/
reference_metric.py
1485 lines (1292 loc) · 72.3 KB
/
reference_metric.py
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# reference_metric.py: Define all needed quantities
# for a reference metric.
# Given uniform (reference metric) coordinate
# (xx[0],xx[1],xx[2]), you must define:
# 1) xxmin[3],xxmax[3]: Valid ranges for each
# uniform coordinate xx0,xx1,xx2
# 2) xxSph[3]: Spherical coordinate (r,theta,phi),
# in terms of uniform coordinate xx0,xx1,xx2
# 3) xx_to_Cart[3]: Cartesian coordinate (x,y,z),
# in terms of uniform coordinate xx0,xx1,xx2
# 4) scalefactor_orthog:
# orthogonal coordinate scale factor
# (positive root of diagonal reference metric
# components)
# 5) Cart_to_xx[3]: Inverse of xx_to_Cart:
# xx0,xx1,xx2 as functions of (x,y,z).
# In the case that there exists no closed-form
# expression, then a root finder might be needed
# 6) UnitVectors[3][3]: Unit vectors of reference
# metric.
# Author: Zachariah B. Etienne
# zachetie **at** gmail **dot* com
import sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends
from outputC import outputC, superfast_uniq, add_to_Cfunction_dict, indent_Ccode # NRPy+: Core C code output module
from outputC import outC_NRPy_basic_defines_h_dict
import NRPy_param_funcs as par # NRPy+: Parameter interface
import grid as gri # NRPy+: Functions having to do with numerical grids
import indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support
import os, sys # Standard Python modules for multiplatform OS-level functions
# Step 0a: Initialize parameters
thismodule = __name__
par.initialize_param(par.glb_param("char", thismodule, "CoordSystem", "Spherical"))
par.initialize_param(par.glb_param("char", thismodule, "enable_rfm_precompute", "False"))
par.initialize_param(par.glb_param("char", thismodule, "rfm_precompute_to_Cfunctions_and_NRPy_basic_defines", "False"))
par.initialize_param(par.glb_param("char", thismodule, "rfm_precompute_Ccode_outdir", "Ccode"))
# Step 0b: Declare global variables
xx = gri.xx
xx_to_Cart = ixp.zerorank1(DIM=4) # Must be set as a function of (xx[0],xx[1],xx[2])
Cart_to_xx = ixp.zerorank1(DIM=4) # Must be set as a function of (xx[0],xx[1],xx[2])
xxSph = ixp.zerorank1(DIM=4) # Must be set as a function of (xx[0],xx[1],xx[2])
scalefactor_orthog = ixp.zerorank1(DIM=4) # Must be set as a function of (xx[0],xx[1],xx[2])
Cartx, Carty, Cartz = sp.symbols("Cartx Carty Cartz", real=True)
Cart = [Cartx, Carty, Cartz]
scalefactor_orthog_funcform = ixp.zerorank1(DIM=4) # Must be set in terms of generic functions of xx[]s
# The following are necessary since SymPy has trouble with its native sinh and cosh functions.
def nrpysinh(x):
return (sp.exp(x) - sp.exp(-x)) * sp.Rational(1, 2)
def nrpycosh(x):
return (sp.exp(x) + sp.exp(-x)) * sp.Rational(1, 2)
have_already_called_reference_metric_function = False
def reference_metric(SymPySimplifyExpressions=True): #, enable_compute_hatted_quantities=True):
global f0_of_xx0_funcform, f1_of_xx1_funcform, f2_of_xx1_funcform, f3_of_xx0_funcform, f4_of_xx2_funcform
global f0_of_xx0, f1_of_xx1, f2_of_xx1, f2_of_xx1, f3_of_xx0, f4_of_xx2
f0_of_xx0_funcform = sp.Function('f0_of_xx0_funcform')(xx[0])
f1_of_xx1_funcform = sp.Function('f1_of_xx1_funcform')(xx[1])
f2_of_xx1_funcform = sp.Function('f2_of_xx1_funcform')(xx[1])
f3_of_xx0_funcform = sp.Function('f3_of_xx0_funcform')(xx[0])
f4_of_xx2_funcform = sp.Function('f4_of_xx2_funcform')(xx[2])
f0_of_xx0, f1_of_xx1, f2_of_xx1, f3_of_xx0, f4_of_xx2 = \
par.Cparameters("REAL", thismodule,
["f0_of_xx0", "f1_of_xx1", "f2_of_xx1", "f3_of_xx0", "f4_of_xx2"], 1e300)
# FIXME: Hack
# return values of par.Cparameters() in the following code block are unused, so we ignore them.
par.Cparameters("REAL", thismodule, ["f0_of_xx0__D0", "f0_of_xx0__DD00","f0_of_xx0__DDD000"], 1e300)
par.Cparameters("REAL", thismodule, ["f1_of_xx1__D1", "f1_of_xx1__DD11","f1_of_xx1__DDD111"], 1e300)
par.Cparameters("REAL", thismodule, ["f2_of_xx1__D1", "f2_of_xx1__DD11","f2_of_xx1__DDD111"], 1e300)
par.Cparameters("REAL", thismodule, ["f3_of_xx0__D0", "f3_of_xx0__DD00"], 1e300)
par.Cparameters("REAL", thismodule, ["f4_of_xx2__D2", "f4_of_xx2__DD22"], 1e300)
global have_already_called_reference_metric_function # setting to global enables other modules to see updated value.
have_already_called_reference_metric_function = True
CoordSystem = par.parval_from_str("reference_metric::CoordSystem")
M_PI, M_SQRT1_2 = par.Cparameters("#define", thismodule, ["M_PI", "M_SQRT1_2"], "")
global xxmin
global xxmax
global UnitVectors
UnitVectors = ixp.zerorank2(DIM=3)
# Set up hatted metric tensor, rescaling matrix, and rescaling vector
#####################################################################
# SPHERICAL-LIKE COORDINATE SYSTEMS WITH & WITHOUT RADIAL RESCALING #
#####################################################################
if CoordSystem in ('Spherical', 'SinhSpherical', 'SinhSphericalv2'):
# Adding assumption real=True can help simplify expressions involving xx[0] & xx[1] below.
xx[0] = sp.symbols("xx0", real=True)
xx[1] = sp.symbols("xx1", real=True)
if CoordSystem == "Spherical":
RMAX = par.Cparameters("REAL", thismodule, ["RMAX"],10.0)
xxmin = [sp.sympify(0), sp.sympify(0), -M_PI]
xxmax = [ RMAX, M_PI, M_PI]
r = xx[0]
th = xx[1]
ph = xx[2]
Cart_to_xx[0] = sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2)
Cart_to_xx[1] = sp.acos(Cartz / Cart_to_xx[0])
Cart_to_xx[2] = sp.atan2(Carty, Cartx)
elif CoordSystem == "SinhSpherical":
xxmin = [sp.sympify(0), sp.sympify(0), -M_PI]
xxmax = [sp.sympify(1), M_PI, M_PI]
AMPL, SINHW = par.Cparameters("REAL",thismodule,["AMPL","SINHW"],[10.0,0.2])
