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#!/usr/bin/env python3
#
# Copyright 2022 S. Eric Clark, LLNL
#
#
# This file is part of WarpX.
#
# License: BSD-3-Clause-LBNL
"""
This script tests the magnetostatic solver in a beampipe by initializing a
uniform electron beam with relativistic gamma of 10, and a beam current of 1 kA.
The MLMG soltution from WarpX is computed and compared to the simple analytic
solution. The analytic and simulated values are plotted over one another for the
electric field and magnetic field solutions after taking the gradient of phi and
the curl of the vector potential A. This runs in 2 dimensions in RZ, has a PEC boundary
at z=0, and a Neumann boundary at z=1. This exercises the domain boundary conditions
as well as the embedded boundary conditions and will fail if the maximum error is too large.
The script additionally outputs png images with the average fields in a subset of the beampipe
as well as the analytical solution.
"""
import matplotlib
matplotlib.use('agg')
import matplotlib.pyplot as plt
import numpy as np
import scipy.constants as con
from pywarpx import fields, picmi
##########################
# physics parameters
##########################
V_domain_boundary = 0.0
V_embedded_boundary = 1.0
##########################
# numerics parameters
##########################
dt = 1e-12
# --- Nb time steps
max_steps = 1
# --- grid
nr = 128
nz = 128
rmin = 0.
rmax = 0.25
zmin = 0.
zmax = 1
r_pipe = 0.2
##########################
# numerics components
##########################
grid = picmi.CylindricalGrid(
number_of_cells = [nr, nz],
lower_bound = [rmin, zmin],
upper_bound = [rmax, zmax],
lower_boundary_conditions = ['none', 'dirichlet'],
upper_boundary_conditions = ['neumann', 'neumann'],
lower_boundary_conditions_particles = ['reflecting', 'absorbing'],
upper_boundary_conditions_particles = ['absorbing', 'absorbing'],
warpx_potential_lo_z = V_domain_boundary,
warpx_blocking_factor=8,
warpx_max_grid_size = 32
)
solver = picmi.ElectrostaticSolver(
grid=grid, method='Multigrid', required_precision=1e-7,warpx_magnetostatic=True,warpx_self_fields_verbosity=3
)
embedded_boundary = picmi.EmbeddedBoundary(
implicit_function="(x**2+y**2-radius**2)",
potential=V_embedded_boundary,
radius = r_pipe
)
# Beam Current Density
current = 1000 # A
beam_r = 0.1 # m
J = current/(np.pi * beam_r**2)
beam_gamma = 10.
vz = con.c*np.sqrt(1. - 1./beam_gamma**2)
n0 = J/(con.e*vz)
beam_dist = picmi.AnalyticDistribution(
density_expression='((x**2+y**2)<rmax**2)*n0',
rmax=beam_r,
n0=n0,
directed_velocity=[0., 0., beam_gamma*vz],
)
beam = picmi.Species(particle_type='electron', name='beam', initial_distribution=beam_dist, warpx_random_theta=False)
beam_layout = picmi.GriddedLayout(n_macroparticle_per_cell=[2,2,2], grid=grid)
##########################
# diagnostics
##########################
field_diag = picmi.FieldDiagnostic(
name = 'diag1',
grid = grid,
period = 1,
data_list = ['Er', 'Bt', 'Az', 'Jz', 'phi', 'rho'],
write_dir = '.',
warpx_file_prefix = 'Python_magnetostatic_eb_rz_plt'
)
##########################
# simulation setup
##########################
sim = picmi.Simulation(
solver = solver,
time_step_size = dt,
max_steps = max_steps,
warpx_embedded_boundary=embedded_boundary,
warpx_field_gathering_algo='momentum-conserving',
warpx_current_deposition_algo='direct',
warpx_use_filter=False,
warpx_serialize_initial_conditions=True,
warpx_do_dynamic_scheduling=False
)
sim.add_species(beam, layout=beam_layout, initialize_self_field=True)
sim.add_diagnostic(field_diag)
##########################
# simulation run
##########################
sim.step(max_steps)
##########################
# Testing
##########################
Er = fields.ExWrapper()
r_vec = Er.mesh('r')
z_vec = Er.mesh('z')
@np.vectorize
def Er_an(r):
if r < beam_r:
er = -current * r / (2.*np.pi*beam_r**2*con.epsilon_0*vz)
elif r >= beam_r and r < r_pipe:
er = -current / (2.*np.pi*r*con.epsilon_0*vz)
else:
er = np.zeros_like(r)
return er
# compare to region from 0.5*zmax to 0.9*zmax
z_idx = ((z_vec >= 0.5*zmax) & (z_vec < 0.9*zmax))
Er_dat = Er[...]
r_idx = (r_vec < 0.95*r_pipe)
r_sub = r_vec[r_idx]
# Average Er along z_sub
Er_mean = Er_dat[:,z_idx].mean(axis=1)
plt.figure(1)
plt.plot(r_vec, Er_an(r_vec))
plt.plot(r_vec, Er_mean,'--')
plt.legend(['Analytical', 'Electrostatic'])
er_err = np.abs(Er_mean[r_idx] - Er_an(r_sub)).max()/np.abs(Er_an(r_sub)).max()
plt.ylabel('$E_r$ (V/m)')
plt.xlabel('r (m)')
plt.title("Max % Error: {} %".format(er_err*100.))
plt.tight_layout()
plt.savefig('er_rz.png')
assert (er_err < 0.02), "Er Error increased above 2%"
########################
# Check B field
########################
Bth = fields.ByWrapper()
r_vec = Bth.mesh('r')
z_vec = Bth.mesh('z')
dr = r_vec[1]-r_vec[0]
r_vec = r_vec + dr/2.
@np.vectorize
def Bth_an(r):
if r < beam_r:
bt = -current * r * con.mu_0 / (2.*np.pi*beam_r**2)
elif r >= beam_r and r < r_pipe:
bt = -current * con.mu_0 / (2.*np.pi*r)
else:
bt = np.zeros_like(r)
return bt
# compare to region from 0.25*zmax to 0.75*zmax
z_idx = ((z_vec >= 0.25*zmax) & (z_vec < 0.75*zmax))
z_sub = z_vec[z_idx]
Bth_dat = Bth[...]
r_idx = (r_vec < 0.95*r_pipe)
r_sub = r_vec[r_idx]
# Average Bth along z_idx
Bth_mean = Bth_dat[:,z_idx].mean(axis=1)
plt.figure(2)
plt.plot(r_vec, Bth_an(r_vec))
plt.plot(r_vec, Bth_mean,'--')
plt.legend(['Analytical', 'Magnetostatic'])
bth_err = np.abs(Bth_mean[r_idx] - Bth_an(r_sub)).max()/np.abs(Bth_an(r_sub)).max()
plt.ylabel('$B_{\Theta}$ (T)')
plt.xlabel('r (m)')
plt.title("Max % Error: {} %".format(bth_err*100.))
plt.tight_layout()
plt.savefig('bth_rz.png')
assert (bth_err < 0.02), "Bth Error increased above 2%"
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