def test_projection():
"""Test whether projection between different orders works"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization import Projector
from hedge.discretization.local import TriangleDiscretization
from hedge.tools import EOCRecorder
from math import sin, pi, sqrt
from numpy import dot
a = numpy.array([1, 3])
def u_analytic(x, el):
return sin(dot(a, x))
mesh = make_disk_mesh(r=pi, max_area=0.5)
discr2 = discr_class(mesh, TriangleDiscretization(2), debug=discr_class.noninteractive_debug_flags())
discr5 = discr_class(mesh, TriangleDiscretization(5), debug=discr_class.noninteractive_debug_flags())
p2to5 = Projector(discr2, discr5)
p5to2 = Projector(discr5, discr2)
u2 = discr2.interpolate_volume_function(u_analytic)
u2_i = p5to2(p2to5(u2))
assert discr2.norm(u2 - u2_i) < 3e-15
def test_projection():
"""Test whether projection between different orders works"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization import Projector
from hedge.discretization.local import TriangleDiscretization
from math import sin, pi
from numpy import dot
a = numpy.array([1, 3])
def u_analytic(x, el):
return sin(dot(a, x))
mesh = make_disk_mesh(r=pi, max_area=0.5)
discr2 = discr_class(mesh,
TriangleDiscretization(2),
debug=discr_class.noninteractive_debug_flags())
discr5 = discr_class(mesh,
TriangleDiscretization(5),
debug=discr_class.noninteractive_debug_flags())
p2to5 = Projector(discr2, discr5)
p5to2 = Projector(discr5, discr2)
u2 = discr2.interpolate_volume_function(u_analytic)
u2_i = p5to2(p2to5(u2))
assert discr2.norm(u2 - u2_i) < 3e-15
def test_tri_mass_mat_trig():
"""Check the integral of some trig functions on a square using the mass matrix"""
from hedge.mesh.generator import make_square_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sqrt, pi, cos, sin
mesh = make_square_mesh(a=-pi, b=pi, max_area=(2 * pi / 10) ** 2 / 2)
discr = discr_class(mesh, TriangleDiscretization(8), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0]) ** 2 * sin(x[1]) ** 2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = numpy.dot(ones, mass_op.apply(discr, f))
num_integral_2 = numpy.dot(f, mass_op.apply(discr, ones))
true_integral = pi ** 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
# print err_1, err_2
assert err_1 < 1e-10
assert err_2 < 1e-10
def test_1d_mass_mat_trig():
"""Check the integral of some trig functions on an interval using the mass
matrix
"""
from hedge.mesh.generator import make_uniform_1d_mesh
from hedge.discretization.local import IntervalDiscretization
from math import pi, cos
from numpy import dot
mesh = make_uniform_1d_mesh(-4 * pi, 9 * pi, 17, periodic=True)
discr = discr_class(mesh,
IntervalDiscretization(8),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0])**2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = dot(ones, mass_op.apply(discr, f))
num_integral_2 = dot(f, mass_op.apply(discr, ones))
num_integral_3 = discr.integral(f)
true_integral = 13 * pi / 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
err_3 = abs(num_integral_3 - true_integral)
assert err_1 < 1e-10
assert err_2 < 1e-10
assert err_3 < 1e-10
def test_1d_mass_mat_trig():
"""Check the integral of some trig functions on an interval using the mass matrix"""
from hedge.mesh.generator import make_uniform_1d_mesh
from hedge.discretization.local import IntervalDiscretization
from math import sqrt, pi, cos, sin
from numpy import dot
mesh = make_uniform_1d_mesh(-4 * pi, 9 * pi, 17, periodic=True)
discr = discr_class(mesh, IntervalDiscretization(8), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: cos(x[0]) ** 2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = dot(ones, mass_op.apply(discr, f))
num_integral_2 = dot(f, mass_op.apply(discr, ones))
num_integral_3 = discr.integral(f)
true_integral = 13 * pi / 2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
err_3 = abs(num_integral_3 - true_integral)
assert err_1 < 1e-10
assert err_2 < 1e-10
assert err_3 < 1e-10
def test_tri_mass_mat_trig():
"""Check the integral of some trig functions on a square using the mass matrix"""
from hedge.mesh.generator import make_square_mesh
from hedge.discretization.local import TriangleDiscretization
from math import pi, cos, sin
mesh = make_square_mesh(a=-pi, b=pi, max_area=(2 * pi / 10)**2 / 2)
discr = discr_class(mesh,
TriangleDiscretization(8),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(
lambda x, el: cos(x[0])**2 * sin(x[1])**2)
ones = discr.interpolate_volume_function(lambda x, el: 1)
from hedge.optemplate import MassOperator
mass_op = MassOperator()
num_integral_1 = numpy.dot(ones, mass_op.apply(discr, f))
num_integral_2 = numpy.dot(f, mass_op.apply(discr, ones))
true_integral = pi**2
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
#print err_1, err_2
assert err_1 < 1e-10
assert err_2 < 1e-10
def test_tri_diff_mat():
"""Check differentiation matrix along the coordinate axes on a disk
Uses sines as the function to differentiate.
