evaluate the accuracy of the
226
magnetic flux density obtained from the αo-MDM by assessing their NRMSEG in groups with the group
227
parameter Na and the function argument Nh. Thus, an increase of the total number of magnetic dipoles
228
ND is caused by an increase of Nh (Fig. 4a). The results show that for each group
229 230 231 232 233
Fig. 4: Results of the optimization of the α-MDM of the cuboidal permanent magnet: (a) the NRMSEG between the αo-MDM and the charge model depending on the number of dipoles along the base edge Na and the number of layers Nh in the αo-MDM, and (b) the optimal parameter αo for the number of dipoles ND in the αo-MDMs corresponding to the minima in (a). The dashed line indicates the dipole positions without optimization.
234
indexed by Na an optimal number of layers Noh,min = {2, 5, 9, 13, 16, 20, 23} with corresponding
235
minimum errors NRMSEoG,min = {1.52, 0.32, 0.07, 0.017, 0.006, 0.002, 0.001} % exists. The NRMSE for
236
these MDM configurations with not optimized dipole positions, i.e., the dipoles are located at to α = 0.5
237
corresponding to the COG of the voxels, equals {3.9, 0.61, 0.1, 0.019, 0.011, 0.003, 0.002} %. Further,
238
an increase of Na yields an increase of Noh,min. Using the results of a least squares fit, the optimal number
239
of layers depends on Na as Noh,min = [1.79Na – 1.71] with [·] denoting the nearest integer function. Further,
240
with increasing Na the edge-to-height ratio of the voxels converges to one. Thus, for high numbers of
241
dipoles the optimal number of layers can be estimated by Noh,min = h/Δa. The optimal parameter αo of
242
the αo-MDMs corresponding to the minimum error in each group is depicted in Fig. 4b as a function of
243
ND. For small ND the optimal position of the magnetic dipoles is lower than the standard choice, i.e., the
244
COG of the voxels indicated by the dashed line. With increasing number of magnetic dipoles αo
245
converges to 0.5. Thus, the optimization has significant influence for MDMs with small numbers of
246
dipoles. 9
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247
Fig. 5 compares the magnetic flux density components Bx and Bz of two αo-MDMs and the charge model.
248
The αo-MDMs are calculated with ND = 22·2 = 8 and ND = 82·13 = 832 dipoles. The flux densities are
249
observed at y = 0, which is the symmetry line of the magnet, and z = -1 mm. This yields a total distance
250
of 2 mm to the permanent magnet.
251 252 253 254
Fig. 5: Comparison of magnetic flux densities obtained from αo-MDMs (BD) and the charge model (BA). The flux densities are calculated at the symmetry line of the permanent magnet (y=0 mm) and z=-1 mm. The αo-MDMs are calculated for ND=22·2=8 with αo=0.4384 (left column) and ND=82·13=832 with αo=0.4992 (right column). The upper and lower rows depict the magnetic flux density components Bx and Bz, respectively.
255
Please note that due to the symmetry of the permanent magnet the By-component equals zero at y = 0. If
256
the αo-MDM consists of eight magnetic dipoles, the NRMSE of the Bx- and Bz-component equals 1.75 %
257
and 1.65 %, respectively. This αo-MDM provides a good approximation of the magnetic flux density at
258
far distances. However, significant discrepancies are observed for -7.5 mm ≤ x ≤ 7.5 mm just below the
259
permanent magnet. The slope of Bx is not smooth and Bz drops to a local minimum. If the cuboidal
260
permanent magnet is represented with 832 dipoles, the NRMSE of both components equals 0.027 %.
