def mandelbrot(data, it, maxval):
C = data
Z = data
mag = af.constant(0, *C.dims())
for ii in range(1, 1 + it):
# Doing the calculation
Z = Z * Z + C
# Get indices where abs(Z) crosses maxval
cond = ((af.abs(Z) > maxval)).as_type(af.Dtype.f32)
mag = af.maxof(mag, cond * ii)
C = C * (1 - cond)
Z = Z * (1 - cond)
af.eval(C)
af.eval(Z)
return mag / maxval
def time_evolution(gv):
'''
'''
# Creating a folder to store hdf5 files. If it doesn't exist.
results_directory = 'results/2d_hdf5_%02d' % (int(params.N_LGL))
if not os.path.exists(results_directory):
os.makedirs(results_directory)
u = gv.u_e_ij
delta_t = gv.delta_t_2d
time = gv.time_2d
u_init = gv.u_e_ij
A_inverse = af.np_to_af_array(np.linalg.inv(np.array(A_matrix(gv))))
for i in trange(time.shape[0]):
L1_norm = af.mean(af.abs(u_init - u))
if (L1_norm >= 100):
break
if (i % 10) == 0:
h5file = h5py.File(
'results/2d_hdf5_%02d/dump_timestep_%06d' %
(int(params.N_LGL), int(i)) + '.hdf5', 'w')
dset = h5file.create_dataset('u_i', data=u, dtype='d')
dset[:, :] = u[:, :]
u += +RK4_timestepping(A_inverse, u, delta_t, gv)
#Implementing second order time-stepping.
#u_n_plus_half = u + af.matmul(A_inverse, b_vector(u))\
# * delta_t / 2
#u += af.matmul(A_inverse, b_vector(u_n_plus_half))\
# * delta_t
return L1_norm
def time_evolution(gv):
'''
'''
# Creating a folder to store hdf5 files. If it doesn't exist.
results_directory = 'results/xi_eta_2d_hdf5_%02d' % (int(params.N_LGL))
if not os.path.exists(results_directory):
os.makedirs(results_directory)
A_inverse = af.np_to_af_array(
np.linalg.inv(np.array(A_matrix_xi_eta(params.N_LGL, gv))))
xi_LGL = lagrange.LGL_points(params.N_LGL)
xi_i = af.flat(af.transpose(af.tile(xi_LGL, 1, params.N_LGL)))
eta_j = af.tile(xi_LGL, params.N_LGL)
u_init_2d = gv.u_e_ij
delta_t = gv.delta_t_2d
time = gv.time
u = u_init_2d
time = gv.time_2d
for i in trange(0, time.shape[0]):
L1_norm = af.mean(af.abs(u_init_2d - u))
if (L1_norm >= 100):
print(L1_norm)
break
if (i % 10) == 0:
h5file = h5py.File(
'results/xi_eta_2d_hdf5_%02d/dump_timestep_%06d' %
(int(params.N_LGL), int(i)) + '.hdf5', 'w')
dset = h5file.create_dataset('u_i', data=u, dtype='d')
dset[:, :] = u[:, :]
u += RK4_timestepping(A_inverse, u, delta_t, gv)
return L1_norm
def test():
print("\nTesting benchmark functions...")
A, b, x0 = setup_input(50) # dense A
Asp = to_sparse(A)
x1, _ = calc_arrayfire(A, b, x0)
x2, _ = calc_arrayfire(Asp, b, x0)
if af.sum(af.abs(x1 - x2) / x2 > 1e-6):
raise ValueError("arrayfire test failed")
if np:
An = to_numpy(A)
bn = to_numpy(b)
x0n = to_numpy(x0)
x3, _ = calc_numpy(An, bn, x0n)
if not np.allclose(x3, x1.to_list()):
raise ValueError("numpy test failed")
if sp:
Asc = to_scipy_sparse(Asp)
x4, _ = calc_scipy_sparse(Asc, bn, x0n)
if not np.allclose(x4, x1.to_list()):
raise ValueError("scipy.sparse test failed")
x5, _ = calc_scipy_sparse_linalg_cg(Asc, bn, x0n)
if not np.allclose(x5, x1.to_list()):
raise ValueError("scipy.sparse.linalg.cg test failed")
print(" all tests passed...")
def check_error(params):
error = np.zeros(N.size)
for i in range(N.size):
domain.N_p1 = int(N[i])
# Defining the physical system to be solved:
system = physical_system(domain, boundary_conditions, params,
initialize, advection_terms,
collision_operator.BGK, moments)
# Declaring a linear system object which will evolve the defined physical system:
nls = nonlinear_solver(system)
# Time parameters:
dt = 0.01 * 32 / nls.N_p1
t_final = 0.2
time_array = np.arange(dt, t_final + dt, dt)
(A_p1, A_p2,
A_p3) = af.broadcast(nls._A_p, nls.f, 0, nls.q1_center, nls.q2_center,
nls.p1_center, nls.p2_center, nls.p3_center,
nls.fields_solver, nls.physical_system.params)
f_analytic = af.broadcast(initialize.initialize_f, nls.q1_center,
nls.q2_center,
add(nls.p1_center, -A_p1 * t_final),
add(nls.p2_center, -A_p2 * t_final),
nls.p3_center, nls.physical_system.params)
for time_index, t0 in enumerate(time_array):
nls.strang_timestep(dt)
error[i] = af.mean(af.abs(nls.f - f_analytic))
return (error)
def test():
print("\nTesting benchmark functions...")
A, b, x0 = setup_input(n=50, sparsity=7) # dense A
Asp = to_sparse(A)
x1, _ = calc_arrayfire(A, b, x0)
x2, _ = calc_arrayfire(Asp, b, x0)
if af.sum(af.abs(x1 - x2)/x2 > 1e-5):
raise ValueError("arrayfire test failed")
if np:
An = to_numpy(A)
bn = to_numpy(b)
x0n = to_numpy(x0)
x3, _ = calc_numpy(An, bn, x0n)
if not np.allclose(x3, x1.to_list()):
raise ValueError("numpy test failed")
if sp:
Asc = to_scipy_sparse(Asp)
x4, _ = calc_scipy_sparse(Asc, bn, x0n)
if not np.allclose(x4, x1.to_list()):
raise ValueError("scipy.sparse test failed")
x5, _ = calc_scipy_sparse_linalg_cg(Asc, bn, x0n)
if not np.allclose(x5, x1.to_list()):
raise ValueError("scipy.sparse.linalg.cg test failed")
print(" all tests passed...")
af.display(c) af.display(-a) af.display(+a) af.display(~a) af.display(a) af.display(af.cast(a, af.c32)) af.display(af.maxof(a,b)) af.display(af.minof(a,b)) af.display(af.rem(a,b)) a = af.randu(3,3) - 0.5 b = af.randu(3,3) - 0.5 af.display(af.abs(a)) af.display(af.arg(a)) af.display(af.sign(a)) af.display(af.round(a)) af.display(af.trunc(a)) af.display(af.floor(a)) af.display(af.ceil(a)) af.display(af.hypot(a, b)) af.display(af.sin(a)) af.display(af.cos(a)) af.display(af.tan(a)) af.display(af.asin(a)) af.display(af.acos(a)) af.display(af.atan(a)) af.display(af.atan2(a, b))
def simple_arith(verbose = False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(3,3,dtype=af.Dtype.u32)
b = af.constant(4, 3, 3, dtype=af.Dtype.u32)
display_func(a)
display_func(b)
c = a + b
d = a
d += b
display_func(c)
display_func(d)
display_func(a + 2)
display_func(3 + a)
c = a - b
d = a
d -= b
display_func(c)
display_func(d)
display_func(a - 2)
display_func(3 - a)
c = a * b
d = a
d *= b
display_func(c * 2)
display_func(3 * d)
display_func(a * 2)
display_func(3 * a)
c = a / b
d = a
d /= b
display_func(c / 2.0)
display_func(3.0 / d)
display_func(a / 2)
display_func(3 / a)
c = a % b
d = a
d %= b
display_func(c % 2.0)
display_func(3.0 % d)
display_func(a % 2)
display_func(3 % a)
c = a ** b
d = a
d **= b
display_func(c ** 2.0)
display_func(3.0 ** d)
display_func(a ** 2)
display_func(3 ** a)
display_func(a < b)
display_func(a < 0.5)
display_func(0.5 < a)
display_func(a <= b)
display_func(a <= 0.5)
display_func(0.5 <= a)
display_func(a > b)
display_func(a > 0.5)
display_func(0.5 > a)
display_func(a >= b)
display_func(a >= 0.5)
display_func(0.5 >= a)
display_func(a != b)
display_func(a != 0.5)
display_func(0.5 != a)
display_func(a == b)
display_func(a == 0.5)
display_func(0.5 == a)
display_func(a & b)
display_func(a & 2)
c = a
c &= 2
display_func(c)
display_func(a | b)
display_func(a | 2)
c = a
c |= 2
display_func(c)
display_func(a >> b)
display_func(a >> 2)
c = a
c >>= 2
display_func(c)
display_func(a << b)
display_func(a << 2)
c = a
c <<= 2
display_func(c)
display_func(-a)
display_func(+a)
display_func(~a)
display_func(a)
display_func(af.cast(a, af.Dtype.c32))
display_func(af.maxof(a,b))
display_func(af.minof(a,b))
display_func(af.rem(a,b))
a = af.randu(3,3) - 0.5
b = af.randu(3,3) - 0.5
display_func(af.abs(a))
display_func(af.arg(a))
display_func(af.sign(a))
display_func(af.round(a))
display_func(af.trunc(a))
display_func(af.floor(a))
display_func(af.ceil(a))
display_func(af.hypot(a, b))
display_func(af.sin(a))
display_func(af.cos(a))
display_func(af.tan(a))
display_func(af.asin(a))
display_func(af.acos(a))
display_func(af.atan(a))
display_func(af.atan2(a, b))
c = af.cplx(a)
d = af.cplx(a,b)
display_func(c)
display_func(d)
display_func(af.real(d))
display_func(af.imag(d))
display_func(af.conjg(d))
display_func(af.sinh(a))
display_func(af.cosh(a))
display_func(af.tanh(a))
display_func(af.asinh(a))
display_func(af.acosh(a))
display_func(af.atanh(a))
a = af.abs(a)
b = af.abs(b)
display_func(af.root(a, b))
display_func(af.pow(a, b))
display_func(af.pow2(a))
display_func(af.exp(a))
display_func(af.expm1(a))
display_func(af.erf(a))
display_func(af.erfc(a))
display_func(af.log(a))
display_func(af.log1p(a))
display_func(af.log10(a))
display_func(af.log2(a))
display_func(af.sqrt(a))
display_func(af.cbrt(a))
a = af.round(5 * af.randu(3,3) - 1)
b = af.round(5 * af.randu(3,3) - 1)
display_func(af.factorial(a))
display_func(af.tgamma(a))
display_func(af.lgamma(a))
display_func(af.iszero(a))
display_func(af.isinf(a/b))
display_func(af.isnan(a/a))
a = af.randu(5, 1)
b = af.randu(1, 5)
c = af.broadcast(lambda x,y: x+y, a, b)
display_func(a)
display_func(b)
display_func(c)
@af.broadcast
def test_add(aa, bb):
return aa + bb
display_func(test_add(a, b))
def test_communicate_f():
obj = test_distribution_function()
communicate_f(obj)
expected = af.sin(2 * np.pi * obj.q1 + 4 * np.pi * obj.q2)
assert (af.mean(af.abs(obj.f - expected)) < 5e-14)
def _initialize(self, params):
"""
Called when the solver object is declared. This function is
used to initialize the distribution function, and the field
quantities using the options as provided by the user. The
quantities are then mapped to the fourier basis by taking FFTs.
