def h_min_from_volume(dcoll: DiscretizationCollection,
dim=None,
dd=None) -> float:
"""Returns a (minimum) characteristic length based on the volume of the
elements. This length may not be representative if the elements have very
high aspect ratios.
:arg dim: an integer denoting topological dimension. If *None*, the
spatial dimension specified by
:attr:`grudge.DiscretizationCollection.dim` is used.
:arg dd: a :class:`~grudge.dof_desc.DOFDesc`, or a value convertible to one.
Defaults to the base volume discretization if not provided.
:returns: a scalar denoting the minimum characteristic length.
"""
from grudge.reductions import nodal_min, elementwise_sum
if dd is None:
dd = DD_VOLUME
dd = as_dofdesc(dd)
if dim is None:
dim = dcoll.dim
ones = dcoll.discr_from_dd(dd).zeros(dcoll._setup_actx) + 1.0
return nodal_min(dcoll, dd, elementwise_sum(dcoll,
op.mass(dcoll, dd,
ones)))**(1.0 / dim)
def test_mass_operator_inverse(actx_factory, name):
actx = actx_factory()
# {{{ cases
import mesh_data
if name == "2-1-ellipse":
# curve
builder = mesh_data.EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
elif name == "spheroid":
# surface
builder = mesh_data.SpheroidMeshBuilder()
elif name.startswith("warped_rect"):
builder = mesh_data.WarpedRectMeshBuilder(dim=int(name[-1]))
else:
raise ValueError("unknown geometry name: %s" % name)
# }}}
# {{{ inv(m) @ m == id
from pytools.convergence import EOCRecorder
eoc = EOCRecorder()
for resolution in builder.resolutions:
mesh = builder.get_mesh(resolution, builder.mesh_order)
dcoll = DiscretizationCollection(actx, mesh, order=builder.order)
volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume_discr.ndofs)
logger.info("nelements: %d", volume_discr.mesh.nelements)
# {{{ compute inverse mass
def f(x):
return actx.np.cos(4.0 * x[0])
dd = dof_desc.DD_VOLUME
x_volm = thaw(volume_discr.nodes(), actx)
f_volm = f(x_volm)
f_inv = op.inverse_mass(dcoll, op.mass(dcoll, dd, f_volm))
inv_error = actx.to_numpy(
op.norm(dcoll, f_volm - f_inv, 2) / op.norm(dcoll, f_volm, 2))
# }}}
# compute max element size
from grudge.dt_utils import h_max_from_volume
h_max = h_max_from_volume(dcoll)
eoc.add_data_point(h_max, inv_error)
logger.info("inverse mass error\n%s", str(eoc))
# NOTE: both cases give 1.0e-16-ish at the moment, but just to be on the
# safe side, choose a slightly larger tolerance
assert eoc.max_error() < 1.0e-14
def mass(self, *args):
return op.mass(self, *args)
def test_mass_mat_trig(actx_factory, ambient_dim, discr_tag):
"""Check the integral of some trig functions on an interval using the mass
matrix.
"""
actx = actx_factory()
nel_1d = 16
order = 4
a = -4.0 * np.pi
b = +9.0 * np.pi
true_integral = 13 * np.pi / 2 * (b - a)**(ambient_dim - 1)
from meshmode.discretization.poly_element import QuadratureSimplexGroupFactory
dd_quad = dof_desc.DOFDesc(dof_desc.DTAG_VOLUME_ALL, discr_tag)
if discr_tag is dof_desc.DISCR_TAG_BASE:
discr_tag_to_group_factory = {}
else:
discr_tag_to_group_factory = {
discr_tag: QuadratureSimplexGroupFactory(order=2 * order)
}
mesh = mgen.generate_regular_rect_mesh(a=(a, ) * ambient_dim,
b=(b, ) * ambient_dim,
nelements_per_axis=(nel_1d, ) *
ambient_dim,
order=1)
dcoll = DiscretizationCollection(
actx,
mesh,
order=order,
discr_tag_to_group_factory=discr_tag_to_group_factory)
def f(x):
return actx.np.sin(x[0])**2
volm_disc = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
x_volm = thaw(volm_disc.nodes(), actx)
f_volm = f(x_volm)
ones_volm = volm_disc.zeros(actx) + 1
quad_disc = dcoll.discr_from_dd(dd_quad)
x_quad = thaw(quad_disc.nodes(), actx)
f_quad = f(x_quad)
ones_quad = quad_disc.zeros(actx) + 1
mop_1 = op.mass(dcoll, dd_quad, f_quad)
num_integral_1 = op.nodal_sum(dcoll, dof_desc.DD_VOLUME, ones_volm * mop_1)
err_1 = abs(num_integral_1 - true_integral)
assert err_1 < 2e-9, err_1
mop_2 = op.mass(dcoll, dd_quad, ones_quad)
num_integral_2 = op.nodal_sum(dcoll, dof_desc.DD_VOLUME, f_volm * mop_2)
err_2 = abs(num_integral_2 - true_integral)
assert err_2 < 2e-9, err_2
if discr_tag is dof_desc.DISCR_TAG_BASE:
# NOTE: `integral` always makes a square mass matrix and
# `QuadratureSimplexGroupFactory` does not have a `mass_matrix` method.
num_integral_3 = op.nodal_sum(dcoll, dof_desc.DD_VOLUME,
f_quad * mop_2)
err_3 = abs(num_integral_3 - true_integral)
assert err_3 < 5e-10, err_3
def test_surface_divergence_theorem(actx_factory, mesh_name, visualize=False):
r"""Check the surface divergence theorem.
