def find_cmax_cmin(field_comp, gamma_faceDD, beta_faceU, alpha_face):
# Inputs: flux direction field_comp, Inverse metric gamma_faceUU, shift beta_faceU,
# lapse alpha_face, metric determinant gammadet_face
# Outputs: maximum and minimum characteristic speeds cmax and cmin
# First, we need to find the characteristic speeds on each face
gamma_faceUU, unusedgammaDET = ixp.generic_matrix_inverter3x3(gamma_faceDD)
find_cp_cm(alpha_face, beta_faceU[field_comp],
gamma_faceUU[field_comp][field_comp])
cpr = cplus
cmr = cminus
find_cp_cm(alpha_face, beta_faceU[field_comp],
gamma_faceUU[field_comp][field_comp])
cpl = cplus
cml = cminus
# The following algorithms have been verified with random floats:
global cmax, cmin
# Now, we need to set cmax to the larger of cpr,cpl, and 0
import Min_Max_and_Piecewise_Expressions as noif
cmax = noif.max_noif(noif.max_noif(cpr, cpl), sp.sympify(0))
# And then, set cmin to the smaller of cmr,cml, and 0
cmin = -noif.min_noif(noif.min_noif(cmr, cml), sp.sympify(0))
def find_cmax_cmin(flux_dirn, gamma_faceDD, beta_faceU, alpha_face):
# Inputs: flux direction flux_dirn, Inverse metric gamma_faceUU, shift beta_faceU,
# lapse alpha_face, metric determinant gammadet_face
# Outputs: maximum and minimum characteristic speeds cmax and cmin
# First, we need to find the characteristic speeds on each face
gamma_faceUU, unusedgammaDET = ixp.generic_matrix_inverter3x3(gamma_faceDD)
find_cp_cm(alpha_face, beta_faceU[flux_dirn],
gamma_faceUU[flux_dirn][flux_dirn])
cpr = cplus
cmr = cminus
find_cp_cm(alpha_face, beta_faceU[flux_dirn],
gamma_faceUU[flux_dirn][flux_dirn])
cpl = cplus
cml = cminus
# The following algorithms have been verified with random floats:
global cmax, cmin
# Now, we need to set cmax to the larger of cpr,cpl, and 0
cmax = sp.Rational(1, 2) * (cpr + cpl + nrpyAbs(cpr - cpl))
cmax = sp.Rational(1, 2) * (cmax + nrpyAbs(cmax))
# And then, set cmin to the smaller of cmr,cml, and 0
cmin = sp.Rational(1, 2) * (cmr + cml - nrpyAbs(cmr - cml))
cmin = -sp.Rational(1, 2) * (cmin - nrpyAbs(cmin))
def Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(
CoordType_in, Sph_r_th_ph_or_Cart_xyz, gammaDD_inSphorCart,
KDD_inSphorCart, alpha_inSphorCart, betaU_inSphorCart, BU_inSphorCart):
# This routine converts the ADM variables
# $$\left\{\gamma_{ij}, K_{ij}, \alpha, \beta^i\right\}$$
# in Spherical or Cartesian basis+coordinates, first to the BSSN variables
# in the chosen reference_metric::CoordSystem coordinate system+basis:
# $$\left\{\bar{\gamma}_{i j},\bar{A}_{i j},\phi, K, \bar{\Lambda}^{i}, \alpha, \beta^i, B^i\right\},$$
# and then to the rescaled variables:
# $$\left\{h_{i j},a_{i j},\phi, K, \lambda^{i}, \alpha, \mathcal{V}^i, \mathcal{B}^i\right\}.$$
# The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations.
# To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in
# the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3.
# Step P1: Set spatial dimension (must be 3 for BSSN)
DIM = 3
# Step P2: Copy gammaSphDD_in to gammaSphDD, KSphDD_in to KSphDD, etc.
# This ensures that the input arrays are not modified below;
# modifying them would result in unexpected effects outside
# this function.
alphaSphorCart = alpha_inSphorCart
betaSphorCartU = ixp.zerorank1()
BSphorCartU = ixp.zerorank1()
gammaSphorCartDD = ixp.zerorank2()
KSphorCartDD = ixp.zerorank2()
for i in range(DIM):
betaSphorCartU[i] = betaU_inSphorCart[i]
BSphorCartU[i] = BU_inSphorCart[i]
for j in range(DIM):
gammaSphorCartDD[i][j] = gammaDD_inSphorCart[i][j]
KSphorCartDD[i][j] = KDD_inSphorCart[i][j]
# Make sure that rfm.reference_metric() has been called.
# We'll need the variables it defines throughout this module.
if rfm.have_already_called_reference_metric_function == False:
print(
"Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without"
)
print(
" first setting up reference metric, by calling rfm.reference_metric()."
)
sys.exit(1)
# Step 1: All input quantitiefs are in terms of r,th,ph or x,y,z. We want them in terms
# of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace
# r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2
# as defined for this particular reference metric in reference_metric.py's
# xxSph[] or xx_to_Cart[], respectively:
# Note that substitution only works when the variable is not an integer. Hence the
# if isinstance(...,...) stuff:
def sympify_integers__replace_rthph_or_Cartxyz(obj, rthph_or_xyz,
rthph_or_xyz_of_xx):
if isinstance(obj, int):
return sp.sympify(obj)
return obj.subs(rthph_or_xyz[0], rthph_or_xyz_of_xx[0]).\
subs(rthph_or_xyz[1], rthph_or_xyz_of_xx[1]).\
subs(rthph_or_xyz[2], rthph_or_xyz_of_xx[2])
r_th_ph_or_Cart_xyz_of_xx = []
if CoordType_in == "Spherical":
r_th_ph_or_Cart_xyz_of_xx = rfm.xxSph
elif CoordType_in == "Cartesian":
r_th_ph_or_Cart_xyz_of_xx = rfm.xx_to_Cart
else:
print(
"Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords."
)
sys.exit(1)
alphaSphorCart = sympify_integers__replace_rthph_or_Cartxyz(
alphaSphorCart, Sph_r_th_ph_or_Cart_xyz, r_th_ph_or_Cart_xyz_of_xx)
for i in range(DIM):
betaSphorCartU[i] = sympify_integers__replace_rthph_or_Cartxyz(
betaSphorCartU[i], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
BSphorCartU[i] = sympify_integers__replace_rthph_or_Cartxyz(
BSphorCartU[i], Sph_r_th_ph_or_Cart_xyz, r_th_ph_or_Cart_xyz_of_xx)
for j in range(DIM):
gammaSphorCartDD[i][
j] = sympify_integers__replace_rthph_or_Cartxyz(
gammaSphorCartDD[i][j], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
KSphorCartDD[i][j] = sympify_integers__replace_rthph_or_Cartxyz(
KSphorCartDD[i][j], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
# Step 2: All ADM initial data quantities are now functions of xx0,xx1,xx2, but
# they are still in the Spherical or Cartesian basis. We can now directly apply
# Jacobian transformations to get them in the correct xx0,xx1,xx2 basis:
# Make these globals so we can recover the ADM quantities in the destination basis if desired.
global alpha, betaU, BU, gammaDD, KDD
# alpha is a scalar, so no Jacobian transformation is necessary.
alpha = alphaSphorCart
Jac_dUSphorCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
# Converting UIUCBlackHole ID to Cartesian coords takes 66.8s if the following isn't simplified:
# Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(r_th_ph_or_Cart_xyz_of_xx[i],rfm.xx[j])
# ... but when we simplify each term, the total conversion time is reduced to 60.93s
Jac_dUSphorCart_dDrfmUD[i][j] = sp.simplify(
sp.diff(r_th_ph_or_Cart_xyz_of_xx[i], rfm.xx[j]))
Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUSphorCart_dDrfmUD)
betaU = ixp.zerorank1()
BU = ixp.zerorank1()
gammaDD = ixp.zerorank2()
KDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j]
BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j]
for k in range(DIM):
for l in range(DIM):
gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][
i] * Jac_dUSphorCart_dDrfmUD[l][j] * gammaSphorCartDD[
k][l]
KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][
i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l]
# Step 3: All ADM quantities were input into this function in the Spherical or Cartesian
# basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above,
# we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2.
# Here we convert ADM quantities in the "rfm" basis to their BSSN Curvilinear
# counterparts:
import BSSN.BSSN_in_terms_of_ADM as BitoA
BitoA.gammabarDD_hDD(gammaDD)
BitoA.trK_AbarDD_aDD(gammaDD, KDD)
BitoA.LambdabarU_lambdaU__exact_gammaDD(gammaDD)
BitoA.cf_from_gammaDD(gammaDD)
BitoA.betU_vetU(betaU, BU)
# Step 4: Return the BSSN Curvilinear variables in the desired xx0,xx1,xx2
# basis, and as functions of the consistent xx0,xx1,xx2 coordinates.
return BitoA.cf, BitoA.hDD, BitoA.lambdaU, BitoA.aDD, BitoA.trK, alpha, BitoA.vetU, BitoA.betU
def write_dfdr_function(self, Ccodesdir, fd_order=2):
# function to write c code to calculate dfdr term in Sommerfeld boundary condition
# Read what # of dimensions being usded
DIM = par.parval_from_str("grid::DIM")
# Set up the chosen reference metric from chosen coordinate system, set within NRPy+
CoordSystem = par.parval_from_str("reference_metric::CoordSystem")
rfm.reference_metric()
# Simplifying the results make them easier to interpret.
do_simplify = True
if "Sinh" in CoordSystem:
# Simplification takes too long on Sinh* coordinate systems
do_simplify = False
# Construct Jacobian matrix, output Jac_dUSph_dDrfmUD[i][j] = \partial x_{Sph}^i / \partial x^j:
Jac_dUSph_dDrfmUD = ixp.zerorank2()
for i in range(3):
for j in range(3):
Jac_dUSph_dDrfmUD[i][j] = sp.diff(rfm.xxSph[i], rfm.xx[j])
