Commit ad8aa1fb authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Merge branch 'disc_patch_revived' into 'master'

Disc patch revived

Implements the changes discussed in #311. 

 - The disc patch example is now 400x400x800 in length with correct internal energy.
 - The accelerations get truncated at |z-z_disc| > 300 with a cosine transition
 - The accelerations are 0 at |z-z_disc| > 380.
 - The calculation is significantly faster thanks to the extraction of constant terms.

See merge request !349
parents 356a3348 1334f2e2
......@@ -18,3 +18,5 @@ output to 'Disc-Patch-dynamic.hdf5'. These are now the ICs for the actual test.
When running SWIFT with the parameters from 'disc-patch.yml' and an
ideal gas EoS on these ICs the disc should stay in equilibrium.
The solution can be checked using the 'plotSolution.py' script.
# Define the system of units to use internally.
InternalUnitSystem:
UnitMass_in_cgs: 1.9885e33 # Grams
UnitLength_in_cgs: 3.0856776e18 # Centimeters
UnitVelocity_in_cgs: 1e5 # Centimeters per second
UnitMass_in_cgs: 1.9885e33 # Grams
UnitLength_in_cgs: 3.08567758149e18 # Centimeters
UnitVelocity_in_cgs: 1e5 # Centimeters per second
UnitCurrent_in_cgs: 1 # Amperes
UnitTemp_in_cgs: 1 # Kelvin
......@@ -11,17 +11,17 @@ TimeIntegration:
time_begin: 0 # The starting time of the simulation (in internal units).
time_end: 968. # The end time of the simulation (in internal units).
dt_min: 1e-4 # The minimal time-step size of the simulation (in internal units).
dt_max: 1. # The maximal time-step size of the simulation (in internal units).
dt_max: 10. # The maximal time-step size of the simulation (in internal units).
# Parameters governing the conserved quantities statistics
Statistics:
delta_time: 1 # Time between statistics output
delta_time: 12. # Time between statistics output
# Parameters governing the snapshots
Snapshots:
basename: Disc-Patch # Common part of the name of output files
time_first: 0. # Time of the first output (in internal units)
delta_time: 12. # Time difference between consecutive outputs (in internal units)
basename: Disc-Patch # Common part of the name of output files
time_first: 0. # Time of the first output (in internal units)
delta_time: 48. # Time difference between outputs (in internal units)
# Parameters for the hydrodynamics scheme
SPH:
......@@ -29,7 +29,7 @@ SPH:
delta_neighbours: 0.1 # The tolerance for the targetted number of neighbours.
CFL_condition: 0.1 # Courant-Friedrich-Levy condition for time integration.
max_ghost_iterations: 30 # Maximal number of iterations allowed to converge towards the smoothing length.
max_smoothing_length: 70. # Maximal smoothing length allowed (in internal units).
h_max: 60. # Maximal smoothing length allowed (in internal units).
# Parameters related to the initial conditions
InitialConditions:
......@@ -39,6 +39,8 @@ InitialConditions:
DiscPatchPotential:
surface_density: 10.
scale_height: 100.
z_disc: 200.
z_disc: 400.
z_trunc: 300.
z_max: 350.
timestep_mult: 0.03
growth_time: 5.
# Define the system of units to use internally.
InternalUnitSystem:
UnitMass_in_cgs: 1.9885e33 # Grams
UnitLength_in_cgs: 3.0856776e18 # Centimeters
UnitVelocity_in_cgs: 1e5 # Centimeters per second
UnitMass_in_cgs: 1.9885e33 # Grams
UnitLength_in_cgs: 3.08567758149e18 # Centimeters
UnitVelocity_in_cgs: 1e5 # Centimeters per second
UnitCurrent_in_cgs: 1 # Amperes
UnitTemp_in_cgs: 1 # Kelvin
......@@ -11,17 +11,17 @@ TimeIntegration:
time_begin: 968 # The starting time of the simulation (in internal units).
time_end: 12000. # The end time of the simulation (in internal units).
dt_min: 1e-4 # The minimal time-step size of the simulation (in internal units).
dt_max: 1. # The maximal time-step size of the simulation (in internal units).
dt_max: 10. # The maximal time-step size of the simulation (in internal units).
