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| author | lnerger <lnerger@89daf98e-ef62-4674-b946-b8ff9de2216c> | 1999-09-22 08:16:21 +0000 |
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| committer | lnerger <lnerger@89daf98e-ef62-4674-b946-b8ff9de2216c> | 1999-09-22 08:16:21 +0000 |
| commit | 510a092611fa9e94944ee8a342e77bf0d775221c (patch) | |
| tree | 5de4b4f75082ce356b44ec6a1062b157fe2ee6f4 /doc | |
| parent | 12e59dbcbdf8cfba97f6f727ce3c5c4239957267 (diff) | |
Documentation added
git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinAnalysis/AHFinder/trunk@9 89daf98e-ef62-4674-b946-b8ff9de2216c
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diff --git a/doc/AHFinder.html b/doc/AHFinder.html new file mode 100644 index 0000000..052ead8 --- /dev/null +++ b/doc/AHFinder.html @@ -0,0 +1,408 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN"> + +<html> + +<head> +<title>Using the thorn AHFinder</title> +</head> + +<BODY TEXT="#333333" BGCOLOR="#FFFFFF" LINK="#006699" VLINK="#999900" ALINK="#999900"> + +<H1 ALIGN=CENTER>Using the thorn AHFinder</H1> + +<CENTER> +<TABLE WIDTH=600 BORDER=0 CELLSPACING=0 CELLPADDDING=0> +<TR><TD> +<CENTER> +<A HREF="#parameters">Parameters</a><BR> +<A HREF="#minimum">Minimal Parameter Settings</A><BR> +<A HREF="#hints">Hints for Parameter Settings</A><BR> +<A HREF="#output">Output to Files</A><BR> +<A HREF="#results">Some Results with the Finder</A><BR> +</CENTER> + +<P><BR> +Thorn_MinimumAHF can be used either with a minimization or a flow algorithm. + +<A NAME="parameters"></A><H3>Parameters</H3> +<UL> + +<LI><B>General parameters</B><BR> + <UL> + <LI>To activate the thorn use:<BR> + <FONT FACE="Courier">ahf_active = "yes"</FONT><BR> + This parameter is set by default to <FONT FACE="Courier"> + "no"</FONT>.<BR><BR> + <LI>By default the minimization algorithm is used. To switch + to the flow algorithm one has to set<BR> + <FONT FACE="Courier">ahf_flow = "yes"</FONT><BR> + </UL> + <BR> + +<LI><B>Parameters used in evolutions</B><BR> + <UL> + <LI>To specify how often the finder is called the parameter:<BR> + <FONT FACE="Courier">ahf_findevery = 1</FONT><BR> + must be specified as the number of iterations. Here 1 is default. + <BR><BR> + <LI>The number of iterations after which the thorn is called the + first time can be specified by:<BR> + <FONT FACE="Courier">ahf_findafter = 0</FONT><BR> + Here the default is 0, calling the thorn also on the initial time + slice.<BR><BR> + <LI>Instead specifying the number of iterations one can specify + after how much coordinate time the thorn is called first. This is + done by using the parameter:<BR> + <FONT FACE="Courier">ahf_findaftertime = 0.0</FONT><BR> + When different from zero, this parameter overrides the value of + <FONT FACE="Courier">ahf_findafter</FONT>. Default here is + also 0.<BR> + </UL> + <BR> + +<LI><B>Parameters specifying the expansion of the surface in sperical + harmonics.</B> +<BR> + <UL> + <LI>The expansion in theta is specified by the maximal number + of terms using:<BR> + <FONT FACE="Courier">ahf_lmax = 8</FONT><BR> + Here 8 is the default. <BR><BR> + <LI>If axisymmetry is expected the surface does not need to be + expanded in phi. This is set by:<BR> + <FONT FACE="Courier">ahf_phi = "no"</FONT>.<BR> + The surface is expanded in theta and phi by default.<BR><BR> + <LI>The position of the center of the expansion is set by the + parameters:<BR> + <FONT FACE="Courier">ahf_xc = 0.0<BR> ahf_yc = 0.0<BR> + ahf_zc = 0.0</FONT>.