def test_interpolation():
'''
'''
threshold = 8e-9
params.N_LGL = 8
gv = gvar.advection_variables(params.N_LGL, params.N_quad,\
params.x_nodes, params.N_Elements,\
params.c, params.total_time, params.wave,\
params.c_x, params.c_y, params.courant,\
params.mesh_file, params.total_time_2d)
N_LGL = 8
xi_LGL = lagrange.LGL_points(N_LGL)
xi_i = af.flat(af.transpose(af.tile(xi_LGL, 1, N_LGL)))
eta_j = af.tile(xi_LGL, N_LGL)
f_ij = np.e**(xi_i + eta_j)
interpolated_f = wave_equation_2d.lag_interpolation_2d(
f_ij, gv.Li_Lj_coeffs)
xi = utils.linspace(-1, 1, 8)
eta = utils.linspace(-1, 1, 8)
assert (af.mean(
af.transpose(utils.polyval_2d(interpolated_f, xi, eta)) -
np.e**(xi + eta)) < threshold)
def test_polyval_2d():
'''
Tests the ``utils.polyval_2d`` function by evaluating the polynomial
.. math:: P_0(\\xi) P_1(\\eta)
here,
.. math:: P_0(\\xi) = 3 \, \\xi^{2} + 2 \, \\xi + 1
.. math:: P_1(\\eta) = 3 \, \\eta^{2} + 2 \, \\eta + 1
at corresponding ``linspace`` points in :math:`\\xi \\in [-1, 1]` and
:math:`\\eta \\in [-1, 1]`.
This value is then compared with the reference value calculated analytically.
The reference values are calculated in
`polynomial_product_two_variables.sagews`_
.. _polynomial_product_two_variables.sagews: https://goo.gl/KwG7k9
'''
threshold = 1e-12
poly_xi_degree = 4
poly_xi = af.flip(af.np_to_af_array(np.arange(1, poly_xi_degree)))
poly_eta_degree = 4
poly_eta = af.flip(af.np_to_af_array(np.arange(1, poly_eta_degree)))
poly_xi_eta = utils.polynomial_product_coeffs(poly_xi, poly_eta)
xi = utils.linspace(-1, 1, 8)
eta = utils.linspace(-1, 1, 8)
polyval_xi_eta = af.transpose(utils.polyval_2d(poly_xi_eta, xi, eta))
polyval_xi_eta_ref = af.np_to_af_array(
np.array([
4.00000000000000, 1.21449396084962, 0.481466055810080,
0.601416076634741, 1.81424406497291, 5.79925031236988,
15.6751353602663, 36.0000000000000
]))
diff = af.abs(polyval_xi_eta - polyval_xi_eta_ref)
assert af.all_true(diff < threshold)
# While evaluating the volume integral using N_LGL
# lobatto quadrature points, The integration can be vectorized
# and in this case the coefficients of the differential of the
# Lagrange polynomials is required
diff_pow = (af.flip(af.transpose(af.range(N_LGL - 1) + 1), 1))
dl_dxi_coeffs = (af.broadcast(utils.multiply, lagrange_coeffs[:, :-1], diff_pow))
# Obtaining an array consisting of the LGL points mapped onto the elements.
element_size = af.sum((x_nodes[1] - x_nodes[0]) / N_Elements)
elements_xi_LGL = af.constant(0, N_Elements, N_LGL)
elements = utils.linspace(af.sum(x_nodes[0]),
af.sum(x_nodes[1] - element_size), N_Elements)
np_element_array = np.concatenate((af.transpose(elements),
af.transpose(elements + element_size)))
element_mesh_nodes = utils.linspace(af.sum(x_nodes[0]),
af.sum(x_nodes[1]),
N_Elements + 1)
element_array = af.transpose(af.interop.np_to_af_array(np_element_array))
element_LGL = wave_equation.mapping_xi_to_x(af.transpose(element_array),
xi_LGL)
# The minimum distance between 2 mapped LGL points.
delta_x = af.min((element_LGL - af.shift(element_LGL, 1, 0))[1:, :])
# dx_dxi for elements of equal size.
def change_parameters(LGL, Elements, quad, wave='sin'):
'''
Changes the parameters of the simulation. Used only for convergence tests.
