# Simulation of Real Defect Geometry and Its Detection Using Passive Magnetic Inspection (PMI) Method
## 3 Simulations and Results  
After converting the rebar mesh to solid in COMSOLR software, the magnetic behavior simulation was undertaken. Considering the variations of Earth's magnetic field in time and location, to obtain consistent and realistic results the average (within a year) of the different components of the magnetic field for the Waterloo, Ontario region (the location of the experiments) was adopted for the simulations (Table 2). Moreover, since the unitless relative magnetic permeability of low-carbon steels (ASTM 1020) range from 50 to 100 [57; 58], a relative magnetic permeability of 75 was selected for this study. The duration of exposure to an external magnetic field will affect the magnetic behavior of ferromagnetic materials. In reality, ferromagnetic materials are affected by the magnetic field of the Earth from the beginning of their production process. There may also be some unknown external magnetic sources in the surrounding environment that affect the magnetic behavior of ferromagnetic objects [59]. However, as accurately as possible, we can apply the magnetic field of the Earth to the object and simulate its magnetic behavior, although some divergence will exist between the simulation and the experimental results. To consider the Earth's magnetic field in the simulation, the rebar was located in a regular space (Figure 2) with dimensions of 100 mm \(\times\) 150 mm \(\times\) 410 mm, which included the magnetic field presented in Table 2 and Figure 3. To have better control of simulation parameters, the box and rebar were meshed separately with tetrahedral meshes according to the specifications in Table 3 (Figure 4a,b). Then, the rebar and box were jointly subjected to the simulation process as a single system (Figure 4c). The values of the different components (X, Y, and Z) of the magnetic flux densities were recorded for the Y direction of the rebar (i.e., the path parallel to the rebar's length). This path is at the surface of the rebar, and extends from one side (Edge A) to the other side of the box (Edge B) (Figure 5). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hole Name** & **Diameter (mm)** & **Depth (mm)** & **Y-Location from the Rebar’s Start Point (mm)** \\ \hline Hole 1 & 0.58 & 1.24 & 57.91 \\ Hole 2 & 0.68 & 0.57 & 282.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the two holes in the rebar. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Background Magnetic Field** & **Background Magnetic Field** & **Background Magnetic Field** \\ (**X-Component)** & **(Y-Component)** & **(Z-Component)** \\ \hline
18 \(\mu\)T & \(-3\)\(\mu\)T & 50 \(\mu\)T \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Background magnetic field (magnetic field of the Earth): from August 2016 to August 2017 (Adapted from Natural Resources Canada ([[http://www.nrcan.gc.ca](http://www.nrcan.gc.ca)]([http://www.nrcan.gc.ca](http://www.nrcan.gc.ca)))). Figure 2: Solid rebar located in a box. As observed in Figure 6, at first, the values of all of the components of magnetic flux densities are equal to the background magnetic flux (the magnetic field of the Earth). When the Y distance reaches  
Figure 4: Initial meshes of the system: (**a**) rebar mesh with its initial sizes; (**b**) box mesh with its initial sizes; (**c**) rebar and box meshes as a single system (front face of the box is removed for better visualization). Figure 3: Box used in analysis; arrows show the resultant vector for X, Y, and Z components of Earth’s magnetic field. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Section Name** & **Rebar** & **Box** \\ \hline Maximum element size (mm) & 2 & 8 \\ Minimum element size (mm) & 1 & 4.1 \\ Maximum element growth rate & 1.45 & 1.45 \\ Curvature factor & 0.5 & 0.5 \\ Resolution of narrow regions & 0.6 & 0.6 \\ Number of degrees of freedom (in total) & 601,773 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mesh specifications of rebar and box in the initial simulation.