"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin, cos, sqrt
from hedge.optemplate import make_nabla
nabla = make_nabla(2)
for coord in [0, 1]:
mesh = make_disk_mesh()
discr = discr_class(make_disk_mesh(), TriangleDiscretization(4), debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: sin(3 * x[coord]))
df = discr.interpolate_volume_function(lambda x, el: 3 * cos(3 * x[coord]))
df_num = nabla[coord].apply(discr, f)
# discr.visualize_vtk("diff-err.vtk",
# [("f", f), ("df", df), ("df_num", df_num), ("error", error)])
linf_error = la.norm(df_num - df, numpy.Inf)
print linf_error
assert linf_error < 4e-5
def test_tri_diff_mat():
"""Check differentiation matrix along the coordinate axes on a disk
Uses sines as the function to differentiate.
"""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin, cos
from hedge.optemplate import make_nabla
nabla = make_nabla(2)
for coord in [0, 1]:
discr = discr_class(make_disk_mesh(),
TriangleDiscretization(4),
debug=discr_class.noninteractive_debug_flags())
f = discr.interpolate_volume_function(lambda x, el: sin(3 * x[coord]))
df = discr.interpolate_volume_function(
lambda x, el: 3 * cos(3 * x[coord]))
df_num = nabla[coord].apply(discr, f)
#discr.visualize_vtk("diff-err.vtk",
#[("f", f), ("df", df), ("df_num", df_num), ("error", error)])
linf_error = la.norm(df_num - df, numpy.Inf)
print linf_error
assert linf_error < 4e-5
def test_quadrature_tri_mass_mat_monomial():
"""Check that quadrature integration on triangles is exact as designed."""
from hedge.mesh.generator import make_square_mesh
mesh = make_square_mesh(a=-1, b=1, max_area=4 * 1 / 8 + 0.001)
order = 4
discr = discr_class(
mesh, order=order, debug=discr_class.noninteractive_debug_flags(), quad_min_degrees={"quad": 3 * order}
)
m, n = 2, 1
f = Monomial((m, n))
f_vec = discr.interpolate_volume_function(lambda x, el: f(x))
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
vis.add_data(visf, [("f", f_vec * f_vec)])
visf.close()
from hedge.optemplate import MassOperator, Field, QuadratureGridUpsampler
f_fld = Field("f")
mass_op = discr.compile(MassOperator()(f_fld * f_fld))
from hedge.optemplate.primitives import make_common_subexpression as cse
f_upsamp = cse(QuadratureGridUpsampler("quad")(f_fld))
quad_mass_op = discr.compile(MassOperator()(f_upsamp * f_upsamp))
num_integral_1 = numpy.dot(ones, mass_op(f=f_vec))
num_integral_2 = numpy.dot(ones, quad_mass_op(f=f_vec))
true_integral = 4 / ((2 * m + 1) * (2 * n + 1))
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
print num_integral_1, num_integral_2, true_integral
print err_1, err_2
assert err_1 > 1e-8
assert err_2 < 1e-14
def test_filter():
"""Exercise mode-based filtering."""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin
mesh = make_disk_mesh(r=3.4, max_area=0.5)
discr = discr_class(mesh,
TriangleDiscretization(5),
debug=discr_class.noninteractive_debug_flags())
from hedge.optemplate.operators import FilterOperator
from hedge.discretization import ExponentialFilterResponseFunction
half_filter = FilterOperator(lambda mid, ldis: 0.5)
for eg in discr.element_groups:
fmat = half_filter.matrix(eg)
n, m = fmat.shape
assert la.norm(fmat - 0.5 * numpy.eye(n, m)) < 2e-15
from numpy import dot
def test_freq(n):
a = numpy.array([1, n])
def u_analytic(x, el):
return sin(dot(a, x))
exp_filter = FilterOperator(ExponentialFilterResponseFunction(0.9, 3)) \
.bind(discr)
u = discr.interpolate_volume_function(u_analytic)
filt_u = exp_filter(u)
int_error = abs(discr.integral(u) - discr.integral(filt_u))
l2_ratio = discr.norm(filt_u) / discr.norm(u)
assert int_error < 1e-14
assert 0.96 < l2_ratio < 0.99999
test_freq(3)
test_freq(5)
test_freq(9)
test_freq(17)
def test_quadrature_tri_mass_mat_monomial():
"""Check that quadrature integration on triangles is exact as designed."""
