...
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
...
the 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*}
|
...
with permittivity
| Latex |
|---|
$\epsilon = \epsilon_0 \epsilon_r $ |
...
...
permeability
...
| Latex |
|---|
...
$\mu = \mu_0 \mu_r$
|
...
...
In
...
the
...
current
...
implementation,
...
a
...
constant
...
value
...
of
...
the
...
effective
...
conductivity
...
| Latex |
|---|
...
$\sigma_{eff} = \tan \delta \cdot 2 \pi f \cdot \epsilon $
|
...
...
assumed
...
by
...
fixing
...
a
...
frequency
...
| Latex |
|---|
...
$f$ |
...
...
and
...
the
...
losses
...
are
...
specified
...
by
...
the
...
loss
...
tangent
...
| Latex |
|---|
...
$\tan \delta $ |
...
...
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 $ |
...
...
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^sum^{N_p}_{i=1} e_i (t) \cdot \emph{N}_i (x) \end{eqnarray*} {latex} |
up to order
| Latex |
|---|
$p$ |
| Latex |
|---|
$N_1 = 6 $ |
| Latex |
|---|
$N_2 = 20 $ |
| Latex |
|---|
$N_6 = 216 $ |