[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 56)
roland.haas at physics.gatech.edu
roland.haas at physics.gatech.edu
Tue Apr 12 07:30:04 CDT 2011
User: rhaas
Date: 2011/04/12 07:30 AM
Added:
/figures/
bbh-boxes.fig, bbh-boxes.pdf, cartoon_plane.fig, cartoon_plane.pdf, faces.fig, faces.pdf, rot180-grid.fig, rot180-grid.pdf
Modified:
/
ET.tex
Log:
add text to boundaries and symmetry section
Directory Changes:
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+ application/octet-stream
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File [modified]: ET.tex
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--- ET.tex 2011-04-12 10:46:47 UTC (rev 55)
+++ ET.tex 2011-04-12 12:30:03 UTC (rev 56)
@@ -1433,6 +1433,7 @@
\paragraph{Vacuum-Matter Coupling}
\paragraph{Object tracking}
+\label{sec:object-tracking}
We provide a module (\codename{PunctureTracker}) for tracking black hole
positions when evolved with the moving puncture technique. This can be used
with (\codename{CarpetTracker}) to have the mesh refinement regions follow the
@@ -1458,7 +1459,7 @@
\subsection{Simulation Domain, Symmetries, Boundaries\pages{3 Roland}}
-\paragraph{Domains and Coordinates.}
+\subsubsection{Domains and Coordinates.}
The Einstein Toolkit provides a module \codename{CoordBase} to facilate
the specification of the simulation domain independend from the actual
evolution module used.
@@ -1484,23 +1485,42 @@
modules to query the domain and description in a uniform way. This
information is used e.g.\ by modules computing the total barionic mass of
the system to determine the region over which to intergrate the mass
-density. Evolution modules use this information to decide which points
-are evolved and therefore require the evaluation of the righ-hand-side
-expression and which ones are set via boundary or symmetry conditions.
+density. \codename{Carpet} queries \codename{CoordBase} for the discrete
+grid when creating the hierarchy of grids. Evolution modules use this
+information to decide which points are evolved and therefore require the
+evaluation of the righ-hand-side expression and which ones are set via
+boundary or symmetry conditions.
-\paragraph{Symmetries, Boundary Conditions.}
+\subsubsection{Symmetries, Boundary Conditions.}
The Einstein Toolkit includes a set of modules, \codename{Boundary}
and \codename{SymBase}, to provide a generic interface to boundary and
symmetry conditions.
The Einstein Toolkit offers a set of reflecting or rotating symmetry
conditions that can be used to reduce the size of the simulation
-domain. These symmetries include periodicity
+domain. These symmetries include periodicity in any of the coordinate
+directions
(via the \codename{Periodic} module), reflections about the
coordinate planes (via the \codename{Reflection} module),
and $90^{\circ}$ and $180^{\circ}$ rotational
(via the \codename{RotatingSymmetry90} and
\codename{RotatingSymmetry180} modules respectively)
-symmetries about the $z$ axis. In applying these symmetries, the
+symmetries about the $z$ axis.
+\codename{Cartoon2D} is a special case since it provides continuous
+rotational symmetry rather than a discrete
+symmetry~\cite{Alcubierre1999ab}. \codename{Cartoon2D} allows fully
+three dimensional codes to be used in axissymmtric problems by evolving
+a slice in the $y=0$ plane and using the rotational symmetry to populate
+ghost points off the plane (see Figure~\ref{fig:cartoon-plane}).
+\begin{figure}[htbp]
+ \begin{center}
+ \includegraphics[width=0.25\textwidth]{cartoon_plane}
+ \end{center}
+ \caption{Grid layout of a simulation using \codename{Cartoon2D}. The
+ $z$-axis is the axis of rotational symmetry. Image courtesy of
+ Denis Pollney.\todo{RH: ask Denis if we can use his figure}}
+ \label{fig:cartoon-plane}
+\end{figure}
+In applying these symmetries, the
transformation behaviour of tensorial quantities (including tensor
densities and non-tensors such as Christoffel symbols) is correctly
taken into account.
@@ -1508,25 +1528,53 @@
take the existing symmetries into account, transparently mapping points
to the numerical domain and respecting the transformation behaviour of
tensorial quantities. Symmetries therefore are handled transparently
-from the point of view of user modules.
-The Einstein Toolkit also offers a set of boundary conditions,
-including Dirichlet, von Neumann, and Robin (fall-off) boundary
-conditions, as well as extrapolation and radiative boundary
-conditions.