.. math::
\int_Sigma \phi \nabla_i f_i =
\int_\Sigma \nabla_i \phi f_i +
\int_\Sigma \kappa \phi f_i n_i +
\int_{\partial \Sigma} \phi f_i m_i
where :math:`n_i` is the surface normal and :class:`m_i` is the
face normal (which should be orthogonal to both the surface normal
and the face tangent).
"""
actx = actx_factory()
# {{{ cases
if mesh_name == "2-1-ellipse":
from mesh_data import EllipseMeshBuilder
builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
elif mesh_name == "spheroid":
from mesh_data import SpheroidMeshBuilder
builder = SpheroidMeshBuilder()
elif mesh_name == "circle":
from mesh_data import EllipseMeshBuilder
builder = EllipseMeshBuilder(radius=1.0, aspect_ratio=1.0)
elif mesh_name == "starfish":
from mesh_data import StarfishMeshBuilder
builder = StarfishMeshBuilder()
elif mesh_name == "sphere":
from mesh_data import SphereMeshBuilder
builder = SphereMeshBuilder(radius=1.0, mesh_order=16)
else:
raise ValueError("unknown mesh name: %s" % mesh_name)
# }}}
# {{{ convergence
def f(x):
return flat_obj_array(
actx.np.sin(3 * x[1]) + actx.np.cos(3 * x[0]) + 1.0,
actx.np.sin(2 * x[0]) + actx.np.cos(x[1]),
3.0 * actx.np.cos(x[0] / 2) + actx.np.cos(x[1]),
)[:ambient_dim]
from pytools.convergence import EOCRecorder
eoc_global = EOCRecorder()
eoc_local = EOCRecorder()
theta = np.pi / 3.33
ambient_dim = builder.ambient_dim
if ambient_dim == 2:
mesh_rotation = np.array([
[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)],
])
else:
mesh_rotation = np.array([
[1.0, 0.0, 0.0],
[0.0, np.cos(theta), -np.sin(theta)],
[0.0, np.sin(theta), np.cos(theta)],
])
mesh_offset = np.array([0.33, -0.21, 0.0])[:ambient_dim]
for i, resolution in enumerate(builder.resolutions):
from meshmode.mesh.processing import affine_map
from meshmode.discretization.connection import FACE_RESTR_ALL
mesh = builder.get_mesh(resolution, builder.mesh_order)
mesh = affine_map(mesh, A=mesh_rotation, b=mesh_offset)
from meshmode.discretization.poly_element import \
QuadratureSimplexGroupFactory
qtag = dof_desc.DISCR_TAG_QUAD
dcoll = DiscretizationCollection(actx,
mesh,
order=builder.order,
discr_tag_to_group_factory={
qtag:
QuadratureSimplexGroupFactory(
2 * builder.order)
})
volume = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
logger.info("ndofs: %d", volume.ndofs)
logger.info("nelements: %d", volume.mesh.nelements)
dd = dof_desc.DD_VOLUME
dq = dd.with_discr_tag(qtag)
df = dof_desc.as_dofdesc(FACE_RESTR_ALL)
ambient_dim = dcoll.ambient_dim
# variables
f_num = f(thaw(dcoll.nodes(dd=dd), actx))
f_quad_num = f(thaw(dcoll.nodes(dd=dq), actx))
from grudge.geometry import normal, summed_curvature
kappa = summed_curvature(actx, dcoll, dd=dq)
normal = normal(actx, dcoll, dd=dq)
face_normal = thaw(dcoll.normal(df), actx)
face_f = op.project(dcoll, dd, df, f_num)
# operators
stiff = op.mass(
dcoll,
sum(
op.local_d_dx(dcoll, i, f_num_i)
for i, f_num_i in enumerate(f_num)))
stiff_t = sum(
op.weak_local_d_dx(dcoll, i, f_num_i)
for i, f_num_i in enumerate(f_num))
kterm = op.mass(dcoll, dq, kappa * f_quad_num.dot(normal))
flux = op.face_mass(dcoll, face_f.dot(face_normal))
# sum everything up
op_global = op.nodal_sum(dcoll, dd, stiff - (stiff_t + kterm))
op_local = op.elementwise_sum(dcoll, dd,
stiff - (stiff_t + kterm + flux))
err_global = abs(op_global)
err_local = op.norm(dcoll, op_local, np.inf)
logger.info("errors: global %.5e local %.5e", err_global, err_local)
# compute max element size
from grudge.dt_utils import h_max_from_volume
h_max = h_max_from_volume(dcoll)
eoc_global.add_data_point(h_max, actx.to_numpy(err_global))
eoc_local.add_data_point(h_max, err_local)
if visualize:
from grudge.shortcuts import make_visualizer
vis = make_visualizer(dcoll)
filename = f"surface_divergence_theorem_{mesh_name}_{i:04d}.vtu"
vis.write_vtk_file(filename, [("r", actx.np.log10(op_local))],
overwrite=True)
# }}}
order = min(builder.order, builder.mesh_order) - 0.5
logger.info("\n%s", str(eoc_global))
logger.info("\n%s", str(eoc_local))
assert eoc_global.max_error() < 1.0e-12 \
or eoc_global.order_estimate() > order - 0.5
assert eoc_local.max_error() < 1.0e-12 \
or eoc_local.order_estimate() > order - 0.5