# Set SinhSpherical radial coordinate by default; overwrite later if CoordSystem == "SinhSphericalv2".
r = AMPL * (sp.exp(xx[0] / SINHW) - sp.exp(-xx[0] / SINHW)) / \
(sp.exp(1 / SINHW) - sp.exp(-1 / SINHW))
th = xx[1]
ph = xx[2]
Cart_to_xx[0] = SINHW*sp.asinh(sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2)*sp.sinh(1/SINHW)/AMPL)
Cart_to_xx[1] = sp.acos(Cartz / sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2))
Cart_to_xx[2] = sp.atan2(Carty, Cartx)
# SinhSphericalv2 adds the parameter "const_dr", which allows for a region near xx[0]=0 to have
# constant radial resolution of const_dr, provided the sinh() term does not dominate near xx[0]=0.
elif CoordSystem == "SinhSphericalv2":
xxmin = [sp.sympify(0), sp.sympify(0), -M_PI]
xxmax = [sp.sympify(1), M_PI, M_PI]
AMPL, SINHW = par.Cparameters("REAL",thismodule,["AMPL","SINHW"],[10.0,0.2])
const_dr = par.Cparameters("REAL",thismodule,["const_dr"],0.0625)
r = AMPL*( const_dr*xx[0] + (sp.exp(xx[0] / SINHW) - sp.exp(-xx[0] / SINHW)) /
(sp.exp(1 / SINHW) - sp.exp(-1 / SINHW)) )
th = xx[1]
ph = xx[2]
# NO CLOSED-FORM EXPRESSION FOR RADIAL INVERSION.
# Cart_to_xx[0] = "NewtonRaphson"
# Cart_to_xx[1] = sp.acos(Cartz / sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2))
# Cart_to_xx[2] = sp.atan2(Carty, Cartx)
xxSph[0] = r
xxSph[1] = th
xxSph[2] = ph
# Now define xCart, yCart, and zCart in terms of x0,xx[1],xx[2].
# Note that the relation between r and x0 is not necessarily trivial in SinhSpherical coordinates. See above.
xx_to_Cart[0] = xxSph[0]*sp.sin(xxSph[1])*sp.cos(xxSph[2])
xx_to_Cart[1] = xxSph[0]*sp.sin(xxSph[1])*sp.sin(xxSph[2])
xx_to_Cart[2] = xxSph[0]*sp.cos(xxSph[1])
scalefactor_orthog[0] = sp.diff(xxSph[0],xx[0])
scalefactor_orthog[1] = xxSph[0]
scalefactor_orthog[2] = xxSph[0]*sp.sin(xxSph[1])
f0_of_xx0 = xxSph[0]
f1_of_xx1 = sp.sin(xxSph[1])
scalefactor_orthog_funcform[0] = sp.diff(f0_of_xx0_funcform,xx[0])
scalefactor_orthog_funcform[1] = f0_of_xx0_funcform
scalefactor_orthog_funcform[2] = f0_of_xx0_funcform*f1_of_xx1_funcform
# Set the unit vectors
UnitVectors = [[sp.sin(xxSph[1])*sp.cos(xxSph[2]), sp.sin(xxSph[1])*sp.sin(xxSph[2]), sp.cos(xxSph[1])],
[sp.cos(xxSph[1])*sp.cos(xxSph[2]), sp.cos(xxSph[1])*sp.sin(xxSph[2]), -sp.sin(xxSph[1])],
[ -sp.sin(xxSph[2]), sp.cos(xxSph[2]), sp.sympify(0) ]]
######################################################################
# SPHERICAL-LIKE COORDINATE SYSTEMS WITH RADIAL AND THETA RESCALINGS #
######################################################################
elif CoordSystem in ('NobleSphericalThetaOptionOne', 'NobleSphericalThetaOptionTwo'):
# WARNING: CANNOT BE USED FOR SENR RUNS;
# THESE DO NOT DEFINE xxmin, xxmax, Cart_to_xx
# ALSO THE RADIAL RESCALINGS ARE NOT ODD FUNCTIONS OF xx0,
# MEANING THAT CURVI. BOUNDARY CONDITIONS WILL NOT WORK.
Rin,R0 = par.Cparameters("REAL", thismodule, ["Rin","R0"],[1.08986052555408,0.0])
x0beg = sp.log(Rin-R0)
xx[0] = sp.symbols("xx0", real=True)
r = R0 + sp.exp(x0beg + xx[0])
# 0.053407075111026485 == 0.017*pi
th_c,xi,x1beg = par.Cparameters("REAL", thismodule, ["th_c","xi","x1beg"],[0.053407075111026485,0.25,0.0])
xx[1] = sp.symbols("xx1", real=True)
x1j = x1beg + xx[1]
if CoordSystem == "NobleSphericalThetaOptionOne":
th = th_c + (M_PI - 2*th_c)*x1j + xi*sp.sin(2*M_PI*x1j)
elif CoordSystem == "NobleSphericalThetaOptionTwo":
x1_n_exponent = par.Cparameters("REAL", thismodule, ["x1_n_exponent"],9.0)
th = M_PI/2 * ( 1 + (1 - xi)*(2*x1j - 1) + (xi - 2*th_c/M_PI)*(2*x1j - 1)**x1_n_exponent )
xx[2] = sp.symbols("xx2", real=True)
ph = xx[2]
xxSph[0] = r
xxSph[1] = th
xxSph[2] = ph
# Now define xCart, yCart, and zCart in terms of x0,xx[1],xx[2].
# Note that the relation between r and x0 is not necessarily trivial in SinhSpherical coordinates. See above.
xx_to_Cart[0] = xxSph[0]*sp.sin(xxSph[1])*sp.cos(xxSph[2])
xx_to_Cart[1] = xxSph[0]*sp.sin(xxSph[1])*sp.sin(xxSph[2])
xx_to_Cart[2] = xxSph[0]*sp.cos(xxSph[1])
scalefactor_orthog[0] = sp.diff(xxSph[0],xx[0])
scalefactor_orthog[1] = xxSph[0]
scalefactor_orthog[2] = xxSph[0]*sp.sin(xxSph[1])
# Set the unit vectors
UnitVectors = [[ sp.sin(xxSph[1])*sp.cos(xxSph[2]), sp.sin(xxSph[1])*sp.sin(xxSph[2]), sp.cos(xxSph[1])],
[ sp.cos(xxSph[1])*sp.cos(xxSph[2]), sp.cos(xxSph[1])*sp.sin(xxSph[2]), -sp.sin(xxSph[1])],
[ -sp.sin(xxSph[2]), sp.cos(xxSph[2]), sp.sympify(0) ]]
##########################################################################
# CYLINDRICAL-LIKE COORDINATE SYSTEMS WITH & WITHOUT RADIAL/Z RESCALINGS #
##########################################################################
elif CoordSystem in ('Cylindrical', 'SinhCylindrical', 'SinhCylindricalv2'):
# Assuming the cylindrical radial coordinate
# is positive makes nice simplifications of
# unit vectors possible.
xx[0] = sp.symbols("xx0", real=True)
if CoordSystem == "Cylindrical":
RHOMAX,ZMIN,ZMAX = par.Cparameters("REAL",thismodule,["RHOMAX","ZMIN","ZMAX"],[10.0,-10.0,10.0])
xxmin = [sp.sympify(0), -M_PI, ZMIN]
xxmax = [ RHOMAX, M_PI, ZMAX]
RHOCYL = xx[0]
PHICYL = xx[1]
ZCYL = xx[2]
Cart_to_xx[0] = sp.sqrt(Cartx ** 2 + Carty ** 2)
Cart_to_xx[1] = sp.atan2(Carty, Cartx)
Cart_to_xx[2] = Cartz
elif CoordSystem == "SinhCylindrical":
xxmin = [sp.sympify(0), -M_PI, sp.sympify(-1)]
xxmax = [sp.sympify(1), M_PI, sp.sympify(+1)]
AMPLRHO, SINHWRHO, AMPLZ, SINHWZ = par.Cparameters("REAL",thismodule,
["AMPLRHO", "SINHWRHO", "AMPLZ", "SINHWZ"],
[ 10.0, 0.2, 10.0, 0.2])