# Parameters governing the conserved quantities statistics
Statistics:
delta_time: 1 # Time between statistics output
delta_time: 24 # Time between statistics output
# Parameters governing the snapshots
Snapshots:
basename: Disc-Patch-dynamic # Common part of the name of output files
time_first: 968. # Time of the first output (in internal units)
delta_time: 24. # Time difference between consecutive outputs (in internal units)
basename: Disc-Patch-dynamic # Common part of the name of output files
time_first: 968. # Time of the first output (in internal units)
delta_time: 96. # Time difference between outputs (in internal units)
# Parameters for the hydrodynamics scheme
SPH:
......@@ -29,7 +29,7 @@ SPH:
delta_neighbours: 0.1 # The tolerance for the targetted number of neighbours.
CFL_condition: 0.1 # Courant-Friedrich-Levy condition for time integration.
max_ghost_iterations: 30 # Maximal number of iterations allowed to converge towards the smoothing length.
max_smoothing_length: 70. # Maximal smoothing length allowed (in internal units).
h_max: 60. # Maximal smoothing length allowed (in internal units).
# Parameters related to the initial conditions
InitialConditions:
......@@ -39,5 +39,7 @@ InitialConditions:
DiscPatchPotential:
surface_density: 10.
scale_height: 100.
z_disc: 200.
z_disc: 400.
z_trunc: 300.
z_max: 380.
timestep_mult: 0.03
;
; test energy / angular momentum conservation of test problem
;
iplot = 1 ; if iplot = 1, make plot of E/Lz conservation, else, simply compare final and initial energy
; set physical constants
@physunits
indir = './'
;basefile = 'Disc-Patch-dynamic_'
basefile = 'Disc-Patch_'
; set properties of potential
uL = phys.pc ; unit of length
uM = phys.msun ; unit of mass
uV = 1d5 ; unit of velocity
; properties of patch
surface_density = 100. ; surface density of all mass, which generates the gravitational potential
scale_height = 100.
z_disk = 200. ;
fgas = 0.1 ; gas fraction
gamma = 5./3.
; derived units
constG = 10.^(alog10(phys.g)+alog10(uM)-2d0*alog10(uV)-alog10(uL)) ;
pcentre = [0.,0.,z_disk] * pc / uL
utherm = !pi * constG * surface_density * scale_height / (gamma-1.)
temp = (utherm*uV^2)*phys.m_h/phys.kb
soundspeed = sqrt(gamma * (gamma-1.) * utherm)
t_dyn = sqrt(scale_height / (constG * surface_density))
rho0 = fgas*(surface_density)/(2.*scale_height)
print,' dynamical time = ',t_dyn,' = ',t_dyn*UL/uV/(1d6*phys.yr),' Myr'
print,' thermal energy per unit mass = ',utherm
print,' central density = ',rho0,' = ',rho0*uM/uL^3/m_h,' particles/cm^3'
print,' central temperature = ',temp
lambda = 2 * !pi * phys.G^1.5 * (scale_height*uL)^1.5 * (surface_density * uM/uL^2)^0.5 * phys.m_h^2 / (gamma-1) / fgas
print,' lambda = ',lambda
stop
;
infile = indir + basefile + '*'
spawn,'ls -1 '+infile,res
nfiles = n_elements(res)
; choose: calculate change of energy and Lz, comparing first and last
; snapshots for all particles, or do so for a subset
; compare all
ifile = 0
inf = indir + basefile + strtrim(string(ifile,'(i3.3)'),1) + '.hdf5'
id = h5rd(inf,'PartType0/ParticleIDs')
nfollow = n_elements(id)
; compute anlytic profile
nbins = 100
zbins = findgen(nbins)/float(nbins-1) * 2 * scale_height
rbins = (surface_density/(2.*scale_height)) / cosh(abs(zbins)/scale_height)^2
; plot analytic profile
wset,0
plot,[0],[0],xr=[0,2*scale_height],yr=[0,max(rbins)],/nodata,xtitle='|z|',ytitle=textoidl('\rho')
oplot,zbins,rbins,color=blue
ifile = 0
nskip = nfiles - 1
isave = 0
nplot = 8192 ; randomly plot particles
color = floor(findgen(nfiles)/float(nfiles-1)*255)
;for ifile=0,nfiles-1,nskip do begin
tsave = [0.]