# Invert Jacobian matrix, output to Jac_dUrfm_dDSphUD.
Jac_dUrfm_dDSphUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUSph_dDrfmUD)
# Jac_dUrfm_dDSphUD[i][0] stores \partial x^i / \partial r
if do_simplify:
for i in range(3):
Jac_dUrfm_dDSphUD[i][0] = sp.simplify(Jac_dUrfm_dDSphUD[i][0])
# Declare \partial_i f, which is actually computed later on
fdD = ixp.declarerank1("fdD") # = [fdD0, fdD1, fdD2]
contraction = sp.sympify(0)
for i in range(3):
contraction += fdD[i] * Jac_dUrfm_dDSphUD[i][0]
contraction = sp.simplify(contraction)
r_str_and_contraction_str = outputC([rfm.xxSph[0], contraction],
["*_r", "*_partial_i_f"],
filename="returnstring",
params="includebraces=False")
def gen_central_fd_stencil_str(intdirn, fd_order):
if fd_order == 2:
if intdirn == 0:
return "(gfs[IDX4S(which_gf,i0+1,i1,i2)]-gfs[IDX4S(which_gf,i0-1,i1,i2)])*0.5" # Does not include the 1/dx multiplication
elif intdirn == 1:
return "(gfs[IDX4S(which_gf,i0,i1+1,i2)]-gfs[IDX4S(which_gf,i0,i1-1,i2)])*0.5" # Does not include the 1/dy multiplication
elif intdirn == 2:
return "(gfs[IDX4S(which_gf,i0,i1,i2+1)]-gfs[IDX4S(which_gf,i0,i1,i2-1)])*0.5" # Does not include the 1/dz multiplication
def output_dfdx(intdirn, fd_order):
dirn = str(intdirn)
dirnp1 = str(
(intdirn + 1) % 3
) # if dirn='0', then we want this to be '1'; '1' then '2'; and '2' then '0'
dirnp2 = str(
(intdirn + 2) % 3
) # if dirn='0', then we want this to be '2'; '1' then '0'; and '2' then '1'
if fd_order == 2:
return """
// On a +x""" + dirn + """ or -x""" + dirn + """ face, do up/down winding as appropriate:
if(abs(FACEXi[""" + dirn + """])==1 || i""" + dirn + """+NGHOSTS >= Nxx_plus_2NGHOSTS""" + dirn + """ || i""" + dirn + """-NGHOSTS <= 0) {
int8_t SHIFTSTENCIL""" + dirn + """ = FACEXi[""" + dirn + """];
if(i""" + dirn + """+NGHOSTS >= Nxx_plus_2NGHOSTS""" + dirn + """) SHIFTSTENCIL""" + dirn + """ = -1;
if(i""" + dirn + """-NGHOSTS <= 0) SHIFTSTENCIL""" + dirn + """ = +1;
SHIFTSTENCIL""" + dirnp1 + """ = 0;
SHIFTSTENCIL""" + dirnp2 + """ = 0;
fdD""" + dirn + """
= SHIFTSTENCIL""" + dirn + """*(-1.5*gfs[IDX4S(which_gf,i0+0*SHIFTSTENCIL0,i1+0*SHIFTSTENCIL1,i2+0*SHIFTSTENCIL2)]
+2.*gfs[IDX4S(which_gf,i0+1*SHIFTSTENCIL0,i1+1*SHIFTSTENCIL1,i2+1*SHIFTSTENCIL2)]
-0.5*gfs[IDX4S(which_gf,i0+2*SHIFTSTENCIL0,i1+2*SHIFTSTENCIL1,i2+2*SHIFTSTENCIL2)]
)*invdx""" + dirn + """;
// Not on a +x""" + dirn + """ or -x""" + dirn + """ face, using centered difference:
} else {
fdD""" + dirn + """ = """ + gen_central_fd_stencil_str(
intdirn, 2) + """*invdx""" + dirn + """;
}
"""
else:
print("Error: fd_order = " + str(fd_order) +
" currently unsupported.")
sys.exit(1)
contraction_term_func = """
void contraction_term(const paramstruct *restrict params, const int which_gf, const REAL *restrict gfs, REAL *restrict xx[3],
const int8_t FACEXi[3], const int i0, const int i1, const int i2, REAL *restrict _r, REAL *restrict _partial_i_f) {
#include "RELATIVE_PATH__set_Cparameters.h" /* Header file containing correct #include for set_Cparameters.h;
* accounting for the relative path */
// Initialize derivatives to crazy values, to ensure that
// we will notice in case they aren't set properly.
REAL fdD0=1e100;
REAL fdD1=1e100;
REAL fdD2=1e100;
REAL xx0 = xx[0][i0];
REAL xx1 = xx[1][i1];
REAL xx2 = xx[2][i2];
int8_t SHIFTSTENCIL0;
int8_t SHIFTSTENCIL1;
int8_t SHIFTSTENCIL2;
"""
for i in range(DIM):
if "fdD" + str(i) in r_str_and_contraction_str:
contraction_term_func += output_dfdx(i, fd_order)
contraction_term_func += "\n" + r_str_and_contraction_str
contraction_term_func += """
} // END contraction_term function
"""
with open(
os.path.join(Ccodesdir,
"boundary_conditions/radial_derivative.h"),
"w") as file:
file.write(contraction_term_func)
def Psi4_tetradsv2():
global l4U, n4U, mre4U, mim4U
# Step 1.c: Check if tetrad choice is implemented:
if par.parval_from_str(thismodule + "::TetradChoice") != "QuasiKinnersley":
print("ERROR: " + thismodule + "::TetradChoice = " +
par.parval_from_str("TetradChoice") + " currently unsupported!")
exit(1)
# Step 1.d: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 1.e: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.f: Import all ADM quantities as written in terms of BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as AB
AB.ADM_in_terms_of_BSSN()
# Step 2.a: Declare the Cartesian x,y,z as input parameters
# and v_1^a, v_2^a, and v_3^a tetrads,
# as well as detgamma and gammaUU from
# BSSN.ADM_in_terms_of_BSSN
x, y, z = par.Cparameters("REAL", thismodule, ["x", "y", "z"])
v1UCart = ixp.zerorank1()
v2UCart = ixp.zerorank1()
# detgamma = AB.detgamma
# gammaUU = AB.gammaUU
# Step 2.b: Define v1U and v2U
v1UCart = [-y, x, sp.sympify(0)]
v2UCart = [x, y, z]
# Step 2.c: Construct the Jacobian d x_Cart^i / d xx^j
Jac_dUCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUCart_dDrfmUD[i][j] = sp.diff(rfm.xxCart[i], rfm.xx[j])
# Step 2.d: Invert above Jacobian to get needed d xx^j / d x_Cart^i
Jac_dUrfm_dDCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUCart_dDrfmUD)
# Step 2.e: Transform gammaDD to Cartesian basis:
gammaCartDD = ixp.zerorank2()
gammaDD = AB.gammaDD
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
gammaCartDD[i][j] += Jac_dUrfm_dDCartUD[k][
i] * Jac_dUrfm_dDCartUD[l][j] * gammaDD[k][l]
gammaCartUU, detgammaCart = ixp.symm_matrix_inverter3x3(gammaCartDD)
# Step 2.e: Transform v1U and v2U from the Cartesian to the xx^i basis
v1U = ixp.zerorank1()
v2U = ixp.zerorank1()
for i in range(DIM):
v1U[i] = v1UCart[i]
v2U[i] = v2UCart[i]
# for i in range(DIM):
# for j in range(DIM):
# v1U[i] += Jac_dUrfm_dDCartUD[i][j]*v1UCart[j]
# v2U[i] += Jac_dUrfm_dDCartUD[i][j]*v2UCart[j]
# Step 2.f: Define the rank-3 version of the Levi-Civita symbol. Amongst
# other uses, this is needed for the construction of the approximate
# quasi-Kinnersley tetrad.
def define_LeviCivitaSymbol_rank3(DIM=-1):
if DIM == -1:
DIM = par.parval_from_str("DIM")
LeviCivitaSymbol = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
# From https://codegolf.stackexchange.com/questions/160359/levi-civita-symbol :
LeviCivitaSymbol[i][j][k] = (i - j) * (j - k) * (k - i) / 2
return LeviCivitaSymbol
# Step 2.g: Define v3U
v3U = ixp.zerorank1()
LeviCivitaSymbolDDD = define_LeviCivitaSymbol_rank3(DIM=3)
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
v3U[a] += sp.sqrt(detgammaCart) * gammaCartUU[a][
d] * LeviCivitaSymbolDDD[d][b][c] * v1U[b] * v2U[c]
# Step 2.h: Define omega_{ij}
omegaDD = ixp.zerorank2()
# Step 2.i: Define e^a_i. Note that:
# omegaDD[0][0] = \omega_{11} above;
# omegaDD[1][1] = \omega_{22} above, etc.
e1U = ixp.zerorank1()
e2U = ixp.zerorank1()
e3U = ixp.zerorank1()
update_omega(omegaDD, v1U, v2U, v3U, gammaCartDD)
for a in range(DIM):
e1U[a] = v1U[a] / sp.sqrt(omegaDD[0][0])
update_omega(omegaDD, e1U, v2U, v3U, gammaCartDD)
for a in range(DIM):
e2U[a] = (v2U[a] - omegaDD[0][1] * e1U[a]) / sp.sqrt(omegaDD[1][1])
update_omega(omegaDD, e1U, e2U, v3U, gammaCartDD)
for a in range(DIM):
e3U[a] = (v3U[a] - omegaDD[0][2] * e1U[a] -
omegaDD[1][2] * e2U[a]) / sp.sqrt(omegaDD[2][2])
# Step 2.j: Construct l^mu, n^mu, and m^mu, based on r^mu, theta^mu, phi^mu, and u^mu:
rCart4U = ixp.zerorank1(DIM=4)
thetaCart4U = ixp.zerorank1(DIM=4)
phiCart4U = ixp.zerorank1(DIM=4)
for a in range(DIM):
rCart4U[a + 1] = e2U[a]
thetaCart4U[a + 1] = e3U[a]
phiCart4U[a + 1] = e1U[a]
r4U = ixp.zerorank1(DIM=4)
theta4U = ixp.zerorank1(DIM=4)
phi4U = ixp.zerorank1(DIM=4)
for a in range(DIM):
for b in range(DIM):
r4U[a + 1] += Jac_dUrfm_dDCartUD[a][b] * rCart4U[b + 1]
theta4U[a + 1] += Jac_dUrfm_dDCartUD[a][b] * thetaCart4U[b + 1]
phi4U[a + 1] += Jac_dUrfm_dDCartUD[a][b] * phiCart4U[b + 1]
u4U = ixp.zerorank1(DIM=4)
# FIXME? assumes alpha=1, beta^i = 0
u4U[0] = 1
l4U = ixp.zerorank1(DIM=4)
n4U = ixp.zerorank1(DIM=4)
mre4U = ixp.zerorank1(DIM=4)
mim4U = ixp.zerorank1(DIM=4)
isqrt2 = 1 / sp.sqrt(2)
for mu in range(4):
l4U[mu] = isqrt2 * (u4U[mu] + r4U[mu])
n4U[mu] = isqrt2 * (u4U[mu] - r4U[mu])
mre4U[mu] = isqrt2 * theta4U[mu]
mim4U[mu] = isqrt2 * phi4U[mu]
def Tmunu_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear(
CoordType_in, Tmunu_input_function_name, pointer_to_ID_inputs=False):
