## 4 Conclusions  
In this work, we introduced FLARE, an open-source MATLAB(r) code developed by the authors for the finite element analysis of electromagnetic interference on earth-return conductors due to the presence of high-voltage transmission lines. We considered a single section of a corridor with a buried steel pipeline underneath a typical 500 kV transmission line and computed the induced current on the pipeline under both sinusoidal steady-state and transient operating conditions of the HVAC line. We described an efficient vectorized implementation of a stationary complex solver and a transient solver with automatic time-stepping in FLARE and verified both of the obtained solutions by implementing the same problem in COMSOL Multiphysics(r), using the AC/DC Module. The results yielded by FLARE and COMSOL Multiphysics(r) were compared in two different cases, i.e., a time-harmonic and a transient simulation. For the 60 Hz time-harmonic problem, the maximum percentage difference between the current magnitudes yielded by the two codes was \(\approx\)0.1% (pipeline). The maximum percentage phase difference was \(\approx\)0.21% (OGW 2). Considering the transient problem, we compared the percentage difference between the two maximum values of pipeline current, which was about \(\approx\)0.41%. Regarding the computation time, FLARE was faster by a factor of \(\approx\)1.4 for the considered time-harmonic problem, where the linear system stemming from the finite element discretization was only solved once. Conversely, COMSOL Multiphysics(r) was faster by a factor of \(\approx\)2 in the transient problem (note that a higher-order backward difference formula is employed by the COMSOL Multiphysics(r) time-marching solver, limiting the number of time steps employed). Subsequently, we discussed an alternative technique for the analysis of transient problems, based on the Laplace transform. This entails solving the problem in the complex domain for multiple values of the complex variable \(s\) and performing a numerical inverse-Laplace transformation to obtain the time-domain solution. We demonstrated this approach by anti-transforming the pipeline response to a current impulse applied to one of the transmission line phase conductors. The transient response of \begin{table}
\begin{tabular}{c c c c c c} \hline
**Algorithm** & **Settings** & & & **Time (s)** & \\ \hline \hline  & & & \(\tau_{1}=2.64\times 10^{-9}\) & \(\tau_{2}=4.55\times 10^{-5}\) & \(\tau_{3}=0.19\) \\  & & & & pipeline current (A) & \\ \hline [PERSON] et al. & Relative tolerance & \(1.00\times 10^{-5}\) & \(-9.419\,87\times 10^{2}\) & \(-7.274\,06\times 10^{4}\) & \(3.076\,32\times 10^{3}\) \\ \hline  & M & n & \multicolumn{2}{c}{\% Error with respect to [PERSON] et al.} \\ \hline  & 4 & 4 & \(1.60\times 10^{-2}\) & \(3.87\times 10^{-2}\) & \(1.23\times 10^{1}\) \\  & 8 & 8 & \(1.45\times 10^{-5}\) & \(3.75\times 10^{-6}\) & \(7.59\times 10^{-3}\) \\ Talbot & 16 & 16 & \(4.18\times 10^{-11}\) & \(1.10\times 10^{-8}\) & \(1.69\times 10^{-4}\) \\  & 32 & 32 & \(3.33\times 10^{-12}\) & \(1.29\times 10^{-7}\) & \(2.50\times 10^{-3}\) \\  & 64 & 64 & \(1.06\times 10^{-6}\) & \(1.77\times 10^{-2}\) & \(3.26\times 10^{2}\) \\ \hline  & 4 & 9 & \(6.05\times 10^{-1}\) & \(2.67\times 10^{-1}\) & \(1.26\) \\  & 8 & 17 & \(1.27\times 10^{-3}\) & \(6.54\times 10^{-4}\) & \(5.12\times 10^{-3}\) \\ [PERSON] & 16 & 33 & \(5.64\times 10^{-9}\) & \(2.89\times 10^{-8}\) & \(4.21\times 10^{-4}\) \\  & 32 & 65 & \(9.50\times 10^{-7}\) & \(5.89\times 10^{-4}\) & \(1.15\times 10^{2}\) \\  & 64 & 129 & \(3.93\times 10^{2}\) & \(5.76\times 10^{7}\) & \(7.57\times 10^{11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Percentage error of the Talbot and Euler inverse-Laplace algorithms with respect to the pipeline current computed with the algorithm by [PERSON] et al. for three time instants. \(M\) is the integer parameter used to control the Talbot and Euler algorithms.