SphericalSurface

Erik Schnetter <schnetter@cct.lsu.edu>

2007-03-06

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.

1 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 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 index. It is up to the user to specify, probably in the parameter file, which thorns use what surfaces for what purpose.

2 Surface Definition

This thorn provides, for each surface, a two-dimensional grid array sf_radius and grid scalars sf_origin_x, sf_origin_y, and sf_origin_z. The number of surfaces is determined by the parameter nsurfaces, which has to be set in the parameter file.

sf_radius should contain the radius of the surface as measured from its origin, where the arrays indices vary in the 𝜃 and ϕ 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 𝜃 and ϕ directions, is available from the grid scalars sf_origin_theta, sf_origin_phi, sf_delta_theta, and sf_delta_phi. These grid scalars are set by the thorn SphericalSurface in the basegrid bin, and are meant to be read-only for other thorns.

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.

2.1 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

sf_mean_radius
Mean of the surface radius. This should be the arithmetic mean where the radii have been weighted with sin𝜃, 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 = A with A =SdΩr2SdΩ.
sf_min_radius, sf_max_radius
Minimum and maximum of the surface radius.
sf_centroid_x, sf_centroid_y, sf_centroid_z
The centre of the surface. While the quantities sf_origin_* denote the point from which the radius of the surface is measured, the quantities 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 Di =SdΩxiA.
sf_quadrupole_xx, sf_quadrupole_xy, sf_quadrupole_xz, sf_quadrupole_yy, sf_quadrupole_yz, 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 Qij =SdΩyiyjA with yi = xi Di.
sf_min_x, sf_min_y, sf_min_z, sf_max_x, sf_max_y, sf_max_z
The bounding box of the surface.

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 qij from the projection of the ADM three-metric γij.

2.2 Validity of Surface Data

There is also an integer flag valid available. Its definition is up to the surface-providing thorn. The following interpretations are suggested:

zero:
No surface is provided at this time step.
negative:
No surface could be found at this time step.
positive:
The surface data are valid.

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.

3 Surface Array Shape

The number of grid points in the radius array sf_radius is determined by the parameters ntheta and nphi. These arrays exist for each surface. (Internally, thorn SphericalSurface stores all surfaces with the same array shape maxntheta and maxnphi, so that the parameters ntheta and nphi must not be used for index calculations. Use the surfaces 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 maxntheta×maxnphi, the actual radius data (of size ntheta[surface_number]×nphi[surface_number] elements) is in general 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 ntheta[surface_number]×nphi[surface_number], or use an interpolator which supports such non-contiguous input arrays. The AEIThorns/AEILocalInterp local interpolation thorn supports these via the input_array_strides parameter-table option. See the AEILocalInterp thorn guide for details.

4 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 sf_symmetry_x, sf_symmetry_y, and 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 𝜃 and ϕ.

5 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.

5.1 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.

6 Parameters




origin_x
Scope: private  REAL



Description: Origin for surface



Range   Default: 0.0
*
origin






origin_y
Scope: private  REAL



Description: Origin for surface



Range   Default: 0.0
*
origin






origin_z
Scope: private  REAL



Description: Origin for surface



Range   Default: 0.0
*
origin






radius
Scope: private  REAL



Description: Radius for surface



Range   Default: 0.0
*
radius






radius_x
Scope: private  REAL



Description: Radius for surface



Range   Default: 0.0
*
radius






radius_y
Scope: private  REAL



Description: Radius for surface



Range   Default: 0.0
*
radius






radius_z
Scope: private  REAL



Description: Radius for surface



Range   Default: 0.0
*
radius






set_elliptic
Scope: private  BOOLEAN



Description: Place surface at a certain radius



  Default: no






set_spherical
Scope: private  BOOLEAN



Description: Place surface at a certain radius



  Default: no






auto_res
Scope: restricted  BOOLEAN



Description: Automatically determine resolution according to radius and Cartesian resolution



  Default: no






auto_res_grid
Scope: restricted  KEYWORD



Description: Choose resolution according to how grids overlap



Range   Default: fully contained
fully contained
SF must be fully contained in Cartesian grid
overlap
SF overlaps with grid
multipatch
SF potentially overlaps with a spherical mutipatch grid






auto_res_ratio
Scope: restricted  REAL



Description: Multiplicative factor by which we want to scale the resolution with respect to Cartesian resolution