The independant modes are then evolved by using the linear
solver.
"""
# af.broadcast(function, *args) performs batched operations on
# function(*args):
f = af.broadcast(self.physical_system.initial_conditions.\
initialize_f, self.q1_center, self.q2_center,
self.p1, self.p2, self.p3, params
)
# Taking FFT:
f_hat = af.fft2(f)
# Since (k_q1, k_q2) = (0, 0) will give the background distribution:
# The division by (self.N_q1 * self.N_q2) is performed since the FFT
# at (0, 0) returns (amplitude * (self.N_q1 * self.N_q2))
self.f_background = af.abs(f_hat[0, 0, :])/ (self.N_q1 * self.N_q2)
# Calculating derivatives of the background distribution function:
self._calculate_dfdp_background()
# Scaling Appropriately:
# Except the case of (0, 0) the FFT returns
# (0.5 * amplitude * (self.N_q1 * self.N_q2)):
f_hat = 2 * f_hat / (self.N_q1 * self.N_q2)
if(self.single_mode_evolution == True):
# Background
f_hat[0, 0, :] = 0
# Finding the indices of the perturbation introduced:
i_q1_max = np.unravel_index(af.imax(af.abs(f_hat))[1],
(self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3
),order = 'F'
)[0]
i_q2_max = np.unravel_index(af.imax(af.abs(f_hat))[1],
(self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3
),order = 'F'
)[1]
# Taking sum to get a scalar value:
params.k_q1 = af.sum(self.k_q1[i_q1_max, i_q2_max])
params.k_q2 = af.sum(self.k_q2[i_q1_max, i_q2_max])
self.f_background = np.array(self.f_background).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self.p1 = np.array(self.p1).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self.p2 = np.array(self.p2).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self.p3 = np.array(self.p3).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self._A_q1 = np.array(self._A_q1).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self._A_q2 = np.array(self._A_q2).\
reshape(self.N_p1, self.N_p2, self.N_p3)
params.rho_background = self.compute_moments('density',
self.f_background
)
params.p1_bulk_background = self.compute_moments('mom_p1_bulk',
self.f_background
) / params.rho_background
params.p2_bulk_background = self.compute_moments('mom_p2_bulk',
self.f_background
) / params.rho_background
params.p3_bulk_background = self.compute_moments('mom_p3_bulk',
self.f_background
) / params.rho_background
params.T_background = ((1 / params.p_dim) * \
self.compute_moments('energy',
self.f_background
)
- params.rho_background *
params.p1_bulk_background**2
- params.rho_background *
params.p2_bulk_background**2
- params.rho_background *
params.p3_bulk_background**2
) / params.rho_background
self.dfdp1_background = np.array(self.dfdp1_background).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self.dfdp2_background = np.array(self.dfdp2_background).\
reshape(self.N_p1, self.N_p2, self.N_p3)
self.dfdp3_background = np.array(self.dfdp3_background).\
reshape(self.N_p1, self.N_p2, self.N_p3)
# Unable to recover pert_real and pert_imag accurately from the input.
# There seems to be some error in recovering these quantities which
# is dependant upon the resolution. Using user-defined quantities instead
delta_f_hat = params.pert_real * self.f_background \
+ params.pert_imag * self.f_background * 1j
self.Y = np.array([delta_f_hat])
compute_electrostatic_fields(self)
self.Y = np.array([delta_f_hat,
self.delta_E1_hat, self.delta_E2_hat, self.delta_E3_hat,
self.delta_B1_hat, self.delta_B2_hat, self.delta_B3_hat
]
)
else:
# Using a vector Y to evolve the system:
self.Y = af.constant(0, self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3,
7, dtype = af.Dtype.c64
)
# Assigning the 0th indice along axis 3 to the f_hat:
self.Y[:, :, :, 0] = f_hat
# Initializing the EM field quantities:
self.E3_hat = af.constant(0, self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3,
dtype = af.Dtype.c64
)
self.B1_hat = af.constant(0, self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3,
dtype = af.Dtype.c64
)
self.B2_hat = af.constant(0, self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3,
dtype = af.Dtype.c64
)
self.B3_hat = af.constant(0, self.N_q1, self.N_q2,
self.N_p1 * self.N_p2 * self.N_p3,
dtype = af.Dtype.c64
)
# Initializing EM fields using Poisson Equation:
if(self.physical_system.params.fields_initialize == 'electrostatic' or
self.physical_system.params.fields_initialize == 'fft'
):
compute_electrostatic_fields(self)
# If option is given as user-defined:
elif(self.physical_system.params.fields_initialize == 'user-defined'):
E1, E2, E3 = \
self.physical_system.initial_conditions.initialize_E(self.q1_center, self.q2_center, self.physical_system.params)
B1, B2, B3 = \
self.physical_system.initial_conditions.initialize_B(self.q1_center, self.q2_center, self.physical_system.params)
# Scaling Appropriately
self.E1_hat = af.tile(2 * af.fft2(E1) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
self.E2_hat = af.tile(2 * af.fft2(E2) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
self.E3_hat = af.tile(2 * af.fft2(E3) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
self.B1_hat = af.tile(2 * af.fft2(B1) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
self.B2_hat = af.tile(2 * af.fft2(B2) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
self.B3_hat = af.tile(2 * af.fft2(B3) / (self.N_q1 * self.N_q2), 1, 1, self.N_p1 * self.N_p2 * self.N_p3)
else:
raise NotImplementedError('Method invalid/not-implemented')
# Assigning other indices along axis 3 to be
# the EM field quantities:
self.Y[:, :, :, 1] = self.E1_hat
self.Y[:, :, :, 2] = self.E2_hat
self.Y[:, :, :, 3] = self.E3_hat
self.Y[:, :, :, 4] = self.B1_hat
self.Y[:, :, :, 5] = self.B2_hat
self.Y[:, :, :, 6] = self.B3_hat
af.eval(self.Y)
return
def Umeda_b1_deposition( charge_electron, positions_x, positions_y, velocity_x, velocity_y,\
x_grid, y_grid, ghost_cells, length_domain_x, length_domain_y, dt ):
'''
A modified Umeda's scheme was implemented to handle a pure one dimensional case.
function Umeda_b1_deposition( charge, x, velocity_x,\
x_grid, ghost_cells, length_domain_x, dt\
)
-----------------------------------------------------------------------
Input variables: charge, x, velocity_required_x, x_grid, ghost_cells, length_domain_x, dt
charge: This is an array containing the charges deposited at the density grid nodes.
positions_x: An one dimensional array of size equal to number of particles taken in the PIC code.
It contains the positions of particles in x direction.
positions_y: An one dimensional array of size equal to number of particles taken in the PIC code.
It contains the positions of particles in y direction.
velocity_x: An one dimensional array of size equal to number of particles taken in the PIC code.
It contains the velocities of particles in y direction.
velocity_y: An one dimensional array of size equal to number of particles taken in the PIC code.
It contains the velocities of particles in x direction.
x_grid: This is an array denoting the position grid in x direction chosen in the PIC simulation
y_grid: This is an array denoting the position grid in y direction chosen in the PIC simulation
ghost_cells: This is the number of ghost cells used in the simulation domain.
length_domain_x: This is the length of the domain in x direction
dt: this is the dt/time step chosen in the simulation
-----------------------------------------------------------------------
returns: Jx_x_indices, Jx_y_indices, Jx_values_at_these_indices,\
Jy_x_indices, Jy_y_indices, Jy_values_at_these_indices
Jx_x_indices: This returns the x indices (columns) of the array where the respective currents stored in
Jx_values_at_these_indices have to be deposited
Jx_y_indices: This returns the y indices (rows) of the array where the respective currents stored in
Jx_values_at_these_indices have to be deposited
Jx_values_at_these_indices: This is an array containing the currents to be deposited.