# Set SinhCylindrical radial & z coordinates by default; overwrite later if CoordSystem == "SinhCylindricalv2".
RHOCYL = AMPLRHO * (sp.exp(xx[0] / SINHWRHO) - sp.exp(-xx[0] / SINHWRHO)) / (sp.exp(1 / SINHWRHO) - sp.exp(-1 / SINHWRHO))
# phi coordinate remains unchanged.
PHICYL = xx[1]
ZCYL = AMPLZ * (sp.exp(xx[2] / SINHWZ) - sp.exp(-xx[2] / SINHWZ)) / (sp.exp(1 / SINHWZ) - sp.exp(-1 / SINHWZ))
Cart_to_xx[0] = SINHWRHO*sp.asinh(sp.sqrt(Cartx ** 2 + Carty ** 2)*sp.sinh(1/SINHWRHO)/AMPLRHO)
Cart_to_xx[1] = sp.atan2(Carty, Cartx)
Cart_to_xx[2] = SINHWZ*sp.asinh(Cartz*sp.sinh(1/SINHWZ)/AMPLZ)
# SinhCylindricalv2 adds the parameters "const_drho", "const_dz", which allows for regions near xx[0]=0
# and xx[2]=0 to have constant rho and z resolution of const_drho and const_dz, provided the sinh() terms
# do not dominate near xx[0]=0 and xx[2]=0.
elif CoordSystem == "SinhCylindricalv2":
xxmin = [sp.sympify(0), -M_PI, sp.sympify(-1)]
xxmax = [sp.sympify(1), M_PI, sp.sympify(+1)]
AMPLRHO, SINHWRHO, AMPLZ, SINHWZ = par.Cparameters("REAL", thismodule,
["AMPLRHO", "SINHWRHO", "AMPLZ", "SINHWZ"],
[ 10.0, 0.2, 10.0, 0.2])
const_drho, const_dz = par.Cparameters("REAL", thismodule, ["const_drho", "const_dz"], [0.0625, 0.0625])
RHOCYL = AMPLRHO * ( const_drho*xx[0] + (sp.exp(xx[0] / SINHWRHO) - sp.exp(-xx[0] / SINHWRHO)) / (sp.exp(1 / SINHWRHO) - sp.exp(-1 / SINHWRHO)) )
PHICYL = xx[1]
ZCYL = AMPLZ * ( const_dz *xx[2] + (sp.exp(xx[2] / SINHWZ ) - sp.exp(-xx[2] / SINHWZ )) / (sp.exp(1 / SINHWZ ) - sp.exp(-1 / SINHWZ )) )
# NO CLOSED-FORM EXPRESSION FOR RADIAL OR Z INVERSION.
# Cart_to_xx[0] = "NewtonRaphson"
# Cart_to_xx[1] = sp.atan2(Carty, Cartx)
# Cart_to_xx[2] = "NewtonRaphson"
xx_to_Cart[0] = RHOCYL*sp.cos(PHICYL)
xx_to_Cart[1] = RHOCYL*sp.sin(PHICYL)
xx_to_Cart[2] = ZCYL
xxSph[0] = sp.sqrt(RHOCYL**2 + ZCYL**2)
xxSph[1] = sp.acos(ZCYL / xxSph[0])
xxSph[2] = PHICYL
scalefactor_orthog[0] = sp.diff(RHOCYL,xx[0])
scalefactor_orthog[1] = RHOCYL
scalefactor_orthog[2] = sp.diff(ZCYL,xx[2])
f0_of_xx0 = RHOCYL
f4_of_xx2 = sp.diff(ZCYL,xx[2])
scalefactor_orthog_funcform[0] = sp.diff(f0_of_xx0_funcform,xx[0])
scalefactor_orthog_funcform[1] = f0_of_xx0_funcform
scalefactor_orthog_funcform[2] = f4_of_xx2_funcform
# Set the unit vectors
UnitVectors = [[ sp.cos(PHICYL), sp.sin(PHICYL), sp.sympify(0)],
[-sp.sin(PHICYL), sp.cos(PHICYL), sp.sympify(0)],
[ sp.sympify(0), sp.sympify(0), sp.sympify(1)]]
elif CoordSystem in ('SymTP', 'SinhSymTP'):
# var1, var2= sp.symbols('var1 var2',real=True)
bScale, SINHWAA, AMAX = par.Cparameters("REAL", thismodule,
["bScale", "SINHWAA", "AMAX"],
[0.5, 0.2, 10.0 ])
# Assuming xx0, xx1, and bScale
# are positive makes nice simplifications of
# unit vectors possible.
xx[0],xx[1] = sp.symbols("xx0 xx1", real=True)
xxmin = [sp.sympify(0), sp.sympify(0), -M_PI]
xxmax = [ AMAX, M_PI, M_PI]
AA = xx[0]
if CoordSystem == "SinhSymTP":
xxmax[0] = sp.sympify(1)
# With xxmax[0] = 1, sinh(xx0/SINHWAA) / sinh(1/SINHWAA) will evaluate to a number between 0 and 1.
# Then AA = AMAX * sinh(xx0/SINHWAA) / sinh(1/SINHWAA) will evaluate to a number between 0 and AMAX.
AA = AMAX * (sp.exp(xx[0] / SINHWAA) - sp.exp(-xx[0] / SINHWAA)) / (sp.exp(1 / SINHWAA) - sp.exp(-1 / SINHWAA))
var1 = sp.sqrt(AA**2 + (bScale * sp.sin(xx[1]))**2)
var2 = sp.sqrt(AA**2 + bScale**2)
RHOSYMTP = AA*sp.sin(xx[1])
PHSYMTP = xx[2]
ZSYMTP = var2*sp.cos(xx[1])
xx_to_Cart[0] = AA *sp.sin(xx[1])*sp.cos(xx[2])
xx_to_Cart[1] = AA *sp.sin(xx[1])*sp.sin(xx[2])
xx_to_Cart[2] = ZSYMTP
xxSph[0] = sp.sqrt(RHOSYMTP**2 + ZSYMTP**2)
xxSph[1] = sp.acos(ZSYMTP / xxSph[0])
xxSph[2] = PHSYMTP
if CoordSystem == "SymTP":
rSph = sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2)
thSph = sp.acos(Cartz / rSph)
phSph = sp.atan2(Carty, Cartx)
# Mathematica script to compute Cart_to_xx[]
# AA = x1;
# var2 = Sqrt[AA^2 + bScale^2];
# RHOSYMTP = AA*Sin[x2];
# ZSYMTP = var2*Cos[x2];
# Solve[{rSph == Sqrt[RHOSYMTP^2 + ZSYMTP^2],
# thSph == ArcCos[ZSYMTP/Sqrt[RHOSYMTP^2 + ZSYMTP^2]],
# phSph == x3},
# {x1, x2, x3}]
Cart_to_xx[0] = sp.sqrt(-bScale**2 + rSph**2 +
sp.sqrt(bScale**4 + 2*bScale**2*rSph**2 + rSph**4 -
4*bScale**2*rSph**2*sp.cos(thSph)**2))*M_SQRT1_2 # M_SQRT1_2 = 1/sqrt(2); define this way for UnitTesting
