T3P is a 3D parallel finite-element time-domain solver to         calculate transient field response of a electromagnetic         structure to imposed fields, and dipole or beam         excitations.

In our approach, Ampere's and Faraday's laws are combined and integrated over time to yield inhomogeneous vector wave equation for the time integral of the electric field *E*:

{latex}\begin{eqnarray*}\left( \epsilon \frac{\partial^2}{\partial t^2} + \sigma_{eff} \frac{\partial}{\partial t} + \nabla \times \mu^{-1}\nabla\times \right) \int^t \emph{E} d \tau = -\emph{J}
\end{eqnarray*}
{latex}\\

with permittivity {latex} $\epsilon = \epsilon_0 \epsilon_r ${latex} and permeability {latex}
$\mu = \mu_0 \mu_r$
{latex} . In the current implementation, a constant value of the effective conductivity {latex}
$\sigma_{eff} = \tan \delta \cdot 2 \pi f \cdot \epsilon $

{latex} is assumed by fixing a frequency{latex}
 $f$ {latex} , and the losses are specified by the loss tangent {latex} $\tan \delta $ {latex} . As is common for Wakefield computations of rigid beams, the electric current source density J is given by a one-dimensional Gaussian particle distribution, moving at the speed of light along the beam line.

The computational domain is discretized into curved tetrahedral elements and {latex} $\int^t \emph{E}d \tau ${latex} in Eq. (1) is represented as an expansion in hierarchical Whitney vector basis functions {latex}$\emph{N}_i \(x) ${latex}

{latex}

\begin{eqnarray*} \int^t \emph{E} (\emph{x}, \tau) d\tau = \sigma^{N_p}_{i=1} e_i(t) \cdot \emph{N}_i \(x) \end{eqnarray*}

{latex}