Parameters
----------
LGL : int
The new N_LGL.
Elements : int
The new N_Elements.
'''
# The domain of the function.
params.x_nodes = af.np_to_af_array(np.array([-1., 1.]))
# The number of LGL points into which an element is split.
params.N_LGL = LGL
# Number of elements the domain is to be divided into.
params.N_Elements = Elements
# The number quadrature points to be used for integration.
params.N_quad = quad
# Array containing the LGL points in xi space.
params.xi_LGL = lagrange.LGL_points(params.N_LGL)
# The weights of the lgl points
params.weight_arr = lagrange.weight_arr_fun(params.xi_LGL)
# N_Gauss number of Gauss nodes.
params.gauss_points = af.np_to_af_array(lagrange.gauss_nodes\
(params.N_quad))
# The Gaussian weights.
params.gauss_weights = lagrange.gaussian_weights(params.N_quad)
# The lobatto nodes to be used for integration.
params.lobatto_quadrature_nodes = lagrange.LGL_points(params.N_quad)
# The lobatto weights to be used for integration.
params.lobatto_weights_quadrature = lagrange.lobatto_weights\
(params.N_quad)
#The b matrix
params.b_matrix = lagrange.b_matrix_eval()
# A list of the Lagrange polynomials in poly1d form.
#params.lagrange_product = lagrange.product_lagrange_poly(params.xi_LGL)
# An array containing the coefficients of the lagrange basis polynomials.
params.lagrange_coeffs = af.np_to_af_array(\
lagrange.lagrange_polynomials(params.xi_LGL)[1])
# Refer corresponding functions.
params.lagrange_basis_value = lagrange.lagrange_function_value\
(params.lagrange_coeffs)
# While evaluating the volume integral using N_LGL
# lobatto quadrature points, The integration can be vectorized
# and in this case the coefficients of the differential of the
# Lagrange polynomials is required
params.diff_pow = (af.flip(af.transpose(af.range(params.N_LGL - 1) + 1),
1))
params.dl_dxi_coeffs = (af.broadcast(utils.multiply,
params.lagrange_coeffs[:, :-1],
params.diff_pow))
# Obtaining an array consisting of the LGL points mapped onto the elements.
params.element_size = af.sum((params.x_nodes[1] - params.x_nodes[0])\
/ params.N_Elements)
params.elements_xi_LGL = af.constant(0, params.N_Elements, params.N_LGL)
params.elements = utils.linspace(af.sum(params.x_nodes[0]),
af.sum(params.x_nodes[1] - params.element_size),\
params.N_Elements)
params.np_element_array = np.concatenate((af.transpose(params.elements),
af.transpose(params.elements +\
params.element_size)))
params.element_mesh_nodes = utils.linspace(af.sum(params.x_nodes[0]),
af.sum(params.x_nodes[1]),\
params.N_Elements + 1)
params.element_array = af.transpose(af.np_to_af_array\
(params.np_element_array))
params.element_LGL = wave_equation.mapping_xi_to_x(af.transpose\
(params.element_array), params.xi_LGL)
# The minimum distance between 2 mapped LGL points.
params.delta_x = af.min(
(params.element_LGL - af.shift(params.element_LGL, 1, 0))[1:, :])
# dx_dxi for elements of equal size.
params. dx_dxi = af.mean(wave_equation.dx_dxi_numerical((params.element_mesh_nodes[0 : 2]),\
params.xi_LGL))
# The value of time-step.
params.delta_t = params.delta_x / (4 * params.c)
# Array of timesteps seperated by delta_t.
params.time = utils.linspace(
0,
int(params.total_time / params.delta_t) * params.delta_t,
int(params.total_time / params.delta_t))