from hedge.mesh.generator import make_square_mesh
mesh = make_square_mesh(a=-1, b=1, max_area=4 * 1 / 8 + 0.001)
order = 4
discr = discr_class(mesh,
order=order,
debug=discr_class.noninteractive_debug_flags(),
quad_min_degrees={"quad": 3 * order})
m, n = 2, 1
f = Monomial((m, n))
f_vec = discr.interpolate_volume_function(lambda x, el: f(x))
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
vis.add_data(visf, [("f", f_vec * f_vec)])
visf.close()
from hedge.optemplate import (MassOperator, Field, QuadratureGridUpsampler)
f_fld = Field("f")
mass_op = discr.compile(MassOperator()(f_fld * f_fld))
from hedge.optemplate.primitives import make_common_subexpression as cse
f_upsamp = cse(QuadratureGridUpsampler("quad")(f_fld))
quad_mass_op = discr.compile(MassOperator()(f_upsamp * f_upsamp))
num_integral_1 = numpy.dot(ones, mass_op(f=f_vec))
num_integral_2 = numpy.dot(ones, quad_mass_op(f=f_vec))
true_integral = 4 / ((2 * m + 1) * (2 * n + 1))
err_1 = abs(num_integral_1 - true_integral)
err_2 = abs(num_integral_2 - true_integral)
print num_integral_1, num_integral_2, true_integral
print err_1, err_2
assert err_1 > 1e-8
assert err_2 < 1e-14
def test_filter():
"""Exercise mode-based filtering."""
from hedge.mesh.generator import make_disk_mesh
from hedge.discretization.local import TriangleDiscretization
from math import sin
mesh = make_disk_mesh(r=3.4, max_area=0.5)
discr = discr_class(mesh, TriangleDiscretization(5),
debug=discr_class.noninteractive_debug_flags())
from hedge.optemplate.operators import FilterOperator
from hedge.discretization import ExponentialFilterResponseFunction
half_filter = FilterOperator(lambda mid, ldis: 0.5)
for eg in discr.element_groups:
fmat = half_filter.matrix(eg)
n, m = fmat.shape
assert la.norm(fmat - 0.5*numpy.eye(n, m)) < 2e-15
from numpy import dot
def test_freq(n):
a = numpy.array([1, n])
def u_analytic(x, el):
return sin(dot(a, x))
exp_filter = FilterOperator(ExponentialFilterResponseFunction(0.9, 3)) \
.bind(discr)
u = discr.interpolate_volume_function(u_analytic)
filt_u = exp_filter(u)
int_error = abs(discr.integral(u) - discr.integral(filt_u))
l2_ratio = discr.norm(filt_u) / discr.norm(u)
assert int_error < 1e-14
assert 0.96 < l2_ratio < 0.99999
test_freq(3)
test_freq(5)
test_freq(9)
test_freq(17)
def test_convergence_advec_2d():
"""Test whether 2D advection actually converges"""
import pyublas # noqa
from hedge.mesh.generator import make_disk_mesh, make_regular_rect_mesh
from hedge.discretization.local import TriangleDiscretization
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin, pi
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([0.27, 0])
norm_a = la.norm(v)
from numpy import dot
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f((-dot(v, x) / norm_a + t * norm_a))
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
for mesh in [
# non-periodic
make_disk_mesh(r=pi, boundary_tagger=boundary_tagger,
max_area=0.5),
# periodic
make_regular_rect_mesh(
a=(0, 0),
b=(2 * pi, 1),
n=(8, 4),
periodicity=(True, False),
boundary_tagger=boundary_tagger,
)
]:
for flux_type in StrongAdvectionOperator.flux_types:
eoc_rec = EOCRecorder()
for order in [1, 2, 3, 4, 5, 6]:
discr = discr_class(
mesh,
TriangleDiscretization(order),
debug=discr_class.noninteractive_debug_flags())
op = StrongAdvectionOperator(
v,
inflow_u=TimeDependentGivenFunction(u_analytic),
flux_type=flux_type)
u = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, 0))
stepper = RK4TimeStepper()
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
for step in range(nsteps):
u = stepper(u, step * dt, dt, rhs)
u_true = discr.interpolate_volume_function(
lambda x, el: u_analytic(x, el, nsteps * dt))
error = u - u_true
error_l2 = discr.norm(error)
eoc_rec.add_data_point(order, error_l2)
if False:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order",
error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 4
assert eoc_rec.estimate_order_of_convergence(2)[-1, 1] > 10
def test_symmetry_preservation_2d():
"""Test whether we preserve symmetry in a symmetric 2D advection problem"""
from numpy import dot
def make_mesh():
array = numpy.array
#
# 1---8---2
# |7 /|\ 1|
# | / | \ |
# |/ 6|0 \|
# 5---4---7
# |\ 5|3 /|
# | \ | / |
# |4 \|/ 2|
# 0---6---3
#
points = [
array([-0.5, -0.5]),
array([-0.5, 0.5]),
array([0.5, 0.5]),
array([0.5, -0.5]),
array([0.0, 0.0]),
array([-0.5, 0.0]),
array([0.0, -0.5]),
array([0.5, 0.0]),
array([0.0, 0.5])
]
elements = [
[8, 7, 4],
[8, 7, 2],