# The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations.
# To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in
# the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3.
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
# Step 1: Define the input variables: the 4D stress-energy tensor, and the ADM 3-metric, lapse, & shift:
T4SphorCartUU = ixp.declarerank2("T4SphorCartUU", "sym01", DIM=4)
gammaSphorCartDD = ixp.declarerank2("gammaSphorCartDD", "sym01")
alphaSphorCart = sp.symbols("alphaSphorCart")
betaSphorCartU = ixp.declarerank1("betaSphorCartU")
# Step 2: All Tmunu initial data quantities are functions of xx0,xx1,xx2, but
# in the Spherical or Cartesian basis.
# We first define the BSSN stress-energy source terms in the Spherical
# or Cartesian basis, respectively.
# To get \gamma_{\mu \nu} = gammabar4DD[mu][nu], we'll need to construct the 4-metric, using Eq. 2.122 in B&S:
# S_{ij} = \gamma_{i \mu} \gamma_{j \nu} T^{\mu \nu}
# S_{i} = -\gamma_{i\mu} n_\nu T^{\mu\nu}
# S = \gamma^{ij} S_{ij}
# rho = n_\mu n_\nu T^{\mu\nu},
# where
# \gamma_{\mu\nu} = g_{\mu\nu} + n_\mu n_\nu
# and
# n_mu = {-\alpha,0,0,0},
# Step 2.1: Construct the 4-metric based on the input ADM quantities.
# This is provided by Eq 4.47 in [Gourgoulhon](https://arxiv.org/pdf/gr-qc/0703035.pdf):
# g_{tt} = -\alpha^2 + \beta^k \beta_k
# g_{ti} = \beta_i
# g_{ij} = \gamma_{ij}
# Eq. 2.121 in B&S
betaSphorCartD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
betaSphorCartD[i] += gammaSphorCartDD[i][j] * betaSphorCartU[j]
# Now compute the beta contraction.
beta2 = sp.sympify(0)
for i in range(DIM):
beta2 += betaSphorCartU[i] * betaSphorCartD[i]
# Eq. 2.122 in B&S
g4SphorCartDD = ixp.zerorank2(DIM=4)
g4SphorCartDD[0][0] = -alphaSphorCart**2 + beta2
for i in range(DIM):
g4SphorCartDD[i + 1][0] = g4SphorCartDD[0][i + 1] = betaSphorCartD[i]
for i in range(DIM):
for j in range(DIM):
g4SphorCartDD[i + 1][j + 1] = gammaSphorCartDD[i][j]
# Step 2.2: Construct \gamma_{mu nu} = g_{mu nu} + n_mu n_nu:
n4SphorCartD = ixp.zerorank1(DIM=4)
n4SphorCartD[0] = -alphaSphorCart
gamma4SphorCartDD = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
gamma4SphorCartDD[mu][nu] = g4SphorCartDD[mu][
nu] + n4SphorCartD[mu] * n4SphorCartD[nu]
# Step 2.3: We now have all we need to construct the BSSN source
# terms in the current basis (Spherical or Cartesian):
# S_{ij} = \gamma_{i \mu} \gamma_{j \nu} T^{\mu \nu}
# S_{i} = -\gamma_{i\mu} n_\nu T^{\mu\nu}
# S = \gamma^{ij} S_{ij}
# rho = n_\mu n_\nu T^{\mu\nu},
SSphorCartDD = ixp.zerorank2()
SSphorCartD = ixp.zerorank1()
SSphorCart = sp.sympify(0)
rhoSphorCart = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
for nu in range(4):
SSphorCartDD[i][j] += gamma4SphorCartDD[
i + 1][mu] * gamma4SphorCartDD[
j + 1][nu] * T4SphorCartUU[mu][nu]
for i in range(DIM):
for mu in range(4):
for nu in range(4):
SSphorCartD[i] += -gamma4SphorCartDD[
i + 1][mu] * n4SphorCartD[nu] * T4SphorCartUU[mu][nu]
gammaSphorCartUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaSphorCartDD)
for i in range(DIM):
for j in range(DIM):
SSphorCart += gammaSphorCartUU[i][j] * SSphorCartDD[i][j]
for mu in range(4):
for nu in range(4):
rhoSphorCart += n4SphorCartD[mu] * n4SphorCartD[
nu] * T4SphorCartUU[mu][nu]
# Step 3: Perform basis conversion to
# Make sure that rfm.reference_metric() has been called.
# We'll need the variables it defines throughout this module.
if rfm.have_already_called_reference_metric_function == False:
print(
"Error. Called Tmunu_Numerical_Spherical_or_Cartesian_to_BSSNCurvilinear() without"
)
print(
" first setting up reference metric, by calling rfm.reference_metric()."
)
exit(1)
# Step 1: All input quantities are in terms of r,th,ph or x,y,z. We want them in terms
# of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace
# r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2
# as defined for this particular reference metric in reference_metric.py's
# xxSph[] or xxCart[], respectively:
r_th_ph_or_Cart_xyz_oID_xx = []
if CoordType_in == "Spherical":
r_th_ph_or_Cart_xyz_oID_xx = rfm.xxSph
elif CoordType_in == "Cartesian":
r_th_ph_or_Cart_xyz_oID_xx = rfm.xxCart
else:
print(
"Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords."
)
exit(1)
# Next apply Jacobian transformations to convert into the (xx0,xx1,xx2) basis
# alpha is a scalar, so no Jacobian transformation is necessary.
alpha = alphaSphorCart
Jac_dUSphorCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(
r_th_ph_or_Cart_xyz_oID_xx[i], rfm.xx[j])
Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUSphorCart_dDrfmUD)
betaU = ixp.zerorank1()
BU = ixp.zerorank1()
gammaSphorCartDD = ixp.zerorank2()
KDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j]
BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j]
for k in range(DIM):
for l in range(DIM):
gammaSphorCartDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * \
gammaSphorCartDD[k][l]
KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][
i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l]
# Step 3: All ADM quantities were input into this function in the Spherical or Cartesian
# basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above,
# we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2.
# Here we convert ADM quantities to their BSSN Curvilinear counterparts:
# Step 3.1: Convert ADM $\gamma_{ij}$ to BSSN $\bar{\gamma}_{ij}$:
# We have (Eqs. 2 and 3 of [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):
# \bar{\gamma}_{i j} = \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \gamma_{ij}.
gammaSphorCartUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaSphorCartDD)
gammabarDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
gammabarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational(
1, 3)) * gammaSphorCartDD[i][j]
# Step 3.2: Convert the extrinsic curvature $K_{ij}$ to the trace-free extrinsic
# curvature $\bar{A}_{ij}$, plus the trace of the extrinsic curvature $K$,
# where (Eq. 3 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)):
# K = \gamma^{ij} K_{ij}, and
# \bar{A}_{ij} &= \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \left(K_{ij} - \frac{1}{3} \gamma_{ij} K \right)
trK = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
trK += gammaSphorCartUU[i][j] * KDD[i][j]
AbarDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
AbarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational(
1, 3)) * (KDD[i][j] -
sp.Rational(1, 3) * gammaSphorCartDD[i][j] * trK)
# Step 3.3: Set the conformal factor variable $\texttt{cf}$, which is set
# by the "BSSN_RHSs::ConformalFactor" parameter. For example if
# "ConformalFactor" is set to "phi", we can use Eq. 3 of
# [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf),
# which in arbitrary coordinates is written:
# \phi = \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right).
# Alternatively if "BSSN_RHSs::ConformalFactor" is set to "chi", then
# \chi = e^{-4 \phi} = \exp\left(-4 \frac{1}{12} \left(\frac{\gamma}{\bar{\gamma}}\right)\right)
# = \exp\left(-\frac{1}{3} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) = \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/3}.
#
# Finally if "BSSN_RHSs::ConformalFactor" is set to "W", then
# W = e^{-2 \phi} = \exp\left(-2 \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) =
# \exp\left(-\frac{1}{6} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) =
# \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/6}.
# First compute gammabarDET:
gammabarUU, gammabarDET = ixp.symm_matrix_inverter3x3(gammabarDD)
cf = sp.sympify(0)
if par.parval_from_str("ConformalFactor") == "phi":
cf = sp.Rational(1, 12) * sp.log(gammaDET / gammabarDET)
elif par.parval_from_str("ConformalFactor") == "chi":
cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 3))
elif par.parval_from_str("ConformalFactor") == "W":
cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 6))
else:
print("Error ConformalFactor type = \"" +
par.parval_from_str("ConformalFactor") + "\" unknown.")