toplot = [1,nfiles-1]
for iplot=0,1 do begin
ifile = toplot[iplot]
inf = indir + basefile + strtrim(string(ifile,'(i3.3)'),1) + '.hdf5'
time = h5ra(inf, 'Header','Time')
tsave = [tsave, time]
print,' time= ',time
p = h5rd(inf,'PartType0/Coordinates')
v = h5rd(inf,'PartType0/Velocities')
id = h5rd(inf,'PartType0/ParticleIDs')
rho = h5rd(inf,'PartType0/Density')
h = h5rd(inf,'PartType0/SmoothingLength')
utherm = h5rd(inf,'PartType0/InternalEnergy')
indx = sort(id)
; substract disk centre
for ic=0,2 do p[ic,*]=p[ic,*] - pcentre[ic]
;; ; if you want to sort particles by ID
;; id = id[indx]
;; rho = rho[indx]
;; utherm = utherm[indx]
;; h = h[indx]
;; for ic=0,2 do begin
;; tmp = reform(p[ic,*]) & p[ic,*] = tmp[indx]
;; tmp = reform(v[ic,*]) & v[ic,*] = tmp[indx]
;; endfor
ip = floor(randomu(ifile+1,nplot)*n_elements(rho))
color = red
if(ifile eq 1) then begin
color=black
endif else begin
color=red
endelse
oplot,abs(p[2,ip]), rho[ip], psym=3, color=color
isave = isave + 1
endfor
; time in units of dynamical time
tsave = tsave[1:*] / t_dyn
label = ['']
for i=0,n_elements(tsave)-1 do label=[label,'time/t_dynamic='+string(tsave[i],format='(f8.0)')]
label = label[1:*]
legend,['analytic',label[0],label[1]],linestyle=[0,0,0],color=[blue,black,red],box=0,/top,/right
; make histograms of particle velocities
xr = 1d-3 * [-1,1]
bsize = 1.d-5
ohist,v[0,*]/soundspeed,x,vx,xr[0],xr[1],bsize
ohist,v[1,*]/soundspeed,y,vy,xr[0],xr[1],bsize
ohist,v[2,*]/soundspeed,z,vz,xr[0],xr[1],bsize
wset,2
plot,x,vx,psym=10,xtitle='velocity/soundspeed',ytitle='pdf',/nodata,xr=xr,/xs
oplot,x,vx,psym=10,color=black
oplot,y,vy,psym=10,color=blue
oplot,z,vz,psym=10,color=red
legend,['vx/c','vy/c','vz/c'],linestyle=[0,0,0],color=[black,blue,red],box=0,/top,/right
end
......@@ -20,139 +20,147 @@
import h5py
import sys
import numpy
import numpy as np
import math
import random
import matplotlib.pyplot as plt
# Generates a disc-patch in hydrostatic equilibrium
# see Creasey, Theuns & Bower, 2013, for the equations:
# disc parameters are: surface density sigma
# scale height b
# density: rho(z) = (sigma/2b) sech^2(z/b)
# isothermal velocity dispersion = <v_z^2? = b pi G sigma
# grad potential = 2 pi G sigma tanh(z/b)
# potential = ln(cosh(z/b)) + const
# Dynamical time = sqrt(b / (G sigma))
# to obtain the 1/ch^2(z/b) profile from a uniform profile (a glass, say, or a uniform random variable), note that, when integrating in z
# \int 0^z dz/ch^2(z) = tanh(z)-tanh(0) = \int_0^x dx = x (where the last integral refers to a uniform density distribution), so that z = atanh(x)
# usage: python makeIC.py 1000
#
# See Creasey, Theuns & Bower, 2013, MNRAS, Volume 429, Issue 3, p.1922-1948
#
#
# Disc parameters are: surface density -- sigma
# scale height -- b
# gas adiabatic index -- gamma
#
# Problem parameters are: Ratio height/width of the box -- z_factor
# Size of the patch -- side_length
# Parameters of the gas disc
surface_density = 10.
scale_height = 100.
gas_gamma = 5./3.
# Parameters of the problem
z_factor = 2
side_length = 400.