[6, 7, 3],
[7, 4, 6],
[5, 6, 0],
[5, 6, 4],
[5, 8, 4],
[1, 5, 8],
]
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
from hedge.mesh import make_conformal_mesh
return make_conformal_mesh(points, elements, boundary_tagger)
from hedge.discretization import SymmetryMap
from hedge.timestep import RK4TimeStepper
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([-1, 0])
mesh = make_mesh()
discr = discr_class(mesh,
order=4,
debug=discr_class.noninteractive_debug_flags())
#ref_discr = DynamicDiscretization(mesh, order=4)
def f(x):
if x < 0.5:
return 0
else:
return (x - 0.5)
#def f(x):
#return sin(5*x)
def u_analytic(x, el, t):
return f(-dot(v, x) + t)
u = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, 0))
sym_map = SymmetryMap(discr, lambda x: numpy.array([x[0], -x[1]]), {
0: 3,
2: 1,
5: 6,
7: 4
})
for flux_type in StrongAdvectionOperator.flux_types:
stepper = RK4TimeStepper()
op = StrongAdvectionOperator(
v,
inflow_u=TimeDependentGivenFunction(u_analytic),
flux_type=flux_type)
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
#test_comp = [ "bflux"]
#test_rhs = op.bind(discr, test_comp)
#ref_rhs = op.bind(ref_discr, test_comp)
for step in xrange(nsteps):
u = stepper(u, step * dt, dt, rhs)
sym_error_u = u - sym_map(u)
sym_error_u_l2 = discr.norm(sym_error_u)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test-%s-%04d" % (flux_type, step))
vis.add_data(
visf,
[
("u", u),
("sym_u", sym_map(u)),
("sym_diff", u - sym_map(u)),
("rhs", rhs(step * dt, u)),
#("rhs_test", test_rhs(step*dt, u)),
#("rhs_ref", ref_rhs(step*dt, u)),
#("rhs_diff", test_rhs(step*dt, u)-ref_rhs(step*dt, u)),
])
print sym_error_u_l2
assert sym_error_u_l2 < 4e-15
def test_interior_fluxes_tet():
"""Check tetrahedron surface integrals computed using interior fluxes
against their known values.
"""
import meshpy.tet as tet
from math import pi, sin, cos
mesh_info = tet.MeshInfo()
# construct a two-box extrusion of this base
base = [(-pi, -pi, 0), (pi, -pi, 0), (pi, pi, 0), (-pi, pi, 0)]
# first, the nodes
mesh_info.set_points(base + [(x, y, z + pi)
for x, y, z in base] + [(x, y, z + pi + 1)
for x, y, z in base])
# next, the facets
# vertex indices for a box missing the -z face
box_without_minus_z = [
[4, 5, 6, 7],
[0, 4, 5, 1],
[1, 5, 6, 2],
[2, 6, 7, 3],
[3, 7, 4, 0],
]
def add_to_all_vertex_indices(facets, increment):
return [[pt + increment for pt in facet] for facet in facets]
mesh_info.set_facets(
[[0, 1, 2, 3]] # base
+ box_without_minus_z # first box
+ add_to_all_vertex_indices(box_without_minus_z, 4) # second box
)
# set the volume properties -- this is where the tet size constraints are
mesh_info.regions.resize(2)
mesh_info.regions[0] = [
0,
0,
pi / 2, # point in volume -> first box
0, # region tag (user-defined number)
0.5, # max tet volume in region
]
mesh_info.regions[1] = [
0,
0,
pi + 0.5, # point in volume -> second box
1, # region tag (user-defined number, arbitrary)
0.1, # max tet volume in region
]
generated_mesh = tet.build(mesh_info,
attributes=True,
volume_constraints=True)
from hedge.mesh import make_conformal_mesh
mesh = make_conformal_mesh(generated_mesh.points, generated_mesh.elements)
from hedge.discretization.local import TetrahedronDiscretization
from hedge.discretization import ones_on_volume
discr = discr_class(mesh,
TetrahedronDiscretization(4),
debug=discr_class.noninteractive_debug_flags())
def f_u(x, el):
if generated_mesh.element_attributes[el.id] == 1:
return cos(x[0] - x[1] + x[2])
else:
return 0
def f_l(x, el):
if generated_mesh.element_attributes[el.id] == 0:
return sin(x[0] - x[1] + x[2])
else:
return 0
u_l = discr.interpolate_volume_function(f_l)
u_u = discr.interpolate_volume_function(f_u)
u = u_l + u_u
# visualize the produced field
#from hedge.visualization import SiloVisualizer
#vis = SiloVisualizer(discr)
#visf = vis.make_file("sandwich")
#vis.add_data(visf,
#[("u_l", u_l), ("u_u", u_u)],
#expressions=[("u", "u_l+u_u")])
# make sure the surface integral of the difference
# between top and bottom is zero
from hedge.flux import make_normal, FluxScalarPlaceholder
from hedge.optemplate import Field, get_flux_operator
fluxu = FluxScalarPlaceholder()
res = discr.compile(
get_flux_operator(
(fluxu.int - fluxu.ext) * make_normal(discr.dimensions)[1]) *
Field("u"))(u=u)
ones = ones_on_volume(discr)
assert abs(numpy.dot(res, ones)) < 5e-14
def test_interior_fluxes_tri():
"""Check triangle surface integrals computed using interior fluxes
against their known values.
