## 3 Results and Discussion
### Numerical Validation of the Inverse-Laplace Approach  
As anticipated, FLARE also allows one to study transient phenomena through an inverse-Laplace approach. Specifically, a numerical inversion routine is used to call the FEM solver for a series of values of the (complex) variables \(s\) in (12) and (13). A complex MVP formulation similar to the one employed for time-harmonic problems is used to perform the finite element analysis in this case. The purely complex factor \(j\omega\) used in the time-harmonic solver is now substituted by \(s\), that may have a nonzero real part. The aim of this section is twofold. First, we want to show how this approach can be applied to efficiently solve the same transient problem considered in the last section. Second, we want to provide verification of the numerical accuracy granted by this inverse-Laplace approach. Without losing any generality, we now focus solely on \(i_{\text{pipe}}(t)\), for the sake of brevity. The most obvious way to obtain \(i_{\text{pipe}}(t)\) using the inverse-Laplace approach would be to Laplace-transform the known current on phase \(1\), \(i_{\text{phase 1}}(s)=\mathcal{L}\big{\{}i_{\text{phase 1}}(t)\big{\}}\), and subsequently use \(i_{\text{phase 1}}(s)\) as the forcing term for the FEM solver in the \(s\) domain. In this way, \(i_{\text{pipe}}(s)\) is obtained, and one can use one of the provided algorithms to perform the inverse transform \(\mathcal{L}^{-1}\big{\{}i_{\text{pipe}}(s)\big{\}}=i_{\text{pipe}}(t)\). Unfortunately, in this way, the calculation must be repeated for every change in \(i_{\text{phase 1}}(t)\). Due to this, we instead opt for using the above described procedure to inverse-transform the pipeline electrical response to the unit impulse (The unit step \(1/s\) can also be used alternatively, knowing that the unit impulse is just the first-derivative of the unit step). In this case, the forcing term for the FEM solver is just \(\mathcal{L}\{\delta\}=1\) and the output \(h_{\text{pipe}}(t)\) of the inverse-Laplace routine is the time-domain pipeline current when the phase conductor is subjected to a unit current impulse. Now the time response of the pipeline to the generic current \(g_{\text{phase 1}}(t)\) applied to phase conductor 1 can be obtained using the convolution theorem:  
\[i_{\text{pipe}}(t)=h_{\text{pipe}}(t)*g_{\text{phase 1}}(t)=\int_{0}^{t}h_{ \text{pipe}}(\tau)g_{\text{phase 1}}(t-\tau)\;d\tau. \tag{21}\]  
The short notation \(*\) is used to identify the convolution integral between two generic time-domain functions from here onward. Three different numerical Laplace inversion algorithms are currently implemented in FLARE. The adaptive approach by [PERSON] and colleagues [34] is used to compute the time-domain impulse response of the system, i.e., \(h_{\text{pipe}}(t)\). We emphasize that \(h_{\text{pipe}}(t)\) (output of the inversion routine) is the pipeline current response when phase conductor 1 is excited by a unit current impulse. Then, by setting \(g_{\text{phase 1}}(t)=i_{\text{phase 1}}(t)\) with the expression of (20), we can find the time-domain pipeline current as a simple convolution. As anticipated, the computational advantage of this approach is that, once \(h_{\text{pipe}}(t)\) has been computed for the given geometry, one can easily find the pipeline response to any given current on phase 1 by updating \(g_{\text{phase 1}}(t)\). More details on the convolution routine are provided in Section 3.6. The default MATLAB(r) function w=conv(u,v) assumes that the two arrays u and v are defined at the same time-instants, which must necessarily be regularly spaced. In other words, the conv function is not particularly practical for working with variables known at unevenly spaced time-instants, such as \(h_{\text{pipe}}(t)\) (computing the impulse response over an evenly spaced time grid would result in a prohibitive number of time steps, even with an implicit time discretization such as the one used in this work).