Konstantin Weise

Optimal Magnet Design for Lorentz Force Eddy-Current Testing

We propose a procedure to determine optimal magnet systems in the framework of the nondestructive evaluation technique Lorentz force eddy-current testing (LET). The underlying optimization problem is clearly defined considering the problem specificity of nondestructive testing scenarios. The quantities involved are classified as design variables, and system and scaling parameters to provide a high level of generality. The objective function is defined as the absolute defect response signal (ADS) of the Lorentz force resulting from an inclusion inside the object under test. Associated constraints are defined according to the applied force sensor technology. A numerical procedure based on the finite-element method is proposed to evaluate the nonlinear objective and constraint functions, and the method of sequential quadratic programming is applied to determine unconstrained and constrained optimal magnet designs. Consequently, we propose a new magnet design based on the Halbach principle in combination with high saturation magnetization iron-cobalt alloys. The proposed magnet system outperforms currently available cylindrical magnets in terms of weight and performance. The corresponding defect response signal is increased up to 180% in the case of small defects located close to the surface of the specimen. The combination of active and passive magnetic materials provides an increase of the ADS by 15% compared with the magnet designs that are built solely from permanent magnet material. The proposed procedure provides a highly adaptive optimization strategy in the framework of LET and proposes new magnet systems with inherently improved characteristics.

Permanent Magnet Modeling for Lorentz Force Evaluation

Lorentz force evaluation (LFE) is a technique to reconstruct defects in electrically conductive materials. The accuracy of the forward and inverse solution highly depends on the applied model of the permanent magnet. The resolution of the technique relies upon the shape and size of the permanent magnet. Furthermore, the application of an existing forward solution requires an analytic integral of the magnetic flux density. Motivated by these aspects, we propose a magnetic dipoles model (MDM), in which the permanent magnet is substituted with an assembly of magnetic dipoles. This approach allows modeling of magnets of arbitrary shape by appropriate positioning of the dipoles, and the integral can be expressed by elementary mathematical functions. We apply the MDM to cuboidal-shaped and cylindrical-shaped magnets and evaluate the obtained magnetic flux density by comparing it to reference solutions. We consider distances of 2-6 mm to the permanent magnet. The representation of a cuboidal magnet with 832 dipoles yields a maximum error of 0.02% between the computed magnetic field of the MDM and the reference solution. Comparable accuracy for the cylindrical magnet is achieved with 1890 dipoles. In addition, we embed the MDM of the cuboidal magnet into an existing forward solution for LFE and find that the errors of the magnetic flux density are partly compensated by the forward calculations. We conclude that our modeling approach can be used to determine the most efficient MDMs for LFE.

Uncertainty Analysis in Transcranial Magnetic Stimulation Using Nonintrusive Polynomial Chaos Expansion

We propose a framework of nonintrusive polynomial chaos methods for transcranial magnetic stimulation (TMS) to investigate the influence of the uncertainty in the electrical conductivity of biological tissues on the induced electric field. The conductivities of three different tissues, namely, cerebrospinal fluid, gray matter (GM), and white matter, are modeled as uniformly distributed random variables. The investigations are performed on a simplified model of a cortical gyrus/sulcus structure. The statistical moments are calculated by means of a generalized polynomial chaos expansion using a regression and cubature approach. Furthermore, the results are compared with the solutions obtained by stochastic collocation. The accuracy of the methods to predict random field distributions was compared by applying different grids and orders of expansion. An investigation on the convergence of the expansion showed that in the present framework, an order 4 expansion is sufficient to determine results with an error of <;1%. The results indicate a major influence of the uncertainty in electrical conductivity on the induced electric field. The standard deviation exceeds values of 20%-40% of the mean induced electric field in the GM. A sensitivity analysis revealed that the uncertainty in electrical conductivity of the GM affects the solution the most. This paper outlines the importance of exact knowledge of the electrical conductivities in TMS in order to provide reliable numerical predictions of the induced electric field. Furthermore, it outlines the performance and the applicability of spectral methods in the framework of TMS for future studies.

