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diff --git a/doc/rotating_sym.tex b/doc/rotating_sym.tex new file mode 100644 index 0000000..e5ef6b7 --- /dev/null +++ b/doc/rotating_sym.tex @@ -0,0 +1,468 @@ +% /*@@ +% @file rotating_sym.tex +% @date 6 June 2002 +% @author Denis Pollney +% @desc +% Description of the implementation of `rotating' +% symmetry conditions in CartGrid3D. +% @enddesc +% @version $Header$ +% @@*/ + +\documentclass{article} + +\newif\ifpdf +\ifx\pdfoutput\undefined +\pdffalse % we are not running PDFLaTeX +\else +\pdfoutput=1 % we are running PDFLaTeX +\pdftrue +\fi + +\ifpdf +\usepackage[pdftex]{graphicx} +\else +\usepackage{graphicx} +\fi + +\parskip = 0 pt +\parindent = 0pt +\oddsidemargin = 0 cm +\textwidth = 16 cm +\topmargin = -1 cm +\textheight = 24 cm + +\begin{document} + +\title{Rotating symmetry conditions} +\author{Denis Pollney} +\date{June 2002} + +\maketitle + +\begin{abstract} +These notes describe the implentation of rotating symmetry conditions +for \emph{bitant} and \emph{quadrant} domains in \texttt{CartGrid3D}. +For these particular domain types, the condition that fields on the +grid are have a rotational symmetry in a plane along one of the +coordinate axes can be simply implemented with minor extensions to the +already existing symmetry mechanism, which copies the components of +a given field to the required ghost-zones with a possible plus-minus +inversion. +\end{abstract} + +%------------------------------------------------------------------------------ +\section{Introduction} +\label{sec:rs_intro} +%------------------------------------------------------------------------------ + +A number of useful physical models involve situations where a +rotational symmetry is present in all of the relevant fields. By +`rotational symmetry' we mean that there exists a pair of half-planes +extending from one of the coordinate axes and separated by an angle +$\theta$ which have the property that on $A$ and near to $A$ can be +mapped onto $B$ and corresponding points near to $B$ (see Figure +\ref{fig:rs_rotation_examples}). +% +\begin{figure} +\centering +\begin{tabular}{ccc} +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_general.eps} +\fi +& +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_bitant.eps} +\fi +& +\ifpdf +\else +\includegraphics[height=25mm]{fig/rotate_octant.eps} +\fi +\\ +(a) & (b) & (c) +\end{tabular} +\caption{Rotational symmetries, looking down the $z$-axis. In the +general case (a), the half-planes $A$ and $B$ are separated by an +angle $\theta$. Data at $A$ can be mapped on to points of $B$ via a +rotation through $\theta$. Particular cases of importance are the +rotation through $\theta=\pi$, so that data on the positive $y$-axis +is mapped onto the negative $y$, and vice versa; and the rotation +through $\theta=3\pi/2$ so that data need be specified only in a +single quarter-plane.} +\label{fig:rs_rotation_examples} +\end{figure} +% +In particular, situations for which such conditions can be useful +include +\begin{itemize} + \item rotating axisymmetric bodies -- satisfies the rotational + symmetry for arbitrary $\theta$. + \item pairs of massive bodies separated by distances $\pm + \mathbf{d}$ from the origin and with identical oppositely directed + momenta $\pm \mathbf{p}$ (Figure \ref{fig:rs_bbh}) -- satisfies + the half-plane rotational symmetry ($\theta=\pi$) or, if + $\mathbf{p}=0$ then also the quarter plane symmetry + ($\theta=3\pi/2$). +\end{itemize} +\begin{figure} +\centering +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_bbh.eps} +\fi +\caption{A physical system consisting of a pair of identical bodies + separated equal distance along a line through the origin, and moving + with equal but opposite momenta, can be modelled using only the + positive half-plane if the rotating symmetry condition is applied to + the $x=0$ plane.