exit(1)
# Step 4: Rescale tensorial quantities according to the prescription described in
# the [BSSN in curvilinear coordinates tutorial module](Tutorial-BSSNCurvilinear.ipynb)
# (also [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):
#
# h_{ij} &= (\bar{\gamma}_{ij} - \hat{\gamma}_{ij})/\text{ReDD[i][j]}\\
# a_{ij} &= \bar{A}_{ij}/\text{ReDD[i][j]}\\
# \lambda^i &= \bar{\Lambda}^i/\text{ReU[i]}\\
# \mathcal{V}^i &= \beta^i/\text{ReU[i]}\\
# \mathcal{B}^i &= B^i/\text{ReU[i]}\\
hDD = ixp.zerorank2()
aDD = ixp.zerorank2()
vetU = ixp.zerorank1()
betU = ixp.zerorank1()
for i in range(DIM):
vetU[i] = betaU[i] / rfm.ReU[i]
betU[i] = BU[i] / rfm.ReU[i]
for j in range(DIM):
hDD[i][j] = (gammabarDD[i][j] - rfm.ghatDD[i][j]) / rfm.ReDD[i][j]
aDD[i][j] = AbarDD[i][j] / rfm.ReDD[i][j]
# Step 5: Output all ADM-to-BSSN expressions to a C function. This function
# must first call the ID_ADM_SphorCart() defined above. Using these
# Spherical or Cartesian data, it sets up all quantities needed for
# BSSNCurvilinear initial data, *except* $\lambda^i$, which must be
# computed from numerical data using finite-difference derivatives.
with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h",
"w") as file:
file.write(
"void ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(const REAL xx0xx1xx2[3],"
)
if pointer_to_ID_inputs == True:
file.write("ID_inputs *other_inputs,")
else:
file.write("ID_inputs other_inputs,")
file.write("""
REAL *hDD00,REAL *hDD01,REAL *hDD02,REAL *hDD11,REAL *hDD12,REAL *hDD22,
REAL *aDD00,REAL *aDD01,REAL *aDD02,REAL *aDD11,REAL *aDD12,REAL *aDD22,
REAL *trK,
REAL *vetU0,REAL *vetU1,REAL *vetU2,
REAL *betU0,REAL *betU1,REAL *betU2,
REAL *alpha, REAL *cf) {
REAL gammaSphorCartDD00,gammaSphorCartDD01,gammaSphorCartDD02,
gammaSphorCartDD11,gammaSphorCartDD12,gammaSphorCartDD22;
REAL KSphorCartDD00,KSphorCartDD01,KSphorCartDD02,
KSphorCartDD11,KSphorCartDD12,KSphorCartDD22;
REAL alphaSphorCart,betaSphorCartU0,betaSphorCartU1,betaSphorCartU2;
REAL BSphorCartU0,BSphorCartU1,BSphorCartU2;
const REAL xx0 = xx0xx1xx2[0];
const REAL xx1 = xx0xx1xx2[1];
const REAL xx2 = xx0xx1xx2[2];
REAL xyz_or_rthph[3];\n""")
outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False"
outputC(r_th_ph_or_Cart_xyz_oID_xx[0:3],
["xyz_or_rthph[0]", "xyz_or_rthph[1]", "xyz_or_rthph[2]"],
"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h",
outCparams + ",CSE_enable=False")
with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h",
"a") as file:
file.write(" " + ADM_input_function_name +
"""(xyz_or_rthph, other_inputs,
&gammaSphorCartDD00,&gammaSphorCartDD01,&gammaSphorCartDD02,
&gammaSphorCartDD11,&gammaSphorCartDD12,&gammaSphorCartDD22,
&KSphorCartDD00,&KSphorCartDD01,&KSphorCartDD02,
&KSphorCartDD11,&KSphorCartDD12,&KSphorCartDD22,
&alphaSphorCart,&betaSphorCartU0,&betaSphorCartU1,&betaSphorCartU2,
&BSphorCartU0,&BSphorCartU1,&BSphorCartU2);
// Next compute all rescaled BSSN curvilinear quantities:\n""")
outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False"
outputC([
hDD[0][0], hDD[0][1], hDD[0][2], hDD[1][1], hDD[1][2], hDD[2][2],
aDD[0][0], aDD[0][1], aDD[0][2], aDD[1][1], aDD[1][2], aDD[2][2], trK,
vetU[0], vetU[1], vetU[2], betU[0], betU[1], betU[2], alpha, cf
], [
"*hDD00", "*hDD01", "*hDD02", "*hDD11", "*hDD12", "*hDD22", "*aDD00",
"*aDD01", "*aDD02", "*aDD11", "*aDD12", "*aDD22", "*trK", "*vetU0",
"*vetU1", "*vetU2", "*betU0", "*betU1", "*betU2", "*alpha", "*cf"
],
"BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h",
params=outCparams)
with open("BSSN/ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs.h",
"a") as file:
file.write("}\n")
# Step 5.A: Output the driver function for the above
# function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs()
# Next write the driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs():
with open("BSSN/ID_BSSN__ALL_BUT_LAMBDAs.h", "w") as file:
file.write(
"void ID_BSSN__ALL_BUT_LAMBDAs(const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],"
)
if pointer_to_ID_inputs == True:
file.write("ID_inputs *other_inputs,")
else:
file.write("ID_inputs other_inputs,")
file.write("REAL *in_gfs) {\n")
file.write(
lp.loop(["i2", "i1", "i0"], ["0", "0", "0"], [
"Nxx_plus_2NGHOSTS[2]", "Nxx_plus_2NGHOSTS[1]",
"Nxx_plus_2NGHOSTS[0]"
], ["1", "1", "1"], [
"#pragma omp parallel for", " const REAL xx2 = xx[2][i2];",
" const REAL xx1 = xx[1][i1];"
], "", """const REAL xx0 = xx[0][i0];
const int idx = IDX3(i0,i1,i2);
const REAL xx0xx1xx2[3] = {xx0,xx1,xx2};
ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(xx0xx1xx2,other_inputs,
&in_gfs[IDX4pt(HDD00GF,idx)],&in_gfs[IDX4pt(HDD01GF,idx)],&in_gfs[IDX4pt(HDD02GF,idx)],
&in_gfs[IDX4pt(HDD11GF,idx)],&in_gfs[IDX4pt(HDD12GF,idx)],&in_gfs[IDX4pt(HDD22GF,idx)],
&in_gfs[IDX4pt(ADD00GF,idx)],&in_gfs[IDX4pt(ADD01GF,idx)],&in_gfs[IDX4pt(ADD02GF,idx)],
&in_gfs[IDX4pt(ADD11GF,idx)],&in_gfs[IDX4pt(ADD12GF,idx)],&in_gfs[IDX4pt(ADD22GF,idx)],
&in_gfs[IDX4pt(TRKGF,idx)],
&in_gfs[IDX4pt(VETU0GF,idx)],&in_gfs[IDX4pt(VETU1GF,idx)],&in_gfs[IDX4pt(VETU2GF,idx)],
&in_gfs[IDX4pt(BETU0GF,idx)],&in_gfs[IDX4pt(BETU1GF,idx)],&in_gfs[IDX4pt(BETU2GF,idx)],
&in_gfs[IDX4pt(ALPHAGF,idx)],&in_gfs[IDX4pt(CFGF,idx)]);
"""))
file.write("}\n")
# Step 6: Compute $\bar{\Lambda}^i$ (Eqs. 4 and 5 of
# [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)),
# from finite-difference derivatives of rescaled metric
# quantities $h_{ij}$:
# \bar{\Lambda}^i = \bar{\gamma}^{jk}\left(\bar{\Gamma}^i_{jk} - \hat{\Gamma}^i_{jk}\right).
# The reference_metric.py module provides us with analytic expressions for
# $\hat{\Gamma}^i_{jk}$, so here we need only compute
# finite-difference expressions for $\bar{\Gamma}^i_{jk}$, based on
# the values for $h_{ij}$ provided in the initial data. Once
# $\bar{\Lambda}^i$ has been computed, we apply the usual rescaling
# procedure:
# \lambda^i = \bar{\Lambda}^i/\text{ReU[i]},
# and then output the result to a C file using the NRPy+
# finite-difference C output routine.
# We will need all BSSN gridfunctions to be defined, as well as
# expressions for gammabarDD_dD in terms of exact derivatives of
# the rescaling matrix and finite-difference derivatives of
# hDD's.
gammabarDD = bssnrhs.gammabarDD
gammabarUU, gammabarDET = ixp.symm_matrix_inverter3x3(gammabarDD)
gammabarDD_dD = bssnrhs.gammabarDD_dD
# Next compute Christoffel symbols \bar{\Gamma}^i_{jk}:
GammabarUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GammabarUDD[i][j][k] += sp.Rational(
1,
2) * gammabarUU[i][l] * (gammabarDD_dD[l][j][k] +
gammabarDD_dD[l][k][j] -
gammabarDD_dD[j][k][l])
# Next evaluate \bar{\Lambda}^i, based on GammabarUDD above and GammahatUDD
# (from the reference metric):
LambdabarU = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LambdabarU[i] += gammabarUU[j][k] * (
GammabarUDD[i][j][k] - rfm.GammahatUDD[i][j][k])
# Finally apply rescaling:
# lambda^i = Lambdabar^i/\text{ReU[i]}
lambdaU = ixp.zerorank1()
for i in range(DIM):
lambdaU[i] = LambdabarU[i] / rfm.ReU[i]
outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False"
lambdaU_expressions = [
lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU0"), rhs=lambdaU[0]),
lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU1"), rhs=lambdaU[1]),
lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU2"), rhs=lambdaU[2])
]
lambdaU_expressions_FDout = fin.FD_outputC("returnstring",
lambdaU_expressions,
outCparams)
with open("BSSN/ID_BSSN_lambdas.h", "w") as file:
file.write("""
void ID_BSSN_lambdas(const int Nxx[3],const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],const REAL dxx[3],REAL *in_gfs) {\n"""
)
file.write(
lp.loop(["i2", "i1", "i0"], ["NGHOSTS", "NGHOSTS", "NGHOSTS"],
["NGHOSTS+Nxx[2]", "NGHOSTS+Nxx[1]", "NGHOSTS+Nxx[0]"],
["1", "1", "1"], [
"const REAL invdx0 = 1.0/dxx[0];\n" +
"const REAL invdx1 = 1.0/dxx[1];\n" +
"const REAL invdx2 = 1.0/dxx[2];\n" +
"#pragma omp parallel for",
" const REAL xx2 = xx[2][i2];",
" const REAL xx1 = xx[1][i1];"
], "", "const REAL xx0 = xx[0][i0];\n" +
lambdaU_expressions_FDout))
file.write("}\n")
def Psi4_tetrads():
global l4U, n4U, mre4U, mim4U
# Step 1.c: Check if tetrad choice is implemented:
if par.parval_from_str(thismodule + "::TetradChoice") != "QuasiKinnersley":
print("ERROR: " + thismodule + "::TetradChoice = " +
par.parval_from_str("TetradChoice") + " currently unsupported!")