# File
fileName = "Disc-Patch.hdf5"
####################################################################
# physical constants in cgs
NEWTON_GRAVITY_CGS = 6.672e-8
NEWTON_GRAVITY_CGS = 6.67408e-8
SOLAR_MASS_IN_CGS = 1.9885e33
PARSEC_IN_CGS = 3.0856776e18
PROTON_MASS_IN_CGS = 1.6726231e24
YEAR_IN_CGS = 3.154e+7
PARSEC_IN_CGS = 3.08567758149e18
PROTON_MASS_IN_CGS = 1.672621898e-24
BOLTZMANN_IN_CGS = 1.38064852e-16
YEAR_IN_CGS = 3.15569252e7
# choice of units
const_unit_length_in_cgs = (PARSEC_IN_CGS)
const_unit_mass_in_cgs = (SOLAR_MASS_IN_CGS)
const_unit_velocity_in_cgs = (1e5)
print "UnitMass_in_cgs: ", const_unit_mass_in_cgs
print "UnitLength_in_cgs: ", const_unit_length_in_cgs
print "UnitVelocity_in_cgs: ", const_unit_velocity_in_cgs
unit_length_in_cgs = (PARSEC_IN_CGS)
unit_mass_in_cgs = (SOLAR_MASS_IN_CGS)
unit_velocity_in_cgs = (1e5)
unit_time_in_cgs = unit_length_in_cgs / unit_velocity_in_cgs
print "UnitMass_in_cgs: %.5e"%unit_mass_in_cgs
print "UnitLength_in_cgs: %.5e"%unit_length_in_cgs
print "UnitVelocity_in_cgs: %.5e"%unit_velocity_in_cgs
print "UnitTime_in_cgs: %.5e"%unit_time_in_cgs
print ""
# Derived units
const_G = NEWTON_GRAVITY_CGS * unit_mass_in_cgs * unit_time_in_cgs**2 * unit_length_in_cgs**-3
const_mp = PROTON_MASS_IN_CGS * unit_mass_in_cgs**-1
const_kb = BOLTZMANN_IN_CGS * unit_mass_in_cgs**-1 * unit_length_in_cgs**-2 * unit_time_in_cgs**2
print "--- Some constants [internal units] ---"
print "G_Newton: %.5e"%const_G
print "m_proton: %.5e"%const_mp
print "k_boltzmann: %.5e"%const_kb
print ""
# derived quantities
temp = math.pi * const_G * surface_density * scale_height * const_mp / const_kb
u_therm = const_kb * temp / ((gas_gamma-1) * const_mp)
v_disp = math.sqrt(2 * u_therm)
soundspeed = math.sqrt(u_therm / (gas_gamma * (gas_gamma-1.)))
t_dyn = math.sqrt(scale_height / (const_G * surface_density))
t_cross = scale_height / soundspeed
print "--- Properties of the gas [internal units] ---"
print "Gas temperature: %.5e"%temp
print "Gas thermal_energy: %.5e"%u_therm
print "Dynamical time: %.5e"%t_dyn
print "Sound crossing time: %.5e"%t_cross
print "Gas sound speed: %.5e"%soundspeed
print "Gas 3D vel_disp: %.5e"%v_disp
print ""
# Problem properties
boxSize_x = side_length
boxSize_y = boxSize_x
boxSize_z = boxSize_x * z_factor
volume = boxSize_x * boxSize_y * boxSize_z
M_tot = boxSize_x * boxSize_y * surface_density * math.tanh(boxSize_z / (2. * scale_height))
density = M_tot / volume
entropy = (gas_gamma - 1.) * u_therm / density**(gas_gamma - 1.)
print "--- Problem properties [internal units] ---"
print "Box: [%.1f, %.1f, %.1f]"%(boxSize_x, boxSize_y, boxSize_z)
print "Volume: %.5e"%volume
print "Total mass: %.5e"%M_tot
print "Density: %.5e"%density
print "Entropy: %.5e"%entropy
print ""
####################################################################
# Read glass pre-ICs
infile = h5py.File('glassCube_32.hdf5', "r")
one_glass_pos = infile["/PartType0/Coordinates"][:,:]
one_glass_h = infile["/PartType0/SmoothingLength"][:]
# Rescale to the problem size
one_glass_pos *= boxSize_x
one_glass_h *= boxSize_x
#print min(one_glass_p[:,0]), max(one_glass_p[:,0])
#print min(one_glass_p[:,1]), max(one_glass_p[:,1])
#print min(one_glass_p[:,2]), max(one_glass_p[:,2])
# Now create enough copies to fill the volume in z
pos = np.copy(one_glass_pos)
h = np.copy(one_glass_h)
for i in range(1, z_factor):
one_glass_pos[:,2] += boxSize_x
pos = np.append(pos, one_glass_pos, axis=0)
h = np.append(h, one_glass_h, axis=0)
# parameters of potential
surface_density = 100. # surface density of all mass, which generates the gravitational potential
scale_height = 100.