# The sign() function in the following expression ensures the correct root is taken.
Cart_to_xx[1] = sp.acos(sp.sign(Cartz)*(
sp.sqrt(1 + rSph**2/bScale**2 -
sp.sqrt(bScale**4 + 2*bScale**2*rSph**2 + rSph**4 -
4*bScale**2*rSph**2*sp.cos(thSph)**2)/bScale**2)*M_SQRT1_2)) # M_SQRT1_2 = 1/sqrt(2); define this way for UnitTesting
Cart_to_xx[2] = phSph
elif CoordSystem == "SinhSymTP":
rSph = sp.sqrt(Cartx ** 2 + Carty ** 2 + Cartz ** 2)
thSph = sp.acos(Cartz / rSph)
phSph = sp.atan2(Carty, Cartx)
# Mathematica script to compute Cart_to_xx[]
# AA = x1;
# var2 = Sqrt[AA^2 + bScale^2];
# RHOSYMTP = AA*Sin[x2];
# ZSYMTP = var2*Cos[x2];
# Solve[{rSph == Sqrt[RHOSYMTP^2 + ZSYMTP^2],
# thSph == ArcCos[ZSYMTP/Sqrt[RHOSYMTP^2 + ZSYMTP^2]],
# phSph == x3},
# {x1, x2, x3}]
Cart_to_xx[0] = sp.sqrt(-bScale**2 + rSph**2 +
sp.sqrt(bScale**4 + 2*bScale**2*rSph**2 + rSph**4 -
4*bScale**2*rSph**2*sp.cos(thSph)**2))*M_SQRT1_2 # M_SQRT1_2 = 1/sqrt(2); define this way for UnitTesting
# The sign() function in the following expression ensures the correct root is taken.
Cart_to_xx[1] = sp.acos(sp.sign(Cartz)*(
sp.sqrt(1 + rSph**2/bScale**2 -
sp.sqrt(bScale**4 + 2*bScale**2*rSph**2 + rSph**4 -
4*bScale**2*rSph**2*sp.cos(thSph)**2)/bScale**2)*M_SQRT1_2)) # M_SQRT1_2 = 1/sqrt(2); define this way for UnitTesting
Cart_to_xx[2] = phSph
scalefactor_orthog[0] = sp.diff(AA,xx[0]) * var1 / var2
scalefactor_orthog[1] = var1
scalefactor_orthog[2] = AA * sp.sin(xx[1])
f0_of_xx0 = AA
f1_of_xx1 = sp.sin(xx[1])
f2_of_xx1 = bScale * sp.sin(xx[1])
f3_of_xx0 = var2
var1_funcform = sp.sqrt(
f0_of_xx0_funcform**2 + f2_of_xx1_funcform**2
)
scalefactor_orthog_funcform[0] = sp.diff(f0_of_xx0_funcform,xx[0]) * var1_funcform/f3_of_xx0_funcform
scalefactor_orthog_funcform[1] = var1_funcform
scalefactor_orthog_funcform[2] = f0_of_xx0_funcform*f1_of_xx1_funcform
# Set the transpose of the matrix of unit vectors
UnitVectors = [[sp.sin(xx[1]) * sp.cos(xx[2]) * var2 / var1,
sp.sin(xx[1]) * sp.sin(xx[2]) * var2 / var1,
AA * sp.cos(xx[1]) / var1],
[AA * sp.cos(xx[1]) * sp.cos(xx[2]) / var1,
AA * sp.cos(xx[1]) * sp.sin(xx[2]) / var1,
-sp.sin(xx[1]) * var2 / var1],
[-sp.sin(xx[2]), sp.cos(xx[2]), sp.sympify(0)]]
#####################################
# CARTESIAN-LIKE COORDINATE SYSTEMS #
#####################################
elif CoordSystem == "Cartesian":
# return values of par.Cparameters() in the following line of code are unused, so we ignore them.
par.Cparameters("REAL",thismodule, ["xmin","xmax","ymin","ymax","zmin","zmax"],
[ -10.0, 10.0, -10.0, 10.0, -10.0, 10.0])
xxmin = ["xmin", "ymin", "zmin"]
xxmax = ["xmax", "ymax", "zmax"]
xx_to_Cart[0] = xx[0]
xx_to_Cart[1] = xx[1]
xx_to_Cart[2] = xx[2]
xxSph[0] = sp.sqrt(xx[0] ** 2 + xx[1] ** 2 + xx[2] ** 2)
xxSph[1] = sp.acos(xx[2] / xxSph[0])
xxSph[2] = sp.atan2(xx[1], xx[0])
Cart_to_xx[0] = Cartx
Cart_to_xx[1] = Carty
Cart_to_xx[2] = Cartz
scalefactor_orthog[0] = sp.sympify(1)
scalefactor_orthog[1] = sp.sympify(1)
scalefactor_orthog[2] = sp.sympify(1)
scalefactor_orthog_funcform[0] = sp.sympify(1)
scalefactor_orthog_funcform[1] = sp.sympify(1)
scalefactor_orthog_funcform[2] = sp.sympify(1)
# Set the transpose of the matrix of unit vectors
UnitVectors = [[sp.sympify(1), sp.sympify(0), sp.sympify(0)],
[sp.sympify(0), sp.sympify(1), sp.sympify(0)],
[sp.sympify(0), sp.sympify(0), sp.sympify(1)]]
elif "SinhCartesian" in CoordSystem:
# SinhCartesian coordinates allows us to push the outer boundary of the
# computational domain a lot further away, while keeping reasonably high
# resolution towards the center of the computational grid.
# Set default values for min and max (x,y,z)
xxmin = [sp.sympify(-1), sp.sympify(-1), sp.sympify(-1)]
xxmax = [sp.sympify(+1), sp.sympify(+1), sp.sympify(+1)]
# Declare basic parameters of the coordinate system and their default values
AMPLXYZ, SINHWXYZ = par.Cparameters("REAL", thismodule,
["AMPLXYZ", "SINHWXYZ"],
[ 10.0, 0.2])
# Compute (xx_to_Cart0,xx_to_Cart1,xx_to_Cart2) from (xx0,xx1,xx2)
for ii in [0, 1, 2]:
xx_to_Cart[ii] = AMPLXYZ*(sp.exp(xx[ii]/SINHWXYZ) - sp.exp(-xx[ii]/SINHWXYZ))/(sp.exp(1/SINHWXYZ) - sp.exp(-1/SINHWXYZ))
# Compute (r,th,ph) from (xx_to_Cart2,xx_to_Cart1,xx_to_Cart2)
xxSph[0] = sp.sqrt(xx_to_Cart[0] ** 2 + xx_to_Cart[1] ** 2 + xx_to_Cart[2] ** 2)
xxSph[1] = sp.acos(xx_to_Cart[2] / xxSph[0])
xxSph[2] = sp.atan2(xx_to_Cart[1], xx_to_Cart[0])
# Compute (xx0,xx1,xx2) from (Cartx,Carty,Cartz)
Cart_to_xx[0] = SINHWXYZ*sp.asinh(Cartx*sp.sinh(1/SINHWXYZ)/AMPLXYZ)
Cart_to_xx[1] = SINHWXYZ*sp.asinh(Carty*sp.sinh(1/SINHWXYZ)/AMPLXYZ)
Cart_to_xx[2] = SINHWXYZ*sp.asinh(Cartz*sp.sinh(1/SINHWXYZ)/AMPLXYZ)
# Compute scale factors
scalefactor_orthog[0] = sp.diff(xx_to_Cart[0], xx[0])
scalefactor_orthog[1] = sp.diff(xx_to_Cart[1], xx[1])
scalefactor_orthog[2] = sp.diff(xx_to_Cart[2], xx[2])
f0_of_xx0 = sp.diff(xx_to_Cart[0], xx[0])
f1_of_xx1 = sp.diff(xx_to_Cart[1], xx[1])
f4_of_xx2 = sp.diff(xx_to_Cart[2], xx[2])
scalefactor_orthog_funcform[0] = f0_of_xx0_funcform
scalefactor_orthog_funcform[1] = f1_of_xx1_funcform
scalefactor_orthog_funcform[2] = f4_of_xx2_funcform
# Set the transpose of the matrix of unit vectors
UnitVectors = [[sp.sympify(1), sp.sympify(0), sp.sympify(0)],
[sp.sympify(0), sp.sympify(1), sp.sympify(0)],
[sp.sympify(0), sp.sympify(0), sp.sympify(1)]]
else:
print("CoordSystem == " + CoordSystem + " is not supported.")
