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# horizon.maple - evaluate $H$ = LHS of horizon function
# $Header$
#
# horizon - overall driver
## non_unit_normal - compute s_d
## non_unit_normal_deriv - compute partial_d_s_d
## expansion - compute Theta_[ABCD],Theta = fn(s_d__fnd) = fn(h)
## expansion_Jacobian - compute Jacobian coeffs for Theta_[ABCD], Theta
#
################################################################################
################################################################################
################################################################################
#
# This function is a driver to compute, and optionally generate C code
# for, the LHS of the apparent horizon equation, H.
#
# Inputs:
# h
#
# Outputs:
# Theta = Theta__fnd(h, ...)
# partial_Theta_wrt_partial_d_h__fnd = partial_Theta_wrt_partial_d_h(h, ...)
# partial_Theta_wrt_partial_dd_h__fnd = partial_Theta_wrt_partial_dd_h(h, ...)
#
horizon :=
proc(cg_flag::boolean)
non_unit_normal();
non_unit_normal_deriv();
expansion(cg_flag);
expansion_Jacobian(cg_flag);
NULL;
end proc;
################################################################################
#
# This function computes the xyz components of the non-unit
# outward-pointing normal (covector) to the horizon, s_d = s_d(h),
# using the equation on p.7.1 of my apparent horizon finding notes.
#
# This function does *NOT* generate any C code.
#
# Outputs:
# s_d = s_d__fnd( x_xyz, X_ud, Diff(h,y_rs) )
#
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 xyz 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.
#
# Inputs:
# s_d = s_d__fnd( x_xyz, X_ud, Diff(h,y_rs) )
#
# Outputs:
# partial_d_s_d = partial_d_s_d__fnd( x_xyz, X_ud, X_udd,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
#
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 expansion $\Theta$ of a trial horizon surface
# as a function of 1st and 2nd angular partial derivatives of the surface
# radius $h$, using equations (14) and (15[abcd]) in my 1996 AH-finding
# paper, but using partial_d_g_uu[k,i,j] instead of Diff(g_uu[i,j], x_xyz[k]).
#
# These equations give Theta_[ABCD] and Theta as functions of s_d, but
# here we use s_d__fnd, which we assume is already given in terms of angular
# derivatives of h. The result is that we compute Theta_[ABCD] and Theta
# directly in terms of 1st and 2nd angular derivatives of h, without s_d
# ever appearing in our final results.
#
# This function also optionally generates C code for the computation
# of Theta_[ABCD]. It does *not* compute C code for the computation of
# Theta itself, since we may want to check that Theta_D > 0 in the C
# code before we compute Theta itself.
#
# Inputs:
# s_d = s_d__fnd( x_xyz, X_ud, Diff(h,y_rs) )
# partial_d_s_d = partial_d_s_d__fnd( x_xyz, X_ud, X_udd,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
#
# Outputs:
# Theta_A = Theta_A__fnd( x_xyz, X_ud, X_udd, g_uu, partial_d_g_uu,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
# Theta_B = Theta_B__fnd( x_xyz, X_ud, X_udd,
# g_uu, partial_d_g_uu, partial_d_ln_sqrt_g,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
# Theta_C = Theta_C__fnd( x_xyz, X_ud, K_uu, Diff(h,y_rs) )
# Theta_D = Theta_D__fnd( x_xyz, X_ud, g_uu, Diff(h,y_rs) )
# ---------------------------------------------------------------
# Theta = Theta__fnd(Theta_A__fnd, Theta_B__fnd, Theta_C__fnd, Theta_D__fnd)
#
expansion :=
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
Theta_A__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
Theta_B__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
Theta_C__fnd := msum('K_uu[i,j]*s_d__fnd[i]*s_d__fnd[j]', 'i'=1..N, 'j'=1..N);
# (15d) in my paper
Theta_D__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
Theta__fnd := + Theta_A__fnd/Theta_D__fnd^(3/2)
+ Theta_B__fnd/Theta_D__fnd^(1/2)
+ Theta_C__fnd/Theta_D__fnd
- K;
if (cg_flag)
then codegen2([ Theta_A__fnd, Theta_B__fnd, Theta_C__fnd, Theta_D__fnd],
['Theta_A', 'Theta_B', 'Theta_C', 'Theta_D' ],
"../gr.cg/expansion.c");
fi;
NULL;
end proc;