261
The irregularities observed in the αo-MDM with eight dipoles vanish. 10
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262
(α,β)-MDM of Cylindrical Permanent Magnet
263
The cylindrical permanent magnet under investigation has the parameters 2R×h = 15×25 mm, δz = 1 mm
264
and the magnetization µ0M = 1.17 T. We optimize the radial and axial positions of the dipoles in a
265
number of (α,β)-MDMs, which are specified by combinations of the number of concentric rings in one
266
slice NR∈{1:1:7}, and the number of slices Nh∈{1:1:25}. The test region G is the same as in the previous
267
section. Similar to the cuboidal permanent magnet, the NRMSEG between the (αo,βo)-MDMs with
268
optimal parameters and the current model are depicted in groups using Nh as the group parameter and
269
NR as the function parameter (Fig. 6a). A minimum for each NR indicates the optimal number of dipole
270
layers Noh,min = {2, 5, 8, 12, 15, 18, 23} for the (αo,βo)-MDMs. The corresponding errors equal
271
NRMSEoG,min={2.19, 0.89, 0.36, 0.122, 0.048, 0.024, 0.012} %. The NRMSE for the MDMs which have
272
the same NR, Nh-combinations but the dipole positions are not optimized, i.e., α = 0.5, equals
273
{4.99, 1.05, 0.41, 0.133, 0.064, 0.051, 0.047} %. Fig. 6b shows the optimal parameters αo and βo for
274
the (αo,βo)-MDMs corresponding to the minimum error in Fig. 6a as a function of ND. For intermediate
275
ND the MDMs tend to smaller z-coordinate (αo), but slightly larger radial coordinate (βo) compared to
276
(α,β) = 0.5. With enlarging ND the parameters converge to 0.5. Therefore, the optimization yields
277
significant improvement for MDMs with small numbers of dipoles.
278 279 280 281 282 283
Fig. 6: Results of the optimization of the (α,β)-MDM of the cylindrical permanent magnet: (a) the NRMSEG between the (αo,βo)-MDM and the current model as a function of the number of concentric rings NR and the number of layers Nh in the (αo,βo)-MDM, and (b) the optimal parameters αo and βo in dependence of ND of the (αo,βo)MDMs corresponding to the minima in (a). The dashed line in (b) indicates the COG of the voxels, i.e., the dipole positions if the optimization is not applied.
284
Instances of the magnetic flux densities obtained from two (αo,βo)-MDMs and the current model are
285
compared in Fig. 7. The flux densities are calculated at y=0 and z = -1 mm yielding a total distance of
286
2 mm to the permanent magnet. If it holds ND = 10 (NR = 2, Noh,min = 2), the NRMSE equal 2.78 % and
287
2.56 % for the Bx- and Bz-component, respectively. Similar to the cuboidal magnet, differences are
288
observed in the region of -7.5 mm ≤ x ≤ 7.5 mm. The Bx-component of the MDM is too small, whereas
289
the Bz-component overshoots the current model. If the (αo,βo)-MDM consists of 1890 magnetic dipoles
11
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290
(NR = 6, Noh,min = 18), the NRMSE of the Bx- and Bz-component result into 0.034 % and 0.039 %,
291
respectively.
292 293 294 295 296
Fig. 7: Comparison of magnetic flux densities obtained from (αo,βo)-MDMs (BD) and the current model (BA). The flux densities are calculated at the symmetry line of the permanent magnet (y = 0 mm) and z = -1 mm. The (αo,βo)MDMs are calculated for ND=10 with αo=0.44, βo=0.3892 (left column) and ND=1890 with αo=0.4992, βo=0.5078 (right column). The upper and lower rows depict the magnetic flux density components Bx and Bz, respectively.
297 298
Comparison of the Optimal Number of Layers and the Accuracy of Optimized and not Optimized
299
Magnetic Dipoles Models
300
For evaluation of the efficiency of the optimization procedure we compare for the cuboidal as well as
301
for the cylindrical permanent magnet MDMs with optimal Nh and optimized dipole positions described
302
in the previous sections to the MDMs with optimal Nh but standard dipole positions. We define that
303
standard positioning is indicated by αs = 0.5 as well as (αs,βs) = (0.5,0.5) for the cuboidal and cylindrical
304
magnets, respectively. We evaluate the NRMSEG of the αs-MDM and (αs,βs)-MDM for same (Na,Nh)-
305
and (NR,Nh)-combinations and test region as in the previous sections. The results show similar curves as 12
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306
for the optimized MDMs, but the minimum errors NRMSEsG,min of the group parameters correspond to
307
different numbers of layers Nsh,min and have different values. For further analysis, we calculate the
308
differences in the number of layers ΔNh,min and in the normalized errors ΔNRMSEG,min between the
309
MDMs corresponding to the minimum errors for optimized and standard positioning of dipoles as
∆N h ,min = N ho,min − N hs,min
310
∆NRMSEG ,min = NRMSEGo ,min − NRMSEGs ,min
.
