% *======================================================================* % Cactus Thorn template for ThornGuide documentation % Author: Ian Kelley % Date: Sun Jun 02, 2002 % $Header$ % % Thorn documentation in the latex file doc/documentation.tex % will be included in ThornGuides built with the Cactus make system. % The scripts employed by the make system automatically include % pages about variables, parameters and scheduling parsed from the % relevant thorn CCL files. % % This template contains guidelines which help to assure that your % documentation will be correctly added to ThornGuides. More % information is available in the Cactus UsersGuide. % % Guidelines: % - Do not change anything before the line % % START CACTUS THORNGUIDE", % except for filling in the title, author, date, etc. fields. % - Each of these fields should only be on ONE line. % - Author names should be separated with a \\ or a comma. % - You can define your own macros, but they must appear after % the START CACTUS THORNGUIDE line, and must not redefine standard % latex commands. % - To avoid name clashes with other thorns, 'labels', 'citations', % 'references', and 'image' names should conform to the following % convention: % ARRANGEMENT_THORN_LABEL % For example, an image wave.eps in the arrangement CactusWave and % thorn WaveToyC should be renamed to CactusWave_WaveToyC_wave.eps % - Graphics should only be included using the graphicx package. % More specifically, with the "\includegraphics" command. Do % not specify any graphic file extensions in your .tex file. This % will allow us to create a PDF version of the ThornGuide % via pdflatex. % - References should be included with the latex "\bibitem" command. % - Use \begin{abstract}...\end{abstract} instead of \abstract{...} % - Do not use \appendix, instead include any appendices you need as % standard sections. % - For the benefit of our Perl scripts, and for future extensions, % please use simple latex. % % *======================================================================* % % Example of including a graphic image: % \begin{figure}[ht] % \begin{center} % \includegraphics[width=6cm]{MyArrangement_MyThorn_MyFigure} % \end{center} % \caption{Illustration of this and that} % \label{MyArrangement_MyThorn_MyLabel} % \end{figure} % % Example of using a label: % \label{MyArrangement_MyThorn_MyLabel} % % Example of a citation: % \cite{MyArrangement_MyThorn_Author99} % % Example of including a reference % \bibitem{MyArrangement_MyThorn_Author99} % {J. Author, {\em The Title of the Book, Journal, or periodical}, 1 (1999), % 1--16. {\tt http://www.nowhere.com/}} % % *======================================================================* % If you are using CVS use this line to give version information % $Header$ \documentclass{article} % Use the Cactus ThornGuide style file % (Automatically used from Cactus distribution, if you have a % thorn without the Cactus Flesh download this from the Cactus % homepage at www.cactuscode.org) \usepackage{../../../../doc/latex/cactus} \begin{document} % The author of the documentation \author{Erik Schnetter \textless schnetter@aei.mpg.de\textgreater} % The title of the document (not necessarily the name of the Thorn) \title{SphericalSurface} % the date your document was last changed, if your document is in CVS, % please use: \date{$ $Date$ $} \maketitle % Do not delete next line % START CACTUS THORNGUIDE % Add all definitions used in this documentation here % \def\mydef etc \begin{abstract} This thorn provides a repository for two-dimensional surfaces with spherical topology. This thorn does not actually do anything with these surfaces, but allows other thorns to store and retrieve the surfaces and some associated information. The exact interpretation of the stored quantities is up to the using thorns, but certain standard definitions are suggested. \end{abstract} \section{Introduction} Many thorns work on manifolds that are two-dimensional, closed surfaces. Examples are apparent and event horizons, or the surfaces on which gravitational waves are extracted. Other such surfaces might be excision or outer boundaries (although these are currently not treated as such). There is a need to have a common representation for such surfaces, so that the