Finite Element Modelling of a Reflection Differential Split-D Eddy Current Probe Scanning Surface Notches

The “Magnetic Field Physics” package in Comsol is selected for FE simulations. The governing equation for the magnetic field interface is based on the time-harmonic Maxwell-Ampere’s law [31]. where µ is the magnetic permeability, σ stands for electrical conductivity, A is the magnetic vector potential, ε is the electrical permittivity, ω is the angular frequency and J_e is externally applied current density. Comsol solves Eq. (2) in the frequency domain.

In the low-frequency regime, where the electromagnetic wavelength is much greater than the size of the system, it is possible to assume that the quasi-static form of Maxwell’s equations can be used (\(\sigma > > \omega \,\varepsilon\)). Although in such a case, the displacement current term \(\left( {\partial {\varvec{D}}/\partial t} \right)\) is generally excluded from the calculations, the equation embedded in the magnetic field package of Comsol takes it into account without introducing any additional computational cost.

Generating an unbounded or infinitely extended air domain in FE simulations has been a long-standing problem for many physics packets. It is beneficial, in terms of resources and execution time, to truncate such a domain to an extent in which the simulation results remain reliable. Fortunately, in eddy current problems, the region of practical interest can be limited to a bounded domain large enough to capture a reasonable amount of the coil’s electromagnetic field. Air domain truncation should be done carefully, so to prevent the edge effects spoil the final results. To this end, the fields are initially solved according to a 2-D-axisymmetric model for a large 500 mm × 500 mm × 12 mm un-defective block surrounded by a 600 mm × 600 mm × 600 mm air domain. Figure 4 shows the contour plots of the φ component of the magnetic vector potential A projected on the r-z plane passing through the center of the coil, where φ, r and z are the components of a cylindrical coordinate system. The air domain is then truncated around the region where the intensity of the magnetic vector potential drops to 0.1% of its maximum value. The width and length of the conductive block domain are also shortened with respect to the truncated size of the air domain such that each of the block’s lateral faces is 10 mm apart from the nearest wall of the air domain. A new simulation is performed with this truncated model and the results do not reveal any changes in the field distribution. This shows no edge effects are introduced by truncating the model’s domains. It shall be noted that the changes in the distribution of the electromagnetic fields caused by the presence of a notch just occur locally around the notch itself. Hence, it is safe to assume that the field distribution at the boundaries of the air domain remains unchanged after introducing the notch. The truncated domain can, therefore, be used for further simulations of this absolute coil model.

A sound understanding of the impact caused by each FE parameter on the model is critical for attaining accurate simulation results. The mesh sizes (i.e. the density of finite elements) as well as the distribution of these elements within the model are certainly two of them. Mesh is a geometrical dependent feature. One should consider the physics used and the dimensions of the domain while generating the elements within each domain. It is also important to have a satisfactory resolution for the solved fields, which is fully dictated by the element’s type and size.

In the current problem, the aluminum block is partitioned using a cylindrical surface in order to create a more concentrated mesh underneath the coil. The diameter of the partitioned region is twice the diameter of the coil. This cylindrical region in the aluminum block moves together with the coil at each displacement step so as to always maintain the concentrated mesh underneath the coil.

Because of the skin effect, eddy currents are almost completely contained within the first three standard penetration depth (δ) as defined by

Therefore, resolving the induced currents within their first three standard penetration depth is crucial for obtaining the correct values of the coil’s complex impedance. The standard penetration depth for the aluminum block is equal to 3.04 mm, as reported by Burke [28]. Initially a boundary mesh composed of 6 equal layers of second-order tetrahedral elements has been generated for this region within the aluminum block. Later, the number of layers has been optimized according to the sensitivity of the simulated signals to this parameter. Apart from these critical regions, free tetrahedral meshes have been created in the rest of the model’s geometry including the coil domain.