(13)
311
The results are shown in Fig. 8 as a function of Na and NR, respectively. Considering the cuboidal
312
permanent magnet, the minimum error of the αo-MDM for low Na is located at a lower Nh than the
313
minimum error of the αs-MDM (Fig. 8a). Despite the lower number of dipoles the error of the αo-MDM
314
is smaller. Due to the optimization the NRMSEoG,min is smaller by 0.39 % than the NRMSEsG,min for
315
Na = 2, and by 0.0018 % for Na = 8. Moreover, the αs-MDMs corresponding to the NRMSEsG,min consist
316
of voxels (∆a×∆a×∆h), which have similar edge and height length. Thus, for the αs-MDMs the optimal
317
number of dipoles can be estimated by Noh,min = h/Δa.
318 319 320
Fig. 8: Comparison of MDMs with optimal number of layers using optimized and standard positioning of magnetic dipoles for (a) the cuboidal and (b) the cylindrical permanent magnets.
321
The results of the cylindrical permanent magnet in Fig. 8b depict that for NR = {1,2,3} the optimal
322
number of layers is less for the (αo,βo)-MDMs than for the (αs,βs)-MDMs. For NR = {5,6,7} the contrary
323
is the case. However, for all NR the error of the (αo,βo)-MDMs is smaller than the error of the (αs,βs)-
324
MDMs. For optimized MDMs the NRMSEoG,min is smaller by 0.55 % and by 0.024 % than the
325
NRMSEsG,min for NR = 2 and NR = 6, respectively. In contrast to the (αo,βo)-MDMs, the optimal number
326
of layers for the (αs,βs)-MDMs can be calculated with respect to the results of a least squares fit as
327
Noh,min = [2.79NR + 0.43].
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328 329
3.2
Lorentz Force Evaluation Using Magnetic Dipoles Models of Cuboidal Permanent Magnet
330
In this section we investigate how the dipole optimization influences the error of the forward calculated
331
Lorentz forces exerted on the permanent magnet. We embed the MDMs with optimized positions (αo-
332
MDMs) corresponding to NRMSEoG,min of the cuboidal permanent magnet obtained in Section 3.1 and
333
the αs-MDMs with the same (Na,Nh)-combinations into an existing approximate forward solution for
334
LFE from [1]. Further, we put the charge model (Section 2.2, Fig. 3a) into the forward solution and use
335
the force signals calculated with this model as a reference solution. Then, we calculate the NRMSE∆F
336
between the three-component Lorentz force perturbations obtained by the α-MDMs and the charge
337
model using a similar formula to (12). This approach ensures that we only investigate signal errors of
338
the α-MDMs and not the total error of the approximate forward solution. We restrict the analysis to the
339
cuboidal permanent magnet, because we were able to find analytical expressions for eddy currents
340
induced in the plate only for the magnet model in the form of a cuboid but not in the form of a cylinder.
341
As a benchmark problem we investigate the LFE problem (Fig. 1) with a cuboidal permanent magnet
342
a×a×h = 15 mm × 15 mm × 25 mm fixed above a moving laminated specimen consisting of 50 aluminum
343
alloy sheets. The sheets have a thickness of 1 mm. Because the sheets are isolated from each other, the
344
conductivity of the specimen can be approximated by an anisotropic conductivity described by a
345
diagonal tensor [σ] = diag(σ0,σ0,0) where σ0 = 30.61 MS/m. A cuboidal defect with the dimensions
346
6 mm × 6 mm × 1 mm is positioned in the third upper sheet, i.e., at depth d = 2 mm. The COG of the
347
defect is located at x = y = 0. The relative velocity equals 0.01 m/s. We examine the profiles of the
348
Lorentz force perturbations exerted on the permanent magnet. The observation points are equidistantly
349
distributed in the range of 0 ≤ x ≤ 20 mm, 0 ≤ y ≤ 10 mm at z = 1 mm. Due to the symmetry of the
350
problem setup the investigated region can be restricted to the first quadrant. The distance between
351
adjacent observation points equals 1 mm.
