...
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
...
:
| Wiki Markup |
|---|
{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
| Wiki Markup |
|---|
{latex} $\epsilon = \epsilon_0 \epsilon_r ${latex} |
...
...
permeability
| Wiki Markup |
|---|
{latex}
$\mu = \mu_0 \mu_r$
{latex} |
...
...
In
...
the
...
current
...
implementation,
...
a
...
constant
...
value
...
of
...
the
...
effective
...
conductivity
| Wiki Markup |
|---|
{latex}
$\sigma_{eff} = \tan \delta \cdot 2 \pi f \cdot \epsilon $
{latex} |
...
...
assumed
...
by
...
fixing
...
a
...
frequency
| Wiki Markup |
|---|
{latex}
$f$ {latex} |
...
...
and
...
the
...
losses
...
are
...
specified
...
by
...
the
...
loss
...
tangent
| Wiki Markup |
|---|
{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
| Wiki Markup |
|---|
{latex} $\int^t \emph{E}d \tau ${latex} |
...
...
Eq.
...
(1)
...
is
...
represented
...
as
...
an
...
expansion
...
in
...
hierarchical
...
Whitney
...
vector
...
basis
...
functions
| Wiki Markup |
|---|
{latex}$ \emph{N}_i (x) ${latex} |
| Wiki Markup |
|---|
{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*} {latex} |
up
...
to
...
order
| Wiki Markup |
|---|
{latex} $p$ {latex} |
...
...
each
...
element.
...
For
...
illustration,
...
the
...
numbers
...
of
...
basis
...
functions
...
for
...
first,
...
second
...
and
...
sixth
...
order
...
are
| Wiki Markup |
|---|
{latex} $N_1 = 6 ${latex} |
...
| Wiki Markup |
|---|
{latex} $N_2 = 20 ${latex} |
...
| Wiki Markup |
|---|
{latex} $N_6 = 216 ${latex} |
...
respectively.
...
Higher
...
order
...
elements
...
(both
...
curved
...
and
...
with
...
higher-order
...
basis
...
functions)
...
not
...
only
...
significantly
...
improve
...
field
...
accuracy
...
and
...
dispersive
...
properties,
...
but
...
also
...
generally
...
lead
...
to
...
higher-order
...
accurate
...
particle-field
...
coupling
...
equivalent
...
to,
...
but
...
much
...
less
...
laborious
...
than,
...
complicated
...
higher-order
...
interpolation
...
schemes
...
commonly
...
found
...
in
...
finite-difference
...
methods.