gamma = 5./3.
fgas = 0.1 # gas fraction
# derived units
const_unit_time_in_cgs = (const_unit_length_in_cgs / const_unit_velocity_in_cgs)
const_G = ((NEWTON_GRAVITY_CGS*const_unit_mass_in_cgs*const_unit_time_in_cgs*const_unit_time_in_cgs/(const_unit_length_in_cgs*const_unit_length_in_cgs*const_unit_length_in_cgs)))
print 'G=', const_G
utherm = math.pi * const_G * surface_density * scale_height / (gamma-1)
v_disp = numpy.sqrt(2 * utherm)
soundspeed = numpy.sqrt(utherm / (gamma * (gamma-1.)))
t_dyn = numpy.sqrt(scale_height / (const_G * surface_density))
t_cross = scale_height / soundspeed
print 'dynamical time = ',t_dyn,' sound crossing time = ',t_cross,' sound speed= ',soundspeed,' 3D velocity dispersion = ',v_disp,' thermal_energy= ',utherm
#print min(pos[:,0]), max(pos[:,0])
#print min(pos[:,1]), max(pos[:,1])
#print min(pos[:,2]), max(pos[:,2])
# Compute further properties of ICs
numPart = np.size(h)
mass = M_tot / numPart
# Parameters
periodic= 1 # 1 For periodic box
boxSize = 400. # [kpc]
Radius = 100. # maximum radius of particles [kpc]
G = const_G
print "--- Particle properties [internal units] ---"
print "Number part.: ", numPart
print "Part. mass: %.5e"%mass
print ""
# File
fileName = "Disc-Patch.hdf5"
# Create additional arrays
u = np.ones(numPart) * u_therm
mass = np.ones(numPart) * mass
vel = np.zeros((numPart, 3))
ids = 1 + np.linspace(0, numPart, numPart, endpoint=False)
#---------------------------------------------------
mass = 1
#--------------------------------------------------
# using glass ICs
# read glass file and generate gas positions and tile it ntile times in each dimension
ntile = 1
inglass = 'glassCube_32.hdf5'
infile = h5py.File(inglass, "r")
one_glass_p = infile["/PartType0/Coordinates"][:,:]
one_glass_h = infile["/PartType0/SmoothingLength"][:]
# scale in [-0.5,0.5]*BoxSize / ntile
one_glass_p[:,:] -= 0.5
one_glass_p *= boxSize / ntile
one_glass_h *= boxSize / ntile
ndens_glass = (one_glass_h.shape[0]) / (boxSize/ntile)**3
h_glass = numpy.amin(one_glass_h) * (boxSize/ntile)
glass_p = []
glass_h = []
for ix in range(0,ntile):
for iy in range(0,ntile):
for iz in range(0,ntile):
shift = one_glass_p.copy()
shift[:,0] += (ix-(ntile-1)/2.) * boxSize / ntile
shift[:,1] += (iy-(ntile-1)/2.) * boxSize / ntile
shift[:,2] += (iz-(ntile-1)/2.) * boxSize / ntile
glass_p.append(shift)
glass_h.append(one_glass_h.copy())
glass_p = numpy.concatenate(glass_p, axis=0)
glass_h = numpy.concatenate(glass_h, axis=0)
# random shuffle of glas ICs
numpy.random.seed(12345)
indx = numpy.random.rand(numpy.shape(glass_h)[0])
indx = numpy.argsort(indx)
glass_p = glass_p[indx, :]
glass_h = glass_h[indx]
# select numGas of them
numGas = 8192
pos = glass_p[0:numGas,:]
h = glass_h[0:numGas]
numGas = numpy.shape(pos)[0]
# compute furthe properties of ICs
column_density = fgas * surface_density * numpy.tanh(boxSize/2./scale_height)
enclosed_mass = column_density * boxSize * boxSize
pmass = enclosed_mass / numGas
meanrho = enclosed_mass / boxSize**3
