#### 2.1. Sample Preparation

For research, the pipe shaped sample made of 13CrMo4-5 constructional steel was considered. This material has relative magnetic permeability of 80 [

16]. The length of the sample was 100.00 mm, its external radius was 10.675 mm, and wall thickness was 0.35 mm. The pipe was sealed with non-magnetic material. Additionally, in the top-side sealing was a valve to regulate the pressure inside the sample. The scheme of the sample is presented in

Figure 1.

Pressure was applied into the sample through a valve with the usage of an air compressor with pressure regulation. The internal pressure caused circumferential stresses in the sample. The value of the stresses were calculated from Barlow’s formula:

where

σ—circumferential stresses in the pipe,

r—outside radius of the pipe,

g—thickness of the pipe wall,

p—pressure inside the pipe.

The sample was measured without any stresses as well as with applied 1 MPa pressure, which resulted in 30 MPa circumferential stresses in the material.

#### 2.2. Measurement Method

The pipe was measured on an eddy current tomography setup (block diagram is presented in

Figure 2., model is presented in

Figure 3), described in detail in [

13]. The tested sample moves linearly between two coaxial coils (driving and measuring) and, for each linear step, fully rotates around its axis in 100 discrete steps. The position of the sample is set by two stepper motors, controlled by an ARM 1114 microcontroller (NXP Semiconductors, Eindhoven, The Netherlands).

Both coils have/consist of 100 turns. The exciting coil has a 7.4 mm internal radius and 17.9 mm external radius, whereas the measuring coil has a 6.1 mm internal radius and 14.8 external radius. The driving coil is powered by a 2 kHz sine current generator and induces an alternating magnetic field, which induces eddy current in the conducting sample. The distribution of eddy current highly depends on the object’s geometry as well as on the electromagnetic parameters of the sample’s material. The magnetic field caused by the eddy current influences the magnetic field in the measuring coil. The changes concern both the amplitude of the measured field as well as phase shift between the exciting and measuring signals. The measurement of the signal’s amplitude signal is done by a 6½ digit multimeter (TH1961, Tonghui, Changzhou, China) and phase shift measurement is done by a digital phase shift meter. The entire measurement procedure is controlled by software developed in LabVIEW (National Instruments, Austin, TX, USA).

#### 2.3. Modelling Method

FEM-based forward tomography transformation was conducted with a set of open-source software. FEM modelling was done in ElmerFEM (CSC–IT Center for Science Ltd, Helsinki, Finland) [

17] with the usage of a magnetodynamics solver. This software solves Maxwell’s equations in frequency domain with the usage of

$\overrightarrow{A}$-V model [

18].

The generation of finite element mesh was done in Netgen 5.3 (Vienna University of Technology, Vienna, Austria). For simulations, the eddy current tomography setup was reduced to 4 elements—the driving coil, measuring coil, sample model, and external ball of air.

The driving coil was modelled as a single turn solenoid, whereas the measuring coil was modelled as a disk, in order to properly represent the phenomena of magnetic induction. In the eddy current tomography setup, measured voltage (induced due to Faraday’s law) is proportional to magnetic flux in the volume of measuring coil. In order to simplify the FEM simulations, the induced voltage was calculated as an integral of magnetic field in the volume of model of measurement coil.

The sample’s actuators were removed from the FEM model. The model of the tested sample was generated automatically in consecutive linear and angular positions.

The fourth object in the model of eddy current tomography setup was an external air ball, which radius significantly exceeds the dimension of any other object in the model. The ball provides finite elements between the other objects in model, in order to properly simulate the distribution of the electromagnetic field. The external surface of the ball (sphere) was used for applying Dirichlet boundary conditions, in order to obtain the uniqueness of the FEM solution, which otherwise would be properly determined only up to constant.

The solid geometry for each measurement point was automatically generated. The Netgen software, based on Delaunay algorithm, created a finite element mesh (example presented on

Figure 4.). The noticeable difference of mesh density for different objects was caused by thin walls of the sample, which require a high-density mesh [

19]. On average, the model of exciting coil consisted of 101,300 elements, the model of measuring coil consisted of 47,200 elements, whereas the sample’s model was formed by 755,000 1st order elements.

The forward tomography transformation is based on conducting FEM simulations for each linear and angular position of the sample. As a result of single simulations, data about distribution of magnetic field is obtained. The magnetic flux density in each finite element is described as a complex number, because utilized magnetodynamics FEM solver provides solution in the frequency domain. Magnetic flux density in the volume of measuring coil’s model is numerically integrated and data proportional to real (in phase) and imaginary (90° phase shifted) parts of inducted voltage are obtained. Thus, the values of signal amplitude (

A) and phase shift between driving and measured signals (

P) can be calculated based on (2) and (3):

where:

V_{re} and

V_{im}—integrated values of real and imaginary part of magnetic flux density in measuring coil volume.

The simulations for different measurement points can be conducted independently, which allows parallelization of the calculations. The entire procedure—generation of finite element mesh, FEM modelling and results computation—is done on a single processor core.

#### 2.4. Method of Inverse Tomography Transformation for Determining the Permeability of the Sample

Inverse tomography transformation is used to reconstruct the properties of the measured object. Due to the fact that phenomena of eddy current induction are highly nonlinear, inverse tomography transformation requires utilization of FEM modelling as well as optimization algorithm. The objective function for the optimization algorithm is minimalization of mean difference between measurement results and FEM-based forward tomography transformation. The diagram of utilized method for inverse tomography transformation is presented in

Figure 5. Initially, the measurement data are acquired. This data is compared with the normalized results of FEM-based forward tomography transformation for the given model and the value of objective function is calculated.

The object’s model may be described either by distribution of material in the cross-section of the sample [

20] or by a cylindrical model with substitute defect [

21]. The optimization algorithm changes the model of the sample for forward transformation in the cycle until it converges. Afterward, the tomography results are obtained as parameters of a best-fitting model.

For determining of the sample’s permeability, a downhill simplex method [

22] was utilized. The sample’s geometry parameters, as well as its electrical conductivity, were set to constant. The magnetic permeability of the sample was only variable for optimization algorithm.