sys.exit(1)
# Finally, call ref_metric__hatted_quantities()
# to construct hatted metric, derivs of hatted
# metric, and Christoffel symbols
ref_metric__hatted_quantities(SymPySimplifyExpressions)
# ref_metric__hatted_quantities(scalefactor_orthog_funcform,SymPySimplifyExpressions)
# ref_metric__hatted_quantities(scalefactor_orthog,SymPySimplifyExpressions)
def ref_metric__hatted_quantities(SymPySimplifyExpressions=True):
enable_rfm_precompute = False
if par.parval_from_str(thismodule+"::enable_rfm_precompute") == "True":
enable_rfm_precompute = True
# Step 0: Set dimension DIM
DIM = par.parval_from_str("grid::DIM")
global ReU, ReD, ReDD, ghatDD
ReU = ixp.zerorank1()
ReD = ixp.zerorank1()
ReDD = ixp.zerorank2()
ghatDD = ixp.zerorank2()
# Step 1: Compute ghatDD (reference metric), ghatUU
# (inverse reference metric), as well as
# rescaling vector ReU & rescaling matrix ReDD
if not enable_rfm_precompute:
for i in range(DIM):
scalefactor_orthog[i] = sp.sympify(scalefactor_orthog[i])
ghatDD[i][i] = scalefactor_orthog[i]**2
ReU[i] = 1/scalefactor_orthog[i]
ReD[i] = scalefactor_orthog[i]
for j in range(DIM):
ReDD[i][j] = scalefactor_orthog[i]*scalefactor_orthog[j]
else:
for i in range(DIM):
scalefactor_orthog_funcform[i] = sp.sympify(scalefactor_orthog_funcform[i])
ghatDD[i][i] = scalefactor_orthog_funcform[i]**2
ReU[i] = 1/scalefactor_orthog_funcform[i]
ReD[i] = scalefactor_orthog_funcform[i]
for j in range(DIM):
ReDD[i][j] = scalefactor_orthog_funcform[i]*scalefactor_orthog_funcform[j]
# Step 1b: Compute ghatUU and detgammahat
global ghatUU
global detgammahat
ghatUU = ixp.zerorank2()
ghatUU, detgammahat = ixp.symm_matrix_inverter3x3(ghatDD)
# Step 1c: Sanity check: verify that ReDD, ghatDD,
# and ghatUU are all symmetric rank-2:
for i in range(DIM):
for j in range(DIM):
if ReDD[i][j] != ReDD[j][i]:
print("Error: ReDD["+ str(i) + "][" + str(j) + "] != ReDD["+ str(j) + "][" + str(i) + ": " + str(ReDD[i][j]) + "!=" + str(ReDD[j][i]))
sys.exit(1)
if ghatDD[i][j] != ghatDD[j][i]:
print("Error: ghatDD["+ str(i) + "][" + str(j) + "] != ghatDD["+ str(j) + "][" + str(i) + ": " + str(ghatDD[i][j]) + "!=" + str(ghatDD[j][i]))
sys.exit(1)
if ghatUU[i][j] != ghatUU[j][i]:
print("Error: ghatUU["+ str(i) + "][" + str(j) + "] != ghatUU["+ str(j) + "][" + str(i) + ": " + str(ghatUU[i][j]) + "!=" + str(ghatUU[j][i]))
sys.exit(1)
# Step 2: Compute det(ghat) and its 1st & 2nd derivatives
global detgammahatdD, detgammahatdDD
detgammahatdD = ixp.zerorank1(DIM)
detgammahatdDD = ixp.zerorank2(DIM)
for i in range(DIM):
detgammahatdD[i] = (sp.diff(detgammahat, xx[i]))
for j in range(DIM):
detgammahatdDD[i][j] = sp.diff(detgammahatdD[i], xx[j])
# Step 3a: Compute 1st & 2nd derivatives of rescaling vectors.
# (E.g., needed in BSSN for betaUdDD computation)
global ReUdD, ReUdDD, ReDdD, ReDdDD
ReUdD = ixp.zerorank2(DIM)
ReUdDD = ixp.zerorank3(DIM)
ReDdD = ixp.zerorank2(DIM)
ReDdDD = ixp.zerorank3(DIM)
for i in range(DIM):
for j in range(DIM):
ReUdD[i][j] = sp.diff(ReU[i], xx[j])
ReDdD[i][j] = sp.diff(ReD[i], xx[j])
for k in range(DIM):
ReUdDD[i][j][k] = sp.diff(ReUdD[i][j], xx[k])
ReDdDD[i][j][k] = sp.diff(ReDdD[i][j], xx[k])
# Step 3b: Compute 1st & 2nd derivatives of rescaling matrix.
global ReDDdD, ReDDdDD
ReDDdD = ixp.zerorank3(DIM)
ReDDdDD = ixp.zerorank4(DIM)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
ReDDdD[i][j][k] = (sp.diff(ReDD[i][j], xx[k]))
for l in range(DIM):
# Simplifying this doesn't appear to help overall NRPy run time.
ReDDdDD[i][j][k][l] = sp.diff(ReDDdD[i][j][k], xx[l])
# Step 3c: Compute 1st & 2nd derivatives of reference metric.
global ghatDDdD, ghatDDdDD
ghatDDdD = ixp.zerorank3(DIM)
ghatDDdDD = ixp.zerorank4(DIM)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
if SymPySimplifyExpressions==True:
# ghatDDdD[i][j][k] = sp.trigsimp(sp.diff(ghatDD[i][j], xx[k])) # FIXME: BAD: MUST BE SIMPLIFIED OR ANSWER IS INCORRECT! Must be some bug in sympy...
ghatDDdD[i][j][k] = sp.simplify(sp.diff(ghatDD[i][j], xx[k])) # FIXME: BAD: MUST BE SIMPLIFIED OR ANSWER IS INCORRECT! Must be some bug in sympy...
else:
ghatDDdD[i][j][k] = (sp.diff(ghatDD[i][j], xx[k])) # FIXME: BAD: MUST BE SIMPLIFIED OR ANSWER IS INCORRECT! Must be some bug in sympy...