sys.exit(1)
# Step 1.d: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 1.e: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.f: Import all ADM quantities as written in terms of BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as AB
AB.ADM_in_terms_of_BSSN()
# Step 2.a: Declare the Cartesian x,y,z in terms of
# xx0,xx1,xx2.
x = rfm.xxCart[0]
y = rfm.xxCart[1]
z = rfm.xxCart[2]
# Step 2.b: Declare detgamma and gammaUU from
# BSSN.ADM_in_terms_of_BSSN;
# simplify detgamma & gammaUU expressions,
# which expedites Psi4 codegen.
detgamma = sp.simplify(AB.detgamma)
gammaUU = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
gammaUU[i][j] = sp.simplify(AB.gammaUU[i][j])
# Step 2.c: Define v1U and v2U
v1UCart = [-y, x, sp.sympify(0)]
v2UCart = [x, y, z]
# Step 2.d: Construct the Jacobian d x_Cart^i / d xx^j
Jac_dUCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUCart_dDrfmUD[i][j] = sp.simplify(
sp.diff(rfm.xxCart[i], rfm.xx[j]))
# Step 2.e: Invert above Jacobian to get needed d xx^j / d x_Cart^i
Jac_dUrfm_dDCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUCart_dDrfmUD)
# Step 2.e.i: Simplify expressions for d xx^j / d x_Cart^i:
for i in range(DIM):
for j in range(DIM):
Jac_dUrfm_dDCartUD[i][j] = sp.simplify(Jac_dUrfm_dDCartUD[i][j])
# Step 2.f: Transform v1U and v2U from the Cartesian to the xx^i basis
v1U = ixp.zerorank1()
v2U = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
v1U[i] += Jac_dUrfm_dDCartUD[i][j] * v1UCart[j]
v2U[i] += Jac_dUrfm_dDCartUD[i][j] * v2UCart[j]
# Step 2.g: Define v3U
v3U = ixp.zerorank1()
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
v3U[a] += sp.sqrt(detgamma) * gammaUU[a][
d] * LeviCivitaSymbolDDD[d][b][c] * v1U[b] * v2U[c]
# Step 2.g.i: Simplify expressions for v1U,v2U,v3U. This greatly expedites the C code generation (~10x faster)
# Drat. Simplification with certain versions of SymPy & coord systems results in a hang. Let's just
# evaluate the expressions so the most trivial optimizations can be performed.
for a in range(DIM):
v1U[a] = v1U[a].doit() # sp.simplify(v1U[a])
v2U[a] = v2U[a].doit() # sp.simplify(v2U[a])
v3U[a] = v3U[a].doit() # sp.simplify(v3U[a])
# Step 2.h: Define omega_{ij}
omegaDD = ixp.zerorank2()
gammaDD = AB.gammaDD
def v_vectorDU(v1U, v2U, v3U, i, a):
if i == 0:
return v1U[a]
if i == 1:
return v2U[a]
if i == 2:
return v3U[a]
print("ERROR: unknown vector!")
sys.exit(1)
def update_omega(omegaDD, i, j, v1U, v2U, v3U, gammaDD):
omegaDD[i][j] = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omegaDD[i][j] += v_vectorDU(v1U, v2U, v3U, i, a) * v_vectorDU(
v1U, v2U, v3U, j, b) * gammaDD[a][b]
# Step 2.i: Define e^a_i. Note that:
# omegaDD[0][0] = \omega_{11} above;
# omegaDD[1][1] = \omega_{22} above, etc.
# First e_1^a: Orthogonalize & normalize:
e1U = ixp.zerorank1()
update_omega(omegaDD, 0, 0, v1U, v2U, v3U, gammaDD)
for a in range(DIM):
e1U[a] = v1U[a] / sp.sqrt(omegaDD[0][0])
# Next e_2^a: First orthogonalize:
e2U = ixp.zerorank1()
update_omega(omegaDD, 0, 1, e1U, v2U, v3U, gammaDD)
for a in range(DIM):
e2U[a] = (v2U[a] - omegaDD[0][1] * e1U[a])
# Then normalize:
update_omega(omegaDD, 1, 1, e1U, e2U, v3U, gammaDD)
for a in range(DIM):
e2U[a] /= sp.sqrt(omegaDD[1][1])
# Next e_3^a: First orthogonalize:
e3U = ixp.zerorank1()
update_omega(omegaDD, 0, 2, e1U, e2U, v3U, gammaDD)
update_omega(omegaDD, 1, 2, e1U, e2U, v3U, gammaDD)
for a in range(DIM):
e3U[a] = (v3U[a] - omegaDD[0][2] * e1U[a] - omegaDD[1][2] * e2U[a])
# Then normalize:
update_omega(omegaDD, 2, 2, e1U, e2U, e3U, gammaDD)
for a in range(DIM):
e3U[a] /= sp.sqrt(omegaDD[2][2])
# Step 2.j: Construct l^mu, n^mu, and m^mu, based on r^mu, theta^mu, phi^mu, and u^mu:
r4U = ixp.zerorank1(DIM=4)
u4U = ixp.zerorank1(DIM=4)
theta4U = ixp.zerorank1(DIM=4)
phi4U = ixp.zerorank1(DIM=4)
for a in range(DIM):
r4U[a + 1] = e2U[a]
theta4U[a + 1] = e3U[a]
phi4U[a + 1] = e1U[a]
# FIXME? assumes alpha=1, beta^i = 0
if par.parval_from_str(thismodule + "::UseCorrectUnitNormal") == "False":
u4U[0] = 1
else:
# Eq. 2.116 in Baumgarte & Shapiro:
# n^mu = {1/alpha, -beta^i/alpha}. Note that n_mu = {alpha,0}, so n^mu n_mu = -1.
import BSSN.BSSN_quantities as Bq
Bq.declare_BSSN_gridfunctions_if_not_declared_already()
Bq.BSSN_basic_tensors()
u4U[0] = 1 / Bq.alpha
for i in range(DIM):
u4U[i + 1] = -Bq.betaU[i] / Bq.alpha
l4U = ixp.zerorank1(DIM=4)
n4U = ixp.zerorank1(DIM=4)
mre4U = ixp.zerorank1(DIM=4)
mim4U = ixp.zerorank1(DIM=4)
# M_SQRT1_2 = 1 / sqrt(2) (defined in math.h on Linux)
M_SQRT1_2 = par.Cparameters("#define", thismodule, "M_SQRT1_2", "")
isqrt2 = M_SQRT1_2 #1/sp.sqrt(2) <- SymPy drops precision to 15 sig. digits in unit tests
for mu in range(4):
l4U[mu] = isqrt2 * (u4U[mu] + r4U[mu])
n4U[mu] = isqrt2 * (u4U[mu] - r4U[mu])
mre4U[mu] = isqrt2 * theta4U[mu]
mim4U[mu] = isqrt2 * phi4U[mu]
def Toroidal():
system = par.parval_from_str(thismodule + "::System_to_use")
DIM = par.parval_from_str("grid::DIM")
dst_basis = par.parval_from_str("reference_metric::CoordSystem")
# Set coordinate system to Cartesian
par.set_parval_from_str("reference_metric::CoordSystem", "Cartesian")
rfm.reference_metric()
global AidU, EidU, psi_ID
x = rfm.xxCart[0]
y = rfm.xxCart[1]
z = rfm.xxCart[2]
AidD_Sph = ixp.zerorank1()
# Set coordinate transformations:
r = sp.sqrt(x * x + y * y + z * z)
sin_theta = z / r
u = time + r
v = time - r
e_lam_u = sp.exp(-lam * u**2)
e_lam_v = sp.exp(-lam * v**2)
# Equation 16 from https://arxiv.org/abs/gr-qc/0201051
AD_phi_hat = (amp*sin_theta)*( ((e_lam_v - e_lam_u)/r**2) - \
2*lam*(v*e_lam_v + u*e_lam_u)/r )
AidD_Sph[2] = AD_phi_hat / (r * sin_theta)
# Coordinate transformation from spherical to Cartesian
AidU_Cart = ixp.zerorank1()
Jac_dxSphU_dxCartD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dxSphU_dxCartD[i][j] = sp.diff(rfm.xxSph[i], rfm.xxCart[j])
# Jac_dxCartU_dxSphD[i][j] = sp.diff(rfm.xxCart[i],rfm.xx[j])
Jac_dxCartU_dxSphD, dummy = ixp.generic_matrix_inverter3x3(
Jac_dxSphU_dxCartD)
for i in range(DIM):
for j in range(DIM):
AidU_Cart[i] += Jac_dxCartU_dxSphD[i][j] * AidD_Sph[j]
for i in range(DIM):
AidU_Cart[i] = sp.simplify(AidU_Cart[i])
# rfm is still defined in Cartesian coordinates
cart_xx = ixp.declarerank1("cart_xx")
for i in range(3):
for k in range(3):
AidU_Cart[i] = AidU_Cart[i].subs(rfm.xx[k], cart_xx[k])
# Set coordinate system to dst_basis
par.set_parval_from_str("reference_metric::CoordSystem", dst_basis)
rfm.reference_metric()
for i in range(3):
for k in range(3):
AidU_Cart[i] = AidU_Cart[i].subs(cart_xx[k], rfm.xxCart[k])
