## 2 Theoretical Analysis and Improvement Research on Finite Element Models
### Theory of Finite Element Analysis for Electromagnetic Fields  
[PERSON]'s equations are crucial for understanding and analyzing the issues of electromagnetic wave propagation in well bores. When studying the short transmission problems of downhole electromagnetic waves, a helical loop antenna is excited by a current source. In this context, the electromagnetic field is a time-harmonic field, and the instantaneous field vectors are differentiated with respect to time. The [PERSON]'s equations can be expressed in the following form:  
\[\
abla\cdot\overset{\rightarrow}{D}=\rho_{0} \tag{1}\]  
\[\
abla\cdot\overset{\rightarrow}{B}=0 \tag{2}\]  
\[\
abla\times\overset{\rightarrow}{E}=-j\omega\mu\overset{\rightarrow}{H} \tag{3}\]  
\[\
abla\times\overset{\rightarrow}{H}=\overset{\rightarrow}{j}_{0}+j\omega \overset{\rightarrow}{D}m \tag{4}\]  
where \(\overset{\rightarrow}{E}\) is the electric field intensity, \(\overset{\rightarrow}{B}\) is the magnetic flux density, \(\overset{\rightarrow}{H}\) is the magnetic field strength, \(\overset{\rightarrow}{D}\) is the electric displacement field, \(\overset{\rightarrow}{j}_{0}\) denotes the conduction current density, \(\omega\) is the angular frequency, \(\mu\) is the magnetic permeability, and \(\rho_{0}\) represents the electric charge distribution in space. By substituting Equation (3) into Equation (4), the wave equation for a time-harmonic field can be obtained, which serves as the governing equation for finite element analysis:  
\[\
abla\times\left(\frac{1}{\mu_{r}}\
abla\times\overset{\rightarrow}{E} \right)-k_{0}^{2}\varepsilon_{r}\cdot\overset{\rightarrow}{E}=-j\omega\mu_{0} \overset{\rightarrow}{J} \tag{5}\]  
where \(k_{0}\) is the wave function in free space, and \(\overset{\rightarrow}{J}\) is the external current density. It is clear that for analyzing low-frequency electromagnetic field problems, obtaining the current density in the loop and the transmission potential is sufficient. The ideal boundary conditions for the drill string are set as follows:  
\[\overset{\rightarrow}{n}\times\overset{\rightarrow}{E}=0 \tag{6}\]  
where \(\overset{\rightarrow}{n}\) is the outward unit normal vector to the boundary. Since the drilling fluid and formation are not ideal conductors, the boundary conditions at the interface are set as follows:  
\[\overset{\rightarrow}{n}\times\frac{1}{\mu}\bigg{(}\
abla\times\overset{ \rightarrow}{E}\bigg{)}=j\sqrt{-\mu(2\pi f\sigma+4j\pi^{2}f^{2}\varepsilon)} \lambda\overset{\rightarrow}{E} \tag{7}\]  
The boundary conditions at the location of the transmitting antenna are as follows:  
\[\overset{\rightarrow}{n}\times\frac{1}{\mu}\bigg{(}\
abla\times\overset{ \rightarrow}{E}\bigg{)}=-2\pi jf\overset{\rightarrow}{J}_{s} \tag{8}\]  
where \(\overset{\rightarrow}{J}_{s}\) represents the imposed current density. When using the finite element method to simulate electromagnetic field problems, it is common to model domains with open boundaries, where the electromagnetic waves pass through the boundaries without reflection. To accomplish this, first-order scattering boundary conditions are set up:  
\[\overset{\rightarrow}{n}\cdot(\
abla\overset{\rightarrow}{E})+ik_{0} \overset{\rightarrow}{E}=0 \tag{9}\]  
By incorporating the aforementioned boundary conditions into Equation (4) and based on the variational principle, the functional expression can be derived as follows:  
\[F(\overset{\rightarrow}{E})=\frac{1}{2}\underset{\underset{V}{\iint}}{ \iint}\bigg{[}\frac{1}{\mu_{r}}\bigg{(}\
abla\times\overset{\rightarrow}{E} \bigg{)}\cdot\bigg{(}\
abla\times\overset{\rightarrow}{E}\bigg{)}-k_{0}^{2} \varepsilon_{r}\cdot\overset{\rightarrow}{E}\bigg{]}dV+j\omega\mu_{0} \underset{\iint}{\iint}\overset{\rightarrow}{E}\cdot\overset{\rightarrow} {J}dV \tag{10}\]  
When solving a functional, it is necessary to discretize the domain of interest into a number of subspaces. For each element, field quantity solution expressions are derived using shape functions within the local coordinate system.