################################################################################
#
# This function computes the Jacobian coefficients for the expansion
# $\Theta$ of a trial horizon surface with respect to 1st and 2nd angular
# partial derivatives of the surface radius $h$. These coefficients
# are (or at least should be :) the same as those in equation (A1) in
# my 1996 apparent horizon finding paper, bue here we compute them in
# a different manner: we have Maple directly differentiate Theta__fnd with
# respect to Diff(h,y_rs[u]) and Diff(h,y_rs[u],y_rs[v]). We use the
# Maple frontend() function to do this.
#
# This function also optionally generates C code for the Jacobian
# coefficients.
#
# Inputs:
# Theta_A = Theta_A__fnd( x_xyz, X_ud, X_udd, g_uu, partial_d_g_uu,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
# Theta_B = Theta_B__fnd( x_xyz, X_ud, X_udd,
# g_uu, partial_d_g_uu, partial_d_ln_sqrt_g,
# Diff(h,y_rs), Diff(h,y_rs,y_rs) )
# Theta_C = Theta_C__fnd( x_xyz, X_ud, K_uu, Diff(h,y_rs) )
# Theta_D = Theta_D__fnd( x_xyz, X_ud, g_uu, Diff(h,y_rs) )
# Theta = Theta__fnd(Theta_A__fnd, Theta_B__fnd, Theta_C__fnd, Theta_D__fnd)
#
# Outputs:
# partial_Theta_A_wrt_partial_d_h partial_Theta_A_wrt_partial_dd_h
# partial_Theta_B_wrt_partial_d_h partial_Theta_B_wrt_partial_dd_h
# partial_Theta_C_wrt_partial_d_h
# partial_Theta_D_wrt_partial_d_h
# ---------------------------------------------------------------
# partial_Theta_wrt_partial_d_h partial_Theta_wrt_partial_dd_h
#
expansion_Jacobian :=
proc(cg_flag::boolean)
global
@include "../maple/coords.minc",
@include "../maple/gfa.minc",
@include "../gr/gr_gfas.minc";
local u,v,
temp;
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(Theta_A);
assert_fnd_exists(Theta_B);
assert_fnd_exists(Theta_C);
assert_fnd_exists(Theta_D);
# Jacobian coefficients of Theta_[ABCD] and Theta wrt Diff(h,y_rs[u])
for u from 1 to N_ang
do
partial_Theta_A_wrt_partial_d_h__fnd[u]
:= frontend('diff', [Theta_A__fnd, Diff(h,y_rs[u])]);
partial_Theta_B_wrt_partial_d_h__fnd[u]
:= frontend('diff', [Theta_B__fnd, Diff(h,y_rs[u])]);
partial_Theta_C_wrt_partial_d_h__fnd[u]
:= frontend('diff', [Theta_C__fnd, Diff(h,y_rs[u])]);
partial_Theta_D_wrt_partial_d_h__fnd[u]
:= frontend('diff', [Theta_D__fnd, Diff(h,y_rs[u])]);
# equation (A1a) in my 1996 apparent horizon finding paper
temp := + (3/2)*Theta_A/Theta_D^(5/2)
+ (1/2)*Theta_B/Theta_D^(3/2)
+ Theta_C/Theta_D^2;
partial_Theta_wrt_partial_d_h__fnd[u]
:= + partial_Theta_A_wrt_partial_d_h__fnd[u] / Theta_D^(3/2)
+ partial_Theta_B_wrt_partial_d_h__fnd[u] / Theta_D^(1/2)
+ partial_Theta_C_wrt_partial_d_h__fnd[u] / Theta_D
- partial_Theta_D_wrt_partial_d_h__fnd[u] * temp;
end do;
# Jacobian coefficients of Theta_[AB] and Theta wrt Diff(h,y_rs[u],y_rs[v])
for u from 1 to N_ang
do
for v from u to N_ang
do
partial_Theta_A_wrt_partial_dd_h__fnd[u,v]
:= frontend('diff', [Theta_A__fnd, Diff(h,y_rs[u],y_rs[v])]);
partial_Theta_B_wrt_partial_dd_h__fnd[u,v]
:= frontend('diff', [Theta_B__fnd, Diff(h,y_rs[u],y_rs[v])]);
# equation (A1b) in my 1996 apparent horizon finding paper
partial_Theta_wrt_partial_dd_h__fnd[u,v]
:= + partial_Theta_A_wrt_partial_dd_h__fnd[u,v] / Theta_D^(3/2)
+ partial_Theta_B_wrt_partial_dd_h__fnd[u,v] / Theta_D^(1/2)
end do;
end do;
if (cg_flag)
then codegen2([partial_Theta_wrt_partial_d_h__fnd,
partial_Theta_wrt_partial_dd_h__fnd],
['partial_Theta_wrt_partial_d_h',
'partial_Theta_wrt_partial_dd_h'],
"../gr.cg/expansion_Jacobian.c");
fi;
NULL;
end proc;
|
