## 3 Sensitivity Optimization
### Finite Element Modelling
#### 3.1.1 Magneto-Static Modelling  
The magnetization level in the sample is a significant parameter to achieve higher sensitivity. While under magnetization may lead to a missed defect signal, oversaturation not only increases the size of the sensor but also the background noise. A 3-D magnetization yoke was modeled in the magnetic fields with no currents module to optimize the size parameters without compromising the sensitivity. The magnetization fields of the permanent magnets in a medium are computed by evaluating the magnetic scalar potential (\(\
abla Vm\)).  
\[H=-\
abla Vm \tag{9}\]  
The constitutive relation for solving the problem in a particular medium is given by Equation (1), i.e., the B-H curve of the mild steel. The BH curve of Carbon Steel 1002 from the material library was used to calculate the magnetic flux density for points equidistant located at 1 mm in the test sample. The material properties defined for different components in the model are listed in Table 1.  
A uniform corrosion of 1 mm up to 4 mm with decrement steps of 1 mm was simulated in the model, and their relative magnetic permeability values were evaluated from the BH curve.  
A mesh convergence study was performed by comparing flux at a fixed point for different mesh sizes. An adaptive meshing technique was used, and mesh parameters were refined until no difference in flux was observed after further mesh refinement. Figure 3 shows the point at which the flux was calculated to carry out the mesh convergence, and the graph between the maximum element size and error between the successive flux computed for the respective mesh sizes. The mesh parameters defined in the model are listed in Table 2.  
It was found that a minimum flux density of 1.4 T in the sample can give good sensitivity using a probe size of 40 \(\times\) 65 \(\times\) 40 mm. The dimensions of the probe are illustrated in Figure 4a. The FE model along with the computed flux densities and relative permeability are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Material** & **Properties** & \\ \hline \multirow{3}{*}{Magnets} & \multirow{3}{*}{Neodymium} & Magnetization: & 835 kA/m \\  & & Relative Permeability: & 1.05 \\  & & Conductivity & 1 \(\times\) 10\({}^{6}\) S/m \\ \hline \multirow{2}{*}{Iron Bar} & \multirow{2}{*}{Cast Iron} & Relative Permeability: & 4000 \\  & & Electrical Conductivity & 11.2 \(\times\) 10\({}^{6}\) S/m \\ \hline \multirow{2}{*}{Plate Sample} & \multirow{2}{*}{Low-carbon steel 1002} & Relative Permeability: & BH curve \\  & & Electrical Conductivity & 8.41 \(\times\) 10\({}^{6}\) S/m \\ \hline \hline \end{tabular}
\end{table}
Table 1: The material properties defined in the FE model.  
Figure 3: (a)The location of the point at which the magnetic flux was computed for mesh convergence study, and (b) the graph showing the decrease in the error between the flux values with mesh refinement.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Element Size Parameters** & **Values** \\ \hline Maximum Element Size & 3 \\ \hline Minimum Element Size & 0.05 \\ \hline Maximum Element growth rate & 1.3 \\ \hline Resolution of narrow regions & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The mesh parameters obtained from mesh convergence study.