# if radial_like_dst_xx0:
# for j in range(3):
# AidU_Cart[j] = sp.refine(sp.simplify(AidU_Cart[j]), sp.Q.positive(rfm.xx[0]))
# Step 3: Transform BSSN tensors in Cartesian basis to destination grid basis, using center of dest. grid as origin
# Step 3.a: Next construct Jacobian and inverse Jacobian matrices:
# _Jac_dUCart_dDrfmUD is unused.
_Jac_dUCart_dDrfmUD, Jac_dUrfm_dDCartUD = rfm.compute_Jacobian_and_inverseJacobian_tofrom_Cartesian(
)
# Step 3.b: Convert basis of all BSSN *vectors* from Cartesian to destination basis
AidU = rfm.basis_transform_vectorU_from_Cartesian_to_rfmbasis(
Jac_dUrfm_dDCartUD, AidU_Cart)
# Define electric field --> E^i = -\partial_t A^i
EidU = ixp.zerorank1()
for j in range(DIM):
EidU[j] = -sp.diff(AidU[j], time)
psi_ID = sp.sympify(0)
if system == "System_II":
global Gamma_ID
Gamma_ID = sp.sympify(0)
print('Currently using ' + system + ' initial data')
elif system == "System_I":
print('Currently using ' + system + ' initial data')
else:
print(
"Invalid choice of system: System_to_use must be either System_I or System_II"
)
sys.exit(1)
def generate_C_code_for_Stilde_flux(
out_dir,
inputs_provided=False,
alpha_face=None,
gamma_faceDD=None,
beta_faceU=None,
Valenciav_rU=None,
B_rU=None,
Valenciav_lU=None,
B_lU=None,
Stilde_flux_HLLED=None,
sqrt4pi=None,
outCparams="outCverbose=False,CSE_sorting=none",
write_cmax_cmin=False,
gamma_faceUU=None,
phi_face=None):
if not inputs_provided:
# We will pass values of the gridfunction on the cell faces into the function. This requires us
# to declare them as C parameters in NRPy+. We will denote this with the _face infix/suffix.
alpha_face = gri.register_gridfunctions("AUXEVOL", "alpha_face")
gamma_faceDD = ixp.register_gridfunctions_for_single_rank2(
"AUXEVOL", "gamma_faceDD", "sym01")
beta_faceU = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "beta_faceU")
# We'll need some more gridfunctions, now, to represent the reconstructions of BU and ValenciavU
# on the right and left faces
Valenciav_rU = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "Valenciav_rU", DIM=3)
B_rU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"B_rU",
DIM=3)
Valenciav_lU = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "Valenciav_lU", DIM=3)
B_lU = ixp.register_gridfunctions_for_single_rank1("AUXEVOL",
"B_lU",
DIM=3)
sqrt4pi = par.Cparameters("REAL", thismodule, "sqrt4pi",
"sqrt(4.0*M_PI)")
# We'll also need to store the results of the HLLE step between functions.
Stilde_flux_HLLED = ixp.register_gridfunctions_for_single_rank1(
"AUXEVOL", "Stilde_flux_HLLED")
if write_cmax_cmin:
# In the staggered case, we will also want to output cmax and cmin
# If we want to write cmax and cmin, we will need to be able to change auxevol_gfs:
input_params_for_Stilde_flux = "const paramstruct *params,REAL *auxevol_gfs,REAL *rhs_gfs"
else:
input_params_for_Stilde_flux = "const paramstruct *params,const REAL *auxevol_gfs,REAL *rhs_gfs"
if gamma_faceUU is None:
gamma_faceUU, unusedgammaDET = ixp.generic_matrix_inverter3x3(
gamma_faceDD)
if write_cmax_cmin:
name_suffixes = ["_x", "_y", "_z"]
for flux_dirn in range(3):
calculate_Stilde_flux(flux_dirn,alpha_face,gamma_faceDD,beta_faceU,\
Valenciav_rU,B_rU,Valenciav_lU,B_lU,sqrt4pi,gamma_faceUU,phi_face)
Stilde_flux_to_print = [
lhrh(lhs=gri.gfaccess("out_gfs", "Stilde_flux_HLLED0"),
rhs=Stilde_fluxD[0]),
lhrh(lhs=gri.gfaccess("out_gfs", "Stilde_flux_HLLED1"),
rhs=Stilde_fluxD[1]),
lhrh(lhs=gri.gfaccess("out_gfs", "Stilde_flux_HLLED2"),
rhs=Stilde_fluxD[2]),
]
if write_cmax_cmin:
Stilde_flux_to_print = Stilde_flux_to_print \
+[
lhrh(lhs=gri.gfaccess("out_gfs","cmax"+name_suffixes[flux_dirn]),rhs=chsp.cmax),
lhrh(lhs=gri.gfaccess("out_gfs","cmin"+name_suffixes[flux_dirn]),rhs=chsp.cmin),
]
desc = "Compute the flux term of all 3 components of tilde{S}_i on the left face in the " + str(
flux_dirn) + "direction for all components."
name = "calculate_Stilde_flux_D" + str(flux_dirn)
Ccode_function = outCfunction(
outfile="returnstring",
desc=desc,
name=name,
params=input_params_for_Stilde_flux,
body=fin.FD_outputC("returnstring",
Stilde_flux_to_print,
params=outCparams).replace("IDX4", "IDX4S"),
loopopts="InteriorPoints",
rel_path_for_Cparams=os.path.join("../")).replace(
"NGHOSTS+Nxx0", "NGHOSTS+Nxx0+1").replace(
"NGHOSTS+Nxx1",
"NGHOSTS+Nxx1+1").replace("NGHOSTS+Nxx2", "NGHOSTS+Nxx2+1")
with open(os.path.join(out_dir, name + ".h"), "w") as file:
file.write(Ccode_function)
pre_body = """// Notice in the loop below that we go from 3 to cctk_lsh-3 for i, j, AND k, even though
// we are only computing the flux in one direction. This is because in the end,
// we only need the rhs's from 3 to cctk_lsh-3 for i, j, and k.
const REAL invdxi[4] = {1e100,invdx0,invdx1,invdx2};
const REAL invdx = invdxi[flux_dirn];"""
FD_body = """const int index = IDX3S(i0,i1,i2);
const int indexp1 = IDX3S(i0+kronecker_delta[flux_dirn][0],i1+kronecker_delta[flux_dirn][1],i2+kronecker_delta[flux_dirn][2]);
rhs_gfs[IDX4ptS(STILDED0GF,index)] += (auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED0GF,index)] - auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED0GF,indexp1)] ) * invdx;
rhs_gfs[IDX4ptS(STILDED1GF,index)] += (auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED1GF,index)] - auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED1GF,indexp1)] ) * invdx;
rhs_gfs[IDX4ptS(STILDED2GF,index)] += (auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED2GF,index)] - auxevol_gfs[IDX4ptS(STILDE_FLUX_HLLED2GF,indexp1)] ) * invdx;"""
desc = "Compute the difference in the flux of StildeD on the opposite faces in flux_dirn for all components."
name = "calculate_Stilde_rhsD"
outCfunction(
outfile=os.path.join(out_dir, name + ".h"),
desc=desc,
name=name,
params=
"const int flux_dirn,const paramstruct *params,const REAL *auxevol_gfs,REAL *rhs_gfs",
preloop=pre_body,
body=FD_body,
loopopts="InteriorPoints",
rel_path_for_Cparams=os.path.join("../"))
def Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(
CoordType_in, Sph_r_th_ph_or_Cart_xyz, gammaDD_inSphorCart,
KDD_inSphorCart, alpha_inSphorCart, betaU_inSphorCart, BU_inSphorCart):
# This routine converts the ADM variables
# $$\left\{\gamma_{ij}, K_{ij}, \alpha, \beta^i\right\}$$
# in Spherical or Cartesian basis+coordinates, first to the BSSN variables
# in the chosen reference_metric::CoordSystem coordinate system+basis:
# $$\left\{\bar{\gamma}_{i j},\bar{A}_{i j},\phi, K, \bar{\Lambda}^{i}, \alpha, \beta^i, B^i\right\},$$
# and then to the rescaled variables:
# $$\left\{h_{i j},a_{i j},\phi, K, \lambda^{i}, \alpha, \mathcal{V}^i, \mathcal{B}^i\right\}.$$
# The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations.
# To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in
# the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3.
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
# Step 0: Copy gammaSphDD_in to gammaSphDD, KSphDD_in to KSphDD, etc.
# This ensures that the input arrays are not modified below;
# modifying them would result in unexpected effects outside
# this function.
alphaSphorCart = alpha_inSphorCart
betaSphorCartU = ixp.zerorank1()
BSphorCartU = ixp.zerorank1()
gammaSphorCartDD = ixp.zerorank2()
KSphorCartDD = ixp.zerorank2()
for i in range(DIM):
betaSphorCartU[i] = betaU_inSphorCart[i]
BSphorCartU[i] = BU_inSphorCart[i]
for j in range(DIM):
gammaSphorCartDD[i][j] = gammaDD_inSphorCart[i][j]
KSphorCartDD[i][j] = KDD_inSphorCart[i][j]
# Make sure that rfm.reference_metric() has been called.
# We'll need the variables it defines throughout this module.
if rfm.have_already_called_reference_metric_function == False:
print(
"Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without"
)
print(
" first setting up reference metric, by calling rfm.reference_metric()."
)
exit(1)
# Step 1: All input quantities are in terms of r,th,ph or x,y,z. We want them in terms
# of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace
# r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2
# as defined for this particular reference metric in reference_metric.py's
# xxSph[] or xxCart[], respectively:
# Note that substitution only works when the variable is not an integer. Hence the
# if isinstance(...,...) stuff:
def sympify_integers__replace_rthph_or_Cartxyz(obj, rthph_or_xyz,
rthph_or_xyz_of_xx):
if isinstance(obj, int):
return sp.sympify(obj)
else:
return obj.subs(rthph_or_xyz[0], rthph_or_xyz_of_xx[0]).\
subs(rthph_or_xyz[1], rthph_or_xyz_of_xx[1]).\
subs(rthph_or_xyz[2], rthph_or_xyz_of_xx[2])
r_th_ph_or_Cart_xyz_of_xx = []
if CoordType_in == "Spherical":
r_th_ph_or_Cart_xyz_of_xx = rfm.xxSph
elif CoordType_in == "Cartesian":
r_th_ph_or_Cart_xyz_of_xx = rfm.xxCart
else:
print(
"Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords."
