def E_field_example():
"""
Example E field solution
:return:
"""
particle = PICParticle(1.6e-27, 1.6e-19, np.asarray([0.1, 0.0, 0.0]),
np.asarray([0.0, 0.1, 0.0]))
field = PointField(-1.6e-19, 0.1, np.zeros(3))
def B_field(x):
B = np.zeros(x.shape)
return B
X = particle.position
V = particle.velocity
Q = np.asarray([particle.charge])
M = np.asarray([particle.mass])
times = np.linspace(0.0, 1, 10000)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(field.e_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
fig = plt.figure(figsize=(20, 10))
ax = fig.add_subplot('111', projection='3d')
ax.plot(x, y, z, label='numerical')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend(loc='best')
ax.set_title("Analytic and Numerical Particle Motion")
plt.show()
def test_uniform_e_field(self):
"""
This test considers a particle moving in a uniform electric field
:return:
"""
seed = 1
num_tests = 100
np.random.seed(seed)
Es = np.random.uniform(low=0.0, high=1.0, size=(3, num_tests))
X_0s = np.random.uniform(low=-1.0, high=1.0, size=(3, num_tests))
V_0s = np.random.uniform(low=-1.0, high=1.0, size=(3, num_tests))
for idx in range(num_tests):
e_field = Es[:, idx]
X_0 = X_0s[:, idx]
V_0 = V_0s[:, idx]
def B_field(x):
B = np.zeros(x.shape)
return B
def E_field(x):
E = np.zeros(x.shape)
for i, b in enumerate(E):
E[i, :] = e_field
return E
X = X_0.reshape((1, 3))
V = V_0.reshape((1, 3))
Q = np.asarray([1.0])
M = np.asarray([1.0])
final_time = 4.0
num_pts = 1000
times = np.linspace(0.0, final_time, num_pts)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(E_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
particle = PICParticle(1.0, Q[0], X_0, V_0)
E = e_field
analytic_times, analytic_positions = solve_E_field(particle,
E,
final_time,
num_pts=num_pts)
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
self.assertLess(np.absolute(
np.average(x - analytic_positions[:, 0])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, e_field))
self.assertLess(np.absolute(
np.average(y - analytic_positions[:, 1])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, e_field))
self.assertLess(np.absolute(
np.average(z - analytic_positions[:, 2])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, e_field))
def test_uniform_B_field(self):
"""
This test considers a particle in a uniform B field, and aims to compare analytic and numerical solution. The
particle can have an arbitrary velocity, and the B field is relatively in an arbitrary direction
:return:
"""
seed = 1
num_tests = 100
np.random.seed(seed)
Bs = np.random.uniform(low=0.0, high=10.0, size=(3, num_tests))
X_0s = np.random.uniform(low=-1.0, high=1.0, size=(3, num_tests))
V_0s = np.random.uniform(low=-1.0, high=1.0, size=(3, num_tests))
for idx in range(num_tests):
b_field = Bs[:, idx]
X_0 = X_0s[:, idx]
V_0 = V_0s[:, idx]
def B_field(x):
B = np.zeros(x.shape)
for i, b in enumerate(B):
B[i, :] = b_field
return B
def E_field(x):
E = np.zeros(x.shape)
return E
X = X_0.reshape((1, 3))
V = V_0.reshape((1, 3))
Q = np.asarray([1.0])
M = np.asarray([1.0])
final_time = 4.0
num_pts = 500
times = np.linspace(0.0, final_time, num_pts)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(E_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
particle = PICParticle(1.0, Q[0], X_0, V_0)
B = b_field
analytic_times, analytic_positions = solve_B_field(particle,
B,
final_time,
num_pts=num_pts)
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
self.assertLess(
np.absolute(np.average(x - analytic_positions[:, 0])), 0.01)
self.assertLess(
np.absolute(np.average(y - analytic_positions[:, 1])), 0.01)
self.assertLess(
np.absolute(np.average(z - analytic_positions[:, 2])), 0.01)
def test_aligned_fields(self):
"""
This test considers a particle moving in a uniform orthogonal electric and magnetic field
:return:
"""
seed = 1
np.random.seed(seed)
for direction in [
np.asarray([1.0, 0.0, 0.0]),
np.asarray([0.0, 1.0, 0.0]),
np.asarray([0.0, 0.0, 1.0])
]:
for sign in [-1, 1]:
# randomise initial conditions
B_mag = np.random.uniform(low=0.0, high=1.0)
E_mag = np.random.uniform(low=0.0, high=1.0)
X_0 = np.random.uniform(low=-1.0, high=1.0, size=(1, 3))
V_0 = np.random.uniform(low=-1.0, high=1.0, size=(1, 3))
def B_field(x):
B = np.zeros(x.shape)
for i, b in enumerate(B):
B[i, :] = B_mag * direction * sign
return B
def E_field(x):
E = np.zeros(x.shape)
for i, b in enumerate(E):