for l in range(DIM):
ghatDDdDD[i][j][k][l] = (sp.diff(ghatDDdD[i][j][k], xx[l]))
# Step 4a: Compute Christoffel symbols of reference metric.
global GammahatUDD
GammahatUDD = ixp.zerorank3(DIM)
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
# GammahatUDD[i][k][l] += sp.trigsimp((sp.Rational(1,2))*ghatUU[i][m]*\
GammahatUDD[i][k][l] += (sp.Rational(1, 2))*ghatUU[i][m]*\
(ghatDDdD[m][k][l] + ghatDDdD[m][l][k] - ghatDDdD[k][l][m])
# Step 4b: Compute derivs of Christoffel symbols of reference metric.
global GammahatUDDdD
GammahatUDDdD = ixp.zerorank4(DIM)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GammahatUDDdD[i][j][k][l] = (sp.diff(GammahatUDD[i][j][k], xx[l]))
# Step 4c: If rfm_precompute is disabled, then we are finished with this function.
# Otherwise continue to Step 5.
if not enable_rfm_precompute:
return
# Step 5: Now that all hatted quantities are written in terms of generic SymPy functions,
# we will now replace SymPy functions with simple variables using rigid NRPy+ syntax,
# and store these variables to globals defined above.
def make_replacements(expr):
sympy_version = sp.__version__.replace("rc", "...").replace("b", "...") # Ignore the rc's and b's
# (release candidates & betas).
sympy_version_decimal = float(int(sympy_version.split(".")[0]) + int(sympy_version.split(".")[1]) / 10.0)
is_old_sympy_version = sympy_version_decimal < 1.2
# The derivative representation changed with SymPy 1.2, forcing version-dependent behavior.
# Example: Derivative(f0_of_xx0_funcform(xx0)(xx0), (xx0, 2)) >> f0_of_xx0__DD00
rule = {} # replacement dictionary
for item in sp.preorder_traversal(expr):
if item.func == sp.Derivative:
# extract function name before '_funcform'
strfunc = str(item.args[0]).split('_funcform(', 1)[0]
if is_old_sympy_version:
# extract differentiation variable and derivative order (SymPy <= 1.1)
var, order = str(item.args[1])[2:], len(item.args) - 1
else:
# extract differentiation variable and derivative order (SymPy >= 1.2)
var, order = str(item.args[1][0])[2:], item.args[1][1]
# build derivative operator with format: __DD...D(var)(var)...(var) where
# D and (var) are repeated for every derivative order
oper = '__D' + 'D' * (order - 1) + var * order
# add replacement rule to dictionary
rule[item] = sp.sympify(strfunc + oper)
expr = expr.xreplace(rule)
rule = {}
# Example: f0_of_xx0_funcform(xx0)(xx0) >> f0_of_xx0
for item in sp.preorder_traversal(expr):
if "_funcform" in str(item.func):
# extract function name before '_funcform'
strfunc = str(item.func).split("_funcform", 1)[0]
# add replacement rule to dictionary
rule[item] = sp.sympify(strfunc)
return expr.xreplace(rule)
# Step 6: enable_rfm_precompute: precompute and store in memory
# expressions related to the reference metric (a.k.a.,
# "hatted quantities"). These expressions may involve
# transcendental functions, which are expensive to compute
# within nested loops in C. Hence we precompute them and
# store the result.
# The precomputed "hatted quantity" expressions will be stored in
# a C struct called rfmstruct. As these expressions generally
# involve computationally expensive transcendental functions
# of xx0,xx1,or xx2, and xx0,xx1, and xx2 remain fixed across
# most (if not all) of a given simulation, setting up the
# rfmstruct can greatly improve performance.
# The core challenge in setting up the rfmstruct is collecting
# all the information needed to automatically generate it.
# Step 5 and onwards implements this algorithm, using the
# *generic functional form* of the hatted quantities (as
# opposed to the exact closed-form expressions of the
# hatted quantities) computed above.
detgammahat = make_replacements(detgammahat)
for i in range(DIM):
ReU[i] = make_replacements(ReU[i])
ReD[i] = make_replacements(ReD[i])
detgammahatdD[i] = make_replacements(detgammahatdD[i])
for j in range(DIM):
ReDD[i][j] = make_replacements(ReDD[i][j])
ReUdD[i][j] = make_replacements(ReUdD[i][j])
ReDdD[i][j] = make_replacements(ReDdD[i][j])
ghatDD[i][j] = make_replacements(ghatDD[i][j])
ghatUU[i][j] = make_replacements(ghatUU[i][j])
detgammahatdDD[i][j] = make_replacements(detgammahatdDD[i][j])
for k in range(DIM):
ReDDdD[i][j][k] = make_replacements(ReDDdD[i][j][k])
ReUdDD[i][j][k] = make_replacements(ReUdDD[i][j][k])
ReDdDD[i][j][k] = make_replacements(ReDdDD[i][j][k])
ghatDDdD[i][j][k] = make_replacements(ghatDDdD[i][j][k])
GammahatUDD[i][j][k] = make_replacements(GammahatUDD[i][j][k])
for l in range(DIM):
ReDDdDD[i][j][k][l] = make_replacements(ReDDdDD[i][j][k][l])
ghatDDdDD[i][j][k][l] = make_replacements(ghatDDdDD[i][j][k][l])
GammahatUDDdD[i][j][k][l] = make_replacements(GammahatUDDdD[i][j][k][l])
# Step 6: At this point, each expression is written in terms of the generic functions
# of xx0, xx1, and/or xx2 and their derivatives. Depending on the functions, some
# of these derivatives may be zero. In this step, we'll evaluate the function
# derivatives exactly and set the expressions to zero. Otherwise in the C code
# we'd be storing performing arithmetic with zeros -- wasteful!
# Step 6.a: Construct the full list of *unique* NRPy+ variables representing the
# SymPy functions and derivatives, so that all zero derivatives can be
# computed.
freevars = []
freevars.extend(detgammahat.free_symbols)
for i in range(DIM):
freevars.extend(ReU[i].free_symbols)
freevars.extend(ReD[i].free_symbols)
freevars.extend(detgammahatdD[i].free_symbols)
for j in range(DIM):
freevars.extend(ReDD[i][j].free_symbols)
freevars.extend(ReUdD[i][j].free_symbols)
freevars.extend(ReDdD[i][j].free_symbols)
freevars.extend(ghatDD[i][j].free_symbols)
freevars.extend(ghatUU[i][j].free_symbols)
freevars.extend(detgammahatdDD[i][j].free_symbols)
for k in range(DIM):
freevars.extend(ReDDdD[i][j][k].free_symbols)
freevars.extend(ReUdDD[i][j][k].free_symbols)
freevars.extend(ReDdDD[i][j][k].free_symbols)
freevars.extend(ghatDDdD[i][j][k].free_symbols)
freevars.extend(GammahatUDD[i][j][k].free_symbols)
for l in range(DIM):
freevars.extend(ReDDdDD[i][j][k][l].free_symbols)
freevars.extend(ghatDDdDD[i][j][k][l].free_symbols)
freevars.extend(GammahatUDDdD[i][j][k][l].free_symbols)
freevars_uniq = superfast_uniq(freevars)
freevars_uniq_xx_indep = []
for i in range(len(freevars_uniq)):
freevars_uniq_xx_indep.append(freevars_uniq[i])
# Step 6.b: Using the expressions f?_of_xx? set in reference_metric(),
# evaluate each needed derivative and, in the case it is zero,
# set the corresponding "freevar" variable to zero.
freevars_uniq_vals = []
for i, var in enumerate(freevars_uniq):
basename = str(var).split("__")[0].replace("_funcform", "")
derivatv = ""
if "__" in str(var):
derivatv = str(var).split("__")[1].replace("_funcform", "")
if basename == "f0_of_xx0":
basefunc = f0_of_xx0
elif basename == "f1_of_xx1":
basefunc = f1_of_xx1
elif basename == "f2_of_xx1":
basefunc = f2_of_xx1
elif basename == "f3_of_xx0":
basefunc = f3_of_xx0
elif basename == "f4_of_xx2":
basefunc = f4_of_xx2
else:
print("Error: function inside " + str(var) + " undefined.")
