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# horizon.maple - evaluate $H$ = LHS of horizon function
# $Id$
#
# horizon - overall driver
## non_unit_normal - compute s_d
## non_unit_normal_deriv - compute partial_d_s_d
## horizon_function - compute HA,HB,HC,HD,H = fn(s_d)
#
################################################################################
################################################################################
################################################################################
#
# This function is a driver to compute, and optionally generate C code
# for, the LHS of the apparent horizon equation, H.
#
# Inputs:
# h
#
# Outputs:
# H = H__fnd(h, ...)
#
horizon :=
proc(cg_flag::boolean)
non_unit_normal();
non_unit_normal_deriv();
horizon_function(cg_flag);
NULL;
end proc;
################################################################################
#
# This function computes the non-unit outward-pointing normal (covector)
# to the horizon, s_d = n_u(h), using the equation on p.7.1 of my apparent
# horizon finding notes. This function does *NOT* generate any C code.
#
non_unit_normal :=
proc()
global
@include "../maple/coords.minc",
@include "../maple/gfa.minc",
@include "../gr/gr_gfas.minc";
local i, u;
printf("%a...\n", procname);
assert_fnd_exists(h);
assert_fnd_exists(X_ud);
assert_fnd_exists(s_d, fnd);
for i from 1 to N
do
s_d__fnd[i] := x_xyz[i]/r
- sum('X_ud[u,i]*Diff(h,y_rs[u])', 'u'=1..N_ang);
end do;
NULL;
end proc;
################################################################################
#
# This function computes the 1st derivatives of the non-unit outward-pointing
# normal (covector) to the horizon, using the equation on p.7.2 of my
# apparent horizon finding notes. This function does *NOT* generate any
# C code.
#
non_unit_normal_deriv :=
proc()
global
@include "../maple/coords.minc",
@include "../maple/gfa.minc",
@include "../gr/gr_gfas.minc";
local temp, i, j, k, u, v;
printf("%a...\n", procname);
assert_fnd_exists(h);
assert_fnd_exists(X_ud);
assert_fnd_exists(X_udd);
assert_fnd_exists(partial_d_s_d, fnd);
for i from 1 to N
do
for j from 1 to N
do
temp := `if`(i = j,
sum('x_xyz[k]^2', 'k'=1..N) - x_xyz[i]^2,
- x_xyz[i] * x_xyz[j]);
# simplify likes to put things over a common denominator,
# so we only apply it to the individual terms, not to
# the whole expression
partial_d_s_d__fnd[i,j]
:= + simplify( temp/r^3 )
- simplify( sum('X_udd[u,i,j]*Diff(h,y_rs[u])', 'u'=1..N_ang) )
- simplify( msum('X_ud[u,i]*X_ud[v,j]*Diff(h,y_rs[u],y_rs[v])',
'u'=1..N_ang, 'v'=1..N_ang) );
end do;
end do;
NULL;
end proc;
################################################################################
#
# This function computes the LHS of the apparent horizon equation
# as a function of 1st and 2nd angular partial derivatives of the
# apparent horizon radius $h$, in the Maple gridfn array H__fnd,
# using equation (15) in my 1996 apparent horizon finding paper,
# and using partial_d_g_uu[k,i,j] instead of Diff(g_uu[i,j], x_xyz[k]).
# This function also optionally generates C code for this computation.
#
horizon_function :=
proc(cg_flag::boolean)
global
@include "../maple/coords.minc",
@include "../maple/gfa.minc",
@include "../gr/gr_gfas.minc";
local i,j,k,l;
printf("%a...\n", procname);
assert_fnd_exists(g_uu);
assert_fnd_exists(K_uu);
assert_fnd_exists(partial_d_ln_sqrt_g);
assert_fnd_exists(partial_d_g_uu);
assert_fnd_exists(s_d, fnd);
assert_fnd_exists(partial_d_s_d, fnd);
#
# n.b. no simplify() in these subexpressions; empirically it makes
# them much *more* complicated (measured by codegen(...,optimized))
#
# (15a) in my paper
HA__fnd := - msum('g_uu[i,k]*s_d__fnd[k] * g_uu[j,l]*s_d__fnd[l]
* partial_d_s_d__fnd[i,j]',
'i'=1..N, 'j'=1..N, 'k'=1..N, 'l'=1..N)
- (1/2)*msum('g_uu[i,j]*s_d__fnd[j]
* partial_d_g_uu[i,k,l] * s_d__fnd[k]*s_d__fnd[l]',
'i'=1..N, 'j'=1..N, 'k'=1..N, 'l'=1..N);
# (15b) in my paper
HB__fnd := + msum('partial_d_g_uu[i,i,j]*s_d__fnd[j]', 'i'=1..N, 'j'=1..N)
+ msum('g_uu[i,j]*partial_d_s_d__fnd[i,j]', 'i'=1..N, 'j'=1..N)
+ msum('g_uu[i,j]*partial_d_ln_sqrt_g[i]*s_d__fnd[j]',
'i'=1..N, 'j'=1..N);
# (15c) in my paper
HC__fnd := msum('K_uu[i,j]*s_d__fnd[i]*s_d__fnd[j]', 'i'=1..N, 'j'=1..N);
# (15d) in my paper
HD__fnd := msum('g_uu[i,j]*s_d__fnd[i]*s_d__fnd[j]', 'i'=1..N, 'j'=1..N);
# n.b. no simplify() here, it would try to put things over a common
# denominator, which would make the equation much more complicated
H__fnd := HA__fnd/HD__fnd^(3/2) + HB__fnd/HD__fnd^(1/2) + HC__fnd/HD__fnd - K;
if (cg_flag)
then codegen2(H__fnd, 'H', "cg/horizon_function.c");
fi;
NULL;
end proc;
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