Uncertainty Analysis in Lorentz Force Eddy Current Testing

The paper addresses the analysis of uncertainties in the framework of the nondestructive evaluation technique Lorentz force eddy current testing. A non-intrusive generalized polynomial chaos expansion is used in order to quantify the impact of multiple unknown input parameters. In this context, the statistics of the velocity and the conductivity of the specimen as well as the magnetic remanence and the lift-off distance of the permanent magnet are determined experimentally and modeled as β-distributed and uniform distributed random variables. The results are compared with Monte Carlo simulations and showed errors <;0.2%. Furthermore, the numerically predicted force profiles are validated with experiments. A sensitivity analysis by means of the Sobol decomposition revealed that the magnetic remanence and the lift-off distance contribute to more than 90% of the total variance of the resulting Lorentz force profile and should be considered first to improve reproducibility.

Lorentz Force Evaluation With Differential Evolution

In the framework of nondestructive testing and evaluation, Lorentz force evaluation (LFE) is a method for reconstructing defects in electrically conducting laminated composites. In this paper, we propose a new inverse calculation strategy for LFE based on a stochastic optimization, the differential evolution (DE) algorithm. We determined the optimal control parameters for the DE and assessed its performance based on simulated and measured data. The results show that the depth of the defect was estimated correctly for all of the data sets that we evaluated. The geometry was reconstructed with errors of less than 4% relative to the size of the defect. The proposed scheme was robust against noise and distortions in the data measurements. We conclude that the proposed reconstruction scheme is a promising method for solving the inverse problem in LFE.

Current Density Reconstructions for Lorentz Force Evaluation

ABSTRACT The detection and reconstruction of fatigue fractures is of great interest in quality assurance. In the framework of nondestructive testing, Lorentz force evaluation (LFE) is an evaluation technique to estimate flaws in electrically conductive materials based on measured Lorentz forces. In the forward solution for LFE, a defect can be interpreted as a distributed current source. This has motivated the authors to propose current density reconstructions (CDRs) calculated with minimum norm estimates to estimate defect geometries. The L1 and L2 norms tend to produce a solution which is either very focused or very smeared. To balance these constraints, the general Lp norm with 1 ≤ p ≤ 2 was used and the inverse solutions compared. This approach was applied to measured data obtained from a laminated composite and simulated data from a monolithic material. The results show that the L1.5 norm provides the most accurate inverse solutions. The location and extent of the defect are determined with an error of 15 % relative to the size of the defect. The depth estimation has a deviation of 50 %. It can be concluded that CDRs are a powerful method to reconstruct and characterize defects in LFE.

Computation of Lorentz Force and 3-D Eddy Current Distribution in Translatory Moving Conductors in the Field of a Permanent Magnet

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

Fast MOR-Based Approach to Uncertainty Quantification in Transcranial Magnetic Stimulation

We propose a new technique based on parametric model order reduction to efficiently calculate the polynomial chaos expansion of the induced electric field in the human brain in the framework of transcranial magnetic stimulation. A comparison to the traditional non-intrusive method based on regression is provided. The relative differences of the mean and the standard deviation less than 0.1% show the accuracy of the proposed method. The new algorithm accelerates the computations by more than two orders of magnitude. In this way, the computational overhead in the case of uncertainty quantification is considerably decreased with respect to the standard methods.

Oscillatory Motion of Permanent Magnets Above a Conducting Slab

This paper provides the 3-D time-dependent analytical solution of the electromagnetic fields and forces emerging if a coil or a permanent magnet moves with a sinusoidal velocity profile relative to a conducting slab of finite thickness. The results can be readily used in application scenarios related to electromagnetic damping, eddy current braking, energy harvesting, or nondestructive testing in order to efficiently analyze diffusion and advection processes in case of harmonic motion. This paper is performed for rectangular and circular coils as well as for cuboidal and cylindrical permanent magnets. The back reaction of the conductor and therewith associated inductive effects are considered. The solutions of the governing equations and the integral expressions for the time-dependent drag and lift force are provided. The analytical results are verified by a comparison with numerical simulations obtained by the finite-element method. The relative difference between the analytically and numerically evaluated force profiles was <;0.1%. Exemplary calculations show that the waveforms of both force components strongly depend on the level of constant nominal velocity

Lorentz Force on Permanent Magnet Rings by Moving Electrical Conductors

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