Range   Default: 2.0
0:*






maxnphi
Scope: restricted  INT



Description: Maximum number of grid points in the phi direction



Range   Default: 38
0:*






maxntheta
Scope: restricted  INT



Description: Maximum number of grid points in the theta direction



Range   Default: 19
0:*






name
Scope: restricted  STRING



Description: User friendly name of spherical surface



Range   Default: (none)
none set
.*
any string






nghostsphi
Scope: restricted  INT



Description: Number of ghost zones in the phi direction



Range   Default: 2
0:*






nghoststheta
Scope: restricted  INT



Description: Number of ghost zones in the theta direction



Range   Default: 2
0:*






nphi
Scope: restricted  INT



Description: Number of grid points in the phi direction



Range   Default: 38
0:*
must be at least 3*nghostsphi






nsurfaces
Scope: restricted  INT



Description: Number of surfaces



Range   Default: (none)
0:42






ntheta
Scope: restricted  INT



Description: Number of grid points in the theta direction



Range   Default: 19
0:*
must be at least 3*nghoststheta






symmetric_x
Scope: restricted  BOOLEAN



Description: Reflection symmetry in the x direction



  Default: no






symmetric_y
Scope: restricted  BOOLEAN



Description: Reflection symmetry in the y direction



  Default: no






symmetric_z
Scope: restricted  BOOLEAN



Description: Reflection symmetry in the z direction



  Default: no






verbose
Scope: restricted  BOOLEAN



Description: Shall I be verbose?



  Default: no



7 Interfaces

General

Implements:

sphericalsurface

Inherits:

grid

Grid Variables

7.0.1 PRIVATE GROUPS




  Group Names     Variable Names     Details   




sf_coordinate_estimators   compact0
sf_delta_theta_estimate  descriptionSurface coordinate estimators
sf_delta_phi_estimate   dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeREAL




7.0.2 PUBLIC GROUPS




  Group Names    Variable Names    Details   




sf_active   compact0
sf_active   dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeINT




sf_valid   compact0
sf_valid   dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeINT




sf_info   compact0
sf_area   descriptionSurface information
sf_mean_radius   dimensions0
sf_centroid_x   distributionCONSTANT
sf_centroid_y   group typeSCALAR
sf_centroid_z   timelevels1
sf_quadrupole_xx  vararray_sizensurfaces
sf_quadrupole_xy  variable typeREAL




sf_minreflevel   compact0
sf_minreflevel   descriptionSurface information
  dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeINT




sf_maxreflevel   compact0
sf_maxreflevel   descriptionSurface information
  dimensions0
  distributionCONSTANT
  group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeINT




sf_radius   compact0
sf_radius   descriptionSurface information
  dimensions2
  distributionCONSTANT
  group typeARRAY
  sizeMAXNTHETA
    sizeMAXNPHI
  timelevels1
 vararray_sizensurfaces
 variable typeREAL








  Group Names     Variable Names    Details   




sf_origin   compact0
sf_origin_x   dimensions0
sf_origin_y   distributionCONSTANT
sf_origin_z   group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeREAL




sf_coordinate_descriptors   compact0
sf_origin_theta   descriptionSurface coordinate descriptors
sf_origin_phi   dimensions0
sf_delta_theta   distributionCONSTANT
sf_delta_phi   group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeREAL




sf_shape_descriptors   compact0
sf_ntheta   descriptionSurface shape descriptors
sf_nphi   dimensions0
sf_nghoststheta   distributionCONSTANT
sf_nghostsphi   group typeSCALAR
  timelevels1
 vararray_sizensurfaces
 variable typeINT




Provides:

sf_IdFromName to

8 Schedule

This section lists all the variables which are assigned storage by thorn CactusNumerical/SphericalSurface. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

 

Always:  
sf_active  
sf_valid  
sf_info  
sf_radius sf_origin  
sf_coordinate_descriptors 
sf_coordinate_estimators  
sf_shape_descriptors  
sf_minreflevel  
sf_maxreflevel  
   

Scheduled Functions

CCTK_BASEGRID

  sphericalsurface_setupres

  set surface resolution automatically

 

 After: spatialcoordinates
   correctcoordinates
  Before: sphericalsurface_setup
 Language:c
 Options: global
   loop-local
 Type: function

CCTK_BASEGRID

  sphericalsurface_setup

  calculate surface coordinate descriptors

 

 Language:c
 Options: global
 Type: function

CCTK_BASEGRID

  sphericalsurface_set

  set surface radii to be used for initial setup in other thorns

 

 Before: sphericalsurface_hasbeenset
 Language:c
 Options: global
 Type: function

CCTK_BASEGRID

  sphericalsurface_hasbeenset

  set the spherical surfaces before this group, and use it afterwards

 

 Type:group

CCTK_POSTSTEP

  sphericalsurface_set

  set surface radii

 

 Before: sphericalsurface_hasbeenset
 Language:c
 Options: global
 Type: function

CCTK_POSTSTEP

  sphericalsurface_hasbeenset

  set the spherical surfaces before this group, and use it afterwards

 

 Type:group

SphericalSurface_HasBeenSet

  sphericalsurface_checkstate

  test the state of the spherical surfaces

 

 Language:c
 Options: global
 Type: function