Jy_x_indices, Jy_y_indices and Jy_values_at_these_indices are similar to Jx_x_indices,
Jx_y_indices and Jx_values_at_these_indices for Jy
For further details on the scheme refer to Umeda's paper provided in the sagemath folder as the
naming conventions used in the function use the papers naming convention.(x_1, x_2, x_r, F_x, )
'''
nx = (x_grid.elements() - 1 - 2 * ghost_cells) # number of zones
ny = (y_grid.elements() - 1 - 2 * ghost_cells) # number of zones
dx = length_domain_x / nx
dy = length_domain_y / ny
# Start location x_1, y_1 at t = n * dt
# Start location x_2, y_2 at t = (n+1) * dt
x_1 = (positions_x).as_type(af.Dtype.f64)
x_2 = (positions_x + (velocity_x * dt)).as_type(af.Dtype.f64)
y_1 = (positions_y).as_type(af.Dtype.f64)
y_2 = (positions_y + (velocity_y * dt)).as_type(af.Dtype.f64)
# Calculation i_1 and i_2, indices of left corners of cells containing the particles
# at x_1 and x_2 respectively and j_1 and j_2: indices of bottoms of cells containing the particles
# at y_1 and y_2 respectively
i_1 = af.arith.floor(((af.abs(x_1 - af.sum(x_grid[0]))) / dx) -
ghost_cells)
j_1 = af.arith.floor(((af.abs(y_1 - af.sum(y_grid[0]))) / dy) -
ghost_cells)
i_2 = af.arith.floor(((af.abs(x_2 - af.sum(x_grid[0]))) / dx) -
ghost_cells)
j_2 = af.arith.floor(((af.abs(y_2 - af.sum(y_grid[0]))) / dy) -
ghost_cells)
i_dx = dx * af.join(1, i_1, i_2)
j_dy = dy * af.join(1, j_1, j_2)
i_dx_x_avg = af.join(1, af.max(i_dx, 1), ((x_1 + x_2) / 2))
j_dy_y_avg = af.join(1, af.max(j_dy, 1), ((y_1 + y_2) / 2))
x_r_term_1 = dx + af.min(i_dx, 1)
x_r_term_2 = af.max(i_dx_x_avg, 1)
y_r_term_1 = dy + af.min(j_dy, 1)
y_r_term_2 = af.max(j_dy_y_avg, 1)
x_r_combined_term = af.join(1, x_r_term_1, x_r_term_2)
y_r_combined_term = af.join(1, y_r_term_1, y_r_term_2)
# Computing the relay point (x_r, y_r)
x_r = af.min(x_r_combined_term, 1)
y_r = af.min(y_r_combined_term, 1)
# Calculating the fluxes and the weights
F_x_1 = charge_electron * (x_r - x_1) / dt
F_x_2 = charge_electron * (x_2 - x_r) / dt
F_y_1 = charge_electron * (y_r - y_1) / dt
F_y_2 = charge_electron * (y_2 - y_r) / dt
W_x_1 = (x_1 + x_r) / (2 * dx) - i_1
W_x_2 = (x_2 + x_r) / (2 * dx) - i_2
W_y_1 = (y_1 + y_r) / (2 * dy) - j_1
W_y_2 = (y_2 + y_r) / (2 * dy) - j_2
# computing the charge densities at the grid nodes using the
# fluxes and the weights
J_x_1_1 = (1 / (dx * dy)) * (F_x_1 * (1 - W_y_1))
J_x_1_2 = (1 / (dx * dy)) * (F_x_1 * (W_y_1))
J_x_2_1 = (1 / (dx * dy)) * (F_x_2 * (1 - W_y_2))
J_x_2_2 = (1 / (dx * dy)) * (F_x_2 * (W_y_2))
J_y_1_1 = (1 / (dx * dy)) * (F_y_1 * (1 - W_x_1))
J_y_1_2 = (1 / (dx * dy)) * (F_y_1 * (W_x_1))
J_y_2_1 = (1 / (dx * dy)) * (F_y_2 * (1 - W_x_2))
J_y_2_2 = (1 / (dx * dy)) * (F_y_2 * (W_x_2))
# concatenating the x, y indices for Jx
Jx_x_indices = af.join(0,\
i_1 + ghost_cells,\
i_1 + ghost_cells,\
i_2 + ghost_cells,\
i_2 + ghost_cells\
)
Jx_y_indices = af.join(0,\
j_1 + ghost_cells,\
(j_1 + 1 + ghost_cells),\
j_2 + ghost_cells,\
(j_2 + 1 + ghost_cells)\
)
# concatenating the currents at x, y indices for Jx
Jx_values_at_these_indices = af.join(0,\
J_x_1_1,\
J_x_1_2,\
J_x_2_1,\
J_x_2_2\
)
# concatenating the x, y indices for Jy
Jy_x_indices = af.join(0,\
i_1 + ghost_cells,\
(i_1 + 1 + ghost_cells),\
i_2 + ghost_cells,\
(i_2 + 1 + ghost_cells)\
)
Jy_y_indices = af.join(0,\
j_1 + ghost_cells,\
j_1 + ghost_cells,\
j_2 + ghost_cells,\
j_2 + ghost_cells\
)
# concatenating the currents at x, y indices for Jx
Jy_values_at_these_indices = af.join(0,\
J_y_1_1,\
J_y_1_2,\
J_y_2_1,\
J_y_2_2\
)
af.eval(Jx_x_indices, Jx_y_indices, Jy_x_indices, Jy_y_indices)
af.eval(Jx_values_at_these_indices, Jy_values_at_these_indices)
return Jx_x_indices, Jx_y_indices, Jx_values_at_these_indices,\
Jy_x_indices, Jy_y_indices, Jy_values_at_these_indices
pl.plot(np.array(numerical_first_moment_p3_term[:, 0, :, 0]).ravel(),
label='N = ' + str(N[i]))
pl.plot(np.array(analytic_first_moment_p3_term_e[:, 0, :, 0]).ravel(),
'--',
color='black')
pl.show()
pl.plot(np.array(numerical_first_moment_p3_term[:, 1, :, 0]).ravel(),
label='N = ' + str(N[i]))
pl.plot(np.array(analytic_first_moment_p3_term_i[:, 0, :, 0]).ravel(),
'--',
color='black')
pl.show()
error_p2_e[i] = af.mean(
af.abs(numerical_first_moment_p2_term[:, 0] -
analytic_first_moment_p2_term_e))
error_p2_i[i] = af.mean(
af.abs(numerical_first_moment_p2_term[:, 1] -
analytic_first_moment_p2_term_i))
error_p3_e[i] = af.mean(
af.abs(numerical_first_moment_p3_term[:, 0] -
analytic_first_moment_p3_term_e))
error_p3_i[i] = af.mean(
af.abs(numerical_first_moment_p3_term[:, 1] -
analytic_first_moment_p3_term_i))
pl.loglog(N, error_p2_e, '-o', label=r'$p_{2e}$')
pl.loglog(N, error_p2_i, '-o', label=r'$p_{2i}$')
pl.loglog(N, error_p3_e, '-o', label=r'$p_{3e}$')
pl.loglog(N, error_p3_i, '-o', label=r'$p_{3i}$')
pl.loglog(N,
def test_shear_y():
t = np.random.rand(1)[0]
N = 2**np.arange(5, 10)
error = np.zeros(N.size)
for i in range(N.size):
domain.N_q1 = int(N[i])
domain.N_q2 = int(N[i])
# Defining the physical system to be solved:
system = physical_system(domain, boundary_conditions_y, params,
initialize_y, advection_terms,
collision_operator.BGK, moment_defs)
L_q1 = domain.q1_end - domain.q1_start
L_q2 = domain.q2_end - domain.q2_start
nls = nonlinear_solver(system)
N_g = nls.N_ghost_q
# For left:
f_reference_bot = af.broadcast(
initialize_y.initialize_f,
af.select(
nls.q1_center - t < domain.q1_start, # Periodic domain
nls.q1_center - t + L_q1,
nls.q1_center - t),
nls.q2_center,
nls.p1_center,
nls.p2_center,
nls.p3_center,
params)[:, N_g:-N_g, -2 * N_g:-N_g]
# For right:
f_reference_top = af.broadcast(
initialize_y.initialize_f,
af.select(nls.q1_center + t > domain.q1_end,
nls.q1_center + t - L_q1, nls.q1_center + t),
nls.q2_center, nls.p1_center, nls.p2_center, nls.p3_center,
params)[:, N_g:-N_g, N_g:2 * N_g]
nls.time_elapsed = t
nls._communicate_f()
nls._apply_bcs_f()
error[i] = af.mean(af.abs(nls.f[:, N_g:-N_g, :N_g] - f_reference_bot)) \
+ af.mean(af.abs(nls.f[:, N_g:-N_g, -N_g:] - f_reference_top))
pl.loglog(N, error, '-o', label='Numerical')
pl.loglog(N,
error[0] * 32**3 / N**3,
'--',
color='black',
label=r'$O(N^{-3})$')
pl.xlabel(r'$N$')
pl.ylabel('Error')
pl.legend()
pl.savefig('plot2.png')
poly = np.polyfit(np.log10(N), np.log10(error), 1)
assert (abs(poly[0] + 3) < 0.3)
af.display(c) af.display(-a) af.display(+a) af.display(~a) af.display(a) af.display(af.cast(a, af.c32)) af.display(af.maxof(a, b)) af.display(af.minof(a, b)) af.display(af.rem(a, b)) a = af.randu(3, 3) - 0.5 b = af.randu(3, 3) - 0.5 af.display(af.abs(a)) af.display(af.arg(a)) af.display(af.sign(a)) af.display(af.round(a)) af.display(af.trunc(a)) af.display(af.floor(a)) af.display(af.ceil(a)) af.display(af.hypot(a, b)) af.display(af.sin(a)) af.display(af.cos(a)) af.display(af.tan(a)) af.display(af.asin(a)) af.display(af.acos(a)) af.display(af.atan(a)) af.display(af.atan2(a, b))
def check_error(params):
error = np.zeros(N.size)
for i in range(N.size):
domain.N_p1 = int(N[i])
domain.N_p2 = int(N[i])
domain.N_p3 = int(N[i])
# Defining the physical system to be solved:
system = physical_system(domain,
boundary_conditions,
params,
initialize,
advection_terms,
collision_operator.BGK,
moment_defs
)
# Declaring a linear system object which will evolve the defined physical system:
nls = nonlinear_solver(system)
N_g = nls.N_ghost_q
# Time parameters:
dt = 0.001 * 32/nls.N_p1
t_final = 0.1
time_array = np.arange(dt, t_final + dt, dt)
if(time_array[-1]>t_final):
time_array = np.delete(time_array, -1)
# Finding final resting point of the blob:
E1 = nls.cell_centered_EM_fields[0]
E2 = nls.cell_centered_EM_fields[1]
E3 = nls.cell_centered_EM_fields[2]
B1 = nls.cell_centered_EM_fields[3]
B2 = nls.cell_centered_EM_fields[4]
B3 = nls.cell_centered_EM_fields[5]
sol = odeint(dp_dt, np.array([0, 0, 0]), time_array,
args = (af.mean(E1), af.mean(E2), af.mean(E3),
af.mean(B1), af.mean(B2), af.mean(B3),
params.charge_electron,
params.mass_particle
),
atol = 1e-12, rtol = 1e-12
)
f_reference = af.broadcast(initialize.initialize_f,
nls.q1_center, nls.q2_center,
nls.p1_center - sol[-1, 0],
nls.p2_center - sol[-1, 1],
nls.p3_center - sol[-1, 2],
params
)
for time_index, t0 in enumerate(time_array):
nls.strang_timestep(dt)
error[i] = af.mean(af.abs( nls.f
- f_reference
)
)
return(error)
# Declaring a linear system object which will evolve the defined physical system:
nls = nonlinear_solver(system)
N_g = nls.N_ghost
# Time parameters:
dt_fvm = params.N_cfl * min(nls.dq1, nls.dq2) \
/ max(domain.p1_end + domain.p2_end + domain.p3_end) # joining elements of the list
dt_fdtd = params.N_cfl * min(nls.dq1, nls.dq2) \
/ params.c # lightspeed
dt = dt_fvm # min(dt_fvm, dt_fdtd)
print('Minimum Value of f:', af.min(nls.f))
print('Error in density:', af.mean(af.abs(nls.compute_moments('density') - 1)))
v2_bulk = nls.compute_moments('mom_v2_bulk') / nls.compute_moments('density')
v3_bulk = nls.compute_moments('mom_v3_bulk') / nls.compute_moments('density')
v2_bulk_ana = params.amplitude * -1.7450858652952794e-15 * af.cos(params.k_q1 * nls.q1_center) \
- params.amplitude * 0.5123323181646575 * af.sin(params.k_q1 * nls.q1_center)
v3_bulk_ana = params.amplitude * 0.5123323181646597 * af.cos(params.k_q1 * nls.q1_center) \
- params.amplitude * 0 * af.sin(params.k_q1 * nls.q1_center)
print('Error in v2_bulk:',
af.mean(af.abs((v2_bulk - v2_bulk_ana) / v2_bulk_ana)))
print('Error in v3_bulk:',
af.mean(af.abs((v3_bulk - v3_bulk_ana) / v3_bulk_ana)))