print 'pmass= ',pmass,' mean(rho) = ', meanrho,' entropy= ', (gamma-1) * utherm / meanrho**(gamma-1)
# desired density
rho = surface_density / (2. * scale_height) / numpy.cosh(abs(pos[:,2])/scale_height)**2
u = (1. + 0 * h) * utherm
entropy = (gamma-1) * u / rho**(gamma-1)
mass = 0.*h + pmass
entropy_flag = 0
vel = 0 + 0 * pos
# move centre of disc to middle of box
pos[:,:] += boxSize/2
# create numPart dm particles
numPart = 0
####################################################################
# Create and write output file
#File
......@@ -160,97 +168,46 @@ file = h5py.File(fileName, 'w')
#Units
grp = file.create_group("/Units")
grp.attrs["Unit length in cgs (U_L)"] = const_unit_length_in_cgs
grp.attrs["Unit mass in cgs (U_M)"] = const_unit_mass_in_cgs
grp.attrs["Unit time in cgs (U_t)"] = const_unit_length_in_cgs / const_unit_velocity_in_cgs
grp.attrs["Unit length in cgs (U_L)"] = unit_length_in_cgs
grp.attrs["Unit mass in cgs (U_M)"] = unit_mass_in_cgs
grp.attrs["Unit time in cgs (U_t)"] = unit_time_in_cgs
grp.attrs["Unit current in cgs (U_I)"] = 1.
grp.attrs["Unit temperature in cgs (U_T)"] = 1.
# Header
grp = file.create_group("/Header")
grp.attrs["BoxSize"] = boxSize
grp.attrs["NumPart_Total"] = [numGas, numPart, 0, 0, 0, 0]
grp.attrs["BoxSize"] = [boxSize_x, boxSize_y, boxSize_z]
grp.attrs["NumPart_Total"] = [numPart, 0, 0, 0, 0, 0]
grp.attrs["NumPart_Total_HighWord"] = [0, 0, 0, 0, 0, 0]
grp.attrs["NumPart_ThisFile"] = [numGas, numPart, 0, 0, 0, 0]
grp.attrs["NumPart_ThisFile"] = [numPart, 0, 0, 0, 0, 0]
grp.attrs["Time"] = 0.0
grp.attrs["NumFilesPerSnapshot"] = 1
grp.attrs["MassTable"] = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
grp.attrs["Flag_Entropy_ICs"] = [entropy_flag]
grp.attrs["Flag_Entropy_ICs"] = [0, 0, 0, 0, 0, 0]
grp.attrs["Dimension"] = 3
#Runtime parameters
grp = file.create_group("/RuntimePars")
grp.attrs["PeriodicBoundariesOn"] = periodic
grp.attrs["PeriodicBoundariesOn"] = 1
# write gas particles
grp0 = file.create_group("/PartType0")
ds = grp0.create_dataset('Coordinates', (numGas, 3), 'f')
ds[()] = pos
ds = grp0.create_dataset('Coordinates', (numPart, 3), 'f', data=pos)
ds = grp0.create_dataset('Velocities', (numPart, 3), 'f')
ds = grp0.create_dataset('Masses', (numPart,), 'f', data=mass)
ds = grp0.create_dataset('SmoothingLength', (numPart,), 'f', data=h)
ds = grp0.create_dataset('InternalEnergy', (numPart,), 'f', data=u)
ds = grp0.create_dataset('ParticleIDs', (numPart, ), 'L', data=ids)
ds = grp0.create_dataset('Velocities', (numGas, 3), 'f')
ds[()] = vel
ds = grp0.create_dataset('Masses', (numGas,), 'f')
ds[()] = mass
ds = grp0.create_dataset('SmoothingLength', (numGas,), 'f')
ds[()] = h
ds = grp0.create_dataset('InternalEnergy', (numGas,), 'f')
u = numpy.full((numGas, ), utherm)
if (entropy_flag == 1):
ds[()] = entropy
else:
ds[()] = u
ids = 1 + numpy.linspace(0, numGas, numGas, endpoint=False)
ds = grp0.create_dataset('ParticleIDs', (numGas, ), 'L')
ds[()] = ids
print "Internal energy:", u[0]
# generate dark matter particles if needed
if(numPart > 0):
# set seed for random number