+from the point of view of user modules (see Figure~\ref{fig:faces} for an
+illustration).
+\begin{figure}[htbp]
+ \begin{center}
+ \includegraphics{faces}
+ \end{center}
+ \caption{Recursive transformation of a point $x$ in quadrant 3 to a the
+ physical point $x''$ for which there is actual data stored. In this
+ example two reflection symmetries along the horizontal and vertical axis
+ are present, notice how the vector components are transformed in
+ transformations $A$ and $B$. Image courtesy of Erik Schnetter\todo{RH:
+ Erik, are you ok with this paper re-using your image from SymBase?}}
+ \label{fig:faces}
+\end{figure}
+Thorn \codename{Boundary} provides basic boundary conditions. A boundary
+condition suitable for matter fields which approach a constant
+(atmosphere) value is provided via either the ``flat'' or ``scalar''
+(Dirichlet) boundary conditions. These boundary conditions set the
+values on the boundary to either those close by in the direction of the
+normal to the boundary or to a fixed given value, respectively.
+The ``copy'' boundary condition uses a secondary grid variable to fill
+in the boundary values, allowing arbitrary boundary conditions to be
+imposed.
+A fall-off (``Robin'') boundary condition is provided suitable for field
+whose behaviour near the boundary is described by
+\begin{equation}
+ f(r) = f_0 + \frac{k}{r^n}\text{,}
+ \label{eqn:robin-falloff}
+\end{equation}
+for a given decay rate $n$, value at infinity $f_0$ and scaling constant
+$k$. $r$ is taken to be the coordinate direction perpendicular to the
+domain boundary under consideration when the boundary condition is
+applied.
+Radiative (Sommerfeld) and extrapolation
+boundary conditions are provided by thorn \codename{NewRad}.
\comment{RH: Christian already describes NewRad's boundary conditions in
his section (these should be used for extrapolation and radiative
boundary conditions I think). Should they eventually be moved here (once
the paper is almost ready)?}
-\paragraph{Adaptive Mesh Refinement}
+\subsubsection{Adaptive Mesh Refinement}
The \codename{Carpet} driver module provides adaptive mesh refinement and
Multi-Block capabilities to modules in the Einstein Toolkit. As in most
adaptive mesh refinement frameworks, care has been taken to separate
driver related code from physics codes. A typical evolution or analysis
code like \codename{McLachlan} or \codename{WeylScal4} is agnostic to
the mesh refinement.
-\todo{RH: add description and motivation of/for Berger-Rigoustos scheme?}
-\todo{RH: add interaction with MoL?}
Carpet supports feature-based mesh refinement, which is based on
extracting e.g.\ the locations of black holes or neutron stars, and
then constructing a mesh hierarchy (stacks of refined regions) based
@@ -1535,6 +1583,67 @@
remove stacks if e.g.\ the number of objects changes. If initial
conditions are constructed outside of Carpet (which is often the
case), then the initial mesh hierarchy has to be defined manually.
+The Einstein toolkit currently provides two ``regridding'' modules in
+the \codename{CarpetRegrid} and \codename{CarpetRegrid2} thorns.
+Both thorns primarily support box-in-box type refined meshes, which is
+well suited to current binary black hole simulations in which the
+high-resolution region is centered on the individual black holes.
+Figure~\ref{fig:bbh-boxes} shows a typical set of nested boxes during
+the inspiral phase of a binary black hole merger simulation.
+\begin{figure}[htbp]
+ \begin{center}
+ \includegraphics{bbh-boxes}
+ \end{center}
+ \caption{Nested boxes following the individual black holes in binary
+ black hole merger simulation (see Section~\ref{sec:bbh-example}),
+ the location of the indivdual black holes is found by
+ \codename{PunctureTracker}. Shown are the innermost three of 9
+ levels of mesh refinement used in this simulation. Notice the use of
+ \codename{RotatingSymmetry180} to reduce the computational domain.}
+ \label{fig:bbh-boxes}
+\end{figure}
+\codename{CarpetRegrid} provides a number of different way to specify
+the refined regions, either as a set of boxes centered around the origin
+or as an explicit list of boxes that make up the grid hierarchy.