352
Fig. 9 presents the resulting NRMSE∆F as a function of ND in the αo- and αs-MDMs. The perturbation
353
force signals calculated with the αo-MDM have a significant smaller error than the signals calculated
354
with the αs-MDMs. NRMSE∆F errors introduced by αo-MDMs equal 0.8 % and 0.002 % for ND = 8 and
355
ND = 832, respectively. For the αs-MDMs with the same number of magnetic dipoles, the error equals
356
8.2 % and 0.019 %, respectively. For large ND the errors approach each other. A comparison to the
357
NRMSEoG,min of the magnetic flux density of the αo-MDMs (Section 3.1, Fig. 4a) shows that the errors
358
of the force signals for the αo-MDMs are smaller by half the value for ND = 8 and one decimal place for
359
ND = 832. However, if the standard models (αs-MDM) are used the errors of force signals are
360
significantly higher for both ND = 8 and for ND = 832 than the NRMSE of the corresponding magnetic
361
flux density outlined in Section 3.1.
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362 363
Fig. 9: NRMSE∆F between the Lorentz force perturbation signals using the αo- and αs-MDMs and the charge model.
364
A comparison of the Lorentz force perturbations caused by a defect using the αo- and the αs-MDM with
365
8 magnetic dipoles and the charge model is shown in Fig. 10.
366 367 368 369 370
Fig. 10: Comparison of Lorentz force perturbations ΔFx, ΔFy, and ΔFz. The perturbations due to the defect are obtained by subtracting the force signals coming from a specimen with and without a defect. The Lorentz forces are computed by applying an existing analytic forward solution including the αo- and αs-MDM with 8 dipoles and the charge model (Ana). The force components are depicted at y = 2 mm.
371
The perturbations of the force signals ΔF are obtained after subtracting the force signals calculated for
372
a specimen with and without a defect. Thus, the profiles of perturbation force components tend to zero
373
far away from the defect region [1]. The signals are depicted at y = 2 mm, because due to the symmetry
374
the side force Fy as well as the ΔFy-component vanish at the symmetry line (y = 0). It can be observed
375
that the signals calculated with the αo-MDM are in excellent agreement with the perturbations calculated
376
with the charge model. The NRMSE of the ΔFx-, ΔFy-, and ΔFz-component calculated for the profile at
377
y = 2 mm equals 0.47 %, 1.54 %, and 1.68 %, respectively. On the contrary, significant deviations are
378
observed in the Lorentz force perturbations calculated with the αs-MDM. Here, the NRMSE of the ΔFx-
379
, ΔFy-, and ΔFz-component equals 7.41 %, 10.11 %, and 6.88 %, respectively.
380
4
381
This contribution was motivated by the necessity to provide an accurate model of the permanent magnet
382
for the forward solution of LFE, which is the basis for successful inverse calculations of defects. We
383
introduced the magnetic dipoles model allowing modeling of permanent magnets of arbitrary shape. The
384
model can be embedded into an analytic forward solution for LFE and the advantage to calculate the
Discussion
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385
flux density and the eddy currents induced in the conductor with elementary analytic mathematics is
386
maintained. Our results showed that 832 magnetic dipoles are necessary to provide a sufficiently
387
accurate model of the magnetic flux density of a cuboidal magnet. A cylindrical magnet is efficiently
388
modeled with 1890 dipoles. In the near field of the permanent magnet the NRMSE of the magnetic flux
389
density of both configurations equals 0.02 %. Since the errors for the (α,β)-MDM of the cylindrically
390
magnet are generally higher than that for an α-MDM of the cuboidal magnet with equal number of
391
dipoles, more dipoles for the (α,β)-MDM have to be considered to achieve a comparable model accuracy.