} +\label{fig:rs_bbh} +\end{figure} + +In order to apply the boundary condition exactly on a numerical grid, +it is necessary that grid points to each side of the mapping planes +$A$ and $B$ can be mapped onto each other exactly. In particular, this +means that the rotating symmetry conditions can be applied exactly if +the half-planes are each aligned with one of the coordinate axes. For +general planes $A$ and $B$, however, interpolation would be required +to put data in the neighbourhood of $A$ onto grid points near +$B$. Only the particular cases described by Figures +\ref{fig:rs_rotation_examples} (b) and (c) will be +considered here. + +%------------------------------------------------------------------------------ +\section{Symmetry conditions} +\label{sec:rs_application} +%------------------------------------------------------------------------------ + +For a given field, the boundary condition is applied as follows. For +each ghost-zone point along the symmetry plane, the corresponding +point on the physical grid (under the rotational $\theta$) is +determined. To determine the value of the ghost-zone point, the +value on the physical grid is simply transformed under the given +rotation.\\ + +The first of these issues is not difficult to resolve. As an example, +consider the half-plane rotational symmetry about the $z$-axis applied +in the $x=0$ plane, as depicted in Figure \ref{fig:rs_grid}. The has +$j=0\ldots m$ in the $y$ direction, an arbitrary number of points in the +$x$ and $z$ directions, and some ghostzones whose $j$ coordinates are +labelled $j=-1,-2,\ldots$. Then for the ghost-zone point $(n-i, -j, +k)$, the corresponding physical point under rotation is (1) $(i,j-1,k)$ +if the $x=0$ plane is on the grid, or (2) $(i,j,k)$ if the $x=0$ plane +is staggered between a grid point and the first ghost-zone +point. There is an implicit assumption here that the grid extends +exactly the same distance to each side of the rotation axis. \\ + +\begin{figure} +\centering +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_grid.eps} +\fi +\caption{The mapping of physical points onto ghost-zone points for a + half-plane rotation about the $z$-axis, where the $x=0$ plane is + staggered between gridpoints, and two ghost zones have been + allocated.} +\label{fig:rs_grid} +\end{figure} + +The remaining issue is to determine how the fields at a point +$(i,j,k)$ are modified under a rotation of a given angle. A set of +basis vectors $(\mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}})$ +can be rotated to an arbitrary direction by applying the rotation +matrix +\begin{equation} + \mathbf{P} = + \left[ + \begin{array}{ccc} + \eta_1(\cos\theta_1\cos\theta_3 - \sin\theta_1\cos\theta_2\sin\theta_3) & + \sin\theta_1\cos\theta_3 + \cos\theta_1\cos\theta_2\sin\theta_3 & + \sin\theta_2\sin\theta_3 \\ + -(\cos\theta_1\sin\theta_3 + \sin\theta_1\cos\theta_2\cos\theta3) & + \eta_2(\sin\theta_1\sin\theta_3 + \cos\theta_1\cos\theta_2\cos\theta_3) & + \sin\theta_2\cos\theta_3 \\ + \sin\theta_1\sin\theta_2 & + -\cos\theta_1\sin\theta_2 & + \eta_3\cos\theta_2 + \end{array} + \right], +\end{equation} +which preserves the orthogonality of the basis. The matrix +$\mathbf{P}$ has been written in terms of the Euler angles +$0\leq\theta_1<2\pi$, $0\leq\theta_2<2\pi$, and $0\leq\theta_3<\pi$, +and the factors $\eta_1=\pm 1$, $\eta_2=\pm 1$, $\eta_3=\pm 1$ are +introduced to allow for reflections of individual axes (ie. changes of +handedness of the basis). +The basis is transformed under $\mathbf{P}$ via the matrix multiplication +\begin{equation} + \left[ + \begin{array}{c} + \mathbf{\hat{x}^\prime} \\ + \mathbf{\hat{y}^\prime} \\ + \mathbf{\hat{z}^\prime} + \end{array} + \right] + = + \mathbf{P} + \left[ + \begin{array}{c} + \mathbf{\hat{x}} \\ + \mathbf{\hat{y}} \\ + \mathbf{\hat{z}} + \end{array} + \right]. +\end{equation} + +Vectors and tensor components are transformed under the given rotation +by applying the $\mathbf{P}$ matrix: +\begin{eqnarray} + \mathbf{v} & \rightarrow & \mathbf{P}^T \mathbf{v}, \\ + \mathbf{G} & \rightarrow & \mathbf{P}^T \mathbf{G} \mathbf{P}, +\end{eqnarray} +where $\mathbf{v}$ is a 3-vector, +\begin{equation} + \mathbf{v} = [v_x, v_y, v_z]^T, +\end{equation} +and $\mathbf{G}$ is a two-index tensor treated as a matrix, +\begin{equation} + \mathbf{G} = + \left[ + \begin{array}{ccc} + g_{xx} & g_{xy} & g_{xz} \\ + g_{yx} & g_{yy} & g_{yz} \\ + g_{zx} & g_{zy} & g_{zz} + \end{array} + \right]. +\end{equation} + +\emph{Example 1.} The special case of a reflection in the $x=0$ plane +corresponds to an inversion of the $x$-axis, so that +$\theta_1=\theta_2=\theta_3=0$ and $\eta_1=-1$, +\begin{equation} + \mathbf{P}_{x-\mathrm{reflect}} = \left[ + \begin{array}{ccc} + -1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{array} + \right]. +\end{equation} + +\emph{Example 2.