+Traditionally groups using \codename{CarpetRegrid} have employed
+auxiliary thorns, that are not part of the Einstein Toolkit, to create
+the list of boxes based on information obtained eg. from apparent
+horizon tracking. \codename{CarpetRegrid2} provides a user friendly
+interface to define sets of nested boxes that follow black holes or
+other tracked objects. \codename{CarpetRegrid2} supports up to 10 sets
+of nested regions which can be either moving or stationary. The number
+of refinement levels in each region as well as the radii of each nested
+box are allowed to change during runtime, making it possible to eg.
+adapt the shape of the refined region to the surface of a star.
+\codename{CarpetRegrid2} contains code to handle the $\pi$-symmetry
+provided by \codename{RotatingSymmetry180}, enforcing the symmetry on
+the resulting grid layout (see Figure~\ref{fig:rot180-grid}).
+\begin{figure}[htbp]
+ \begin{center}
+ \includegraphics[width=0.3\textwidth]{rot180-grid}
+ \end{center}
+ \caption{Grid layout created by \codename{CarpetRegrid2}. In this
+ example we use one ghost point, one boundary, and two buffer points
+ as well as \codename{RotatingSymmetry180}. There are two refinement
+ levels present, a coarse one represented by big red circles and a
+ fine one represented by small black circles. The
+ filled black circle is the single point specified by the user.
+ \codename{CarpetRegrid2} surrounded it with a layer of buffer
+ points, indicated by the cyan filled circles. The open circles are
+ ghost and boundary points which are maintained by \codename{Carpet}.
+ The presence of the $\pi$-symmetry forces \codename{CarpetRegrid2}
+ to create the tiny region to the bottom left of the grid. They serve
+ only as source points for the boundary condition.}
+ \label{fig:rot180-grid}
+\end{figure}
+\codename{CarpetTracker} provides a simple interface to slave
+\codename{CarpetRegrid2}'s regions to the object trackers
+\codename{PunctureTracker} and \codename{NSTracker} (see
+section~\ref{sec:object-tracking}) by copying their
+positions out of a \codename{SphericalSurface} into
+\codename{CarpetRegrid2}.
+%Figure~\ref{fig:bbh-boxes} shows a typical
+%grid layout during the inspiral phase of a binary black hole simulation.
@@ -1543,6 +1652,7 @@
\subsection{Spinning BH\pages{2 Peter}}
\subsection{BBH\pages{2 Bruno}}
+\label{sec:bbh-example}
\subsection{TOV\pages{2 Frank}}
Show stable TOV evolution
Directory: /figures/
====================
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+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 8400 150 150 4800 8400 4800 8550
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 9600 150 150 4800 9600 4800 9750
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 0 150 150 4800 0 4800 150
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 0 150 150 6000 0 6000 150
+1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 6000 150 150 7200 6000 7200 6150
+1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 7200 150 150 7200 7200 7200 7350
+1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 8400 150 150 7200 8400 7200 8550
+1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 9600 150 150 7200 9600 7350 9600
+1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 6000 6000 150 150 6000 6000 6000 6150
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 9600 150 150 6000 9600 6150 9600
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 8400 150 150 6000 8400 6000 8550
+1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 7200 150 150 6000 7200 6000 7350
+1 3 0 4 0 0 49 -1 20 0.000 1 0.0000 7200 0 150 150 7200 0 7200 150
+2 1 1 4 0 16 51 -1 -1 10.000 0 0 -1 1 0 2
+ 1 1 1.00 60.00 120.00
+ 2400 4800 12000 4800
+2 1 1 4 0 16 51 -1 -1 10.000 0 0 -1 1 0 2
+ 1 1 1.00 60.00 120.00
+ 4800 9600 4800 0
+2 1 0 4 0 16 49 -1 -1 0.000 0 0 -1 1 0 2
+ 1 1 4.00 240.00 480.00
+ 300 4800 1950 4800
+2 1 0 4 0 16 49 -1 20 0.000 0 0 -1 1 0 2
+ 1 1 4.00 240.00 480.00
+ 4800 11550 4800 9900
+3 0 1 4 0 7 48 -1 -1 10.000 0 1 0 3
+ 2 0 4.00 180.00 480.00
+ 6000 9600 2400 4800 3600 0
+ 0.000 1.000 0.000
+4 1 0 49 -1 0 48 0.0000 4 735 1650 450 5700 y = 0\001
+4 1 0 49 -1 0 48 0.0000 4 570 1650 4800 12150 x = 0\001
File [added]: rot180-grid.pdf
Delta lines: +0 -0
===================================================================
(Binary files differ)
Property changes on: figures/rot180-grid.pdf
___________________________________________________________________
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