"""
from math import pi, sin, cos
def round_trip_connect(start, end):
for i in range(start, end):
yield i, i + 1
yield end, start
a = -pi
b = pi
points = [(a, 0), (b, 0), (a, -1), (b, -1), (a, 1), (b, 1)]
import meshpy.triangle as triangle
mesh_info = triangle.MeshInfo()
mesh_info.set_points(points)
mesh_info.set_facets([(0, 1), (1, 3), (3, 2), (2, 0), (0, 4), (4, 5),
(1, 5)])
mesh_info.regions.resize(2)
mesh_info.regions[0] = [
0,
-0.5, # coordinate
1, # lower element tag
0.1, # max area
]
mesh_info.regions[1] = [
0,
0.5, # coordinate
2, # upper element tag
0.01, # max area
]
generated_mesh = triangle.build(mesh_info,
attributes=True,
volume_constraints=True)
#triangle.write_gnuplot_mesh("mesh.dat", generated_mesh)
def element_tagger(el):
if generated_mesh.element_attributes[el.id] == 1:
return ["upper"]
else:
return ["lower"]
from hedge.mesh import make_conformal_mesh
mesh = make_conformal_mesh(generated_mesh.points, generated_mesh.elements)
from hedge.discretization.local import TriangleDiscretization
from hedge.discretization import ones_on_volume
discr = discr_class(mesh,
TriangleDiscretization(4),
debug=discr_class.noninteractive_debug_flags())
def f_u(x, el):
if generated_mesh.element_attributes[el.id] == 1:
return cos(x[0] - x[1])
else:
return 0
def f_l(x, el):
if generated_mesh.element_attributes[el.id] == 0:
return sin(x[0] - x[1])
else:
return 0
# u_l = discr.interpolate_volume_function(f_l)
u_u = discr.interpolate_volume_function(f_u)
u = u_u + u_u
#discr.visualize_vtk("dual.vtk", [("u", u)])
from hedge.flux import make_normal, FluxScalarPlaceholder
from hedge.optemplate import Field, get_flux_operator
fluxu = FluxScalarPlaceholder()
res = discr.compile(
get_flux_operator(
(fluxu.int - fluxu.ext) * make_normal(discr.dimensions)[1]) *
Field("u"))(u=u)
ones = ones_on_volume(discr)
err = abs(numpy.dot(res, ones))
#print err
assert err < 5e-14
def test_2d_gauss_theorem():
"""Verify Gauss's theorem explicitly on a mesh"""
from hedge.mesh.generator import make_disk_mesh
from math import sin, cos, sqrt, exp, pi
from numpy import dot
mesh = make_disk_mesh()
order = 2
discr = discr_class(mesh, order=order, debug=discr_class.noninteractive_debug_flags())
ref_discr = discr_class(mesh, order=order)
from hedge.flux import make_normal, FluxScalarPlaceholder
normal = make_normal(discr.dimensions)
flux_f_ph = FluxScalarPlaceholder(0)
one_sided_x = flux_f_ph.int * normal[0]
one_sided_y = flux_f_ph.int * normal[1]
def f1(x, el):
return sin(3 * x[0]) + cos(3 * x[1])
def f2(x, el):
return sin(2 * x[0]) + cos(x[1])
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
f1_v = discr.interpolate_volume_function(f1)
f2_v = discr.interpolate_volume_function(f2)
from hedge.optemplate import BoundaryPair, Field, make_nabla, get_flux_operator
nabla = make_nabla(discr.dimensions)
diff_optp = nabla[0] * Field("f1") + nabla[1] * Field("f2")
divergence = nabla[0].apply(discr, f1_v) + nabla[1].apply(discr, f2_v)
int_div = discr.integral(divergence)
flux_optp = get_flux_operator(one_sided_x)(BoundaryPair(Field("f1"), Field("fz"))) + get_flux_operator(one_sided_y)(
BoundaryPair(Field("f2"), Field("fz"))
)
from hedge.mesh import TAG_ALL
bdry_val = discr.compile(flux_optp)(f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
ref_bdry_val = ref_discr.compile(flux_optp)(f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
boundary_int = dot(bdry_val, ones)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
from hedge.tools import make_obj_array
from hedge.mesh import TAG_ALL
vis.add_data(
visf,
[
("bdry", bdry_val),
("ref_bdry", ref_bdry_val),
("div", divergence),
("f", make_obj_array([f1_v, f2_v])),
("n", discr.volumize_boundary_field(discr.boundary_normals(TAG_ALL), TAG_ALL)),
],
expressions=[("bdiff", "bdry-ref_bdry")],
)
# print abs(boundary_int-int_div)
assert abs(boundary_int - int_div) < 5e-15
def test_interior_fluxes_tri():
"""Check triangle surface integrals computed using interior fluxes
against their known values.