#!/usr/bin/python
#######################################################
# Copyright (c) 2015, ArrayFire
# All rights reserved.
#
# This file is distributed under 3-clause BSD license.
# The complete license agreement can be obtained at:
# http://arrayfire.com/licenses/BSD-3-Clause
########################################################
import arrayfire as af
af.info()
POINTS = 30
N = 2 * POINTS
x = (af.iota(d0 = N, d1 = 1, tile_dims = (1, N)) - POINTS) / POINTS
y = (af.iota(d0 = 1, d1 = N, tile_dims = (N, 1)) - POINTS) / POINTS
win = af.Window(800, 800, "3D Surface example using ArrayFire")
t = 0
while not win.close():
t = t + 0.07
z = 10*x*-af.abs(y) * af.cos(x*x*(y+t))+af.sin(y*(x+t))-1.5;
win.surface(x, y, z)
def af_assert(left, right, eps=1E-6):
if (af.max(af.abs(left - right)) > eps):
raise ValueError("Arrays not within dictated precision")
return
# Defining the physical system to be solved:
system = physical_system(domain, boundary_conditions, params, initialize,
advection_terms, collision_operator.BGK,
moment_defs)
# Declaring a linear system object which will evolve the defined physical system:
nls = nonlinear_solver(system)
N_g = nls.N_ghost_q
# Time parameters:
dt = 0.01 * 32 / nls.N_q1
t_final = 0.1
time_array = np.arange(dt, t_final + dt, dt)
from initialize import initialize_f
f_initial = 0.01 * af.exp(-500 * (nls.q1_center - 0.6)**2 - 500 *
(nls.q2_center - 0.6)**2)
add = lambda a, b: a + b
#f_initial = af.broadcast(intialize_f, af.broadcast(nls.q1_center, nls.q2_center,
for time_index, t0 in enumerate(time_array):
nls.strang_timestep(dt)
error[i] = af.mean(
af.abs(nls.f[10, N_g:-N_g, N_g:-N_g] -
f_initial[:, N_g:-N_g, N_g:-N_g]))
print(error)
print(np.polyfit(np.log10(N), np.log10(error), 1))
def simple_arith(verbose=False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(3, 3)
b = af.constant(4, 3, 3)
display_func(a)
display_func(b)
c = a + b
d = a
d += b
display_func(c)
display_func(d)
display_func(a + 2)
display_func(3 + a)
c = a - b
d = a
d -= b
display_func(c)
display_func(d)
display_func(a - 2)
display_func(3 - a)
c = a * b
d = a
d *= b
display_func(c * 2)
display_func(3 * d)
display_func(a * 2)
display_func(3 * a)
c = a / b
d = a
d /= b
display_func(c / 2.0)
display_func(3.0 / d)
display_func(a / 2)
display_func(3 / a)
c = a % b
d = a
d %= b
display_func(c % 2.0)
display_func(3.0 % d)
display_func(a % 2)
display_func(3 % a)
c = a**b
d = a
d **= b
display_func(c**2.0)
display_func(3.0**d)
display_func(a**2)
display_func(3**a)
display_func(a < b)
display_func(a < 0.5)
display_func(0.5 < a)
display_func(a <= b)
display_func(a <= 0.5)
display_func(0.5 <= a)
display_func(a > b)
display_func(a > 0.5)
display_func(0.5 > a)
display_func(a >= b)
display_func(a >= 0.5)
display_func(0.5 >= a)
display_func(a != b)
display_func(a != 0.5)
display_func(0.5 != a)
display_func(a == b)
display_func(a == 0.5)
display_func(0.5 == a)
a = af.randu(3, 3, dtype=af.Dtype.u32)
b = af.constant(4, 3, 3, dtype=af.Dtype.u32)
display_func(a & b)
display_func(a & 2)
c = a
c &= 2
display_func(c)
display_func(a | b)
display_func(a | 2)
c = a
c |= 2
display_func(c)
display_func(a >> b)
display_func(a >> 2)
c = a
c >>= 2
display_func(c)
display_func(a << b)
display_func(a << 2)
c = a
c <<= 2
display_func(c)
display_func(-a)
display_func(+a)
display_func(~a)
display_func(a)
display_func(af.cast(a, af.Dtype.c32))
display_func(af.maxof(a, b))
display_func(af.minof(a, b))
display_func(af.rem(a, b))
a = af.randu(3, 3) - 0.5
b = af.randu(3, 3) - 0.5
display_func(af.abs(a))
display_func(af.arg(a))
display_func(af.sign(a))
display_func(af.round(a))
display_func(af.trunc(a))
display_func(af.floor(a))
display_func(af.ceil(a))
display_func(af.hypot(a, b))
display_func(af.sin(a))
display_func(af.cos(a))
display_func(af.tan(a))
display_func(af.asin(a))
display_func(af.acos(a))
display_func(af.atan(a))
display_func(af.atan2(a, b))
c = af.cplx(a)
d = af.cplx(a, b)
display_func(c)
display_func(d)
display_func(af.real(d))
display_func(af.imag(d))
display_func(af.conjg(d))
display_func(af.sinh(a))
display_func(af.cosh(a))
display_func(af.tanh(a))
display_func(af.asinh(a))
display_func(af.acosh(a))
display_func(af.atanh(a))
a = af.abs(a)
b = af.abs(b)
display_func(af.root(a, b))
display_func(af.pow(a, b))
display_func(af.pow2(a))
display_func(af.sigmoid(a))
display_func(af.exp(a))
display_func(af.expm1(a))
display_func(af.erf(a))
display_func(af.erfc(a))
display_func(af.log(a))
display_func(af.log1p(a))
display_func(af.log10(a))
display_func(af.log2(a))
display_func(af.sqrt(a))
display_func(af.cbrt(a))
a = af.round(5 * af.randu(3, 3) - 1)
b = af.round(5 * af.randu(3, 3) - 1)
display_func(af.factorial(a))
display_func(af.tgamma(a))
display_func(af.lgamma(a))
display_func(af.iszero(a))
display_func(af.isinf(a / b))
display_func(af.isnan(a / a))
a = af.randu(5, 1)
b = af.randu(1, 5)
c = af.broadcast(lambda x, y: x + y, a, b)
display_func(a)
display_func(b)
display_func(c)
@af.broadcast
def test_add(aa, bb):
return aa + bb
display_func(test_add(a, b))
def abserr(predicted, target):
return 100 * af.sum(af.abs(predicted - target)) / predicted.elements()
def indices_and_currents_TSC_2D( charge_electron, positions_x, positions_y, velocity_x, velocity_y,\
x_grid, y_grid, ghost_cells, length_domain_x, length_domain_y, dt ):
"""
function indices_and_currents_TSC_2D( charge_electron, positions_x,\
positions_y, velocity_x, velocity_y,\
x_grid, y_grid, ghost_cells,\
length_domain_x, length_domain_y, dt\
)
return the x and y indices for Jx and Jy and respective currents associated with these indices
"""
positions_x_new = positions_x + velocity_x * dt
positions_y_new = positions_y + velocity_y * dt
base_indices_x = af.data.constant(0,
positions_x.elements(),
dtype=af.Dtype.u32)
base_indices_y = af.data.constant(0,
positions_x.elements(),
dtype=af.Dtype.u32)
dx = af.sum(x_grid[1] - x_grid[0])
dy = af.sum(y_grid[1] - y_grid[0])
# Computing S0_x and S0_y
###########################################################################################
# Determining the grid cells containing the respective particles
x_zone = (((af.abs(positions_x - af.sum(x_grid[0]))) / dx).as_type(
af.Dtype.u32))
y_zone = (((af.abs(positions_y - af.sum(y_grid[0]))) / dy).as_type(
af.Dtype.u32))
# Determing the indices of the closest grid node in x direction
temp = af.where(af.abs(positions_x-x_grid[x_zone]) < \
af.abs(positions_x-x_grid[x_zone + 1])\
)
if (temp.elements() > 0):
base_indices_x[temp] = x_zone[temp]
temp = af.where(af.abs(positions_x - x_grid[x_zone]) >= \
af.abs(positions_x-x_grid[x_zone + 1])\
)
if (temp.elements() > 0):
base_indices_x[temp] = (x_zone[temp] + 1).as_type(af.Dtype.u32)
# Determing the indices of the closest grid node in y direction
temp = af.where(af.abs(positions_y-y_grid[y_zone]) < \
af.abs(positions_y-y_grid[y_zone + 1])\
)
if (temp.elements() > 0):
base_indices_y[temp] = y_zone[temp]
temp = af.where(
af.abs(positions_y - y_grid[y_zone]) >= af.abs(positions_y -
x_grid[y_zone + 1]))
if (temp.elements() > 0):
base_indices_y[temp] = (y_zone[temp] + 1).as_type(af.Dtype.u32)
# Concatenating the index list for near by grid nodes in x direction
# TSC affect 5 nearest grid nodes around in 1 Dimensions
base_indices_minus_two = (base_indices_x - 2).as_type(af.Dtype.u32)
base_indices_minus = (base_indices_x - 1).as_type(af.Dtype.u32)
base_indices_plus = (base_indices_x + 1).as_type(af.Dtype.u32)
base_indices_plus_two = (base_indices_x + 2).as_type(af.Dtype.u32)
index_list_x = af.join( 1,\
af.join(1, base_indices_minus_two, base_indices_minus, base_indices_x),\
af.join(1, base_indices_plus, base_indices_plus_two),\
)
# Concatenating the index list for near by grid nodes in y direction
# TSC affect 5 nearest grid nodes around in 1 Dimensions
base_indices_minus_two = (base_indices_y - 2).as_type(af.Dtype.u32)
base_indices_minus = (base_indices_y - 1).as_type(af.Dtype.u32)
base_indices_plus = (base_indices_y + 1).as_type(af.Dtype.u32)
base_indices_plus_two = (base_indices_y + 2).as_type(af.Dtype.u32)
index_list_y = af.join( 1,\
af.join(1, base_indices_minus_two, base_indices_minus, base_indices_y),\
af.join(1, base_indices_plus, base_indices_plus_two),\
)
# Concatenating the positions_x for determining weights for near by grid nodes in y direction
# TSC affect 5 nearest grid nodes around in 1 Dimensions
positions_x_5x = af.join( 0,\