# Initializing the amplitudes. Change u_init to required initial conditions.
if (wave == 'sin'):
params.u_init = af.sin(2 * np.pi * params.element_LGL)
if (wave == 'gaussian'):
params.u_init = np.e**(-(params.element_LGL)**2 / 0.4**2)
params.u = af.constant(0, params.N_LGL, params.N_Elements, params.time.shape[0],\
dtype = af.Dtype.f64)
params.u[:, :, 0] = params.u_init
return
def __init__(self, N_LGL, N_quad, x_nodes, N_elements, c, total_time, wave,
c_x, c_y, courant, mesh_file, total_time_2d):
'''
Initializes the variables using the user parameters.
Parameters
----------
N_LGL : int
Number of LGL points(for both :math:`2D` and :math:`1D` wave
equation solver).
N_quad : int
Number of the quadrature points to use in Gauss-Lobatto or
Gauss-Legendre quadrature.
x_nodes : af.Array [2 1 1 1]
:math:`x` nodes for the :math:`1D` wave equation elements.
N_elements : int
Number of elements in a :math:`1D` domain.
c : float64
Wave speed for 1D wave equation.
total_time : float64
Total time for which :math:`1D` wave equation is to be
evolved.
wave : str
Used to set u_init to ``sin`` or ``cos``.
c_x : float64
:math:`x` component of wave speed for a :math:`2D` wave.
c_y : float64
:math:`y` component of wave speed for a :math:`2D` wave.
courant : float64
Courant parameter used for the time evolution of the wave.
mesh_file : str
Path of the mesh file for the 2D wave equation.
total_time_2d : float64
Total time for which the wave is to propogated.
Returns
-------
None
'''
self.xi_LGL = lagrange.LGL_points(N_LGL)
# N_Gauss number of Gauss nodes.
self.gauss_points = af.np_to_af_array(lagrange.gauss_nodes(N_quad))
# The Gaussian weights.
self.gauss_weights = lagrange.gaussian_weights(N_quad)
# The lobatto nodes to be used for integration.
self.lobatto_quadrature_nodes = lagrange.LGL_points(N_quad)
# The lobatto weights to be used for integration.
self.lobatto_weights_quadrature = lagrange.lobatto_weights(N_quad)
# An array containing the coefficients of the lagrange basis polynomials.
self.lagrange_coeffs = lagrange.lagrange_polynomial_coeffs(self.xi_LGL)
self.lagrange_basis_value = lagrange.lagrange_function_value(
self.lagrange_coeffs, self.xi_LGL)
self.diff_pow = af.flip(af.transpose(af.range(N_LGL - 1) + 1), 1)
self.dl_dxi_coeffs = af.broadcast(utils.multiply,
self.lagrange_coeffs[:, :-1],
self.diff_pow)
self.element_size = af.sum((x_nodes[1] - x_nodes[0]) / N_elements)
self.elements_xi_LGL = af.constant(0, N_elements, N_LGL)
self.elements = utils.linspace(af.sum(x_nodes[0]),
af.sum(x_nodes[1] - self.element_size),
N_elements)
self.np_element_array = np.concatenate(
(af.transpose(self.elements),
af.transpose(self.elements + self.element_size)))
self.element_mesh_nodes = utils.linspace(af.sum(x_nodes[0]),
af.sum(x_nodes[1]),
N_elements + 1)
self.element_array = af.transpose(
af.interop.np_to_af_array(self.np_element_array))
self.element_LGL = wave_equation.mapping_xi_to_x(
af.transpose(self.element_array), self.xi_LGL)
# The minimum distance between 2 mapped LGL points.
self.delta_x = af.min(
(self.element_LGL - af.shift(self.element_LGL, 1, 0))[1:, :])
# dx_dxi for elements of equal size.
self.dx_dxi = af.mean(
wave_equation.dx_dxi_numerical(self.element_mesh_nodes[0:2],
self.xi_LGL))
