Chris Liebold
The detection and evaluation of defects in solid conductive materials is important in quality assurance. Non-destructive testing is a quality control tool widely used in the metal industry for inspecting electrically conductive materials. One of the most extensively used techniques in the non-destructive testing inspection is the eddy current testing. Here, an alternating current produces a time-varying magnetic flux density, which according to the Faraday's electromagnetic induction, induces currents in an electrical conductor. Variations in electrical conductivity inside an object under test or the presence of defects cause changes in eddy currents and consequently a change in the impedance of the coil. Measuring this impedance, a defect can be detected. However, due to the skin effect, eddy current testing is limited to surface cracks and cracks close to the surface.
With the intent to increase the depth of the inspection in conductive materials, Lorentz force eddy current testing has been introduced [1]. An electromotive force is dynamically induced into the specimen under test, by applying a DC magnetic field and a relative motion between an electrically conductive specimen and a permanent magnet, Fig. 1.
The electromotive force leads to induced eddy currents, which counteract with the magnetic field of the permanent magnet. This interaction leads to Lorentz forces which are measured on force sensors mounted to the permanent magnet. A defect that is present in the conductive material or even a conducting inhomogeneity disturbs the eddy current distribution in the material and generates a perturbation of the Lorentz force signal. Using the perturbed force data, defect properties could be determined. In order to find the defect properties, the inverse problem has to be solved. The solution of an inverse problem requires solving of a forward problem, which mainly involves the calculation of the induced eddy current field.
Over the past two decades, the finite element method (FEM) has been established as a very powerful numerical tool for determining eddy current fields in solid conductive materials. However, the usage of the FEM solution requires the discretization of the whole volume conductor and is therefore computationally intensive. High quality meshes are needed to avoid mesh-related artifacts in reconstruction of defects. Moreover, in electromagnetic field problems which involve moving conductors, remeshing is required. Additionally, FEM requires numerical integration which is a dominant task in the computation process. All these result in complex and not flexible codes.
We have proposed Boundary element source method (BESM) [2], for fast calculation of velocity induced eddy currents, based on the determination of surface charges that occur at all the surfaces where electrical conductivity of the material changes. This method does not involve any assumptions on the eddy current flow and can be used for the determination of the induced eddy current distribution in solid conductive materials with the defects of any conductivity. Multiple defects can be considered as well, and each defect can have its own conductivity. The position and orientation of a defect inside a specimen has no influence on the implementation of this method. The BESM does not have the difficulties of numerical integration. The accuracy of the method is verified by comparing the results with the solutions obtained from a finite-element model. L2 relative error of the eddy current profiles obtained by the BESM with respect to the reference FEM solution calculated at 13237 points inside the volume of the conductor are 0.37 %, 0,29 % and 2,4 % for x-, y- and z-component of eddy currents, respectively. Lorentz force exerting the permanent magnet obtained by BESM is in a very good agreement with FEM solution, Fig. 2. The proposed BESM approach is shown to be simple, robust, and computationally accurate.
In order to avoid mesh generation for calculating velocity induced eddy currents inside conductive materials, we have proposed mesh-free method of fundamental solutions (MFS) [3]. The basic idea of the MFS is to approximate the solution in terms of a set of fundamental solutions, placing fictitious point sources outside the region where the eddy current field is calculated and satisfying appropriate boundary conditions at a number of points on the boundary of the domain. The mesh-free property makes the MFS very attractive for irregular geometries and inverse procedures. The algorithm eliminates the need to treat singularities. Since the numerical evaluation of integrals for the matrix coefficients is omitted, computational requirements are reduced. The procedure is easy to implement. Our numerical experiments show that the MFS can produce highly accurate solutions. We obtain mean normalized root mean square errors below 1% for defect-free specimen and specimen containing a defect. FEM and MFS models contain 936993 and 17000 elements, respectively. Memory of 13.27 GB required by FEM is reduced to 6 GB in MFS model. Total eddy current profile obtained by mesh-free MFS is presented in Fig. 3.
A final step of a forward Lorentz force eddy current testing problem is calculation of the Lorentz force acting on the permanent magnet. This is commonly done by calculating the force acting on the conductor, using the triple integral over the whole volume of the specimen. All proposed methods [1]-[3] require discretization of the entire volume for solving the triple integral. Since one has to evaluate the integral for each specific position of a moving permanent magnet, this procedure is very time consuming, especially for large conductors. One way to avoid the necessity of volume discretization is a transition from a volume to a surface integral. This method is known as Maxwell stress tensor approach. However, in this approach there exists the problem of selecting an integration surface. When the integration surface is chosen close to the object, functions being integrated experience discontinuities due to their definition based on the electric field. This implies a large sensitivity of the force calculations to the field discontinuity at the boundary. On the other hand, when an integration surface far away from the object is considered, the field is affected by larger errors and numerical noise. Therefore, one usually uses multiple integration surfaces and averages the results, which produces larger effort compared to using a single surface. In order to avoid the impact of this field discontinuity on the stability of the results, we derived new vector functions, based on the electric scalar potential instead on the electric field. Electric scalar potential can be determined completely mesh-free [3]. Using this novel substitution method, we convert the triple integral over the volume of the conductor to a double integral over its surface, where the sub-integral functions are continuous through the different compartments of the system [4]. Given these functions, we calculate the Lorentz force performing just one integration over the physical surface of the conductor. The transition from a volume to a surface integral benefits greatly from a strongly reduced number of discretization elements and therefore contributes to the less calculation time. In the scope of force calculation, the determination of the eddy current distribution inside the conductor is not required. The method is referred to as the Lorentz force surface integration method (LFSIM). We exemplify our approach on a parallelepipedical specimen and a magnetic dipole, where we obtain normalized root mean square errors below 0.6 % with respect to a reference finite element solution.
All the developed methods can handle arbitrary geometry of the specimen and arbitrary orientation of the magnetization vector.
The work was supported by the DFG Grant PE 2389 and the Postdoctoral fellowship program to promote the academic work of women at the university awarded by the TU Ilmenau.
[1] B. Petković, J. Haueisen, M. Zec, R. P. Uhlig, H. Brauer, and M. Ziolkowski, “Lorentz force evaluation: A new approximation method for defect reconstruction,” NDT&E Int., vol. 59, pp. 57–67, Oct. 2013.
[2] B. Petković, K. Weise, and J. Haueisen, “Computation of Lorentz force and 3-D eddy current distribution in translatory moving conductors in the field of a permanent magnet,” IEEE Trans. Magn., vol. 53, Art. no. 7000109, Feb. 2017.
[3] B. Petković, E.-M. Dölker, R. Schmidt, and J. Haueisen, “Method of fundamental solutions applied to 3-D velocity induced eddy current problems,” IEEE Trans. Magn., vol. 54, no. 8, 2018, Art. ID 6201610.
[4] B. Petković, E.M. Dölker, R. Schmidt, H. Toepfer, and J. Haueisen, “Lorentz force surface integration method: calculation of Lorentz force by means of surface integrals,” submitted to IEEE Trans. Magn.