surface-finding thorns and the thorns working with these surfaces can be independent. A common representation will also facilitate visualisation. This thorn \texttt{SphericalSurface} provides just such a common representation. This thorn is not meant to do anything else but be a ``repository'' for surfaces. It is up to the surface-finding and surface-using thorns to agree on the details of the information stored. Of course, standard definitions for the stored quantities are suggested. (For example, there is no exact definition of the quadrupole moment, because this definition will depend on the kind of surface that is stored. However, it is specified that the quadrupole moment should be calculate with respect to the centroid, and that it should not be trace-free.) This thorn provides storage for several independent surfaces, identified by an \emph{index}. It is up to the user to specify, probably in the parameter file, which thorns use what surfaces for what purpose. \section{Surface Definition} This thorn provides, for each surface, a two-dimensional grid array \texttt{sf\_radius} and grid scalars \texttt{sf\_origin\_x}, \texttt{sf\_origin\_y}, and \texttt{sf\_origin\_z}. The number of surfaces is determined by the parameter \texttt{nsurfaces}, which has to be set in the parameter file. \texttt{sf\_radius} should contain the radius of the surface as measured from its origin, where the arrays indices vary in the $\theta$ and $\phi$ direction, respectively. Both the radius array and the surface origin are supposed to be set when a surface is stored. The coordinates on the surface, i.e.\ the grid origin and spacing in the $\theta$ and $\phi$ directions, is available from the grid scalars \texttt{sf\_origin\_theta}, \texttt{sf\_origin\_phi}, \texttt{sf\_delta\_theta}, and \texttt{sf\_delta\_phi}. These grid scalars are set by the thorn SphericalSurface in the \texttt{basegrid} bin, and are meant to be read-only for other thorns. \begin{quotation} \subsection*{A note on vector grid variables} A relatively new addition to Cactus (in November 2003) are vector grid variables. These are essentially arrays of grid variables. Thorn SphericalSurface makes use of these by storing the surfaces in such arrays. That means that in order to access data from a single surface, on has to use the corresponding surface index as array index. In a similar manner, thorn SurfaceIndex uses array parameters for its parameters (except certain global ones). This should be kept in mind when writing source code. C has the unfortunate property of converting arrays into meaningless integers if an array subscript is accidentally omitted. Fortran knows whole-array expressions, meaning that it would act on all surfaces instead of a single one if an array subscript is accidentally omitted. Each element of a vector grid function is a grid function. (The term ``grid function vector'' might have been more appropriate.) As such, it has a name, which can be used e.g.\ for output. The name consists of the vector grid function name to which the surface index in square brackets has been appended. \end{quotation} \subsection{Global Surface Quantities} In many cases, only some abstract information about the surface is of interest, such as its mean radius or its quadrupole moment. For that purpose there are additional grid scalars that carry this information. These grid scalars are also supposed to be set when a surface is stored. These grid scalars are \begin{description} \item[\texttt{sf\_mean\_radius}] Mean of the surface radius. This should be the arithmetic mean where the radii have been weighted with $\sin\theta$, or a suitable generalisation thereof. This quantity is also supposed to be a measure of the surface's monopole moment. One suggested expression is $M=\sqrt{A}$ with $A = \int_S d\Omega\, r^2 / \int_S d\Omega$. \item[\texttt{sf\_min\_radius}, \texttt{sf\_max\_radius}] Minimum and maximum of the surface radius. \item[\texttt{sf\_centroid\_x}, \texttt{sf\_centroid\_y}, \texttt{sf\_centroid\_z}] The centre of the surface. While the quantities \texttt{sf\_origin\_*} denote the point from which the radius of the surface is measured, the quantities \texttt{sf\_centroid\_*} should contain the point which is ``logically'' the centre of the surface. This quantity