<BR> + Here the origin is default. The center of the expansion should be + set the expected center of the apparent horizon. But it works also + with a non-centered horizon as long as the center of expansion lies + inside the horizon (otherwise the algorithm will fail).<BR><BR> + + <LI>The center of the expansion can also be allowed to move. This + is controlled by:<BR> + <FONT FACE="Courier">ahf_wander = "yes"</FONT>.<BR> + However, at the moment this only works with the minimization + algorithm.<BR><BR> + + <LI>Finally the radius of the initial sphere can also be specified. + This is controlled by the parameter:<BR> + <FONT FACE="Courier">ahf_r0 = 0.0</FONT>.<BR> + The default is 0.0, forcing the largest sphere possible in the grid. + <BR><BR> + </UL> + +<LI><B>Parameters for the initial guess</B><BR> + The initial guess can be specified by some parameters which are + set to <FONT FA=CE="Courier">"no"</FONT> by default.<BR><BR> + <UL> + + <LI>To use on old horizon as initial guess one must set:<BR> + <FONT FACE="Courier">ahf_guessold = "yes"</FONT>.<BR> + However, if in the evolutionof the apparent horizon jumps + discontinuously it might be lost by using this.<BR><BR> + + <LI>If no old horizon is used the inital guess can be specified + further for the minimization algorithm. This algorithm is sensitive + to the initial guess, so this is important. The initial guess is + set up by an expansion in spherical harmonics in the first two + coefficients (l=0,l=2). For both of these a number of subdivisions + can be specified by:<BR> + <FONT FACE="Courier">ahf_nn0 = 10</FONT><BR> + <FONT FACE="Courier">ahf_nn2 = 10</FONT><BR> + The algorithm then tests in this case 100 different combinations to + find the best initial guess.<BR><BR> + + <LI>It is also possible to use only a sphere as initial guess. + This is much faster and is done by setting:<BR> + <FONT FACE="Courier">ahf_sloppyguess = "yes"</FONT>.<BR> + In this case a number of spheres (specified by + <FONT FACE="Courier">ahf_nn0</FONT>) with different radii are + tested for the initial guess.<BR><BR> + + <LI>If one want to look for an inner horizon instead of an outer one, + this can be done by setting<BR> + <FONT FACE="Courier">ahf_inner = "yes"</FONT>.<BR> + This only works with the minimization algorithm.<BR><BR> + </UL> + +<LI><B>Parameters for surface intergrals</B><BR> + The number of subdivisions in theta and phi are specified by:<BR> + <FONT FACE="Courier">ahf_ntheta = 200</FONT><BR> + <FONT FACE="Courier">ahf_nphi = 200</FONT>.<BR> + 200 is the default for both parameters.<BR><BR> + +<LI><B>Parameters indicating symmetries</B><BR> + Symmetries can either be specified by reflection symmetry on single + planes or by octant symmetry. Please notice that this symmetries + refer to the symmetries of the surface, and not of the computational + grid.<BR><BR> + <UL> + <LI>Reflection symmetry on the yz-plane, i.e. symmetry x -> -x, + is indicated by:<BR> + <FONT FACE="Courier">ahf_refx = "yes"</FONT>.<BR> + For the symmetries y -> -y and z -> -z this is done in an analogous + way by<BR> + <FONT FACE="Courier">ahf_refy = "yes"</FONT><BR> + <FONT FACE="Courier">ahf_refz = "yes"</FONT><BR> + By default these parameters are set to + <FONT FACE="Courier">"no"</FONT>.<BR><BR> + + <LI>Octant symmetry is specified by the parameter:<BR> + <FONT FACE="Courier">ahf_octant = "yes"</FONT>.<BR> + This is set to <FONT FACE="Courier">no</FONT> by default. + Possible parmeter settings are <FONT FACE="Courier">"yes"</FONT> + for reflection symmetries on all three coordinate planes and + <FONT FACE="Courier">"high"</FONT> for an additional rotational + symmetry of pi/2 around the z axis.