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#!/usr/bin/env python3
#
# This is a script that analyses the multimode simulation results.
# This simulates a RZ multimode periodic plasma wave.
# The electric field from the simulation is compared to the analytic value
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np
from pywarpx import fields, picmi
constants = picmi.constants
##########################
# physics parameters
##########################
density = 2.e24
epsilon0 = 0.001*constants.c
epsilon1 = 0.001*constants.c
epsilon2 = 0.001*constants.c
w0 = 5.e-6
n_osc_z = 3
# Plasma frequency
wp = np.sqrt((density*constants.q_e**2)/(constants.m_e*constants.ep0))
kp = wp/constants.c
##########################
# numerics parameters
##########################
nr = 64
nz = 200
rmin = 0.e0
zmin = 0.e0
rmax = +20.e-6
zmax = +40.e-6
# Wave vector of the wave
k0 = 2.*np.pi*n_osc_z/(zmax - zmin)
diagnostic_intervals = 40
##########################
# physics components
##########################
uniform_plasma = picmi.UniformDistribution(density = density,
upper_bound = [+18e-6, +18e-6, None],
directed_velocity = [0., 0., 0.])
momentum_expressions = ["""+ epsilon0/kp*2*x/w0**2*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
- epsilon1/kp*2/w0*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
+ epsilon1/kp*4*x**2/w0**3*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
- epsilon2/kp*8*x/w0**2*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
+ epsilon2/kp*8*x*(x**2-y**2)/w0**4*exp(-(x**2+y**2)/w0**2)*sin(k0*z)""",
"""+ epsilon0/kp*2*y/w0**2*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
+ epsilon1/kp*4*x*y/w0**3*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
+ epsilon2/kp*8*y/w0**2*exp(-(x**2+y**2)/w0**2)*sin(k0*z)
+ epsilon2/kp*8*y*(x**2-y**2)/w0**4*exp(-(x**2+y**2)/w0**2)*sin(k0*z)""",
"""- epsilon0/kp*k0*exp(-(x**2+y**2)/w0**2)*cos(k0*z)
- epsilon1/kp*k0*2*x/w0*exp(-(x**2+y**2)/w0**2)*cos(k0*z)
- epsilon2/kp*k0*4*(x**2-y**2)/w0**2*exp(-(x**2+y**2)/w0**2)*cos(k0*z)"""]
analytic_plasma = picmi.AnalyticDistribution(density_expression = density,
upper_bound = [+18e-6, +18e-6, None],
epsilon0 = epsilon0,
epsilon1 = epsilon1,
epsilon2 = epsilon2,
kp = kp,
k0 = k0,
w0 = w0,
momentum_expressions = momentum_expressions)
electrons = picmi.Species(particle_type='electron', name='electrons', initial_distribution=analytic_plasma)
protons = picmi.Species(particle_type='proton', name='protons', initial_distribution=uniform_plasma)
##########################
# numerics components
##########################
grid = picmi.CylindricalGrid(number_of_cells = [nr, nz],
n_azimuthal_modes = 3,
lower_bound = [rmin, zmin],
upper_bound = [rmax, zmax],
lower_boundary_conditions = ['none', 'periodic'],
upper_boundary_conditions = ['none', 'periodic'],
lower_boundary_conditions_particles = ['absorbing', 'periodic'],
upper_boundary_conditions_particles = ['absorbing', 'periodic'],
moving_window_velocity = [0.,0.],
warpx_max_grid_size=64)
solver = picmi.ElectromagneticSolver(grid=grid, cfl=1.)
##########################
# diagnostics
##########################
field_diag1 = picmi.FieldDiagnostic(name = 'diag1',
grid = grid,
period = diagnostic_intervals,
data_list = ['Ex', 'Ez', 'By', 'Jx', 'Jz', 'part_per_cell'],
write_dir = '.',
warpx_file_prefix = 'Python_Langmuir_rz_multimode_plt')
part_diag1 = picmi.ParticleDiagnostic(name = 'diag1',
period = diagnostic_intervals,
species = [electrons],
data_list = ['weighting', 'momentum'])
##########################
# simulation setup
##########################
sim = picmi.Simulation(solver = solver,
max_steps = 40,
verbose = 1,
warpx_current_deposition_algo = 'esirkepov',
warpx_field_gathering_algo = 'energy-conserving',
warpx_particle_pusher_algo = 'boris',
warpx_use_filter = 0)
sim.add_species(electrons, layout=picmi.GriddedLayout(n_macroparticle_per_cell=[2,16,2], grid=grid))
sim.add_species(protons, layout=picmi.GriddedLayout(n_macroparticle_per_cell=[2,16,2], grid=grid))
sim.add_diagnostic(field_diag1)
sim.add_diagnostic(part_diag1)
