## 4 Numerical Simulation and Experimental Work
### Numerical Simulation
Providing the environment and necessary system parameters the mesh is generated in the model as shown in Figure 4b. Figure 4c shows the solved back-to-back triangular capacitive sensor model for a sound specimen. The contour lines on top of the specimen represent electric potential values. In the coordinate system illustrated in Figure 5, based on which the results will be shown henceforth, the surface of the electrodes is centred at (\(y=0\) or \(z=0\)). \(Y=0\) plane (or \(xz\) plane) is the cross-sectional plane along the long axis of symmetry of the capacitive sensor, as shown in Figure 5a, and \(z=0\) plane (or \(xy\) plane) is the parallel plane under the probe surface at the sample surface, as shown in Figure 5b. To clarify the results, the plane coordinate systems for the two types of cross-sections are presented in Figure 5c. Figure 4: 3D model of the finite element modeling: (**a**) the computational domain (120 mm \(\times\) 120 mm \(\times\) 120 mm) with a coplanar capacitive probe, (**b**) model after finite element mesh generation, and (**c**) The electric potential contour lines on the surface of the sound specimen. Figure 6 shows a sample of the electric field distribution prediction of the 3D finite element models for the coplanar capacitive electrodes and how these fields behave under different conditions. This figure is a contour plot of the electric field in the \(y=0\) plane. Figure 6a shows this example for a sound zone of a specimen, Figure 6b shows a sample of output for a defective zone filled with air, and Figure 6c shows the electric field distribution for a specimen with a narrow crack on the surface. It can be seen that the presence of defects distorts the electric field within and around the regions of discontinuity in the sample and this leads to a variation in the detectable signal on the sensing electrode. This is because of the different dielectric properties such as permittivity (or dielectric constant) presumed for the sample and defects. Therefore, the model shows how the capacitive sensor detects the defects due to the different dielectric properties between the sample and defect. Figure 5: The cross-sectional plane along the long axis of symmetry of the capacitive probe: (**a**) \(y=0\) plane (\(xz\) plane), (**b**) \(z=0\) plane (\(xy\) plane), and (**c**) plane coordinate systems for the two kinds of cross sections. Simulations were conducted to determine the effect of various depths of subsurface defects on the electric field strength. The real nonconducting sample, a Plexiglas plate containing three subsurface air-filled cylindrical voids at different depths, was simulated. Figure 7 shows a surface plot of the electric field of the back-to-back triangular electrodes for the different defect depth in the \(z=0\) planes, including Figure 7a a sound sample, (b) a defect in 0.5 mm depth, (c) a defect in 1.0 mm depth, and (d) a defect in 1.5 mm depth. By comparing the results, it can be inferred the presence of the air-filled defect at different depths causes a different distortion of the electric field and this would lead to a detectable change in the output. This is caused by the different values of permittivity (or dielectric constant) assumed for the solid sample (Plexiglas, permittivity = 3.4) and air (permittivity = 1.0). Thus, the model shows how the capacitive probe detects subsurface defects due to a permittivity difference between the sample and defect. It is obvious from these results that the electric field strength decreases as the depth of the defect increases. In other words, the penetration depth and strength of the electric field vary for different depths of the defect. This is confirmed by the experimental result to be shown later in this work. Moreover, these figures show that different depths of a defect create a different electric field pattern which can be used to estimate the defect depth.