sys.exit(1)
diff_result = basefunc
if derivatv == "":
pass
else:
derivorder = derivatv.replace("d", "").replace("D", "").\
replace("0", "0 ").replace("1", "1 ").replace("2", "2 ").split(" ")
for derivdirn in derivorder:
if derivdirn != "":
derivwrt = xx[int(derivdirn)]
diff_result = sp.diff(diff_result, derivwrt)
freevars_uniq_vals.append(diff_result)
frees_uniq = superfast_uniq(diff_result.free_symbols)
has_xx_dependence = False
for dirn in range(3):
if gri.xx[dirn] in frees_uniq:
has_xx_dependence = True
if not has_xx_dependence:
freevars_uniq_xx_indep[i] = diff_result
# Step 6.c: Finally, substitute integers for all functions & derivatives that evaluate to integers
for varidx, freevar in enumerate(freevars_uniq):
detgammahat = detgammahat.subs(freevar, freevars_uniq_xx_indep[varidx])
for i in range(DIM):
ReU[i] = ReU[i].subs(freevar, freevars_uniq_xx_indep[varidx])
ReD[i] = ReD[i].subs(freevar, freevars_uniq_xx_indep[varidx])
detgammahatdD[i] = detgammahatdD[i].subs(freevar, freevars_uniq_xx_indep[varidx])
for j in range(DIM):
ReDD[i][j] = ReDD[i][j].subs(freevar, freevars_uniq_xx_indep[varidx])
ReUdD[i][j] = ReUdD[i][j].subs(freevar, freevars_uniq_xx_indep[varidx])
ReDdD[i][j] = ReDdD[i][j].subs(freevar, freevars_uniq_xx_indep[varidx])
ghatDD[i][j] = ghatDD[i][j].subs(freevar, freevars_uniq_xx_indep[varidx])
ghatUU[i][j] = ghatUU[i][j].subs(freevar, freevars_uniq_xx_indep[varidx])
detgammahatdDD[i][j] = detgammahatdDD[i][j].subs(freevar,
freevars_uniq_xx_indep[varidx])
for k in range(DIM):
ReDDdD[i][j][k] = ReDDdD[i][j][k].subs(freevar, freevars_uniq_xx_indep[varidx])
ReUdDD[i][j][k] = ReUdDD[i][j][k].subs(freevar, freevars_uniq_xx_indep[varidx])
ReDdDD[i][j][k] = ReDdDD[i][j][k].subs(freevar, freevars_uniq_xx_indep[varidx])
ghatDDdD[i][j][k] = ghatDDdD[i][j][k].subs(freevar, freevars_uniq_xx_indep[varidx])
GammahatUDD[i][j][k] = GammahatUDD[i][j][k].subs(freevar,
freevars_uniq_xx_indep[varidx])
for l in range(DIM):
ReDDdDD[i][j][k][l] = ReDDdDD[i][j][k][l].subs(freevar,
freevars_uniq_xx_indep[varidx])
ghatDDdDD[i][j][k][l] = ghatDDdDD[i][j][k][l].subs(freevar,
freevars_uniq_xx_indep[varidx])
GammahatUDDdD[i][j][k][l] = GammahatUDDdD[i][j][k][l].subs(freevar,
freevars_uniq_xx_indep[varidx])
# Step 7: Construct needed C code for declaring rfmstruct, allocating storage for
# rfmstruct arrays, defining each element in each array, reading the
# rfmstruct data from memory (both with and without SIMD enabled), and
# freeing allocated memory for the rfmstruct arrays.
# struct_str: String that declares the rfmstruct struct.
struct_str = "typedef struct __rfmstruct__ {\n"
define_str = ""
# rfmstruct stores pointers to (so far) 1D arrays. The malloc_str string allocates space for the arrays.
malloc_str = ""
if par.parval_from_str("reference_metric::rfm_precompute_to_Cfunctions_and_NRPy_basic_defines") == "False":
malloc_str = "rfm_struct rfmstruct;\n"
freemm_str = ""
# readvr_str reads the arrays from memory as needed
readvr_str = ["", "", ""]
readvr_SIMD_outer_str = ["", "", ""]
readvr_SIMD_inner_str = ["", "", ""]
