...
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 $ |
Latex |
---|
$ \emph{N}_i (x) $ |
Latex |
---|
\begin{eqnarray*} \int^t \emph{E} (\emph{x}, \tau) d \tau = \sum^{N_p}_{i=1} e_i (t) \cdot \emph{N}_i (x) \end{eqnarray*}
|
up to order
Latex |
---|
$p$ |
Latex |
---|
$N_1 = 6 $ |
Latex |
---|
$N_2 = 20 $ |
Latex |
---|
$N_6 = 216 $ |