392
The increased number of dipoles compensates the fact, that the approximation with one magnetic dipole
393
is, because of the more complex geometry for cylindrical voxels and hollow cylinder segments, less
394
accurate than for cuboidal voxels [4].
395
The comparison of the magnetic flux density (Section 3.1) and the forward calculated Lorentz forces
396
(Section 3.3) obtained from the αo-MDM and αs-MDMs shows, that the use of optimized dipole positions
397
yields a significant improvement for MDMs with small numbers of dipoles. With increasing number of
398
dipoles the influence of the optimization is reduced. This is especially valid for inverse calculations,
399
since the computational costs can be reduced by using optimized instead of a larger number of dipoles
400
whereas both approaches yield a reduction of the modeling error. The optimization is performed only
401
once before any forward and inverse calculation and, thus, the computational demand is comparatively
402
low. Therefore, we also recommend using optimal dipole positions even if the enhancement is small.
403
Further, the comparison of the error differences of the magnetic flux density and the Lorentz force
404
signals (Fig. 4a and 9) show that for the αo-MDMs the error in the magnetic flux density is partly
405
compensated by the analytic forward calculations. However, this is not the case for the αs-MDMs. For
406
ND = 8 a large error in the amplitude of the Lorentz forces can be observed. This is likely to be explained
407
by the large differences in the α-parameter (αo = 0.41) determining a difference of 0.8 mm in the z-
408
position of the dipoles. This aspect strongly supports the use of the αo-MDMs. In order to evaluate the
409
outlined behavior in detail it is necessary to consider the models of the specimen and the relative
410
movement in the forward solution, which will be a subject of future investigations.
411
Without showing the results here, we included in our study a cubic permanent magnet with dimensions
412
a = h = 15 mm. We optimized the same α-MDM-configurations as for the cuboidal magnet and evaluated
413
the forward computed Lorentz forces. The results show that for Na = {2,4,6} the optimal number of
414
layers equals Noh.min = Na-1, whereas for larger Na it holds Noh.min = Na. The ΔNRMSEG,min is
415
monotonically increasing and the NRMSEF is monotonically decreasing for increasing number of
416
dipoles. Further, the dependence of the optimal number of layers and Na is similar as for the cuboidal
417
magnet.
418
The results show an approximate linear dependence between the dipole distribution in one layer and the
419
optimal number of layers for the αo- and αs-MDM of the cubic and cuboidal and the (αs,βs)-MDM of the
420
cylindrical magnet. No similar relationship was observed for the (αo,βo)-MDM. These results are likely 16
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421
explained by the structure of the MDMs. In our study the dipole distributions in the MDMs are
422
symmetric. Because the definition of the MDMs implies that the dipoles for MDMs with varying number
423
of dipoles are positioned according to the same principle, the symmetry lines are equal for all evaluated
424
MDMs. Further, the dipoles represent an equal volume of the permanent magnet and have the same
425
moment. Merely the (αo,βo)-MDM depends on two parameters. These show a greater and non-
426
monotonous variation in its values than the one parameter of the cuboidal permanent magnet (Fig. 4b
427
and 6b). Apart from symmetric dipole distributions non-symmetric distributions can be applied. The
428
author in [22] presents a variety of non-symmetric distributions for the dipoles in the layers with the
429
required weighting coefficients, which are different for the individual dipole moments.
430
The Bz-component of the magnetic flux density of the αo-MDM with ND = 8 depicted in Fig. 5 shows a
431
drop at x = 0. In Fig. 7, the Bz-component of the cylindrical magnet shows a higher maximum than the
432
reference solution. Further, the extremal values of the Bx-component are closer to the origin of the
433
coordinate system and the slopes are steeper. These effects can be attributed to the close distance of the
434
respective dipoles in the lower plane of the MDMs and the test region (near field of the dipoles). In the
435
near field the inter-dipole distances have a stronger influence on the resulting magnetic flux density for
436
MDMs when a small number of dipoles is considered. In the (αo,βo)-MDM with ND = 10, the distance
437
between the ring dipoles and the central dipole is reduced, because the parameter βo is smaller than 0.5
438
(Fig. 2b and 6b). Since the dipoles in the αo-MDM with ND=8 are fixed to the COG of the bottom and
439
top face, the dipoles in the (αo,βo)-MDM are closer to the symmetry axis (y=0), at which the magnetic
440
flux density is evaluated.