af.join(0, positions_x, positions_x, positions_x),\
af.join(0, positions_x, positions_x),\
)
positions_y_5x = af.join( 0,\
af.join(0, positions_y, positions_y, positions_y),\
af.join(0, positions_y, positions_y),\
)
# Determining S0 for positions at t = n * dt
distance_nodes_x = x_grid[af.flat(index_list_x)]
distance_nodes_y = y_grid[af.flat(index_list_y)]
W_x = 0 * distance_nodes_x.copy()
W_y = 0 * distance_nodes_y.copy()
# Determining weights in x direction
temp = af.where(af.abs(distance_nodes_x - positions_x_5x) < (0.5 * dx))
if (temp.elements() > 0):
W_x[temp] = 0.75 - (
af.abs(distance_nodes_x[temp] - positions_x_5x[temp]) / dx)**2
temp = af.where((af.abs(distance_nodes_x - positions_x_5x) >= (0.5*dx) )\
* (af.abs(distance_nodes_x - positions_x_5x) < (1.5 * dx) )\
)
if (temp.elements() > 0):
W_x[temp] = 0.5 * (
1.5 -
(af.abs(distance_nodes_x[temp] - positions_x_5x[temp]) / dx))**2
# Determining weights in y direction
temp = af.where(af.abs(distance_nodes_y - positions_y_5x) < (0.5 * dy))
if (temp.elements() > 0):
W_y[temp] = 0.75 - (
af.abs(distance_nodes_y[temp] - positions_y_5x[temp]) / dy)**2
temp = af.where((af.abs(distance_nodes_y - positions_y_5x) >= (0.5*dy) )\
* (af.abs(distance_nodes_y - positions_y_5x) < (1.5 * dy) )\
)
if (temp.elements() > 0):
W_y[temp] = 0.5 * (
1.5 -
(af.abs(distance_nodes_y[temp] - positions_y_5x[temp]) / dy))**2
# Restructering W_x and W_y for visualization and ease of understanding
W_x = af.data.moddims(W_x, positions_x.elements(), 5)
W_y = af.data.moddims(W_y, positions_y.elements(), 5)
# Tiling the S0_x and S0_y for the 25 indices around the particle
S0_x = af.tile(W_x, 1, 1, 5)
S0_y = af.tile(W_y, 1, 1, 5)
S0_y = af.reorder(S0_y, 0, 2, 1)
#Computing S1_x and S1_y
###########################################################################################
positions_x_5x_new = af.join( 0,\
af.join(0, positions_x_new, positions_x_new, positions_x_new),\
af.join(0, positions_x_new, positions_x_new),\
)
positions_y_5x_new = af.join( 0,\
af.join(0, positions_y_new, positions_y_new, positions_y_new),\
af.join(0, positions_y_new, positions_y_new),\
)
# Determining S0 for positions at t = n * dt
W_x = 0 * distance_nodes_x.copy()
W_y = 0 * distance_nodes_y.copy()
# Determining weights in x direction
temp = af.where(af.abs(distance_nodes_x - positions_x_5x_new) < (0.5 * dx))
if (temp.elements() > 0):
W_x[temp] = 0.75 - (
af.abs(distance_nodes_x[temp] - positions_x_5x_new[temp]) / dx)**2
temp = af.where((af.abs(distance_nodes_x - positions_x_5x_new) >= (0.5*dx) )\
* (af.abs(distance_nodes_x - positions_x_5x_new) < (1.5 * dx) )\
)
if (temp.elements() > 0):
W_x[temp] = 0.5 * (1.5 - (af.abs(distance_nodes_x[temp] \
- positions_x_5x_new[temp])/dx\
)\
)**2
# Determining weights in y direction
temp = af.where(af.abs(distance_nodes_y - positions_y_5x_new) < (0.5 * dy))
if (temp.elements() > 0):
W_y[temp] = 0.75 - (af.abs(distance_nodes_y[temp] \
- positions_y_5x_new[temp]\
)/dy\
)**2
temp = af.where((af.abs(distance_nodes_y - positions_y_5x_new) >= (0.5*dy) )\
* (af.abs(distance_nodes_y - positions_y_5x_new) < (1.5 * dy) )\
)
if (temp.elements() > 0):
W_y[temp] = 0.5 * (1.5 - (af.abs(distance_nodes_y[temp] \
- positions_y_5x_new[temp])/dy\
)\
)**2
# Restructering W_x and W_y for visualization and ease of understanding
W_x = af.data.moddims(W_x, positions_x.elements(), 5)
W_y = af.data.moddims(W_y, positions_x.elements(), 5)
# Tiling the S0_x and S0_y for the 25 indices around the particle
S1_x = af.tile(W_x, 1, 1, 5)
S1_y = af.tile(W_y, 1, 1, 5)
S1_y = af.reorder(S1_y, 0, 2, 1)
###########################################################################################
# Determining the final weight matrix for currents in 3D matrix form factor
W_x = (S1_x - S0_x) * (S0_y + (0.5 * (S1_y - S0_y)))
W_y = (S1_y - S0_y) * (S0_x + (0.5 * (S1_x - S0_x)))
###########################################################################################
# Assigning Jx and Jy according to Esirkepov's scheme
Jx = af.data.constant(0, positions_x.elements(), 5, 5, dtype=af.Dtype.f64)
Jy = af.data.constant(0, positions_x.elements(), 5, 5, dtype=af.Dtype.f64)
Jx[:, 0, :] = -1 * charge_electron * (dx / dt) * W_x[:, 0, :].copy()
Jx[:,
1, :] = Jx[:, 0, :] + -1 * charge_electron * (dx /
dt) * W_x[:, 1, :].copy()
Jx[:,
2, :] = Jx[:, 1, :] + -1 * charge_electron * (dx /
dt) * W_x[:, 2, :].copy()
Jx[:,
3, :] = Jx[:, 2, :] + -1 * charge_electron * (dx /
dt) * W_x[:, 3, :].copy()
Jx[:,
4, :] = Jx[:, 3, :] + -1 * charge_electron * (dx /
dt) * W_x[:, 4, :].copy()
# Computing current density using currents
Jx = (1 / (dx * dy)) * Jx
Jy[:, :, 0] = -1 * charge_electron * (dy / dt) * W_y[:, :, 0].copy()
Jy[:, :,
1] = Jy[:, :,
0] + -1 * charge_electron * (dy / dt) * W_y[:, :, 1].copy()
Jy[:, :,
2] = Jy[:, :,
1] + -1 * charge_electron * (dy / dt) * W_y[:, :, 2].copy()
Jy[:, :,
3] = Jy[:, :,
2] + -1 * charge_electron * (dy / dt) * W_y[:, :, 3].copy()
Jy[:, :,
4] = Jy[:, :,
3] + -1 * charge_electron * (dy / dt) * W_y[:, :, 4].copy()
# Computing current density using currents
Jy = (1 / (dx * dy)) * Jy
# Preparing the final index and current vectors
###########################################################################################
# Determining the x indices for charge deposition
index_list_x_Jx = af.flat(af.tile(index_list_x, 1, 1, 5))
# Determining the y indices for charge deposition
y_current_zone = af.tile(index_list_y, 1, 1, 5)
index_list_y_Jx = af.flat(af.reorder(y_current_zone, 0, 2, 1))
currents_Jx = af.flat(Jx)
# Determining the x indices for charge deposition
index_list_x_Jy = af.flat(af.tile(index_list_x, 1, 1, 5))
# Determining the y indices for charge deposition
y_current_zone = af.tile(index_list_y, 1, 1, 5)
index_list_y_Jy = af.flat(af.reorder(y_current_zone, 0, 2, 1))
# Flattenning the Currents array
currents_Jy = af.flat(Jy)
af.eval(index_list_x_Jx, index_list_y_Jx)
af.eval(index_list_x_Jy, index_list_y_Jy)
af.eval(currents_Jx, currents_Jy)
return index_list_x_Jx, index_list_y_Jx, currents_Jx,\
index_list_x_Jy, index_list_y_Jy, currents_Jy
def f0_ee_constant_T(f, p_x, p_y, p_z, params):
"""
Return the local equilibrium distribution corresponding to the tau_ee
relaxation time when lattice temperature, T, is set to constant.
Parameters:
-----------
f : Distribution function array
shape:(N_v, N_s, N_q1, N_q2)
p_x : The array that holds data for the v1 dimension in v-space
shape:(N_v, N_s, 1, 1)
p_y : The array that holds data for the v2 dimension in v-space
shape:(N_v, N_s, 1, 1)
p_z : The array that holds data for the v3 dimension in v-space
shape:(N_v, N_s, 1, 1)
params: The parameters file/object that is originally declared by the user.
This can be used to inject other functions/attributes into the function
"""
# Initial guess
mu_ee = params.mu_ee
T_ee = params.T_ee
vel_drift_x = params.vel_drift_x
vel_drift_y = params.vel_drift_y
for n in range(params.collision_nonlinear_iters):
E_upper = params.E_band
k = params.boltzmann_constant
tmp1 = (E_upper - mu_ee - p_x*vel_drift_x - p_y*vel_drift_y)
tmp = (tmp1/(k*T_ee))
denominator = (k*T_ee**2.*(af.exp(tmp) + 2. + af.exp(-tmp)) )
a_0 = T_ee / denominator
a_1 = tmp1 / denominator
a_2 = T_ee * p_x / denominator
a_3 = T_ee * p_y / denominator
af.eval(a_0, a_1, a_2, a_3)
# TODO: Multiply with the integral measure dp_x * dp_y
a_00 = integral_over_p(a_0, params.integral_measure)
##a_01 = af.sum(a_1, 0)
a_02 = integral_over_p(a_2, params.integral_measure)
a_03 = integral_over_p(a_3, params.integral_measure)
#a_10 = af.sum(E_upper * a_0, 0)
#a_11 = af.sum(E_upper * a_1, 0)
#a_12 = af.sum(E_upper * a_2, 0)
#a_13 = af.sum(E_upper * a_3, 0)
a_20 = integral_over_p(p_x * a_0, params.integral_measure)
##a_21 = af.sum(p_x * a_1, 0)
a_22 = integral_over_p(p_x * a_2, params.integral_measure)
a_23 = integral_over_p(p_x * a_3, params.integral_measure)
a_30 = integral_over_p(p_y * a_0, params.integral_measure)
##a_31 = af.sum(p_y * a_1, 0)
a_32 = integral_over_p(p_y * a_2, params.integral_measure)
a_33 = integral_over_p(p_y * a_3, params.integral_measure)
A = [ [a_00, a_02, a_03], \
[a_20, a_22, a_23], \
[a_30, a_32, a_33] \
]
fermi_dirac = 1./(af.exp( ( E_upper - mu_ee
- vel_drift_x*p_x - vel_drift_y*p_y
)/(k*T_ee)
) + 1.