In order to get accurate simulation results in the shortest computational time, a sensitivity study is performed with respect to mesh size. For this study, the mesh is generated manually for each simulation step in order to have better control over its distribution within different regions. The sensitivity of the simulation results in terms of mesh sizes is studied in two stages. First, the number of boundary layer meshes is kept at a constant value (six across the 3δ depth within the aluminum block) while the maximum element to the coil thickness (6 mm) ratio is changed (i.e., the element’s size in the coil and aluminum block domains is refined in 3 steps). The computed impedance data are then compared to Burke’s measurements [28]. In each step, regardless of the total number of elements, the mesh sizes in the coil and aluminum block domains are equally scaled such that the corresponding maximum size ratio remains constant. This approach enables to keep the mesh quality within an acceptable level and avoid the adverse influence of low element quality in the solution. In the second stage of this study, the best maximum element size to the coil’s thickness ratio (S/T) obtained previously is applied to all domains and kept constant. The purpose of introducing the ratio is to relate the element size to the probe’s physical geometry since the two different absolute and differential configurations are investigated in the study, where the coil geometry can vary significantly from one configuration to another. Then, the number of boundary layer meshes is varied from 1 to 3 per δ depth within the aluminum block. It is noteworthy that for mesh sensitivity studies the conductivity of air is set to 1 S/m.

Figure 5a and b show the variations of the coil’s resistance and inductive reactance for different S/T values as the coil scans the aluminum block containing a notch, respectively. Resistance and inductive reactance values are normalized by the impedance measurements of Burke in these figures [28]. In the normalization process, at each scanning position, the coil’s inductive reactance and resistance measured by Burke are subtracted from the estimated values by simulations, and the results are divided by the maximum variations of the reported impedance measurements. Looking at the resistive part of impedance presented in Fig. 5a, \(\Delta R_{Comsol}\) is associated with the calculated probe’s resistance at each position. This value is subtracted by the measured resistance (\(\Delta R_{Burke} )\) at the same scan position. Finally, the subtraction results for all scan positions were divided by the maximum impedance value measured by Burke (i.e. \(\left| {\Delta Z_{Burke\_1} - \Delta Z_{Burke\_2} } \right|\)). The point of this normalization scheme is to subtract the measured impedance values, provided by Burke, from our computed impedance values, and report the difference (i.e. error in simulation) in percentage. This normalization strategy also allows examining the convergence of the solver as different S/T ratios are selected. For better understanding, the coil scanning positions relative to the notch are shown schematically for 5 different coil displacements in Fig. 5c. According to Fig. 5a, changing the S/T ratio from 0.23 to 0.33 does not introduce any significant change in the resistive part of the impedance and, regardless of the scan position, the error always remains below 2%. However, the solver becomes unstable for an S/T ratio of 0.66 resulting in unreliable resistance values for the coil. This means that S/T ratio needs to be small enough to capture the resistive losses caused by eddy currents inside the conductor. On the other hand, the error connected to the coil’s inductive reactance has the same values for S/T ratios of 0.33 and 0.66 position wise. Furthermore, as the S/T ratio is further reduced to 0.23, the inductive error is decreased by only 0.5%. Although impedance estimations with the lowest level of error are achieved through meshing the block with the smallest S/T of 0.23, such a parameter increases the computational expenses unreasonably. Therefore, to avoid lengthy computational time, a S/T ratio of 0.33 is selected knowing that the solver converges to consistent impedance estimations.

