Commit 3dfed95a authored by snuverink_j's avatar snuverink_j
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replace engordnumbers

parent 18d5cf22
......@@ -1557,7 +1557,7 @@ transverse dependence on electric field.
\index{RFCavity!Time Dependence}
\index{Cavity!Time Dependence}
The \texttt{POLYNOMIAL\_TIME\_DEPENDENCE} element is used to define time dependent
parameters in RF cavities in terms of a \engordnumber{4} order polynomial.
parameters in RF cavities in terms of a fourth order polynomial.
\begin{kdescription}
\item[P0]
Constant term in the polynomial expansion.
......
......@@ -126,7 +126,7 @@ Field maps in \textit{OPAL-t} come in three basic types:
\item 2D or 3D field map. For this type of map, a field is specified on a grid and linear interpolation is used to find field
values at intermediate points.
\item 1D on axis field map. For this type of map, one on-axis field profile is specified. \textit{OPAL-t} calculates a Fourier series
from this profile and then uses the \engordnumber{1}, \engordnumber{2} and \engordnumber{3} derivatives of the series to
from this profile and then uses the first, second and third derivatives of the series to
compute the off-axis field values. (This type of field is very smooth numerically, but can be inaccurate far from the
field axis.) Only a few (user specified) terms from the Fourier series are used.
\item Enge function \ref{enge} field map. This type of field map uses Enge functions to describe the fringe fields of
......@@ -385,8 +385,8 @@ AstraMagnetostatic 40
\caption[Example of an ASTRA compatible magnetostatic field map]{A 1D field map describing a magnetostatic field using $N$
non-equidistant grid points in the longitudinal direction. From these values $N$ equidistant field values are computed from
which in turn $N/2$ complex Fourier coefficients are calculated. In this example only 40 Fourier coefficients are kept
when calculating field values during a simulation. The z-position of each field sampling is in the \engordnumber{1}
column (in meters), the corresponding longitudinal on-axis magnetic field amplitude is in the \engordnumber{2} column.
when calculating field values during a simulation. The z-position of each field sampling is in the first
column (in meters), the corresponding longitudinal on-axis magnetic field amplitude is in the second column.
As with the 1DMagnetoStatic see~Section~\ref{1DMagnetoStatic} field maps, \textit{OPAL-t} normalizes the field values to
$\max(|B_{\text{on axis}}|) = {1.0}{T}$. In the header only the first line is needed since the information on the
longitudinal dimension is contained in the first column of the data. (\textit{OPAL-t} does not provide a \texttt{FAST} version of
......@@ -527,8 +527,8 @@ AstraDynamic 40
non-equidistant grid points in longitudinal direction. From these $N$ non-equidistant field values $N$ equidistant
field values are computed from which in turn $N/2$ complex Fourier coefficients are calculated. In this example
only 40 Fourier coefficients are kept when calculating field values during the simulation. The z-position of each
sampling is in the \engordnumber{1} column (in meters), the corresponding longitudinal on-axis electric field amplitude
is in the \engordnumber{2} column. \textit{OPAL-t} normalizes the field values such that $\max(|E_{\text{on axis}}|) = {1}{MV/m}$.
sampling is in the first column (in meters), the corresponding longitudinal on-axis electric field amplitude
is in the second column. \textit{OPAL-t} normalizes the field values such that $\max(|E_{\text{on axis}}|) = {1}{MV/m}$.
The frequency of this field is {2997.924}{MHz}. (\textit{OPAL-t} does not provide a \texttt{FAST} version of this map type.)
}
\label{fig:AstraDynamic}
......
......@@ -40,7 +40,7 @@ According to the number of particles defined by the argument \texttt{npart} in \
The number in the first line is the total number of particles.
In the second line the data represents $x, p_x, y,$$ p_y, z, p_z$ in the local reference frame. Their units are described in Section~\ref{variablesopalcycl}.
Please don't try to run this mode in parallel environment. You should believe that a single processor of the \engordnumber{21} century is capable of doing
Please don't try to run this mode in parallel environment. You should believe that a single processor of the 21^{st} century is capable of doing
the single particle tracking.
\subsection{Tune Calculation mode}
......@@ -164,7 +164,7 @@ There are two possible situations. One is the real field map on median plane of
Limited by the narrow gaps of magnets, in most cases with cyclotrons, only vertical field $B_z$ on the median plane ($z=0$) is measured.
Since the magnetic field data off the median plane field components is necessary for those particles with $z \neq 0$, the field need to be expanded in $Z$ direction.
According to the approach given by Gordon and Taivassalo, by using a magnetic potential and measured $B_z$ on the median plane,
at the point $(r,\theta, z)$ in cylindrical polar coordinates, the \engordnumber{3} order field can be written as
at the point $(r,\theta, z)$ in cylindrical polar coordinates, the third order field can be written as
\begin{eqnarray}\label{eq:Bfield}
B_r(r,\theta, z) & = & z\diffp{B_z}{ r}-\frac{1}{6}z^3 C_r, \\
B_\theta(r,\theta, z) & = & \frac{z}{r}\diffp{B_z}{\theta}-\frac{1}{6}\frac{z^3}{r} C_{\theta}, \\
......@@ -403,7 +403,7 @@ Please note that in this case, the E field is treated as a part of \texttt{CYCLO
\section{Particle Tracking and Acceleration}
The precision of the tracking methods is vital for the entire simulation process, especially for long distance tracking jobs.
\textit{OPAL-cycl} uses a \engordnumber{4} order Runge-Kutta algorithm and the \engordnumber{2} order Leap-Frog scheme. The \engordnumber{4} order Runge-Kutta algorithm needs four external magnetic field evaluation in each time step $\tau$ .
\textit{OPAL-cycl} uses a fourth order Runge-Kutta algorithm and the second order Leap-Frog scheme. The fourth order Runge-Kutta algorithm needs four external magnetic field evaluation in each time step $\tau$ .
During the field interpolation process, for an arbitrary given point the code first interpolates Formula $B_z$
for its counterpart on the median plane and then expands to this give point using Equation~\ref{Bfield}.
......@@ -419,7 +419,7 @@ and then tracks it for $t_2 = \tau - t_1$.
\textit{OPAL-cycl} uses the same solvers as \textit{OPAL-t} to calculate the space charge effects see~Chapter~\ref{fieldsolver}.
Typically, the space charge field is calculated once per time step. This is no surprise for the \engordnumber{2} order Boris-Buneman time integrator (leapfrog-like scheme) which has per default only one force evaluation per step. The \engordnumber{4} order Runge-Kutta integrator keeps the space charge field constant for one step, although there are four external field evaluations. There is an experimental multiple-time-stepping (MTS) variant of the Boris-Buneman/leapfrog-method, which evaluates space charge only every N\engordletters{th} step, thus greatly reducing computation time while usually being still accurate enough.
Typically, the space charge field is calculated once per time step. This is no surprise for the second order Boris-Buneman time integrator (leapfrog-like scheme) which has per default only one force evaluation per step. The fourth order Runge-Kutta integrator keeps the space charge field constant for one step, although there are four external field evaluations. There is an experimental multiple-time-stepping (MTS) variant of the Boris-Buneman/leapfrog-method, which evaluates space charge only every N^{th} step, thus greatly reducing computation time while usually being still accurate enough.
% -- -- -- -- -- -- Section -- -- -- -- -- --
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