)
af.eval(fermi_dirac)
zeroth_moment = (f - fermi_dirac)
#second_moment = E_upper*(f - fermi_dirac)
first_moment_x = p_x*(f - fermi_dirac)
first_moment_y = p_y*(f - fermi_dirac)
eqn_mass_conservation = integral_over_p(zeroth_moment, params.integral_measure)
#eqn_energy_conservation = af.sum(second_moment, 0)
eqn_mom_x_conservation = integral_over_p(first_moment_x, params.integral_measure)
eqn_mom_y_conservation = integral_over_p(first_moment_y, params.integral_measure)
residual = [eqn_mass_conservation, \
eqn_mom_x_conservation, \
eqn_mom_y_conservation]
error_norm = np.max([af.max(af.abs(residual[0])),
af.max(af.abs(residual[1])),
af.max(af.abs(residual[2]))
]
)
print(" rank = ", params.rank,
"||residual_ee|| = ", error_norm
)
# if (error_norm < 1e-13):
# params.mu_ee = mu_ee
# params.T_ee = T_ee
# params.vel_drift_x = vel_drift_x
# params.vel_drift_y = vel_drift_y
# return(fermi_dirac)
b_0 = eqn_mass_conservation
#b_1 = eqn_energy_conservation
b_2 = eqn_mom_x_conservation
b_3 = eqn_mom_y_conservation
b = [b_0, b_2, b_3]
# Solve Ax = b
# where A == Jacobian,
# x == delta guess (correction to guess),
# b = -residual
A_inv = inverse_3x3_matrix(A)
x_0 = A_inv[0][0]*b[0] + A_inv[0][1]*b[1] + A_inv[0][2]*b[2]
#x_1 = A_inv[1][0]*b[0] + A_inv[1][1]*b[1] + A_inv[1][2]*b[2] + A_inv[1][3]*b[3]
x_2 = A_inv[1][0]*b[0] + A_inv[1][1]*b[1] + A_inv[1][2]*b[2]
x_3 = A_inv[2][0]*b[0] + A_inv[2][1]*b[1] + A_inv[2][2]*b[2]
delta_mu = x_0
#delta_T = x_1
delta_vx = x_2
delta_vy = x_3
mu_ee = mu_ee + delta_mu
#T_ee = T_ee + delta_T
vel_drift_x = vel_drift_x + delta_vx
vel_drift_y = vel_drift_y + delta_vy
af.eval(mu_ee, vel_drift_x, vel_drift_y)
# Solved for (mu_ee, T_ee, vel_drift_x, vel_drift_y). Now store in params
params.mu_ee = mu_ee
#params.T_ee = T_ee
params.vel_drift_x = vel_drift_x
params.vel_drift_y = vel_drift_y
fermi_dirac = 1./(af.exp( ( E_upper - mu_ee
- vel_drift_x*p_x - vel_drift_y*p_y
)/(k*T_ee)
) + 1.
)
af.eval(fermi_dirac)
zeroth_moment = f - fermi_dirac
#second_moment = E_upper*(f - fermi_dirac)
first_moment_x = p_x*(f - fermi_dirac)
first_moment_y = p_y*(f - fermi_dirac)
eqn_mass_conservation = integral_over_p(zeroth_moment, params.integral_measure)
#eqn_energy_conservation = af.sum(second_moment, 0)
eqn_mom_x_conservation = integral_over_p(first_moment_x, params.integral_measure)
eqn_mom_y_conservation = integral_over_p(first_moment_y, params.integral_measure)
residual = [eqn_mass_conservation, \
eqn_mom_x_conservation, \
eqn_mom_y_conservation
]
error_norm = np.max([af.max(af.abs(residual[0])),
af.max(af.abs(residual[1])),
af.max(af.abs(residual[2]))
]
)
print(" rank = ", params.rank,
"||residual_ee|| = ", error_norm
)
N_g = domain.N_ghost
print(" rank = ", params.rank,
"mu_ee = ", af.mean(params.mu_ee[0, 0, N_g:-N_g, N_g:-N_g]),
"T_ee = ", af.mean(params.T_ee[0, 0, N_g:-N_g, N_g:-N_g]),
"<v_x> = ", af.mean(params.vel_drift_x[0, 0, N_g:-N_g, N_g:-N_g]),
"<v_y> = ", af.mean(params.vel_drift_y[0, 0, N_g:-N_g, N_g:-N_g])
)
PETSc.Sys.Print(" ------------------")
return(fermi_dirac)
pl.gca().set_aspect('equal')
pl.colorbar()
pl.title('Time = %.2f' % (index * 10 * delta_t_2d))
fig.savefig('results/2D_Wave_images/%04d' % (index) + '.png')
pl.close('all')
return
for i in trange(201):
h5py_data = h5py.File(
'results/xi_eta_2d_hdf5_%02d/dump_timestep_%06d' %
(int(params.N_LGL), int(10 * i)) + '.hdf5', 'r')
u_LGL = af.np_to_af_array(h5py_data['u_i'][:])
contour_2d(u_LGL, i)
if i > 199:
print(af.mean(af.abs(u_LGL - params.u_e_ij)))
# Creating a folder to store hdf5 files. If it doesn't exist.
results_directory = 'results/1D_Wave_images'
if not os.path.exists(results_directory):
os.makedirs(results_directory)
# The directory where h5py files are stored.
#h5py_directory = 'results/hdf5_%02d' %int(params.N_LGL)
#
#path, dirs, files = os.walk(h5py_directory).__next__()
#file_count = len(files)
#print(file_count)
#
#
def check_error(params):
error = np.zeros(N.size)
for i in range(N.size):
domain.N_p1 = int(N[i])
domain.N_p2 = int(N[i])
# Defining the physical system to be solved:
system = physical_system(domain,
boundary_conditions,
params,
initialize,
advection_terms,
collision_operator.BGK,
moments
)
nls = nonlinear_solver(system)
# Time parameters:
dt = 0.001 * 32/nls.N_p1
# First, we check when the blob returns to (0, 0)
E1 = nls.fields_solver.cell_centered_EM_fields[0]
E2 = nls.fields_solver.cell_centered_EM_fields[1]
B3 = nls.fields_solver.cell_centered_EM_fields[5]
sol = odeint(dp_dt, np.array([0, 0]), time_array_odeint,
args = (af.mean(E1), af.mean(E2), af.mean(B3),
af.sum(params.charge[0]),
af.sum(params.mass[0])
),
atol = 1e-12, rtol = 1e-12
)
dist_from_origin = abs(sol[:, 0]) + abs(sol[:, 1])
# The time when the distance is minimum apart from the start is the time
# when the blob returns back to the center:
# However, this is an approximate solution. To get a more accurate solution,
# we provide this guess to our root finder scipy.optimize.root
t_final_approx = time_array_odeint[np.argmin(dist_from_origin[1:])]
t_final = root(residual, t_final_approx,
args = (af.mean(E1), af.mean(E2), af.mean(B3),
params.charge[0],
params.mass[0]
),
method = 'lm', tol = 1e-12
).x
time_array = np.arange(dt, float("{0:.3f}".format(t_final[0])) + dt, dt)
if(time_array[-1]>t_final):
time_array = np.delete(time_array, -1)
f_reference = nls.f
for time_index, t0 in enumerate(time_array):
nls.strang_timestep(dt)
error[i] = af.mean(af.abs( nls.f
- f_reference
)
)
return(error)
#!/usr/bin/python
#######################################################
# Copyright (c) 2015, ArrayFire
# All rights reserved.
#
# This file is distributed under 3-clause BSD license.
# The complete license agreement can be obtained at:
# http://arrayfire.com/licenses/BSD-3-Clause
########################################################
import arrayfire as af
af.info()
POINTS = 30
N = 2 * POINTS
x = (af.iota(d0=N, d1=1, tile_dims=(1, N)) - POINTS) / POINTS
y = (af.iota(d0=1, d1=N, tile_dims=(N, 1)) - POINTS) / POINTS
win = af.Window(800, 800, "3D Surface example using ArrayFire")
t = 0
while not win.close():
t = t + 0.07
z = 10 * x * -af.abs(y) * af.cos(x * x * (y + t)) + af.sin(y *
(x + t)) - 1.5
win.surface(x, y, z)
def test_fdtd_mode1_periodic():
N = np.array([128]) #2**np.arange(5, 8)
error_B1 = np.zeros(N.size)
error_B2 = np.zeros(N.size)
error_E3 = np.zeros(N.size)
for i in range(N.size):
af.device_gc()
dt = (1 / int(N[i])) * 1 / 2
time = np.arange(dt, 10 * 1 + dt, dt)
params.dt = dt
obj = test_periodic(int(N[i]), initialize_fdtd_mode1, params)
N_g = obj.fields_solver.N_g
E1_initial = obj.fields_solver.yee_grid_EM_fields[0].copy()
E2_initial = obj.fields_solver.yee_grid_EM_fields[1].copy()
E3_initial = obj.fields_solver.yee_grid_EM_fields[2].copy()
B1_initial = obj.fields_solver.yee_grid_EM_fields[3].copy()
B2_initial = obj.fields_solver.yee_grid_EM_fields[4].copy()
B3_initial = obj.fields_solver.yee_grid_EM_fields[5].copy()
# electric_energy = 1/4 * ( E3_initial**2 # Ez(i+1/2, j+1/2)
# + af.shift(E3_initial, 0, 0, 1, 0)**2 # Ez(i-1/2, j+1/2)
# + af.shift(E3_initial, 0, 0, 0, 1)**2 # Ez(i+1/2, j-1/2)
# + af.shift(E3_initial, 0, 0, 1, 1)**2 # Ez(i-1/2, j-1/2)
# )
# magnetic_energy_x = 0.5 * ( B1_n_plus_half * B1_n_minus_half # (i+1/2, j)
# + af.shift(B1_n_plus_half * B1_n_minus_half, 0, 0, 1, 0) # (i-1/2, j)
# )
# magnetic_energy_y = 0.5 * ( B2_n_plus_half * B2_n_minus_half # (i, j+1/2)
# + af.shift(B2_n_plus_half * B2_n_minus_half, 0, 0, 0, 1) # (i, j-1/2)
# )
energy = np.zeros([time.size])
B1_at_n_minus_half_i = obj.fields_solver.yee_grid_EM_fields[3].copy()
B1_at_n_minus_half_i_plus_1 = af.shift(B1_at_n_minus_half_i, 0, 0, 0,
-1)
for time_index, t0 in enumerate(time):
B1_at_n_plus_half_i = obj.fields_solver.yee_grid_EM_fields[3].copy(
