## 4 Results and Discussion  
In the current application, the numerical and experimental investigation of weld thin cracks in joined AISI 304 pipe were carried out, using SFEM code analysis implemented under MATLAB. The pipe radius is 160 mm, far greater than the probe size, so the pipe wall can be considered as a conducting plate [37; 38]. According to this, the studied problem is simplified and becomes two-dimensional (2D) in the (x, y) plane as shown in Figure 5.  
The solving domain resolution concerns the studied electromagnetic device and the air; it is divided into six regions with different physical properties, with physical boundary conditions of homogeneous Dirichlet type applied at the fields of study. The scheme is illustrated in Figure 5, which shows that the mesh air domain that we have taken into consideration is large enough to contain the zone of influence of the probe, so that the emitted field is negligible at the border of the field of study. The field of study is covered by finite element mesh as illustrated by Figure 6a. To reproduce the real geometry of the studied device and to approach the measurement results, it is discretized by subdividing it into subdomains, with 38,400 triangular elements and 19,269 nodes generated automatically. Eddy current distributes locally near the coil [39], to consider this fact in the simulation work. A remeshing is done at each probe displacement. This technique allows obtaining dense and fine mesh around the probe with good quality elements. The triangular mesh quality as a function of the probe displacement is shown in Figure 6b. The mesh quality is over than 0.75 and consequently more accurate simulation results. Figure 5: Solving domain and boundary conditions. As explained above, after a step of random-numbers generation, a computational technique was introduced by which was obtained the relative magnetic permeability. Depending on the experimental measurement for each pitch measurement, this magnetic property distribution is not the same and it varies from 1 to 20, hence the assimilation to a delta-ferrite structure. The simulated relative magnetic permeability distribution (to reproduce the behavior as the experimental) is illustrated by Figure 7, covering a displacement of 20 mm from the middle length of the inspected devise, by cause of symmetry distribution. The ECT probe moves along the direction of the cracks, from the position x = 0 to x = 40 mm, in step of 1 mm. The objective here is to conduct a qualitative study relating to the presence or absence of the most frequent thin cracks in circumferential girth weld, considering the influence of the heat welding on the HAZ and the weld area. The HAZ length is to be assumed 12.7 mm (\(\frac{1}{2}\) inches), refer to the KEPIC MI Technical Standard [40]. To achieve both larger skin depth and to control the surface of inspected pipe, the measurements have been realized using six frequencies, operating frequency 20 kHz, detection frequency 40 kHz, optimum frequency 50 kHz and resonant frequency 300 kHz. In addition, 10 kHz and 100 kHz were added for comparison. The experiments were performed by an ASNT Level II qualified examiner by following ISO 7912 instructions [25]. The measurement of the eddy currents resulting from experimentation and simulation was exploited by the measurement of the related quantity, which results in the impedance measurement. The results of the comparisons were normalized and given in Figures 8-13 for different crack depths. The impedance variation was analyzed for weld cracks and  
Figure 6: (**a**) Mesh of the solving domain and (**b**) Mesh quality according to the displacement. Figure 7: The magnetic relative permeability distribution. stored with the corresponding displacement coordinate of the ECT probe. The normalized impedance was computed by \(Delta\)\(Z(\%)=(Z-Z_{0})/(Z-Z_{0})_{max}\). With \(Z_{0}\) and \(Z\) are the impedances of the raw material without crack and the impedance from the HAZ and the weld with crack, respectively.