## 4 Multi-Physics Simulation Analyses of GMM-FBG Current Sensor
### Ferrite-GMM Magnetostrictive Field Analyses  
The magnetic field vectors of the ferrite-GMM assemblage and the magnetic induction intensity distribution on the longitudinal and lateral cross-sections of the GMM rod without current excitation and thermal strain (\(i=0\), \(T=273\) K) are shown in Figure 2. It is indicated that the residual magnetic moments of the ferrite magnetic material in a proper integrated structure with the GMM material can provide a sufficient magnetic field and effectively aggregate the magnetic flux to introduce a uniform magnetic field of 17 kA/m as an appropriate initial working point for the GMM-FBG current sensor into the GMM rod except for a small local area at the air gap and ferrite junction. In contrast to the dominant part of the GMM rod and the ferrite magnetizer, the magnetic induction intensity changes obviously in the region caved into ferrite at the rod bottom with a high-density magnetic flux being introduced from the cylinder side of the GMM rod through the locations near the surface of the ferrite magnetic pole, as shown in Figure 2a. The high magnetic induction intensity originates from the abrupt change in geometry at the circular intersection between the side of the GMM cylinder and the surface of the ferrite pole, which accounts for the major introduction of magnetic flux into the GMM rod. Despite the high magnetic induction intensity around this circular intersection in the connection of the GMM rod and ferrite magnetizer, the magnetic field vectors are axisymmetrically distributed with the cylinder axis of the GMM rod, as shown in top panel of Figure 2a, will not cause the lateral swing of \begin{table}
\begin{tabular}{c c c c c} \hline
**GMM** & **Young’s Modulus** & \begin{tabular}{c} **Thermal Expansion** \\ **Coefficient** \\ \end{tabular} & \begin{tabular}{c} **Easy-Axis Conversion** \\ **Temperature** \\ \end{tabular} & **Curie Temperature** \\ \hline \begin{tabular}{c} Symbol/unit \\ value \\ \end{tabular} & \(\chi\) & \(\lambda_{\mathrm{S}}\); \(\beta\)/K\({}^{-1}\) & \(\sigma_{\mathrm{s}}\) & \(\chi_{\mathrm{m}}\) \\ \end{tabular} & 0.5 & \(7.0\times 10^{-4}\); \(-6.5\times 10^{-6}\) & 4.2 \(\times\) 10\({}^{6}\) & 29 \\ \hline \begin{tabular}{c} Ferrite magnetizer \\ \end{tabular} & \begin{tabular}{c} Thermal expansion \\ coefficient \\ \end{tabular} & \
\begin{tabular}{c} Residual magnetic flux \\ density \\ \end{tabular} & Relative permeability \\ \hline
\begin{tabular}{c} Symbol/unit \\ value \\ \end{tabular} & \(\alpha\)/K\({}^{-1}\) & \(M_{\mathrm{r}}\)/T & \(\mu_{\mathrm{r}}\) \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._the GMM rod by the non-axisymmetric strains during current detection, and the strains generated by magnetostriction are negligible relative to the air-gap width, leading to a uniformly distributed magnetic field along the GMM rod axis. It is especially noted that the hemispherical surface at the tip of the GMM rod in contact with the air gap facilitates the magnetic flux aggregation in the GMM rod and avoids flux leakage at the air gap. Figure 2: (**a**) Magnetic field vector (top panel) in ferrite-GMM assemblage and magnetic induction intensity (bottom panel, contoured on GMM surface and axial longitudinal section) in GMM rod; (**b**) Magnetostriction (top panel) and magnetic flux density (bottom panel) versus magnetic field on the center of the GMM rod; (**c**) Longitudinal stretching component of strain tensor field in the GMM rod contoured on the longitude (**left**) and cross (**right**) sections. The changes in magnetic flux and magnetostriction of the GMM rod in the quasi-static process of slowly raising the direct current magnitude are simulated through continuously changing the excitation current magnitude in the ferrite-GMM center by setting the direct current magnitude as a variable with the parameter-scanning function of COMSOL Multiphysics software.