Figure 6a and b depict the variations of the normalized coil’s resistance and inductive reactance as a function of the coil’s position for three different numbers of boundary layer meshes. The normalization scheme is similar to the one used in the previous case. As it can be observed in Fig. 6a, the coil’s resistance is underestimated up to the scan position of 10 mm. This error is larger for 3 boundary layers as compared to the one obtained by 6 and 9 boundary layers. The estimation error becomes positive for the remaining probe positions where 6 boundary element layers provide the best estimation. In Fig. 6b, computed inductive reactance values are overestimated at all scan positions, which is similar to the behavior seen in the previous sensitivity study. It is evident from the figure that using 6 and 9 boundary layers in the model result in very close computed values at each scan position. This suggests the solver already converges with 6 boundary layers and that applying additional boundary layers would undoubtedly result in an undesired increase of the computational time. Accordingly, 6 boundary layered elements should be sufficient across the first 3δ depth. The total number of elements used in the model can vary between 400,000 to 800,000, approximately, depending on the number of boundary layers defined in the sample. Solving such a model for one scanning position in a direct frequency-domain study of Comsol, takes from 15 to 40 min when using a desktop personal computer (PC) configured with an Intel© Core i7 processor with base frequency of 2.60 GHz, and a double data rate 3 (DDR3)—32 Gigabytes (GB) of random access memory (RAM).

The optimized simulation parameters are applied to the model, as concluded from the previous sensitivity studies. Subsequently, the estimated values for the resistive and the inductive parts of the impedance are superimposed on Burke’s measurements in Fig. 7a and b, respectively. Although Fig. 7 shows a quite good fit with experimental data, a small discrepancy between the calculated and the measured impedance components is observed at each scan position. The computed inductive reactance is always at least 1.5% higher than the measured values, regardless of the parameters in use. A deeper investigation into the model indicates that the reference edge specified within the circular coil’s model, which determines the direction of current flow in a relatively thick-sectioned coil, can affect the computed coil’s impedance. Based on the instructions provided by Comsol for modelling 3-D coils [32], choosing a closed-loop running through the middle of the circular coil’s thickness is expected to provide the best-computed impedance values. In an attempt to improve the accuracy of the model, the effect of the position of the reference edge would thus require further investigation. Fortunately, this issue is not a concern in the modelling of split-D probes, where the coils’ cross-sections are far thinner (i.e., one or two layers of AWG 42 wires). Additionally, they are modelled through numeric multi-turn coil domains instead of circular multi-turn coil domains. In such a modelling tool, there is no need to specify reference edges since the direction of the current is determined by defining an additional step within the solver.

The effective probe footprint may vary depending on the probe lift-off, the operating frequency, the probe shielding, the coils' geometry, and their configuration. Accordingly, the truncation sizes of the air and aluminum block domains, as deduced in Sect. 3, are no longer applicable to the case of the split-D probe since the operating frequency is significantly higher and the probe geometry much smaller. However, the air and aluminum block domains are still truncated following the same strategy based on the magnitude of the magnetic vector potential in order to reduce the computation time. Figure 9 illustrates the contour map of the magnetic field potential component perpendicular to the symmetry plane at 500 kHz along with the model’s initial mesh distribution. The outermost contour shows the region at which the amplitude of vector potential reduces to approximately one-thousandth of its maximum value. Consequently, the air domain can be truncated up to twice the size of the shielding diameter as the vector potential field is concentrated inside the shielding.

Following the methodology used in Sect. 3.3, the maximum element size in the aluminum block domain to driver coil thickness ratio (i.e. S/T ratio) is changed in 4 steps while the element size of other domains is scaled proportionally. The thickness of the driver coil for the split-D probe is 0.32 mm considering the wire diameter and its stacking order in the coil. Worth to mention that an air conductivity value of 1 S/m is used for these simulations and the standard penetration depth in aluminum block is 0.16 S/m.

Figures 10 and 11 show the variations of the real and imaginary parts of the normalized probe’s impedance scanning over the 0.188 mm and the 1.008 mm deep notches, respectively. These figures are plotted for 4 different S/T values. The same normalization scheme used in Sect. 3 is applied here however, the curves are normalized by the impedance measurements of the notches using the split-D probe. The figures show that imaginary and real parts of the computed impedance converge as the ratio of 1.87 is used in the model. In order to leave some room for higher resolution results, the ratio of 1.25 will be used for this probe model in the following simulations. Given the ratio selected here, the model consists of approximately 900,000 s order elements. It takes the solver about 45 min to process each scan step of the simulation using the PC configurations mentioned earlier.