# Sort freevars_uniq_vals and freevars_uniq_xx_indep, according to alphabetized freevars_uniq_xx_indep.
# Without this step, the ordering of elements in rfmstruct would be random, and would change each time
# this function was called.
if len(freevars_uniq_xx_indep) > 0:
freevars_uniq_xx_indep, freevars_uniq_vals = (list(x) for x in zip(*sorted(zip(freevars_uniq_xx_indep, freevars_uniq_vals),key=str)))
# Tease out how many variables each function in freevars_uniq_vals
which_freevar = 0
for expr in freevars_uniq_vals:
if "_of_xx" in str(freevars_uniq_xx_indep[which_freevar]):
frees = expr.free_symbols
frees_uniq = superfast_uniq(frees)
xx_list = []
malloc_size = 1
for i in range(3):
if gri.xx[i] in frees_uniq:
xx_list.append(gri.xx[i])
malloc_size *= gri.Nxx_plus_2NGHOSTS[i]
struct_str += "\tREAL *restrict " + str(freevars_uniq_xx_indep[which_freevar]) + ";\n"
malloc_str += "rfmstruct." + str(
freevars_uniq_xx_indep[which_freevar]) + " = (REAL *)malloc(sizeof(REAL)*" + str(malloc_size) + ");\n"
freemm_str += "free(rfmstruct." + str(freevars_uniq_xx_indep[which_freevar]) + ");\n"
output_define_and_readvr = False
for dirn in range(3):
if (gri.xx[dirn] in frees_uniq) and not (gri.xx[(dirn+1)%3] in frees_uniq) and not (gri.xx[(dirn+2)%3] in frees_uniq):
define_str += "for(int i"+str(dirn)+"=0;i"+str(dirn)+"<Nxx_plus_2NGHOSTS"+str(dirn)+";i"+str(dirn)+"++) {\n"
define_str += " const REAL xx"+str(dirn)+" = xx["+str(dirn)+"][i"+str(dirn)+"];\n"
define_str += " rfmstruct." + str(freevars_uniq_xx_indep[which_freevar]) + "[i"+str(dirn)+"] = " + str(sp.ccode(freevars_uniq_vals[which_freevar])) + ";\n"
define_str += "}\n\n"
readvr_str[dirn] += "const REAL " + str(freevars_uniq_xx_indep[which_freevar]) + " = rfmstruct->" + \
str(freevars_uniq_xx_indep[which_freevar]) + "[i"+str(dirn)+"];\n"
readvr_SIMD_outer_str[dirn] += "const double NOSIMD" + str(
freevars_uniq_xx_indep[which_freevar]) + " = rfmstruct->" + str(freevars_uniq_xx_indep[which_freevar]) + "[i"+str(dirn)+"]; "
readvr_SIMD_outer_str[dirn] += "const REAL_SIMD_ARRAY " + str(freevars_uniq_xx_indep[which_freevar]) + \
" = ConstSIMD(NOSIMD" + str(freevars_uniq_xx_indep[which_freevar]) + ");\n"
readvr_SIMD_inner_str[dirn] += "const REAL_SIMD_ARRAY " + str(freevars_uniq_xx_indep[which_freevar]) + \
" = ReadSIMD(&rfmstruct->" + str(freevars_uniq_xx_indep[which_freevar]) + "[i"+str(dirn)+"]);\n"
output_define_and_readvr = True
if (not output_define_and_readvr) and (gri.xx[0] in frees_uniq) and (gri.xx[1] in frees_uniq):
define_str += """
for(int i1=0;i1<Nxx_plus_2NGHOSTS1;i1++) for(int i0=0;i0<Nxx_plus_2NGHOSTS0;i0++) {
const REAL xx0 = xx[0][i0];
const REAL xx1 = xx[1][i1];
rfmstruct.""" + str(freevars_uniq_xx_indep[which_freevar]) + """[i0 + Nxx_plus_2NGHOSTS0*i1] = """ + str(sp.ccode(freevars_uniq_vals[which_freevar])) + """;
}\n\n"""
readvr_str[0] += "const REAL " + str(freevars_uniq_xx_indep[which_freevar]) + " = rfmstruct->" + \
str(freevars_uniq_xx_indep[which_freevar]) + "[i0 + Nxx_plus_2NGHOSTS0*i1];\n"
readvr_SIMD_outer_str[0] += "const double NOSIMD" + str(freevars_uniq_xx_indep[which_freevar]) + \
" = rfmstruct->" + str(freevars_uniq_xx_indep[which_freevar]) + "[i0 + Nxx_plus_2NGHOSTS0*i1]; "
readvr_SIMD_outer_str[0] += "const REAL_SIMD_ARRAY " + str(freevars_uniq_xx_indep[which_freevar]) + \
" = ConstSIMD(NOSIMD" + str(freevars_uniq_xx_indep[which_freevar]) + ");\n"
readvr_SIMD_inner_str[0] += "const REAL_SIMD_ARRAY " + str(freevars_uniq_xx_indep[which_freevar]) + \
" = ReadSIMD(&rfmstruct->" + str(freevars_uniq_xx_indep[which_freevar]) + "[i0 + Nxx_plus_2NGHOSTS0*i1]);\n"
output_define_and_readvr = True
if not output_define_and_readvr:
print("ERROR: Could not figure out the (xx0,xx1,xx2) dependency within the expression for "+str(freevars_uniq_xx_indep[which_freevar])+":")
print(str(freevars_uniq_vals[which_freevar]))
sys.exit(1)
which_freevar += 1
struct_str += "} rfm_struct;\n"
# Step 8: Output needed C code to files
outdir = par.parval_from_str(thismodule+"::rfm_precompute_Ccode_outdir")
if par.parval_from_str(thismodule+"::rfm_precompute_to_Cfunctions_and_NRPy_basic_defines") == "False":
with open(os.path.join(outdir, "rfm_struct__declare.h"), "w") as file:
file.write(struct_str)
with open(os.path.join(outdir, "rfm_struct__malloc.h"), "w") as file:
file.write(malloc_str)
with open(os.path.join(outdir, "rfm_struct__define.h"), "w") as file:
file.write(define_str)
with open(os.path.join(outdir, "rfm_struct__freemem.h"), "w") as file:
file.write(freemm_str)
else:
global NRPy_basic_defines_str
NRPy_basic_defines_str = struct_str
global rfm_struct__malloc, rfm_struct__define, rfm_struct__freemem
rfm_struct__malloc = malloc_str
rfm_struct__define = define_str
rfm_struct__freemem = freemm_str
for i in range(3):
with open(os.path.join(outdir, "rfm_struct__read" + str(i) + ".h"), "w") as file:
file.write(readvr_str[i])
with open(os.path.join(outdir, "rfm_struct__SIMD_outer_read" + str(i) + ".h"), "w") as file:
file.write(readvr_SIMD_outer_str[i])
with open(os.path.join(outdir, "rfm_struct__SIMD_inner_read" + str(i) + ".h"), "w") as file:
file.write(readvr_SIMD_inner_str[i])
####################################################
# Core Jacobian (basis) transformation functions,
# for reference metric basis to/from the
# Cartesian basis.
# We define Jacobians relative to the reference metric
# basis at a point x^j_rfm=(xx0,xx1,xx2)_rfm on the source grid:
#
# Jac_dUCart_dDrfmUD[i][j] = dx^i_Cart / dx^j_rfm
#
# via exact differentiation (courtesy SymPy), and the inverse Jacobian
#
# Jac_dUrfm_dDCartUD[i][j] = dx^i_rfm / dx^j_Cart
#
# using NRPy+'s generic_matrix_inverter3x3() function
def compute_Jacobian_and_inverseJacobian_tofrom_Cartesian():
# Step 2.a: First construct Jacobian matrix:
Jac_dUCart_dDrfmUD = ixp.zerorank2()
for i in range(3):
for j in range(3):
Jac_dUCart_dDrfmUD[i][j] = sp.diff(xx_to_Cart[i], xx[j])
Jac_dUrfm_dDCartUD, dummyDET = ixp.generic_matrix_inverter3x3(Jac_dUCart_dDrfmUD)
return Jac_dUCart_dDrfmUD, Jac_dUrfm_dDCartUD
def basis_transform_vectorU_from_rfmbasis_to_Cartesian(Jac_dUCart_dDrfmUD, src_vectorU):
Cart_dst_vectorU = ixp.zerorank1()
for i in range(3):
for l in range(3):
Cart_dst_vectorU[i] += Jac_dUCart_dDrfmUD[i][l] * src_vectorU[l]
return Cart_dst_vectorU
def basis_transform_vectorD_from_rfmbasis_to_Cartesian(Jac_dUrfm_dDCartUD, src_vectorD):
Cart_dst_vectorD = ixp.zerorank1()
for i in range(3):
for l in range(3):
Cart_dst_vectorD[i] += Jac_dUrfm_dDCartUD[l][i] * src_vectorD[l]
return Cart_dst_vectorD
def basis_transform_tensorDD_from_rfmbasis_to_Cartesian(Jac_dUrfm_dDCartUD, src_tensorDD):
Cart_dst_tensorDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
for l in range(3):
for m in range(3):
Cart_dst_tensorDD[i][j] += Jac_dUrfm_dDCartUD[l][i]*Jac_dUrfm_dDCartUD[m][j]*src_tensorDD[l][m]
return Cart_dst_tensorDD
def basis_transform_vectorU_from_Cartesian_to_rfmbasis(Jac_dUrfm_dDCartUD, Cart_src_vectorU):
rfm_dst_vectorU = ixp.zerorank1()
for i in range(3):
for l in range(3):
rfm_dst_vectorU[i] += Jac_dUrfm_dDCartUD[i][l] * Cart_src_vectorU[l]
return rfm_dst_vectorU