# The value of time-step.
self.delta_t = self.delta_x / (4 * c)
# Array of timesteps seperated by delta_t.
self.time = utils.linspace(
0,
int(total_time / self.delta_t) * self.delta_t,
int(total_time / self.delta_t))
# Initializing the amplitudes. Change u_init to required initial conditions.
if (wave == 'sin'):
self.u_init = af.sin(2 * np.pi * self.element_LGL)
if (wave == 'gaussian'):
self.u_init = np.e**(-(self.element_LGL)**2 / 0.4**2)
self.test_array = af.np_to_af_array(np.array(self.u_init))
# The parameters below are for 2D advection
# -----------------------------------------
########################################################################
#######################2D Wave Equation#################################
########################################################################
self.xi_i = af.flat(af.transpose(af.tile(self.xi_LGL, 1, N_LGL)))
self.eta_j = af.tile(self.xi_LGL, N_LGL)
self.dLp_xi_ij = af.moddims(
af.reorder(
af.tile(utils.polyval_1d(self.dl_dxi_coeffs, self.xi_i), 1, 1,
N_LGL), 1, 2, 0), N_LGL**2, 1, N_LGL**2)
self.Lp_xi_ij = af.moddims(
af.reorder(
af.tile(utils.polyval_1d(self.lagrange_coeffs, self.xi_i), 1,
1, N_LGL), 1, 2, 0), N_LGL**2, 1, N_LGL**2)
self.dLq_eta_ij = af.tile(
af.reorder(utils.polyval_1d(self.dl_dxi_coeffs, self.eta_j), 1, 2,
0), 1, 1, N_LGL)
self.Lq_eta_ij = af.tile(
af.reorder(utils.polyval_1d(self.lagrange_coeffs, self.eta_j), 1,
2, 0), 1, 1, N_LGL)
self.dLp_Lq = self.Lq_eta_ij * self.dLp_xi_ij
self.dLq_Lp = self.Lp_xi_ij * self.dLq_eta_ij
self.Li_Lj_coeffs = wave_equation_2d.Li_Lj_coeffs(N_LGL)
self.delta_y = self.delta_x
self.delta_t_2d = courant * self.delta_x * self.delta_y \
/ (self.delta_x * c_x + self.delta_y * c_y)
self.c_lax_2d_x = c_x
self.c_lax_2d_y = c_y
self.nodes, self.elements = msh_parser.read_order_2_msh(mesh_file)
self.x_e_ij = af.np_to_af_array(
np.zeros([N_LGL * N_LGL, len(self.elements)]))
self.y_e_ij = af.np_to_af_array(
np.zeros([N_LGL * N_LGL, len(self.elements)]))
for element_tag, element in enumerate(self.elements):
self.x_e_ij[:, element_tag] = isoparam.isoparam_x_2D(
self.nodes[element, 0], self.xi_i, self.eta_j)
self.y_e_ij[:, element_tag] = isoparam.isoparam_y_2D(
self.nodes[element, 1], self.xi_i, self.eta_j)
self.u_e_ij = af.sin(self.x_e_ij * 2 * np.pi + self.y_e_ij * 4 * np.pi)
# Array of timesteps seperated by delta_t.
self.time_2d = utils.linspace(
0,
int(total_time_2d / self.delta_t_2d) * self.delta_t_2d,
int(total_time_2d / self.delta_t_2d))
self.sqrt_det_g = wave_equation_2d.sqrt_det_g(self.nodes[self.elements[0]][:, 0], \
self.nodes[self.elements[0]][:, 1], np.array(self.xi_i), np.array(self.eta_j))
self.elements_nodes = (af.reorder(
af.transpose(af.np_to_af_array(self.nodes[self.elements[:]])), 0,
2, 1))
self.sqrt_g = af.reorder(wave_equation_2d.trial_sqrt_det_g(self.elements_nodes[:, 0, :],\
self.elements_nodes[:, 1, :], self.xi_i, self.eta_j), 0, 2, 1)
return