)
B1_at_n_plus_half_i_plus_1 = af.shift(B1_at_n_plus_half_i, 0, 0, 0,
-1)
E3_n = obj.fields_solver.yee_grid_EM_fields[2].copy()
J1 = J2 = J3 = 0 * obj.fields_solver.q1_center**0
obj.fields_solver.evolve_electrodynamic_fields(J1, J2, J3, dt)
E3_n_plus_1 = obj.fields_solver.yee_grid_EM_fields[2].copy()
energy[time_index] = af.sum(
(E3_n_plus_1 * B1_at_n_minus_half_i_plus_1
)[:, :, N_g:-N_g,
N_g:-N_g]) * obj.fields_solver.dq1 * obj.fields_solver.dq2
B1_at_n_minus_half_i = B1_at_n_plus_half_i.copy()
B1_at_n_minus_half_i_plus_1 = af.shift(B1_at_n_minus_half_i, 0, 0,
0, -1)
# electric_energy = E3_at_n**2
# magnetic_energy_x = B1_n_plus_half * B1_n_minus_half
# magnetic_energy_y = B2_n_plus_half * B2_n_minus_half
# pl.plot(np.array(obj.fields_solver.q1_center).reshape(134, 9)[3:-3, 0],
# np.array(obj.fields_solver.yee_grid_EM_fields[1]).reshape(134, 9)[3:-3, 0],
# label = r'$Ey_{i+1/2}^n$')
# pl.plot(np.array(obj.fields_solver.q1_center).reshape(134, 9)[3:-3, 0],
# np.array(obj.fields_solver.yee_grid_EM_fields[5]).reshape(134, 9)[3:-3, 0], '--',
# label = r'${Bz}_{i}^{n+1/2}$')
# pl.legend(fontsize = 20)
# pl.ylim([-2, 2])
# pl.title('Time = %.2f'%t0)
# pl.savefig('images/%04d'%time_index + '.png')
# pl.clf()
# pl.contourf(np.array(obj.fields_solver.yee_grid_EM_fields[2]).reshape(134, 134), 40)
# pl.savefig('images/%04d'%time_index + '.png')
# pl.clf()
# energy = af.sum((electric_energy + magnetic_energy_x + magnetic_energy_y)[:, :, N_g:-N_g, N_g:-N_g])
# if(time_index == 0):
# error[time_index] = energy * obj.fields_solver.dq1 * obj.fields_solver.dq2
# else:
# error[time_index] = abs((energy) * obj.fields_solver.dq1 * obj.fields_solver.dq2 - error[0])
import h5py
h5f = h5py.File('data/Bi+n-_Ei+n+.h5', 'w')
h5f.create_dataset('data', data=energy[1:])
h5f.create_dataset('time', data=time[1:])
h5f.close()
# pl.plot(time[1:], energy[1:])
# pl.show()
# # pl.ylabel(r'$B_z^{n+1/2}(i) \times (E_y^{n+1}(i+1/2) + E_y^{n}(i+1/2))$')
# pl.xlabel('Time')
# # pl.ylim([1e-15, 1e-9])
# pl.savefig('plot.png', bbox_inches = 'tight')
error_B1[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[3, :, N_g:-N_g,
N_g:-N_g] -
B1_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B1_initial.elements())
error_B2[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[4, :, N_g:-N_g,
N_g:-N_g] -
B2_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B2_initial.elements())
error_E3[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[2, :, N_g:-N_g,
N_g:-N_g] -
E3_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E3_initial.elements())
poly_B1 = np.polyfit(np.log10(N), np.log10(error_B1), 1)
poly_B2 = np.polyfit(np.log10(N), np.log10(error_B2), 1)
poly_E3 = np.polyfit(np.log10(N), np.log10(error_E3), 1)
print(error_B1)
print(error_B2)
print(error_E3)
print(poly_B1)
print(poly_B2)
print(poly_E3)
pl.loglog(N, error_B1, '-o', label=r'$B_x$')
pl.loglog(N, error_B2, '-o', label=r'$B_y$')
pl.loglog(N, error_E3, '-o', label=r'$E_z$')
pl.loglog(N,
error_B2[0] * 32**2 / N**2,
'--',
color='black',
label=r'$O(N^{-2})$')
pl.xlabel(r'$N$')
pl.ylabel('Error')
pl.legend()
pl.savefig('convergenceplot.png')
assert (abs(poly_B1[0] + 2) < 0.2)
assert (abs(poly_B2[0] + 2) < 0.2)
assert (abs(poly_E3[0] + 2) < 0.2)
def af_assert(left, right, eps=1E-6):
if (af.max(af.abs(left -right)) > eps):
raise ValueError("Arrays not within dictated precision")
return
def test_fdtd_mode2_periodic():
N = np.array([64]) #2**np.arange(5, 8)
error_E1 = np.zeros(N.size)
error_E2 = np.zeros(N.size)
error_B3 = np.zeros(N.size)
for i in range(N.size):
dt = (1 / int(N[i])) * np.sqrt(9 / 5) / 2
time = np.arange(dt, 10 * np.sqrt(9 / 5) + dt, dt)
params.dt = dt
obj = test_periodic(N[i], initialize_fdtd_mode2, params)
N_g = obj.fields_solver.N_g
B3_initial = obj.fields_solver.yee_grid_EM_fields[5].copy()
E1_initial = obj.fields_solver.yee_grid_EM_fields[0].copy()
E2_initial = obj.fields_solver.yee_grid_EM_fields[1].copy()
error = np.zeros([time.size])
for time_index, t0 in enumerate(time):
B3_n_minus_half = obj.fields_solver.yee_grid_EM_fields[5].copy()
J1 = J2 = J3 = 0 * obj.fields_solver.q1_center**0
obj.fields_solver.evolve_electrodynamic_fields(J1, J2, J3, dt)
B3_n_plus_half = obj.fields_solver.yee_grid_EM_fields[5].copy()
E1_at_n = obj.fields_solver.yee_grid_EM_fields[0].copy()
E2_at_n = obj.fields_solver.yee_grid_EM_fields[1].copy()
electric_energy_x = E1_at_n**2
electric_energy_y = E2_at_n**2
magnetic_energy = B3_n_plus_half * B3_n_minus_half
energy = af.sum((magnetic_energy + electric_energy_x +
electric_energy_y)[:, :, N_g:-N_g, N_g:-N_g])
if (time_index == 0):
error[
time_index] = energy * obj.fields_solver.dq1 * obj.fields_solver.dq2
else:
error[time_index] = abs((energy) * obj.fields_solver.dq1 *
obj.fields_solver.dq2 - error[0])
pl.semilogy(time[1:], error[1:])
pl.ylabel('Error')
pl.xlabel('Time')
pl.ylim([1e-15, 1e-9])
pl.savefig('plot.png', bbox_inches='tight')
error_E1[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[0, :, N_g:-N_g,
N_g:-N_g] -
E1_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E1_initial.elements())
error_E2[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[1, :, N_g:-N_g,
N_g:-N_g] -
E2_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E2_initial.elements())
error_B3[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[5, :, N_g:-N_g,
N_g:-N_g] -
B3_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B3_initial.elements())
print(error_E1)
print(error_E2)
print(error_B3)
poly_E1 = np.polyfit(np.log10(N), np.log10(error_E1), 1)
poly_E2 = np.polyfit(np.log10(N), np.log10(error_E2), 1)
poly_B3 = np.polyfit(np.log10(N), np.log10(error_B3), 1)
pl.loglog(N, error_E1, '-o', label=r'$E_x$')
pl.loglog(N, error_E2, '-o', label=r'$E_y$')
pl.loglog(N, error_B3, '-o', label=r'$B_z$')
pl.loglog(N,
error_E1[0] * 32**2 / N**2,
'--',
color='black',
label=r'$O(N^{-2})$')
pl.xlabel(r'$N$')
pl.ylabel('Error')
pl.legend()
pl.savefig('convergenceplot.png')
print(poly_E1)
print(poly_E2)
print(poly_B3)
assert (abs(poly_E1[0] + 2) < 0.3)
assert (abs(poly_E2[0] + 2) < 0.3)
assert (abs(poly_B3[0] + 2) < 0.3)
def f0_defect_constant_T(f, p_x, p_y, p_z, params):
"""
Return the local equilibrium distribution corresponding to the tau_D
relaxation time when lattice temperature, T, is set to constant.
Parameters:
-----------
f : Distribution function array
shape:(N_v, N_s, N_q1, N_q2)
p_x : The array that holds data for the v1 dimension in v-space
shape:(N_v, N_s, 1, 1)
p_y : The array that holds data for the v2 dimension in v-space
shape:(N_v, N_s, 1, 1)
p_z : The array that holds data for the v3 dimension in v-space
shape:(N_v, N_s, 1, 1)
params: The parameters file/object that is originally declared by the user.
This can be used to inject other functions/attributes into the function
"""
mu = params.mu
T = params.T
for n in range(params.collision_nonlinear_iters):
E_upper = params.E_band
k = params.boltzmann_constant
tmp = ((E_upper - mu)/(k*T))
denominator = (k*T**2.*(af.exp(tmp) + 2. + af.exp(-tmp)) )
# TODO: Multiply with the integral measure dp_x * dp_y
a00 = integral_over_p(T/denominator, params.integral_measure)
fermi_dirac = 1./(af.exp( (E_upper - mu)/(k*T) ) + 1.)
af.eval(fermi_dirac)
zeroth_moment = f - fermi_dirac
eqn_mass_conservation = integral_over_p(zeroth_moment,
params.integral_measure
)
N_g = domain.N_ghost
error_mass_conservation = af.max(af.abs(eqn_mass_conservation)[0, 0, N_g:-N_g, N_g:-N_g])
print(" rank = ", params.rank,
"||residual_defect|| = ", error_mass_conservation
)
res = eqn_mass_conservation
dres_dmu = -a00
delta_mu = -res/dres_dmu
mu = mu + delta_mu
af.eval(mu)
# Solved for mu. Now store in params
params.mu = mu
# Print final residual
fermi_dirac = 1./(af.exp( (E_upper - mu)/(k*T) ) + 1.)