<BR><BR> + </UL> + +<LI><B>Parameters for the minimization algorithm</B><BR> + <UL> + <LI>The tolerance for the minimization algorithm can be specified + by:<BR> + <FONT FACE="Courier">ahf_tol = 0.1</FONT><BR> + with 0.1 being default.<BR><BR> + + <LI>The number if iterations for each step (POWELL algorithm) is + specified by:<BR> + <FONT FACE="Courier">ahf_maxiter = 10</FONT> .<BR> + 10 is the default.<BR><BR> + + <LI>usually the square of the expansion is minimized. To switch + to minimization of the area one can use:<BR> + <FONT FACE="Courier">ahf_minarea = "yes"</FONT><BR> + (default is <FONT FACE="Courier">"no"</FONT>). Only for time + symmetric data the surface found by area minimization corresponds + to an apparent horizon.<BR><BR> + </UL> + +<LI><B>Parameters for the flow algorithm</B><BR> + <UL> + <LI>The maximum number of iterations for the flow algorithm is + specified by:<BR> + <FONT FACE="Courier">ahf_flowiter = 200</FONT>,<BR> + with 200 being default.<BR><BR> + + <LI>The tolerance can be set by<BR> + <FONT FACE="Courier">ahf_flowtol = 0.0001</FONT>.<BR> + The default is 0.0001.<BR><BR> + + <LI>Further, the flow algorithm can be tuned by some additional + parameters (see Carsten's paper). The default values are shown + here:<BR> + <FONT FACE="Courier">ahf_flowa = 0.01</FONT>, + gives the alpha parameter for the flow.<BR> + <FONT FACE="Courier">ahf_flowb = 0.5</FONT>, + sets the beta parameter for the flow.<BR> + <FONT FACE="Courier">ahf_flowh = 0.0</FONT>, + specifies the weight of H flow.<BR> + <FONT FACE="Courier">ahf_flowc = 1.0</FONT>, + specifies the weight of C flow.<BR> + <FONT FACE="Courier">ahf_flown = 0.0</FONT>, + specifies the weight of N flow (this is not yet implemented)<BR><BR> + </UL> + + The character of the differnet flows and the alpha and beta + parameters are described in Carsten Gundlach's paper on his + pseudo-spectral apparent horizon finder (gr-qc/9707050).<BR><BR> + +<LI><B>Parameters for output</B><BR> + <UL> + <LI>By default no logfile for MinimumAHF is written. To obtain + a log file one must set:<BR> + <FONT FACE="Courier">ahf_logfile = "yes"</FONT><BR><BR> + + <LI>There are two modes for verbosity:<BR> + <FONT FACE="Courier">ahf_verbose = "yes"</FONT><BR> + prints messages to screen at the beginning and the end of the + algorithm, and<BR> + <FONT FACE="Courier">ahf_veryverbose = "yes"</FONT><BR> + prints messages also during the iteration process. By default + the verbose mode is set to <FONT FACE="Courier">"yes"</FONT>, + and the veryverbose mode to <FONT FACE="Courier">"no"</FONT>.<BR><BR> + + <LI>Up to now the thorn only supports 2D output of grid functions. + The output is controlled be the thorn itself, not by cactus standard + output. 2D output can be switched off by<BR> + <FONT FACE="Courier">ahf_2Doutput = "no"</FONT>.<BR> + In future versions of the thorn a 3D output will be possible by setting + <BR> + <FONT FACE="Courier">ahf_3Doutput = "yes"</FONT>.<BR><BR> + + <LI>Finally an area map can be found by setting<BR> + <FONT FACE="Courier">ahf_areamap = "yes"</FONT>.<BR> + This is useful for looking at the behaviour of the area for + surfaces close to the horizon.<BR><BR> + </UL> + +<LI><B>Parameters for mask</B><BR> + <UL> + <LI>The mask is 0 inside the horizon and 1 outside, and is used in + black-hole excision (AHBC) techniques. By default the mask is off. + It can be enabled by setting:<BR> + <FONT FACE="Courier">ahf_mask = "yes"</FONT>.<BR> + This sets the mask only if the finder is sure that a horizon was + found. Setting also:<BR> + <FONT FACE="Courier">ahf_weakmask = "yes"</FONT>.