The response of the split-D probe is here investigated numerically and experimentally as it scans three EDM notches of different depths in an aluminum block. Imaginary and real parts of the differential probe impedance are measured and numerically calculated. These values are plotted together, on an impedance plane, in Fig. 12 for 1.008 mm, 0.503 mm, and 0.188 mm deep notches. All the three impedance planes of Fig. 12 are shown with the same scale to ease comparison between the given signals. Again, only a single loop of the 8-shaped signal is shown because of the symmetry of the model.

Referring to Figs. 13 it is observed that the shape of simulated and measured signals is in good agreement for all the notches. The most significant discrepancy appears in the imaginary component of the impedance. From Fig. 10 (the shallowest notch), it could be seen that the imaginary part is underestimated by 6% at a probe position of about 0.5 mm. This discrepancy in the imaginary part of the probe impedance is reflected in the signal peak of the simulated complex impedance loops in Fig. 12c. Moreover, referring to Fig. 11 (the deepest notch) the signal is underestimated by 4% at the scan position of 0.8 mm. This deviation of simulation form measurement appears as a widening of the simulated signal in Fig. 12a. Otherwise, most of the simulated impedance values are contained within the measurement error. To derive the error, each of these notches is scanned 5 times using the split-D probe and the impedance variations in each scan are recorded. Afterward, the mean value and the standard deviation (\(\sigma\)) of the 5 impedance measurements are calculated for each notch. Accordingly, the measurement error is presented by \(\pm \sigma\) at each probe’s position.

Discrepancies between simulated and measured impedances are believed to be related to the deviation of the geometry of the manufactured notches from the ideal simulated ones (rectangular slot). In an EDM process, thin electrodes are used to erode narrow surface notches. As the notch gets deeper, the electrode’s lateral faces may further remove metal from the notch walls. Consequently, the resulting notch is wider in the vicinity of its opening than at its tip. This nonuniformity grows by increasing the nominal depth of a notch. Correspondingly, referring to Fig. 12, we find that the largest discrepancy in the width of the complex impedance loops is occurring with the deepest notch. Dimensions and properties of the probe components used in the model could also contribute to the observed discrepancies. In addition, the sensitivity of the probe’s signal to geometrical imperfections grows significantly as the notch gets shallower. Hence, the largest discrepancy between simulations and measurements is observed for the shallowest notch.

The desired precision level for each feature of a signal, such as amplitude, phase, and shape, is strongly application oriented. For instance, in order to have a good estimation of the crack characteristics in inversion approaches, the crack signals shall be accurately reproduced using modelling. However, the importance of signal shape and phase is undoubtedly less in reliability studies (POD studies) where the signal amplitude is the primary influential factor [33, 34]. Nonetheless, results reported in this work confirm that Comsol model can reasonably predict probe impedance variations (in shape and amplitude) as it is scanned across a notch.

The use of such a model could also help to properly design a specific probe as the impact of design parameters on probe performances could be accurately simulated. Indeed, in addition to probe impedance, magnetic and electric vector fields could be displayed to understand the probe behavior. As an example of this, Fig. 13 shows the surface distribution, using the vector field representation, for the z and x components of \({\varvec{J}}_{in}\) (induced current density) in the vicinity of 0.188 mm deep notch for three different probe positions over this notch. Moreover, the surface distribution for the y and x components of \(H\)(magnetic field intensity) is depicted beneath these figures as well. As demonstrated, the surface current density is significantly perturbed at the scan positions of 0 mm and 0.6 mm. At the scan position of 0 mm, the perturbation is seen by both of the receiver coils equally. However, the notch and its corresponding perturbation zone are directly located underneath one of the receiver coils at the scan position of 0.6 mm. This results in the maximum amplitude of the differential impedance of the receiver coils for the shallowest notch.