)
exit(1)
alphaSphorCart = sympify_integers__replace_rthph_or_Cartxyz(
alphaSphorCart, Sph_r_th_ph_or_Cart_xyz, r_th_ph_or_Cart_xyz_of_xx)
for i in range(DIM):
betaSphorCartU[i] = sympify_integers__replace_rthph_or_Cartxyz(
betaSphorCartU[i], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
BSphorCartU[i] = sympify_integers__replace_rthph_or_Cartxyz(
BSphorCartU[i], Sph_r_th_ph_or_Cart_xyz, r_th_ph_or_Cart_xyz_of_xx)
for j in range(DIM):
gammaSphorCartDD[i][
j] = sympify_integers__replace_rthph_or_Cartxyz(
gammaSphorCartDD[i][j], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
KSphorCartDD[i][j] = sympify_integers__replace_rthph_or_Cartxyz(
KSphorCartDD[i][j], Sph_r_th_ph_or_Cart_xyz,
r_th_ph_or_Cart_xyz_of_xx)
# Step 2: All ADM initial data quantities are now functions of xx0,xx1,xx2, but
# they are still in the Spherical or Cartesian basis. We can now directly apply
# Jacobian transformations to get them in the correct xx0,xx1,xx2 basis:
# alpha is a scalar, so no Jacobian transformation is necessary.
alpha = alphaSphorCart
Jac_dUSphorCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(
r_th_ph_or_Cart_xyz_of_xx[i], rfm.xx[j])
Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUSphorCart_dDrfmUD)
betaU = ixp.zerorank1()
BU = ixp.zerorank1()
gammaDD = ixp.zerorank2()
KDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j]
BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j]
for k in range(DIM):
for l in range(DIM):
gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][
i] * Jac_dUSphorCart_dDrfmUD[l][j] * gammaSphorCartDD[
k][l]
KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][
i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l]
# Step 3: All ADM quantities were input into this function in the Spherical or Cartesian
# basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above,
# we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2.
# Here we convert ADM quantities to their BSSN Curvilinear counterparts:
# Step 3.1: Convert ADM $\gamma_{ij}$ to BSSN $\bar{\gamma}_{ij}$:
# We have (Eqs. 2 and 3 of [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):
# \bar{\gamma}_{i j} = \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \gamma_{ij}.
gammaUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaDD)
gammabarDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
gammabarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational(
1, 3)) * gammaDD[i][j]
# Step 3.2: Convert the extrinsic curvature $K_{ij}$ to the trace-free extrinsic
# curvature $\bar{A}_{ij}$, plus the trace of the extrinsic curvature $K$,
# where (Eq. 3 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)):
# K = \gamma^{ij} K_{ij}, and
# \bar{A}_{ij} &= \left(\frac{\bar{\gamma}}{\gamma}\right)^{1/3} \left(K_{ij} - \frac{1}{3} \gamma_{ij} K \right)
trK = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
trK += gammaUU[i][j] * KDD[i][j]
AbarDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
AbarDD[i][j] = (rfm.detgammahat / gammaDET)**(sp.Rational(
1, 3)) * (KDD[i][j] - sp.Rational(1, 3) * gammaDD[i][j] * trK)
# Step 3.3: Define $\bar{\Lambda}^i$ (Eqs. 4 and 5 of [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)):
# \bar{\Lambda}^i = \bar{\gamma}^{jk}\left(\bar{\Gamma}^i_{jk} - \hat{\Gamma}^i_{jk}\right).
gammabarUU, gammabarDET = ixp.symm_matrix_inverter3x3(gammabarDD)
# First compute \bar{\Gamma}^i_{jk}:
GammabarUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GammabarUDD[i][j][k] += sp.Rational(
1, 2) * gammabarUU[i][l] * (
sp.diff(gammabarDD[l][j], rfm.xx[k]) +
sp.diff(gammabarDD[l][k], rfm.xx[j]) -
sp.diff(gammabarDD[j][k], rfm.xx[l]))
# Next evaluate \bar{\Lambda}^i, based on GammabarUDD above and GammahatUDD
# (from the reference metric):
LambdabarU = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LambdabarU[i] += gammabarUU[j][k] * (GammabarUDD[i][j][k] -
rfm.GammahatUDD[i][j][k])
# Step 3.4: Set the conformal factor variable $\texttt{cf}$, which is set
# by the "BSSN_quantities::ConformalFactor" parameter. For example if
# "ConformalFactor" is set to "phi", we can use Eq. 3 of
# [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf),
# which in arbitrary coordinates is written:
# \phi = \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right).
# Alternatively if "BSSN_quantities::ConformalFactor" is set to "chi", then
# \chi = e^{-4 \phi} = \exp\left(-4 \frac{1}{12} \left(\frac{\gamma}{\bar{\gamma}}\right)\right)
# = \exp\left(-\frac{1}{3} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) = \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/3}.
#
# Finally if "BSSN_quantities::ConformalFactor" is set to "W", then
# W = e^{-2 \phi} = \exp\left(-2 \frac{1}{12} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) =
# \exp\left(-\frac{1}{6} \log\left(\frac{\gamma}{\bar{\gamma}}\right)\right) =
# \left(\frac{\gamma}{\bar{\gamma}}\right)^{-1/6}.
cf = sp.sympify(0)
if par.parval_from_str("ConformalFactor") == "phi":
cf = sp.Rational(1, 12) * sp.log(gammaDET / gammabarDET)
elif par.parval_from_str("ConformalFactor") == "chi":
cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 3))
elif par.parval_from_str("ConformalFactor") == "W":
cf = (gammaDET / gammabarDET)**(-sp.Rational(1, 6))
else:
print("Error ConformalFactor type = \"" +
par.parval_from_str("ConformalFactor") + "\" unknown.")
exit(1)
# Step 4: Rescale tensorial quantities according to the prescription described in
# the [BSSN in curvilinear coordinates tutorial module](Tutorial-BSSNCurvilinear.ipynb)
# (also [Ruchlin *et al.*](https://arxiv.org/pdf/1712.07658.pdf)):
#
# h_{ij} &= (\bar{\gamma}_{ij} - \hat{\gamma}_{ij})/\text{ReDD[i][j]}\\
# a_{ij} &= \bar{A}_{ij}/\text{ReDD[i][j]}\\
# \lambda^i &= \bar{\Lambda}^i/\text{ReU[i]}\\
# \mathcal{V}^i &= \beta^i/\text{ReU[i]}\\
# \mathcal{B}^i &= B^i/\text{ReU[i]}\\
hDD = ixp.zerorank2()
aDD = ixp.zerorank2()
lambdaU = ixp.zerorank1()
vetU = ixp.zerorank1()
betU = ixp.zerorank1()
for i in range(DIM):
lambdaU[i] = LambdabarU[i] / rfm.ReU[i]
vetU[i] = betaU[i] / rfm.ReU[i]
betU[i] = BU[i] / rfm.ReU[i]
for j in range(DIM):
hDD[i][j] = (gammabarDD[i][j] - rfm.ghatDD[i][j]) / rfm.ReDD[i][j]
aDD[i][j] = AbarDD[i][j] / rfm.ReDD[i][j]
#print(sp.mathematica_code(hDD[0][0]))
# Step 5: Return the BSSN Curvilinear variables in the desired xx0,xx1,xx2
# basis, and as functions of the consistent xx0,xx1,xx2 coordinates.
return cf, hDD, lambdaU, aDD, trK, alpha, vetU, betU
def Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear(CoordType_in, ADM_input_function_name,
Ccodesdir = "BSSN", pointer_to_ID_inputs=False,loopopts=",oldloops"):
# The ADM & BSSN formalisms only work in 3D; they are 3+1 decompositions of Einstein's equations.
# To implement axisymmetry or spherical symmetry, simply set all spatial derivatives in
# the relevant angular directions to zero; DO NOT SET DIM TO ANYTHING BUT 3.
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
# Step 1: All ADM initial data quantities are now functions of xx0,xx1,xx2, but
# they are still in the Spherical or Cartesian basis. We can now directly apply
# Jacobian transformations to get them in the correct xx0,xx1,xx2 basis:
# All input quantities are in terms of r,th,ph or x,y,z. We want them in terms
# of xx0,xx1,xx2, so here we call sympify_integers__replace_rthph() to replace
# r,th,ph or x,y,z, respectively, with the appropriate functions of xx0,xx1,xx2
# as defined for this particular reference metric in reference_metric.py's
# xxSph[] or xx_to_Cart[], respectively:
# Define the input variables:
gammaSphorCartDD = ixp.declarerank2("gammaSphorCartDD", "sym01")
KSphorCartDD = ixp.declarerank2("KSphorCartDD", "sym01")
alphaSphorCart = sp.symbols("alphaSphorCart")
betaSphorCartU = ixp.declarerank1("betaSphorCartU")
BSphorCartU = ixp.declarerank1("BSphorCartU")
# Make sure that rfm.reference_metric() has been called.
# We'll need the variables it defines throughout this module.
if rfm.have_already_called_reference_metric_function == False:
print("Error. Called Convert_Spherical_ADM_to_BSSN_curvilinear() without")
print(" first setting up reference metric, by calling rfm.reference_metric().")
sys.exit(1)
r_th_ph_or_Cart_xyz_oID_xx = []
if CoordType_in == "Spherical":
r_th_ph_or_Cart_xyz_oID_xx = rfm.xxSph
elif CoordType_in == "Cartesian":
r_th_ph_or_Cart_xyz_oID_xx = rfm.xx_to_Cart
else:
print("Error: Can only convert ADM Cartesian or Spherical initial data to BSSN Curvilinear coords.")
sys.exit(1)
# Step 2: All ADM initial data quantities are now functions of xx0,xx1,xx2, but
# they are still in the Spherical or Cartesian basis. We can now directly apply
# Jacobian transformations to get them in the correct xx0,xx1,xx2 basis:
# alpha is a scalar, so no Jacobian transformation is necessary.
alpha = alphaSphorCart
Jac_dUSphorCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUSphorCart_dDrfmUD[i][j] = sp.diff(r_th_ph_or_Cart_xyz_oID_xx[i], rfm.xx[j])
Jac_dUrfm_dDSphorCartUD, dummyDET = ixp.generic_matrix_inverter3x3(Jac_dUSphorCart_dDrfmUD)
betaU = ixp.zerorank1()
BU = ixp.zerorank1()
gammaDD = ixp.zerorank2()
KDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
betaU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * betaSphorCartU[j]
BU[i] += Jac_dUrfm_dDSphorCartUD[i][j] * BSphorCartU[j]
for k in range(DIM):
for l in range(DIM):
gammaDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * \
gammaSphorCartDD[k][l]
KDD[i][j] += Jac_dUSphorCart_dDrfmUD[k][i] * Jac_dUSphorCart_dDrfmUD[l][j] * KSphorCartDD[k][l]