441
Since in this paper we have focused only on analytical models of ideal permanent magnets described by
442
a constant magnetization vector, we aim to evaluate models based on measurement data in future
443
investigations. These include an analysis of the experimental setup.
444
Especially in inverse LFE calculations with noisy measurement data, an accurate and fast forward solver
445
with an efficient model of the permanent magnet is required to obtain valuable results. In conclusion,
446
we recommend embedding the αo-MDM with 832 and the (αo,βo)-MDM with 1890 magnetic dipoles in
447
inverse calculation in LFE in order to obtain enhanced reconstruction results. Further, our approach can
448
be used in other applications, in which precise simulations of permanent magnets are required.
449
Appendix A
450
Magnetic Flux Density of Cuboidal Permanent Magnet
451
The magnetic flux density components Bx, By, and Bz at the point P = [x, y, z]T outside a cuboidal
452
permanent magnet, which has dimensions 2a×2b×2c, a homogenous magnetization along the z-axis
453
M = Mez, a COG located at the origin of the coordinate system and all edges parallel to the coordinate
454
axis, can be calculated as [15] 17
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455
Bx ( x, y, z ) =
µ0 M F2 (− x, y, − z ) F2 ( x, y, z ) ln , 4π F2 (− x, y, z ) F2 ( x, y, − z )
(14)
456
B y ( x, y , z ) =
µ0 M F2 (− y, x, − z ) F2 ( y, x, z ) ln , 4π F2 (− y, x, z ) F2 ( y, x, − z )
(15)
= Bz ( x, y, z )
457
µ0 M [ F1 ( x, y, z ) + F1 ( x, y, − z ) + F1 (− x, − y, z ) + F1 (− x, − y, z ) + . 4π
(16)
F1 (− x, y, z ) + F1 (− x, y, − z ) + F1 ( x, − y, z ) + F1 ( x, − y, z )] 458
The functions F1(⋅) and F2(⋅) are defined as
459
F1 ( x, y, z ) = arctan
460
F2 ( x, y, z ) =
( x + a )( y + b) ( z + c ) ( x + a ) 2 + ( y + b) 2 + ( z + c ) 2
( x + a ) 2 + ( y − b) 2 + ( z + c ) 2 + b − y ( x + a ) 2 + ( y + b) 2 + ( z + c ) 2 − b − y
,
.
(17)
(18)
461
Appendix B
462
Magnetic Flux Density of Cylindrical Permanent Magnet
463
A cylindrical permanent magnet with radius R, height h = 2b, and homogenous magnetization along the
464
cylinder axis M = Mez is positioned in such a way that its COG coincides with the origin of the coordinate
465
system. In cylindrical coordinates, the components of the magnetic flux density Br and Bz can be
466
calculated by [16]
468 = Bz 469
470
471
µ0 M [α +C (k+ ,1,1, −1) − α −C (k− ,1,1, −1)] , π
(19)
µ0 M R β + C (k+ , γ 2 ,1, γ ) − β −C (k− , γ 2 ,1, γ ) , π R+r
(20)
Br =
467
with
z±2 + ( R − r ) 2 z± R−r R z± =± z b, γ = , k± = 2 , α± = , β± = . 2 2 2 2 R+r z± + ( R + r ) z± + ( R + r ) z± + ( R + r ) 2 The function C(⋅) denotes the generalized complete elliptic integral π /2
472
C ( k c , p , c, s ) =
∫ 0
c cos 2 ϕ + s sin 2 ϕ (cos 2 ϕ + p sin 2 ϕ ) cos 2 ϕ + kc2 sin 2 ϕ
dϕ .
18
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(21)
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473
Acknowledgment
474
The presented work was supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework
475
of the Research Training Group “Lorentz force velocimetry and Lorentz force eddy current testing” (GK
476
1567) at the Technische Universität Ilmenau.
477
5
478
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