af.eval(fermi_dirac)
zeroth_moment = f - fermi_dirac
eqn_mass_conservation = integral_over_p(zeroth_moment,
params.integral_measure
)
N_g = domain.N_ghost
error_mass_conservation = af.max(af.abs(eqn_mass_conservation)[0, 0, N_g:-N_g, N_g:-N_g])
print(" rank = ", params.rank,
"||residual_defect|| = ", error_mass_conservation
)
print(" rank = ", params.rank,
"mu = ", af.mean(params.mu[0, 0, N_g:-N_g, N_g:-N_g]),
"T = ", af.mean(params.T[0, 0, N_g:-N_g, N_g:-N_g])
)
PETSc.Sys.Print(" ------------------")
return(fermi_dirac)
def test_fdtd_mode1_mirror():
N = 2**np.arange(5, 8)
error_B1 = np.zeros(N.size)
error_B2 = np.zeros(N.size)
error_E3 = np.zeros(N.size)
for i in range(N.size):
dt = (1 / int(N[i])) * np.sqrt(9 / 5) / 2
time = np.arange(dt, 4 * np.sqrt(9 / 5) + dt, dt)
params.dt = dt
obj = test_mirror(N[i], initialize_fdtd_mode2, params)
N_g = obj.fields_solver.N_g
B1_initial = obj.fields_solver.yee_grid_EM_fields[3].copy()
B2_initial = obj.fields_solver.yee_grid_EM_fields[4].copy()
E3_initial = obj.fields_solver.yee_grid_EM_fields[2].copy()
for time_index, t0 in enumerate(time):
J1 = J2 = J3 = 0 * obj.fields_solver.q1_center**0
obj.fields_solver.evolve_electrodynamic_fields(J1, J2, J3, dt)
error_B1[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[3, :, N_g:-N_g,
N_g:-N_g] -
B1_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B1_initial.elements())
error_B2[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[4, :, N_g:-N_g,
N_g:-N_g] -
B2_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B2_initial.elements())
error_E3[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[2, :, N_g:-N_g,
N_g:-N_g] -
E3_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E3_initial.elements())
print(error_B1)
print(error_B2)
print(error_E3)
poly_B1 = np.polyfit(np.log10(N), np.log10(error_B1), 1)
poly_B2 = np.polyfit(np.log10(N), np.log10(error_B2), 1)
poly_E3 = np.polyfit(np.log10(N), np.log10(error_E3), 1)
pl.loglog(N, error_B1, '-o', label=r'$B_x$')
pl.loglog(N, error_B2, '-o', label=r'$B_y$')
pl.loglog(N, error_E3, '-o', label=r'$E_z$')
pl.loglog(N,
error_B1[0] * 32**2 / N**2,
'--',
color='black',
label=r'$O(N^{-2})$')
pl.xlabel(r'$N$')
pl.ylabel('Error')
pl.legend()
pl.savefig('convergenceplot.png')
assert (abs(poly_B1[0] + 1) < 0.2)
assert (abs(poly_B2[0] + 1) < 0.2)
assert (abs(poly_E3[0] + 1) < 0.2)
def initialize_f(q1, q2, p1, p2, p3, params):
f = af.select(af.abs(p1)<2.5, q1**0 * p1**0, 0)
af.eval(f)
return (f)
def test_fdtd_mode2_mirror():
N = 2**np.arange(5, 11)
error_E1 = np.zeros(N.size)
error_E2 = np.zeros(N.size)
error_B3 = np.zeros(N.size)
for i in range(N.size):
dt = (1 / int(N[i])) * 1 / 2
time = np.arange(dt, 4 * 1 + dt, dt)
params.dt = dt
obj = test_mirror(N[i], initialize_fdtd_mode2, params)
N_g = obj.fields_solver.N_g
E1_initial = obj.fields_solver.yee_grid_EM_fields[0].copy()
E2_initial = obj.fields_solver.yee_grid_EM_fields[1].copy()
B3_initial = obj.fields_solver.yee_grid_EM_fields[5].copy()
for time_index, t0 in enumerate(time):
# pl.subplot(3, 1, 1)
# pl.gca().axes.xaxis.set_ticklabels([])
# pl.gca().axes.yaxis.set_ticklabels([])
# pl.title(r'$E_x$')
# pl.contourf(np.array(obj.fields_solver.q1_center).reshape(38, 38),
# np.array(obj.fields_solver.q2_center).reshape(38, 38),
# np.array(obj.fields_solver.yee_grid_EM_fields[0, :, :, :]).reshape(38, 38),
# 100
# )
# pl.subplot(3, 1, 2)
# pl.gca().axes.xaxis.set_ticklabels([])
# pl.gca().axes.yaxis.set_ticklabels([])
# pl.title(r'$E_y$')
# pl.contourf(np.array(obj.fields_solver.q1_center).reshape(38, 38),
# np.array(obj.fields_solver.q2_center).reshape(38, 38),
# np.array(obj.fields_solver.yee_grid_EM_fields[1, :, :, :]).reshape(38, 38),
# 100
# )
# pl.subplot(3, 1, 3)
# pl.title(r'$B_z$')
# pl.contourf(np.array(obj.fields_solver.q1_center).reshape(38, 38),
# np.array(obj.fields_solver.q2_center).reshape(38, 38),
# np.array(obj.fields_solver.yee_grid_EM_fields[5, :, :, :]).reshape(38, 38),
# 100
# )
# pl.xlabel(r'$x$')
# pl.ylabel(r'$y$')
# pl.savefig('images/%04d'%time_index + '.png')
# pl.clf()
J1 = J2 = J3 = 0 * obj.fields_solver.q1_center**0
obj.fields_solver.evolve_electrodynamic_fields(J1, J2, J3, dt)
error_E1[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[0, :, N_g:-N_g,
N_g:-N_g] -
E1_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E1_initial.elements())
error_E2[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[1, :, N_g:-N_g,
N_g:-N_g] -
E2_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
E2_initial.elements())
error_B3[i] = af.sum(
af.abs(obj.fields_solver.yee_grid_EM_fields[5, :, N_g:-N_g,
N_g:-N_g] -
B3_initial[:, :, N_g:-N_g, N_g:-N_g])) / (
B3_initial.elements())
poly_E1 = np.polyfit(np.log10(N), np.log10(error_E1), 1)
poly_E2 = np.polyfit(np.log10(N), np.log10(error_E2), 1)
poly_B3 = np.polyfit(np.log10(N), np.log10(error_B3), 1)
print(error_E1)
print(error_E2)
print(error_B3)
print(poly_E1)
print(poly_E2)
print(poly_B3)
pl.loglog(N, error_E1, '-o', label=r'$E_x$')
pl.loglog(N, error_E2, '-o', label=r'$E_y$')
pl.loglog(N, error_B3, '-o', label=r'$B_z$')
pl.loglog(N,
error_B3[0] * N[0]**2 / N**2,
'--',
color='black',
label=r'$\mathcal{O}(N^{-2})$')
pl.xlabel(r'$N$')
pl.ylabel('Error')
pl.legend(framealpha=0)
pl.savefig('convergenceplot.png', bbox_inches='tight')
assert (abs(poly_E1[0] + 2) < 0.4)
assert (abs(poly_E2[0] + 2) < 0.4)
assert (abs(poly_B3[0] + 2) < 0.4)
def __abs__(self):
s = arrayfire.abs(self.d_array)
# dtype is wrong for complex types
return ndarray(self.shape, dtype=pu.typemap(s.dtype()), af_array=s)
def reconstruct_weno5(input_array, axis):
"""
Reconstructs the input array using a WENO5 reconstruction.
Parameters
----------
input_array: af.Array
Array holding the cells data.
axis: int
Axis along which the reconstruction method is to be applied.
"""
eps = 1e-17
if (axis == 0):
x0_shift = 2
y0_shift = 0
z0_shift = 0
w0_shift = 0
x1_shift = 1
y1_shift = 0
z1_shift = 0
w1_shift = 0
x3_shift = -1
y3_shift = 0
z3_shift = 0
w3_shift = 0
x4_shift = -2
y4_shift = 0
z4_shift = 0
w4_shift = 0
elif (axis == 1):
x0_shift = 0
y0_shift = 2
z0_shift = 0
w0_shift = 0
x1_shift = 0
y1_shift = 1
z1_shift = 0
w1_shift = 0
x3_shift = 0
y3_shift = -1
z3_shift = 0
w3_shift = 0
x4_shift = 0
y4_shift = -2
z4_shift = 0
w4_shift = 0
elif (axis == 2):
x0_shift = 0
y0_shift = 0
z0_shift = 2
w0_shift = 0
x1_shift = 0
y1_shift = 0
z1_shift = 1
w1_shift = 0
x3_shift = 0
y3_shift = 0
z3_shift = -1
w3_shift = 0
x4_shift = 0
y4_shift = 0
z4_shift = -2
w4_shift = 0
elif (axis == 3):
x0_shift = 0
y0_shift = 0
z0_shift = 0
w0_shift = 2
x1_shift = 0
y1_shift = 0
z1_shift = 0
w1_shift = 1
x3_shift = 0
y3_shift = 0
z3_shift = 0
w3_shift = -1
x4_shift = 0
y4_shift = 0
z4_shift = 0
w4_shift = -2
else:
raise Exception('Invalid choice for axis')
y0 = af.shift(input_array, x0_shift, y0_shift, z0_shift, w0_shift)
y1 = af.shift(input_array, x1_shift, y1_shift, z1_shift, w1_shift)
y2 = input_array
y3 = af.shift(input_array, x3_shift, y3_shift, z3_shift, w3_shift)
y4 = af.shift(input_array, x4_shift, y4_shift, z4_shift, w4_shift)
# Compute smoothness operators
beta1 = ((4 / 3) * y0 * y0 - (19 / 3) * y0 * y1 + (25 / 3) * y1 * y1 +
(11 / 3) * y0 * y2 - (31 / 3) * y1 * y2 + (10 / 3) * y2 *
y2) + eps * (1.0 + af.abs(y0) + af.abs(y1) + af.abs(y2))
beta2 = ((4 / 3) * y1 * y1 - (19 / 3) * y1 * y2 + (25 / 3) * y2 * y2 +
(11 / 3) * y1 * y3 - (31 / 3) * y2 * y3 + (10 / 3) * y3 *
y3) + eps * (1.0 + af.abs(y1) + af.abs(y2) + af.abs(y3))
beta3 = ((4 / 3) * y2 * y2 - (19 / 3) * y2 * y3 + (25 / 3) * y3 * y3 +
(11 / 3) * y2 * y4 - (31 / 3) * y3 * y4 + (10 / 3) * y4 *
y4) + eps * (1.0 + af.abs(y2) + af.abs(y3) + af.abs(y4))
# Compute weights
w1r = 1 / (16 * beta1 * beta1)
w2r = 5 / (8 * beta2 * beta2)
w3r = 5 / (16 * beta3 * beta3)
w1l = 5 / (16 * beta1 * beta1)
w2l = 5 / (8 * beta2 * beta2)
w3l = 1 / (16 * beta3 * beta3)
denl = w1l + w2l + w3l
denr = w1r + w2r + w3r
# Substencil Interpolations
u1r = 0.375 * y0 - 1.25 * y1 + 1.875 * y2
u2r = -0.125 * y1 + 0.75 * y2 + 0.375 * y3
u3r = 0.375 * y2 + 0.75 * y3 - 0.125 * y4
u1l = -0.125 * y0 + 0.75 * y1 + 0.375 * y2
u2l = 0.375 * y1 + 0.75 * y2 - 0.125 * y3
u3l = 1.875 * y2 - 1.25 * y3 + 0.375 * y4
# Reconstruction:
left_value = (w1l * u1l + w2l * u2l + w3l * u3l) / denl
right_value = (w1r * u1r + w2r * u2r + w3r * u3r) / denr
return (left_value, right_value)
dt = min(dt_fvm, dt_fdtd)
params.dt = dt
# Defining the physical system to be solved:
system = physical_system(domain, boundary_conditions, params, initialize,
advection_terms, collision_operator.BGK, moments)
# Declaring a linear system object which will evolve the defined physical system:
nls = nonlinear_solver(system)
N_g = nls.N_ghost
print('Minimum Value of f_e:', af.min(nls.f[:, 0]))
print('Minimum Value of f_i:', af.min(nls.f[:, 1]))
print('Error in density_e:',
af.mean(af.abs(nls.compute_moments('density')[:, 0] - 1)))
print('Error in density_i:',
af.mean(af.abs(nls.compute_moments('density')[:, 1] - 1)))
v2_bulk = nls.compute_moments('mom_v2_bulk') / nls.compute_moments('density')
v3_bulk = nls.compute_moments('mom_v3_bulk') / nls.compute_moments('density')
v2_bulk_i = params.amplitude * -4.801714581503802e-15 * af.cos(params.k_q1 * nls.q1_center) \
- params.amplitude * -0.3692429960259134 * af.sin(params.k_q1 * nls.q1_center)
v2_bulk_e = params.amplitude * -4.85722573273506e-15 * af.cos(params.k_q1 * nls.q1_center) \
- params.amplitude * - 0.333061857862197* af.sin(params.k_q1 * nls.q1_center)
v3_bulk_i = params.amplitude * -0.3692429960259359 * af.cos(params.k_q1 * nls.q1_center) \
- params.amplitude * 1.8041124150158794e-16 * af.sin(params.k_q1 * nls.q1_center)
def __abs__(self):
s = arrayfire.abs(self.d_array)
# dtype is wrong for complex types
return ndarray(self.shape, dtype=pu.typemap(s.dtype()), af_array=s)