# Step 3: All ADM quantities were input into this function in the Spherical or Cartesian
# basis, as functions of r,th,ph or x,y,z, respectively. In Steps 1 and 2 above,
# we converted them to the xx0,xx1,xx2 basis, and as functions of xx0,xx1,xx2.
# Here we convert ADM quantities in the "rfm" basis to their BSSN Curvilinear
# counterparts, for all BSSN quantities *except* lambda^i:
import BSSN.BSSN_in_terms_of_ADM as BitoA
BitoA.gammabarDD_hDD(gammaDD)
BitoA.trK_AbarDD_aDD(gammaDD, KDD)
BitoA.cf_from_gammaDD(gammaDD)
BitoA.betU_vetU(betaU, BU)
hDD = BitoA.hDD
trK = BitoA.trK
aDD = BitoA.aDD
cf = BitoA.cf
vetU = BitoA.vetU
betU = BitoA.betU
# Step 4: Compute $\bar{\Lambda}^i$ (Eqs. 4 and 5 of
# [Baumgarte *et al.*](https://arxiv.org/pdf/1211.6632.pdf)),
# from finite-difference derivatives of rescaled metric
# quantities $h_{ij}$:
# \bar{\Lambda}^i = \bar{\gamma}^{jk}\left(\bar{\Gamma}^i_{jk} - \hat{\Gamma}^i_{jk}\right).
# The reference_metric.py module provides us with analytic expressions for
# $\hat{\Gamma}^i_{jk}$, so here we need only compute
# finite-difference expressions for $\bar{\Gamma}^i_{jk}$, based on
# the values for $h_{ij}$ provided in the initial data. Once
# $\bar{\Lambda}^i$ has been computed, we apply the usual rescaling
# procedure:
# \lambda^i = \bar{\Lambda}^i/\text{ReU[i]},
# and then output the result to a C file using the NRPy+
# finite-difference C output routine.
# We will need all BSSN gridfunctions to be defined, as well as
# expressions for gammabarDD_dD in terms of exact derivatives of
# the rescaling matrix and finite-difference derivatives of
# hDD's. This functionality is provided by BSSN.BSSN_unrescaled_and_barred_vars,
# which we call here to overwrite above definitions of gammabarDD,gammabarUU, etc.
Bq.gammabar__inverse_and_derivs() # Provides gammabarUU and GammabarUDD
gammabarUU = Bq.gammabarUU
GammabarUDD = Bq.GammabarUDD
# Next evaluate \bar{\Lambda}^i, based on GammabarUDD above and GammahatUDD
# (from the reference metric):
LambdabarU = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LambdabarU[i] += gammabarUU[j][k] * (GammabarUDD[i][j][k] - rfm.GammahatUDD[i][j][k])
# Finally apply rescaling:
# lambda^i = Lambdabar^i/\text{ReU[i]}
lambdaU = ixp.zerorank1()
for i in range(DIM):
lambdaU[i] = LambdabarU[i] / rfm.ReU[i]
if ADM_input_function_name == "DoNotOutputADMInputFunction":
return hDD,aDD,trK,vetU,betU,alpha,cf,lambdaU
# Step 5.A: Output files containing finite-differenced lambdas.
outCparams = "preindent=1,outCfileaccess=a,outCverbose=False,includebraces=False"
lambdaU_expressions = [lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU0"), rhs=lambdaU[0]),
lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU1"), rhs=lambdaU[1]),
lhrh(lhs=gri.gfaccess("in_gfs", "lambdaU2"), rhs=lambdaU[2])]
desc = "Output lambdaU[i] for BSSN, built using finite-difference derivatives."
name = "ID_BSSN_lambdas"
params = "const paramstruct *restrict params,REAL *restrict xx[3],REAL *restrict in_gfs"
preloop = ""
enableCparameters=True
if "oldloops" in loopopts:
params = "const int Nxx[3],const int Nxx_plus_2NGHOSTS[3],REAL *xx[3],const REAL dxx[3],REAL *in_gfs"
enableCparameters=False
preloop = """
const REAL invdx0 = 1.0/dxx[0];
const REAL invdx1 = 1.0/dxx[1];
const REAL invdx2 = 1.0/dxx[2];
"""
outCfunction(
outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params,
preloop=preloop,
body=fin.FD_outputC("returnstring", lambdaU_expressions, outCparams),
loopopts="InteriorPoints,Read_xxs"+loopopts, enableCparameters=enableCparameters)
# Step 5: Output all ADM-to-BSSN expressions to a C function. This function
# must first call the ID_ADM_SphorCart() defined above. Using these
# Spherical or Cartesian data, it sets up all quantities needed for
# BSSNCurvilinear initial data, *except* $\lambda^i$, which must be
# computed from numerical data using finite-difference derivatives.
ID_inputs_param = "ID_inputs other_inputs,"
if pointer_to_ID_inputs == True:
ID_inputs_param = "ID_inputs *other_inputs,"
desc = "Write BSSN variables in terms of ADM variables at a given point xx0,xx1,xx2"
name = "ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs"
enableCparameters=True
params = "const paramstruct *restrict params, "
if "oldloops" in loopopts:
enableCparameters=False
params = ""
params += "const int i0i1i2[3], const REAL xx0xx1xx2[3]," + ID_inputs_param + """
REAL *hDD00,REAL *hDD01,REAL *hDD02,REAL *hDD11,REAL *hDD12,REAL *hDD22,
REAL *aDD00,REAL *aDD01,REAL *aDD02,REAL *aDD11,REAL *aDD12,REAL *aDD22,
REAL *trK,
REAL *vetU0,REAL *vetU1,REAL *vetU2,
REAL *betU0,REAL *betU1,REAL *betU2,
REAL *alpha, REAL *cf"""
outCparams = "preindent=1,outCverbose=False,includebraces=False"
outCfunction(
outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params,
body="""
REAL gammaSphorCartDD00,gammaSphorCartDD01,gammaSphorCartDD02,
gammaSphorCartDD11,gammaSphorCartDD12,gammaSphorCartDD22;
REAL KSphorCartDD00,KSphorCartDD01,KSphorCartDD02,
KSphorCartDD11,KSphorCartDD12,KSphorCartDD22;
REAL alphaSphorCart,betaSphorCartU0,betaSphorCartU1,betaSphorCartU2;
REAL BSphorCartU0,BSphorCartU1,BSphorCartU2;
const REAL xx0 = xx0xx1xx2[0];
const REAL xx1 = xx0xx1xx2[1];
const REAL xx2 = xx0xx1xx2[2];
REAL xyz_or_rthph[3];\n""" +
outputC(r_th_ph_or_Cart_xyz_oID_xx[0:3], ["xyz_or_rthph[0]", "xyz_or_rthph[1]", "xyz_or_rthph[2]"],
"returnstring",
outCparams + ",CSE_enable=False") + " " + ADM_input_function_name + """(params,i0i1i2, xyz_or_rthph, other_inputs,
&gammaSphorCartDD00,&gammaSphorCartDD01,&gammaSphorCartDD02,
&gammaSphorCartDD11,&gammaSphorCartDD12,&gammaSphorCartDD22,
&KSphorCartDD00,&KSphorCartDD01,&KSphorCartDD02,
&KSphorCartDD11,&KSphorCartDD12,&KSphorCartDD22,
&alphaSphorCart,&betaSphorCartU0,&betaSphorCartU1,&betaSphorCartU2,
&BSphorCartU0,&BSphorCartU1,&BSphorCartU2);
// Next compute all rescaled BSSN curvilinear quantities:\n""" +
outputC([hDD[0][0], hDD[0][1], hDD[0][2], hDD[1][1], hDD[1][2], hDD[2][2],
aDD[0][0], aDD[0][1], aDD[0][2], aDD[1][1], aDD[1][2], aDD[2][2],
trK, vetU[0], vetU[1], vetU[2], betU[0], betU[1], betU[2],
alpha, cf],
["*hDD00", "*hDD01", "*hDD02", "*hDD11", "*hDD12", "*hDD22",
"*aDD00", "*aDD01", "*aDD02", "*aDD11", "*aDD12", "*aDD22",
"*trK", "*vetU0", "*vetU1", "*vetU2", "*betU0", "*betU1", "*betU2",
"*alpha", "*cf"], "returnstring", params=outCparams),
enableCparameters=enableCparameters)
# Step 5.a: Output the driver function for the above
# function ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs()
# Next write the driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs():
desc = """Driver function for ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(),
which writes BSSN variables in terms of ADM variables at a given point xx0,xx1,xx2"""
name = "ID_BSSN__ALL_BUT_LAMBDAs"
params = "const paramstruct *restrict params,REAL *restrict xx[3]," + ID_inputs_param + "REAL *in_gfs"
enableCparameters = True
funccallparams = "params, "
idx3replace = "IDX3S"
idx4ptreplace = "IDX4ptS"
if "oldloops" in loopopts:
params = "const int Nxx_plus_2NGHOSTS[3],REAL *xx[3]," + ID_inputs_param + "REAL *in_gfs"
enableCparameters = False
funccallparams = ""
idx3replace = "IDX3"
idx4ptreplace = "IDX4pt"
outCfunction(
outfile=os.path.join(Ccodesdir, name + ".h"), desc=desc, name=name, params=params,
body="""
const int idx = IDX3(i0,i1,i2);
const int i0i1i2[3] = {i0,i1,i2};
const REAL xx0xx1xx2[3] = {xx0,xx1,xx2};
ID_ADM_xx0xx1xx2_to_BSSN_xx0xx1xx2__ALL_BUT_LAMBDAs(""".replace("IDX3",idx3replace)+funccallparams+"""i0i1i2,xx0xx1xx2,other_inputs,
&in_gfs[IDX4pt(HDD00GF,idx)],&in_gfs[IDX4pt(HDD01GF,idx)],&in_gfs[IDX4pt(HDD02GF,idx)],
&in_gfs[IDX4pt(HDD11GF,idx)],&in_gfs[IDX4pt(HDD12GF,idx)],&in_gfs[IDX4pt(HDD22GF,idx)],
&in_gfs[IDX4pt(ADD00GF,idx)],&in_gfs[IDX4pt(ADD01GF,idx)],&in_gfs[IDX4pt(ADD02GF,idx)],
&in_gfs[IDX4pt(ADD11GF,idx)],&in_gfs[IDX4pt(ADD12GF,idx)],&in_gfs[IDX4pt(ADD22GF,idx)],
&in_gfs[IDX4pt(TRKGF,idx)],
&in_gfs[IDX4pt(VETU0GF,idx)],&in_gfs[IDX4pt(VETU1GF,idx)],&in_gfs[IDX4pt(VETU2GF,idx)],
&in_gfs[IDX4pt(BETU0GF,idx)],&in_gfs[IDX4pt(BETU1GF,idx)],&in_gfs[IDX4pt(BETU2GF,idx)],
&in_gfs[IDX4pt(ALPHAGF,idx)],&in_gfs[IDX4pt(CFGF,idx)]);
""".replace("IDX4pt",idx4ptreplace),
loopopts="AllPoints,Read_xxs"+loopopts, enableCparameters=enableCparameters)