E[i, :] = E_mag * direction * sign
return E
X = X_0.reshape((1, 3))
V = V_0.reshape((1, 3))
Q = np.asarray([1.0])
M = np.asarray([1.0])
final_time = 4.0
num_pts = 1000
times = np.linspace(0.0, final_time, num_pts)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(E_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
particle = PICParticle(1.0, Q[0], X_0[0], V_0[0])
E = E_mag * direction * sign
B = B_mag * direction * sign
analytic_times, analytic_positions = solve_aligned_fields(
particle, E, B, final_time, num_pts=num_pts)
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
self.assertLess(np.absolute(
np.average(x - analytic_positions[:, 0])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, E, B))
self.assertLess(np.absolute(
np.average(y - analytic_positions[:, 1])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, E, B))
self.assertLess(np.absolute(
np.average(z - analytic_positions[:, 2])),
0.01,
msg="{}, {}, {}".format(X_0, V_0, E, B))
def test_aligned_fields():
"""
This test considers a particle moving in a uniform orthogonal electric and magnetic field
:return:
"""
seed = 1
np.random.seed(seed)
for direction in [
np.asarray([1.0, 0.0, 0.0]),
np.asarray([0.0, 1.0, 0.0]),
np.asarray([0.0, 0.0, 1.0])
]:
for sign in [-1, 1]:
# randomise initial conditions
B_mag = np.random.uniform(low=0.0, high=1.0)
E_mag = np.random.uniform(low=0.0, high=1.0)
X_0 = np.random.uniform(low=-1.0, high=1.0, size=(1, 3))
V_0 = np.random.uniform(low=-1.0, high=1.0, size=(1, 3))
def B_field(x):
B = np.zeros(x.shape)
for i, b in enumerate(B):
B[i, :] = B_mag * direction * sign
return B
def E_field(x):
E = np.zeros(x.shape)
for i, b in enumerate(E):
E[i, :] = E_mag * direction * sign
return E
X = X_0.reshape((1, 3))
V = V_0.reshape((1, 3))
Q = np.asarray([1.0])
M = np.asarray([1.0])
final_time = 4.0
num_pts = 1000
times = np.linspace(0.0, final_time, num_pts)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(E_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
particle = PICParticle(1.0, Q[0], X_0[0], V_0[0])
E = E_mag * direction * sign
B = B_mag * direction * sign
analytic_times, analytic_positions = solve_aligned_fields(
particle, E, B, final_time, num_pts=num_pts)
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
fig = plt.figure(figsize=(20, 10))
ax = fig.add_subplot('111', projection='3d')
ax.plot(x, y, z, label='numerical')
ax.plot(analytic_positions[:, 0],
analytic_positions[:, 1],
analytic_positions[:, 2],
label='analytic')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend(loc='best')
ax.set_title("Analytic and Numerical Particle Motion")
plt.show()
def single_particle_example(b_field=np.asarray([0.0, 0.0, 1.0]),
X_0=np.asarray([[0.0, 1.0, 0.0]]),
V_0=np.asarray([[-2.0, 0.0, 1.0]])):
"""
Example solution to simple magnetic field case
:return:
"""
def B_field(x):
B = np.zeros(x.shape)
for i, b in enumerate(B):
B[i, :] = b_field
return B
def E_field(x):
E = np.zeros(x.shape)
return E
X = X_0
V = V_0
Q = np.asarray([1.0])
M = np.asarray([1.0])
times = np.linspace(0.0, 4.0, 1000)
positions = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
velocities = np.zeros((times.shape[0], X.shape[0], X.shape[1]))
for i, t in enumerate(times):
if i == 0:
positions[i, :, :] = X
velocities[i, :, :] = V
continue
dt = times[i] - times[i - 1]
x, v = boris_solver(E_field, B_field, X, V, Q, M, dt)
positions[i, :, :] = x
velocities[i, :, :] = v
X = x
V = v
particle = PICParticle(1.0, Q[0], X_0[0], V_0[0])
B = b_field
analytic_times, analytic_positions = solve_B_field(particle, B, 4.0)
x = positions[:, :, 0].flatten()
y = positions[:, :, 1].flatten()
z = positions[:, :, 2].flatten()
fig = plt.figure(figsize=(20, 10))
ax = fig.add_subplot('111', projection='3d')
ax.plot(x, y, z, label='numerical')
ax.plot(analytic_positions[:, 0],
analytic_positions[:, 1],
analytic_positions[:, 2],
label='analytic')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend(loc='best')
ax.set_title("Analytic and Numerical Particle Motion")
plt.show()
fig, axes = plt.subplots(3, figsize=(20, 10))
axes[0].plot(x)
axes[1].plot(y)
axes[2].plot(z)
fig.suptitle("Numerical Solution")
plt.show()
fig, axes = plt.subplots(3, figsize=(20, 10))
axes[0].plot(x - analytic_positions[:, 0])
axes[1].plot(y - analytic_positions[:, 1])
axes[2].plot(z - analytic_positions[:, 2])
fig.suptitle("Deviation of Numerical Solution from the Analytic")
plt.show()