##########################
# simulation run
##########################
# write_inputs will create an inputs file that can be used to run
# with the compiled version.
#sim.write_input_file(file_name='inputsrz_from_PICMI')
# Alternatively, sim.step will run WarpX, controlling it from Python
sim.step()
# Below is WarpX specific code to check the results.
def calcEr( z, r, k0, w0, wp, t, epsilons) :
"""
Return the radial electric field as an array
of the same length as z and r, in the half-plane theta=0
"""
Er_array = (
epsilons[0] * constants.m_e*constants.c/constants.q_e * 2*r/w0**2 *
np.exp( -r**2/w0**2 ) * np.sin( k0*z ) * np.sin( wp*t )
- epsilons[1] * constants.m_e*constants.c/constants.q_e * 2/w0 *
np.exp( -r**2/w0**2 ) * np.sin( k0*z ) * np.sin( wp*t )
+ epsilons[1] * constants.m_e*constants.c/constants.q_e * 4*r**2/w0**3 *
np.exp( -r**2/w0**2 ) * np.sin( k0*z ) * np.sin( wp*t )
- epsilons[2] * constants.m_e*constants.c/constants.q_e * 8*r/w0**2 *
np.exp( -r**2/w0**2 ) * np.sin( k0*z ) * np.sin( wp*t )
+ epsilons[2] * constants.m_e*constants.c/constants.q_e * 8*r**3/w0**4 *
np.exp( -r**2/w0**2 ) * np.sin( k0*z ) * np.sin( wp*t ))
return( Er_array )
def calcEz( z, r, k0, w0, wp, t, epsilons) :
"""
Return the longitudinal electric field as an array
of the same length as z and r, in the half-plane theta=0
"""
Ez_array = (
- epsilons[0] * constants.m_e*constants.c/constants.q_e * k0 *
np.exp( -r**2/w0**2 ) * np.cos( k0*z ) * np.sin( wp*t )
- epsilons[1] * constants.m_e*constants.c/constants.q_e * k0 * 2*r/w0 *
np.exp( -r**2/w0**2 ) * np.cos( k0*z ) * np.sin( wp*t )
- epsilons[2] * constants.m_e*constants.c/constants.q_e * k0 * 4*r**2/w0**2 *
np.exp( -r**2/w0**2 ) * np.cos( k0*z ) * np.sin( wp*t ))
return( Ez_array )
# Current time of the simulation
t0 = sim.extension.gett_new(0)
# Get the raw field data. Note that these are the real and imaginary
# parts of the fields for each azimuthal mode.
Ex_sim_wrap = fields.ExWrapper()
Ez_sim_wrap = fields.EzWrapper()
Ex_sim_modes = Ex_sim_wrap[...]
Ez_sim_modes = Ez_sim_wrap[...]
rr_Er = Ex_sim_wrap.mesh('r')
zz_Er = Ex_sim_wrap.mesh('z')
rr_Ez = Ez_sim_wrap.mesh('r')
zz_Ez = Ez_sim_wrap.mesh('z')
rr_Er = rr_Er[:,np.newaxis]*np.ones(zz_Er.shape[0])[np.newaxis,:]
zz_Er = zz_Er[np.newaxis,:]*np.ones(rr_Er.shape[0])[:,np.newaxis]
rr_Ez = rr_Ez[:,np.newaxis]*np.ones(zz_Ez.shape[0])[np.newaxis,:]
zz_Ez = zz_Ez[np.newaxis,:]*np.ones(rr_Ez.shape[0])[:,np.newaxis]
# Sum the real components to get the field along x-axis (theta = 0)
Er_sim = Ex_sim_modes[:,:,0] + np.sum(Ex_sim_modes[:,:,1::2], axis=2)
Ez_sim = Ez_sim_modes[:,:,0] + np.sum(Ez_sim_modes[:,:,1::2], axis=2)
# The analytical solutions
Er_th = calcEr(zz_Er, rr_Er, k0, w0, wp, t0, [epsilon0, epsilon1, epsilon2])
Ez_th = calcEz(zz_Ez, rr_Ez, k0, w0, wp, t0, [epsilon0, epsilon1, epsilon2])
max_error_Er = abs(Er_sim - Er_th).max()/abs(Er_th).max()
max_error_Ez = abs(Ez_sim - Ez_th).max()/abs(Ez_th).max()
print("Max error Er %e"%max_error_Er)
print("Max error Ez %e"%max_error_Ez)
# Plot the last field from the loop (Er at iteration 40)
fig, ax = plt.subplots(3)
im = ax[0].imshow( Er_sim, aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[0], orientation='vertical')
ax[0].set_title('Er, last iteration (simulation)')
ax[1].imshow( Er_th, aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[1], orientation='vertical')
ax[1].set_title('Er, last iteration (theory)')
im = ax[2].imshow( (Er_sim - Er_th)/abs(Er_th).max(), aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[2], orientation='vertical')
ax[2].set_title('Er, last iteration (difference)')
plt.savefig('langmuir_multi_rz_multimode_analysis_Er.png')
fig, ax = plt.subplots(3)
im = ax[0].imshow( Ez_sim, aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[0], orientation='vertical')
ax[0].set_title('Ez, last iteration (simulation)')
ax[1].imshow( Ez_th, aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[1], orientation='vertical')
ax[1].set_title('Ez, last iteration (theory)')
im = ax[2].imshow( (Ez_sim - Ez_th)/abs(Ez_th).max(), aspect='auto', origin='lower' )
fig.colorbar(im, ax=ax[2], orientation='vertical')
ax[2].set_title('Ez, last iteration (difference)')
plt.savefig('langmuir_multi_rz_multimode_analysis_Ez.png')
assert max(max_error_Er, max_error_Ez) < 0.02
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