is supposed to be a measure of the dipole moment of the surface. One suggested expression is $D^i = \int_S d\Omega\, x^i / A$. \item[\texttt{sf\_quadrupole\_xx}, \texttt{sf\_quadrupole\_xy}, \texttt{sf\_quadrupole\_xz}, \texttt{sf\_quadrupole\_yy}, \texttt{sf\_quadrupole\_yz}, \texttt{sf\_quadrupole\_zz}] The quadrupole moment of the surface. This should be the full quadrupole moment and not a trace-free quantity. One suggested expression is $Q^{ij} = \int_S d\Omega\, y^i y^j / A$ with $y^i = x^i - D^i$. \item[\texttt{sf\_min\_x}, \texttt{sf\_min\_y}, \texttt{sf\_min\_z}, \texttt{sf\_max\_x}, \texttt{sf\_max\_y}, \texttt{sf\_max\_z}] The bounding box of the surface. \end{description} Note that the integral expressions are only suggestions which should be adapted to whatever is natural for the stored surface. The suggested integral expressions also depend on the metric which is used; this should be a ``natural'' metric for the surface. E.g.\ for apparent horizons, this might be the induced two-metric $q_{ij}$ from the projection of the ADM three-metric $\gamma_{ij}$. \subsection{Validity of Surface Data} There is also an integer flag \texttt{valid} available. Its definition is up to the surface-providing thorn. The following interpretations are \emph{suggested}: \begin{description} \item[zero:] No surface is provided at this time step. \item[negative:] No surface could be found at this time step. \item[positive:] The surface data are valid. \end{description} Note that, if this flag is used, it is necessary to set this flag at every iteration. This flag is not automatically reset to zero. \section{Surface Array Shape} The number of grid points in the radius array \texttt{sf\_radius} is determined by the parameters \texttt{ntheta} and \texttt{nphi}. These arrays exist for each surface. (Internally, thorn SphericalSurface stores all surfaces with the same array shape \texttt{maxntheta} and \texttt{maxnphi}, so that the parameters \texttt{ntheta} and \texttt{nphi} must not be used for index calculations. Use the surfaces \texttt{lsh} instead.) The surface array shape includes ghost or boundary zones at the array edges. These ghost zones have the same size for all surfaces. Note that because the radius arrays are stored with larger size $\texttt{maxntheta} \times \texttt{maxnphi}$, the actual radius data (of size $\texttt{ntheta[surface\_number]} \times \texttt{nphi[surface\_number]}$ elements) is in general {\em not contiguous in memory.\/} If you want to interpolate a SphericalSurface surface radius, you need to either copy the radius data to a contiguous 2-D array of size $\texttt{ntheta[surface\_number]} \times \texttt{nphi[surface\_number]}$, or use an interpolator which supports such non-contiguous input arrays. The AEIThorns/AEILocalInterp local interpolation thorn supports these via the \texttt{input\_array\_strides} parameter-table option. See the AEILocalInterp thorn guide for details. \section{Surface Symmetries} It is often the case that one uses symmetries to reduce the size of the simulation domain, such as octant or quadrant mode. Whenever a symmetry plane intersects a surface, only part of the surface is actually stored. The user has to define the symmetries of each surface in the parameter file via the parameters \texttt{sf\_symmetry\_x}, \texttt{sf\_symmetry\_y}, and \texttt{sf\_symmetry\_z}. They indicate that a reflection symmetry exists in the corresponding direction. The surface origin is required to lie in the corresponding symmetry planes. Thorn SphericalSurface takes these symmetries into account when it calculates the grid spacing and the origin of the surface coordinates $\theta$ and $\phi$. \section{Input and Output} As the surfaces are stored as grid variables, the standard input and output routines will work for them. The standard visualisation tools will be able to visualise them. The surfaces will also automatically be checkpointed and restored. \subsection{Acknowledgements} This thorn was suggested during meetings of the numerical relativity group at the AEI. Jonathan Thornburg provided many detailed and useful suggestions. Ed Evans, Carsten Gundlach, Ian Hawke, and Denis Pollney contributed comments and suggestions. % Do not delete next line % END CACTUS THORNGUIDE \end{document}