} For a rotation about the $z$ axis by an angle of +$\theta=\pi$ (corresponding to Figure \ref{fig:rs_grid}), the only +non-zero rotation angle is $\theta_1=\pi$, so that +\begin{equation} + \mathbf{P}_{z-\mathrm{rotate}} = \left[ + \begin{array}{ccc} + -1 & 0 & 0 \\ + 0 & -1 & 0 \\ + 0 & 0 & 1 + \end{array} + \right]. +\end{equation} + +\emph{Example 3.} A quarter-plane grid with positive $y$-axis values +mapped onto the positive $x$-axis (as in Figure\nobreak~ +\ref{fig:rs_rotation_examples}) corresponds non-zero angle +$\theta_1=3\pi/2$, with the resulting rotation matrix +\begin{equation} + \mathbf{P}_{z-\mathrm{rotate(3/2)}} = \left[ + \begin{array}{ccc} + 0 & -1 & 0 \\ + 1 & 0 & 0 \\ + 0 & 0 & 1 + \end{array} + \right]. +\end{equation} + +%------------------------------------------------------------------------------ +\section{Implementation} +\label{sec:rs_implementation} +%------------------------------------------------------------------------------ + +In order to implement the bitant, quadrant and octant +\emph{reflection} symmetries which already exist in +\texttt{CartGrid3D}, it was necessary to attach to each grid function +information corresponding to how the field transforms under +reflections in each of the $x$, $y$, and $z$ axes. These are +specified as an array of three integers, each taking the value $+1$ or +$-1$, which is passed to the symmetry registration function. For +example, +\begin{verbatim} + static int one=1; + int sym[3]; + sym[0] = -one; + sym[1] = -one; + sym[2] = one; + SetCartSymVN(cctkGH, sym,"einstein::gxy"); +\end{verbatim} +specifies that the grid function \texttt{einstein::gxy} should be +negated under reflections in $x=0$ and $y=0$, but keeps the same value +under reflections in the $z=0$ plane.\\ + +This is the only information required for the reflection symmetry +since the value of the field at the new (reflected) point does not +require information from any other fields. For example, for a vector +$v = (v_x, v_y, v_z)^T$ reflected in $x=0$, we have +\begin{equation} + v_x \rightarrow -v_x, \qquad v_y \rightarrow v_y, \qquad + v_z \rightarrow v_z. +\end{equation} +That is, the transformation of $v_x$ only needs to know the values of +$v_x$, and does not need to know anything about the values of $v_y$ or +$v_z$. On the other hand, for the $3\pi/2$-rotation corresponding to +Example 3, above, the vector would transform as: +\begin{equation} + v_x \rightarrow -v_y, \qquad v_y \rightarrow v_x, \qquad + v_z \rightarrow v_z. +\end{equation} +In this case, to determine the rotated $v_x$ component it is necessary +to know the value of $v_y$. However, there is no concept of vector or +tensor within Cactus, so that given a grid function corresponding to +$v_x$, it is not generally possible to know which grid function +corresponds to $v_y$ which should be used to determine the rotated +values.\\ + +As a result, the only symmetries which can easily be implemented +within the current mechanism are those which require only the grid +function itself in order to determine the ghost zone points. This +includes the $\pi$-rotation symmetries (`half-plane', Figure +\ref{fig:rs_rotation_examples} (b)) but not the $3\pi/2$-rotation +symmetries (`quarter-plane', Figure \ref{fig:rs_rotation_examples} +(c)).\\ + +Table \ref{tbl:rs_xform} lists the transformations of the components +of an arbitrary scalar, vector and two-index tensor under reflections +in each plane, and rotations about each axis. +\begin{table} +\centering +\begin{tabular}{r|rrr|rrr}\hline\hline + & \multicolumn{3}{c|}{reflection in} + & \multicolumn{3}{c}{rotation about} \\ + & $x$ & $y$ & $z$ & $x$ & $y$ & $z$ \\ \hline +$\phi$ & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline +$v_x$ & -1 & 1 & 1 & 1 & -1 & -1 \\ +$v_y$ & 1 & -1 & 1 & -1 & 1 & -1 \\ +$v_z$ & 1 & 1 & -1 & -1 & -1 & 1 \\ \hline +$g_{xx}$ & 1 & 1 & 1 & 1 & 1 & 1 \\ +$g_{xy}$ & -1 & -1 & 1 & -1 & -1 & 1 \\ +$g_{xz}$ & -1 & 1 & -1 & -1 & 1 & -1 \\ +$g_{yy}$ & 1 & 1 & 1 & 1 & 1 & 1 \\ +$g_{yz}$ & 1 & -1 & -1 & 1 & -1 & -1 \\ +$g_{zz}$ & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline\hline +\end{tabular} +\caption{Transformation factors for reflection and half-plane rotation +symmetries.