"""
from math import pi, sin, cos
def round_trip_connect(start, end):
for i in range(start, end):
yield i, i + 1
yield end, start
a = -pi
b = pi
points = [(a, 0), (b, 0), (a, -1), (b, -1), (a, 1), (b, 1)]
import meshpy.triangle as triangle
mesh_info = triangle.MeshInfo()
mesh_info.set_points(points)
mesh_info.set_facets([(0, 1), (1, 3), (3, 2), (2, 0), (0, 4), (4, 5), (1, 5)])
mesh_info.regions.resize(2)
mesh_info.regions[0] = [0, -0.5, 1, 0.1] # coordinate # lower element tag # max area
mesh_info.regions[1] = [0, 0.5, 2, 0.01] # coordinate # upper element tag # max area
generated_mesh = triangle.build(mesh_info, attributes=True, volume_constraints=True)
# triangle.write_gnuplot_mesh("mesh.dat", generated_mesh)
def element_tagger(el):
if generated_mesh.element_attributes[el.id] == 1:
return ["upper"]
else:
return ["lower"]
from hedge.mesh import make_conformal_mesh
mesh = make_conformal_mesh(generated_mesh.points, generated_mesh.elements)
from hedge.discretization.local import TriangleDiscretization
from hedge.discretization import ones_on_volume
discr = discr_class(mesh, TriangleDiscretization(4), debug=discr_class.noninteractive_debug_flags())
def f_u(x, el):
if generated_mesh.element_attributes[el.id] == 1:
return cos(x[0] - x[1])
else:
return 0
def f_l(x, el):
if generated_mesh.element_attributes[el.id] == 0:
return sin(x[0] - x[1])
else:
return 0
u_l = discr.interpolate_volume_function(f_l)
u_u = discr.interpolate_volume_function(f_u)
u = u_u + u_u
# discr.visualize_vtk("dual.vtk", [("u", u)])
from hedge.flux import make_normal, FluxScalarPlaceholder
from hedge.optemplate import Field, get_flux_operator
fluxu = FluxScalarPlaceholder()
res = discr.compile(get_flux_operator((fluxu.int - fluxu.ext) * make_normal(discr.dimensions)[1]) * Field("u"))(u=u)
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
err = abs(numpy.dot(res, ones))
# print err
assert err < 5e-14
def test_2d_gauss_theorem():
"""Verify Gauss's theorem explicitly on a mesh"""
from hedge.mesh.generator import make_disk_mesh
from math import sin, cos
from numpy import dot
mesh = make_disk_mesh()
order = 2
discr = discr_class(mesh,
order=order,
debug=discr_class.noninteractive_debug_flags())
ref_discr = discr_class(mesh, order=order)
from hedge.flux import make_normal, FluxScalarPlaceholder
normal = make_normal(discr.dimensions)
flux_f_ph = FluxScalarPlaceholder(0)
one_sided_x = flux_f_ph.int * normal[0]
one_sided_y = flux_f_ph.int * normal[1]
def f1(x, el):
return sin(3 * x[0]) + cos(3 * x[1])
def f2(x, el):
return sin(2 * x[0]) + cos(x[1])
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
f1_v = discr.interpolate_volume_function(f1)
f2_v = discr.interpolate_volume_function(f2)
from hedge.optemplate import BoundaryPair, Field, make_nabla, \
get_flux_operator
nabla = make_nabla(discr.dimensions)
divergence = nabla[0].apply(discr, f1_v) + nabla[1].apply(discr, f2_v)
int_div = discr.integral(divergence)
flux_optp = (
get_flux_operator(one_sided_x)(BoundaryPair(Field("f1"),
Field("fz"))) +
get_flux_operator(one_sided_y)(BoundaryPair(Field("f2"), Field("fz"))))
from hedge.mesh import TAG_ALL
bdry_val = discr.compile(flux_optp)(f1=f1_v,
f2=f2_v,
fz=discr.boundary_zeros(TAG_ALL))
ref_bdry_val = ref_discr.compile(flux_optp)(
f1=f1_v, f2=f2_v, fz=discr.boundary_zeros(TAG_ALL))
boundary_int = dot(bdry_val, ones)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test")
from hedge.tools import make_obj_array
from hedge.mesh import TAG_ALL
vis.add_data(visf, [
("bdry", bdry_val),
("ref_bdry", ref_bdry_val),
("div", divergence),
("f", make_obj_array([f1_v, f2_v])),
("n",
discr.volumize_boundary_field(discr.boundary_normals(TAG_ALL),
TAG_ALL)),
],
expressions=[("bdiff", "bdry-ref_bdry")])
#print abs(boundary_int-int_div)
assert abs(boundary_int - int_div) < 5e-15
def test_interior_fluxes_tet():
"""Check tetrahedron surface integrals computed using interior fluxes
against their known values.