<BR> + makes the finder set the mask also if a horizon is probably there + but the resolution or lmax are to low to really resolve it.<BR><BR> + + <LI>There can be a buffer zone between the region inside the + horizon and the region where the mask is 0. This is controlled + by the shrink factor:<BR> + <FONT FACE="Courier">ahf_maskshrink = -2.0</FONT>.<BR> + which here sets the region with where the mask is zero to be + 2 gridpoints inside the horizon.<BR><BR> + </UL> + +</UL> +<A NAME="minimum"></A> +<H3>Minimal Parameter Settings</H3> + +Usually only a few of the parameters described above are needed in the +parameter file. The simplest parameter settings for using the flow +algorithm for a full 3D horizon with a sphere as initial guess is +<PRE> + intorder = 2 # Second order interpolation + + ahf_active = "yes" + ahf_flow = "yes" + + ahf_phi = "yes" + ahf_sloppyguess = "yes" +</PRE> +This looks for a horizon around the origin with lmax = 8. It starts with +a sphere of maximum radius and outputs 2D grid functions. The other +parameters can be used if needed.<BR> + +<A NAME="hints"></A> +<H3>Hints for Parameter Settings</H3> + +In full 3D the flow algorithm is faster than the minimization algorithm. +However, in cases where the terms in the expansion in spherical harmonics +is less than about the minimization can be faster. In axisymmetry this +can happen for l<sub>max</sub><=10.<BR> + +<P> +While the default settings usually work fine, they can be changed to meet +special purposes:<BR> +<UL> + <LI>If the horizon is expected to be far from spherical the parameter + <FONT FACE="Courier">ahf_lmax</FONT> can be set to a higher value. + <FONT FACE="Courier">12</FONT> should be high enough. However values + up to 20 are supported.<BR><BR> + <LI>If the latter parameter is set to a values higher then + <FONT FACE="Courier">8</FONT> the parameter + <FONT FACE="Courier">ahf_maxiter</FONT> can be raised to e.g. + <FONT FACE="Courier">14</FONT>. This can be useful since more + iterations can be necessary for higher coefficients of the expansion. + <BR><BR> + <LI>Usually I suffices to use a sphere for the initial guell of the + surface. So it can be set<BR> + <FONT FACE="Courier">ahf_sloppyguess = "yes"</FONT>.<BR><BR> + <LI>The finder is not running with the convergence mode of cactus, + since there are confusions with the names of the finder's output + files.<BR> +</UL> +<A NAME="output"></A> +<H3>Output to Files</H3> + +The output of the thorn consists of two gridfunctions and several one dimensional output files. +<UL> + <LI>To depict the position of the horizon most important are the + files <FONT FACE="Courier">ahfgrid_2d_...ieee</FONT>. This files + contain a 2D gridfunction. The zero level of this locates the horizon. + <BR><BR> + <LI>The files <FONT FACE="Courier">ahf_exp_2d_...ieee</FONT> show the + expansion of outgoing photoons on the level set of the gridfunction + ahfgrid. The horizon generally coincides with zeros of the expansion. + <BR><BR> + <LI>The surface area of the horizon is given in <FONT FACE="Courier"> + ahf_area.tl</FONT><BR><BR> + <LI><FONT FACE="Courier">ahf_mass.tl</FONT> contains the mass of the + surface.<BR><BR> + <LI>The coefficients of the expansion in spherical harmonics are given + in <FONT FACE="Courier">ahf_coeff.alm</FONT>.<BR><BR> + <LI>The files <FONT FACE="Courier">ahf_circ_eq.tl</FONT>, + <FONT FACE="Courier">ahf_meri_p1.tl</FONT> and <FONT FACE="Courier"> + ahf_meri_p2.tl</FONT> contain the equatorial circumference of the + surface, the length of the meridian at phi=0, and the length of the + meridian at phi=pi/2 of the surface, respectively.