} +\label{tbl:rs_xform} +\end{table} +We note the following useful fact: The transformation factor $s_i$ for +a rotation about the axis $i$ is given by $s_j \times s_k$ where +$i\neq j\neq k$. For example, for a rotation about the $z$-axis, the +transformation factor for the is given by +\begin{equation} + s_x \times s_y = -1 \times -1 = 1. +\end{equation} +Intuitively, this is clear since the rotation of the axes by $\pi$ +radians about $z$ is equivalent to a reflection in $y$ followed by +a reflection in $x$ (Figure \ref{fig:rs_rotate_reflect}).\\ + +\begin{figure} +\centering +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_reflect.eps} +\fi +\caption{A rotation by $\pi$ radians about the $z$-axis is equivalent + to successive reflections about the $y$- and $x$-axes.} +\label{fig:rs_rotate_reflect} +\end{figure} + +Since the transformation values for reflection symmetries are +already specified for each Cactus grid function, we can use this +information to unambiguously determine the $\pi$-rotation +transformation coefficients, without requiring that any new +information be added if the reflection symmetries have already +been defined (as is the case for any Cactus GFs which are able to +use the current \texttt{bitant}, \texttt{quadrant} and \texttt{octant} +domains). + +%------------------------------------------------------------------------------ +\section{\texttt{CartGrid3D} Notes} +\label{sec:rs_cartgrid3d} +%------------------------------------------------------------------------------ + +Two types of rotating symmetry conditions have been implemented in +\texttt{CartGrid3D} as extensions to the \texttt{grid::domain} +parameter.\\ + +The first, \texttt{bitant\_rotate}, defines a grid which is half-sized +along one axis, and assumes a field that is rotating about a +perpendicular axis. The half-axis is chosen using the +\texttt{bitant\_plane} parameter, which specifies the plane at which +the grid is to be cut as either ``\texttt{xy}'', ``\texttt{xz}'' or +``\texttt{yz}''. The rotation axis is chosen using the +\texttt{rotation\_axis} parameter, which takes values of +``\texttt{x}'', ``\texttt{y}'' or ``\texttt{z}''. For example, to +specify a bitant domain along the positive $y$-axis, on which fields +are rotating about the $z$-axis, the following parameters would do the +job: +\begin{verbatim} + grid::domain = "bitant_rotate" + grid::bitant_plane = "xz" + grid::rotation_axis = "z" +\end{verbatim} +This setup is illustrated in Figure \ref{fig:rs_bitant_example} (a).\\ + +\begin{figure} +\centering +\begin{tabular}{cc} +\ifpdf +\else +\includegraphics[height=40mm]{fig/rotate_bitant_example.eps} +\fi +& +\ifpdf +\else +\includegraphics[height=30mm]{fig/rotate_quadrant_example.eps} +\fi +\\ +(a) & (b) +\end{tabular} +\caption{Active grids for the \texttt{bitant\_rotate} and + \texttt{quadrant\_reflect\_rotate} domains for the examples given in + the text. (a) The given \texttt{bitant\_rotate} domain corresponds + to the $y>0$ half-grid, with fields rotating about the $z$-axis. (b) + The \texttt{quadrant\_reflect\_rotate} example is similar, except + the active grid is only along the positive $z$-axis and a reflection + symmetry is assumed in the $z=0$ plane.} +\label{fig:rs_bitant_example} +\end{figure} + +The \texttt{quadrant\_reflect\_rotate} symmetry cuts two of the axes +in half so that only a quadrant of the full domain is active. A +standard reflection symmetry is applied to one of the half-planes, +while the physical fields are assumed to rotate in the other plane. To +set up such a grid which rotates about the $z$-axis and which is +reflection symmetric in the $z=0$ plane, the following parameters +could be used: +\begin{verbatim} + grid::domain = "quadrant_rotate_reflect" + grid::quadrant_direction = "x" + grid::rotation_axis = "z" +\end{verbatim} +Note that the \texttt{quadrant\_direction} parameter follows the +current \texttt{CartGrid3D} standard, which defines the domain on the +\emph{positive} quadrant with the long edge aligned with the specified +axis. It is currently not possible to choose the negative quadrant.\\ + +Octant rotation symmetries, or the alternate quadrant symmetry (for +which the data on one half-plane is rotated onto the other), are more +difficult to implement without some generalisation of the existing +\texttt{CartGrid3D} specification of symmetry boundaries for the +reasons mentioned in the previous section. + +%------------------------------------------------------------------------------ +\end{document} |