"""
import meshpy.tet as tet
from math import pi, sin, cos
mesh_info = tet.MeshInfo()
# construct a two-box extrusion of this base
base = [(-pi, -pi, 0), (pi, -pi, 0), (pi, pi, 0), (-pi, pi, 0)]
# first, the nodes
mesh_info.set_points(base + [(x, y, z + pi) for x, y, z in base] + [(x, y, z + pi + 1) for x, y, z in base])
# next, the facets
# vertex indices for a box missing the -z face
box_without_minus_z = [[4, 5, 6, 7], [0, 4, 5, 1], [1, 5, 6, 2], [2, 6, 7, 3], [3, 7, 4, 0]]
def add_to_all_vertex_indices(facets, increment):
return [[pt + increment for pt in facet] for facet in facets]
mesh_info.set_facets(
[[0, 1, 2, 3]] # base
+ box_without_minus_z # first box
+ add_to_all_vertex_indices(box_without_minus_z, 4) # second box
)
# set the volume properties -- this is where the tet size constraints are
mesh_info.regions.resize(2)
mesh_info.regions[0] = [
0,
0,
pi / 2, # point in volume -> first box
0, # region tag (user-defined number)
0.5, # max tet volume in region
]
mesh_info.regions[1] = [
0,
0,
pi + 0.5, # point in volume -> second box
1, # region tag (user-defined number, arbitrary)
0.1, # max tet volume in region
]
generated_mesh = tet.build(mesh_info, attributes=True, volume_constraints=True)
from hedge.mesh import make_conformal_mesh
mesh = make_conformal_mesh(generated_mesh.points, generated_mesh.elements)
from hedge.discretization.local import TetrahedronDiscretization
from hedge.discretization import ones_on_volume
discr = discr_class(mesh, TetrahedronDiscretization(4), debug=discr_class.noninteractive_debug_flags())
def f_u(x, el):
if generated_mesh.element_attributes[el.id] == 1:
return cos(x[0] - x[1] + x[2])
else:
return 0
def f_l(x, el):
if generated_mesh.element_attributes[el.id] == 0:
return sin(x[0] - x[1] + x[2])
else:
return 0
u_l = discr.interpolate_volume_function(f_l)
u_u = discr.interpolate_volume_function(f_u)
u = u_l + u_u
# visualize the produced field
# from hedge.visualization import SiloVisualizer
# vis = SiloVisualizer(discr)
# visf = vis.make_file("sandwich")
# vis.add_data(visf,
# [("u_l", u_l), ("u_u", u_u)],
# expressions=[("u", "u_l+u_u")])
# make sure the surface integral of the difference
# between top and bottom is zero
from hedge.flux import make_normal, FluxScalarPlaceholder
from hedge.optemplate import Field, get_flux_operator
fluxu = FluxScalarPlaceholder()
res = discr.compile(get_flux_operator((fluxu.int - fluxu.ext) * make_normal(discr.dimensions)[1]) * Field("u"))(u=u)
from hedge.discretization import ones_on_volume
ones = ones_on_volume(discr)
assert abs(numpy.dot(res, ones)) < 5e-14
def test_convergence_advec_2d():
"""Test whether 2D advection actually converges"""
import pyublas
from hedge.mesh.generator import make_disk_mesh, make_regular_rect_mesh
from hedge.discretization.local import TriangleDiscretization
from hedge.timestep import RK4TimeStepper
from hedge.tools import EOCRecorder
from math import sin, pi, sqrt
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([0.27, 0])
norm_a = la.norm(v)
from numpy import dot
def f(x):
return sin(x)
def u_analytic(x, el, t):
return f((-dot(v, x) / norm_a + t * norm_a))
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
for mesh in [
# non-periodic
make_disk_mesh(r=pi, boundary_tagger=boundary_tagger, max_area=0.5),
# periodic
make_regular_rect_mesh(
a=(0, 0), b=(2 * pi, 1), n=(8, 4), periodicity=(True, False), boundary_tagger=boundary_tagger
),
]:
for flux_type in StrongAdvectionOperator.flux_types:
eoc_rec = EOCRecorder()
for order in [1, 2, 3, 4, 5, 6]:
discr = discr_class(mesh, TriangleDiscretization(order), debug=discr_class.noninteractive_debug_flags())
op = StrongAdvectionOperator(v, inflow_u=TimeDependentGivenFunction(u_analytic), flux_type=flux_type)
u = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, 0))
stepper = RK4TimeStepper()
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
for step in range(nsteps):
u = stepper(u, step * dt, dt, rhs)
u_true = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, nsteps * dt))
error = u - u_true
error_l2 = discr.norm(error)
eoc_rec.add_data_point(order, error_l2)
if False:
print "%s\n%s\n" % (flux_type.upper(), "-" * len(flux_type))
print eoc_rec.pretty_print(abscissa_label="Poly. Order", error_label="L2 Error")
assert eoc_rec.estimate_order_of_convergence()[0, 1] > 4
assert eoc_rec.estimate_order_of_convergence(2)[-1, 1] > 10
def test_symmetry_preservation_2d():
"""Test whether we preserve symmetry in a symmetric 2D advection problem"""
from numpy import dot
def make_mesh():
array = numpy.array
#
# 1---8---2
# |7 /|\ 1|
# | / | \ |
# |/ 6|0 \|
# 5---4---7
# |\ 5|3 /|
# | \ | / |
# |4 \|/ 2|
# 0---6---3
#
points = [
array([-0.5, -0.5]),
array([-0.5, 0.5]),
array([0.5, 0.5]),
array([0.5, -0.5]),
array([0.0, 0.0]),
array([-0.5, 0.0]),
array([0.0, -0.5]),
array([0.5, 0.0]),
array([0.0, 0.5]),
]
elements = [[8, 7, 4], [8, 7, 2], [6, 7, 3], [7, 4, 6], [5, 6, 0], [5, 6, 4], [5, 8, 4], [1, 5, 8]]
def boundary_tagger(vertices, el, face_nr, all_v):
if dot(el.face_normals[face_nr], v) < 0:
return ["inflow"]
else:
return ["outflow"]
from hedge.mesh import make_conformal_mesh
return make_conformal_mesh(points, elements, boundary_tagger)
from hedge.discretization import SymmetryMap
from hedge.discretization.local import TriangleDiscretization
from hedge.timestep import RK4TimeStepper
from math import sqrt, sin
from hedge.models.advection import StrongAdvectionOperator
from hedge.data import TimeDependentGivenFunction
v = numpy.array([-1, 0])
mesh = make_mesh()
discr = discr_class(mesh, order=4, debug=discr_class.noninteractive_debug_flags())
# ref_discr = DynamicDiscretization(mesh, order=4)
def f(x):
if x < 0.5:
return 0
else:
return x - 0.5
# def f(x):
# return sin(5*x)
def u_analytic(x, el, t):
return f(-dot(v, x) + t)
u = discr.interpolate_volume_function(lambda x, el: u_analytic(x, el, 0))
sym_map = SymmetryMap(discr, lambda x: numpy.array([x[0], -x[1]]), {0: 3, 2: 1, 5: 6, 7: 4})
for flux_type in StrongAdvectionOperator.flux_types:
stepper = RK4TimeStepper()
op = StrongAdvectionOperator(v, inflow_u=TimeDependentGivenFunction(u_analytic), flux_type=flux_type)
dt = op.estimate_timestep(discr, stepper=stepper)
nsteps = int(1 / dt)
rhs = op.bind(discr)
# test_comp = [ "bflux"]
# test_rhs = op.bind(discr, test_comp)
# ref_rhs = op.bind(ref_discr, test_comp)
for step in xrange(nsteps):
u = stepper(u, step * dt, dt, rhs)
sym_error_u = u - sym_map(u)
sym_error_u_l2 = discr.norm(sym_error_u)
if False:
from hedge.visualization import SiloVisualizer
vis = SiloVisualizer(discr)
visf = vis.make_file("test-%s-%04d" % (flux_type, step))
vis.add_data(
visf,
[
("u", u),
("sym_u", sym_map(u)),
("sym_diff", u - sym_map(u)),
("rhs", rhs(step * dt, u)),
("rhs_test", test_rhs(step * dt, u)),
("rhs_ref", ref_rhs(step * dt, u)),
("rhs_diff", test_rhs(step * dt, u) - ref_rhs(step * dt, u)),
],
)
print sym_error_u_l2
assert sym_error_u_l2 < 4e-15