<BR><BR> + <LI>If an output of a logfile is set in the parameters, the log file + for the last time the horizon was called is <FONT FACE="Courier"> + ahf_logfile</FONT>. +</UL> +<A NAME="results"></A> +<H3>Some Results with the Finder</H3> + +The finder algorithms have been examined with puncture initial data for +single and binary-black hole scenarios. +<P> + +Calculations with different grid spacings but constant grid size show +convergence of the horizon area. The convergence is of the order of 4, +as should be expected if the radius converges with second order.<P> + +This has been checked with different linear momenta in z direction p<sub> +z</sub>= (0M, 2M, 5M) and vanishing spin. Also for p<sub>z</sub>=2M +and a spin of 5M in x direction the horizon converges. The figure shows +the case with p<sub>z</sub>=2M and vanishing spin. +<CENTER><IMG SRC="p2_areacomp.gif"> +</CENTER> +<P> + +Further not only the area converges but also the shape of the horizon. For +both the minimization and the flow algorithm the horizon converges to +the same shape, as can be seen from the coefficients fo the expansion. The +order of convergence for the coefficients is between 1.4 and 1.7. +<P> +By using the parameters <FONT FACE="Courier">ahf_xc, ahf_yc, +ahf_zc</FONT> it can also be shown that the finder also locates +horizons which are not centered around the point given by these parameters. +This works in general as long as the surface can be expanded in spherical +harmonics around this point but the error increases with the off-centering. +<P> + +The parameter <FONT FACE="Courier">ahf_r0</FONT> can be used +e.g. when dealing with two black holes. If one searches for separate +horizons one can center the finder on one of the locations of the +holes and use an initial radius <FONT FACE="Courier">ahf_r0</FONT> +smaller than the coordinate distance of the holes. With this parameter +settings the single horizon can be found faster. But also a setup with +an initial sphere of maximum radius should work at least for the flow +algorithm. This has been checked with puncture data for two holes with +vanishing linear and angular momentum for each hole (equivalent to +Brill-Lindquist data). Here for a coordinate distance of the holes of +1.6M the separated horizons for the holes are found but no common +horizon. For a coordinate distance of 1.5M a common horizon is found +and also single ones, which are inner surfaces in this case. This coincides +with other work where the critical coordinate distance for a single horizon +is between 1.53M and 1.56M (gr-qc/9809004).<BR> +<CENTER> +<P> +<IMG SRC="hori_bl.gif"> +<P> +</CENTER> +The dashed lines show inner trapped surfaces in the left figure and +the surface where the algorithm stopped without finding a horizon in +the right figure. +<P> + +Also the Misner case was checked. Here for mu = 1.35 a common horizon +is found. For mu = 1.37 separated horizons are found. From the +literature we know that (e.g. gr-qc/9809004) the critical value of mu +is 1.36. This is confirmed by the horizon finder. +<P> + +The information of when a horizon was found can be seen in the cactus-logfile. +There will be output from the thorn even if no horizon was found. +<P> + +</TD></TR> +</TABLE> +</CENTER> +</BODY> +</html> diff --git a/doc/hori_bl.gif b/doc/hori_bl.gif Binary files differnew file mode 100644 index 0000000..861e40a --- /dev/null +++ b/doc/hori_bl.gif diff --git a/doc/p2_areacomp.gif b/doc/p2_areacomp.gif Binary files differnew file mode 100644 index 0000000..e3d75c5 --- /dev/null +++ b/doc/p2_areacomp.gif |
