def test_tensor_product_nodes(dim):
nnodes = 10
nodes_1d = np.arange(nnodes)
nodes = nd.tensor_product_nodes(dim, nodes_1d)
assert np.allclose(nodes[0],
np.array(nodes_1d.tolist() * nnodes**(dim - 1)))
def test_tensor_product_nodes(dim):
from modepy.nodes import tensor_product_nodes
nnodes = 10
nodes_1d = np.arange(nnodes)
nodes = tensor_product_nodes(dim, nodes_1d)
assert np.allclose(nodes[-1],
np.array(nodes_1d.tolist() * nnodes**(dim - 1)))
def __init__(self, quads):
"""
:arg quad: a :class:`tuple` of :class:`Quadrature` for one-dimensional
intervals, one for each dimension of the tensor product.
"""
from modepy.nodes import tensor_product_nodes
x = tensor_product_nodes([quad.nodes for quad in quads])
w = np.prod(tensor_product_nodes([quad.weights for quad in quads]),
axis=0)
assert w.size == x.shape[1]
try:
exact_to = min(quad.exact_to for quad in quads)
except AttributeError:
# e.g. FejerQuadrature does not have any 'exact_to'
exact_to = None
super().__init__(x, w, exact_to=exact_to)
def unit_nodes(self):
from modepy.nodes import tensor_product_nodes
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
return tensor_product_nodes(self.dim,
legendre_gauss_lobatto_nodes(self.order))
def make_group_from_vertices(vertices, vertex_indices, order,
group_factory=None):
# shape: (dim, nelements, nvertices)
el_vertices = vertices[:, vertex_indices]
from meshmode.mesh import SimplexElementGroup, TensorProductElementGroup
if group_factory is None:
group_factory = SimplexElementGroup
if issubclass(group_factory, SimplexElementGroup):
el_origins = el_vertices[:, :, 0][:, :, np.newaxis]
# ambient_dim, nelements, nspan_vectors
spanning_vectors = (
el_vertices[:, :, 1:] - el_origins)
nspan_vectors = spanning_vectors.shape[-1]
dim = nspan_vectors
# dim, nunit_nodes
if dim <= 3:
unit_nodes = mp.warp_and_blend_nodes(dim, order)
else:
unit_nodes = mp.equidistant_nodes(dim, order)
unit_nodes_01 = 0.5 + 0.5*unit_nodes
nodes = np.einsum(
"si,des->dei",
unit_nodes_01, spanning_vectors) + el_origins
elif issubclass(group_factory, TensorProductElementGroup):
nelements, nvertices = vertex_indices.shape
dim = 0
while True:
if nvertices == 2**dim:
break
if nvertices < 2**dim:
raise ValueError("invalid number of vertices for tensor-product "
"elements, must be power of two")
dim += 1
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
from modepy.nodes import tensor_product_nodes
unit_nodes = tensor_product_nodes(dim, legendre_gauss_lobatto_nodes(order))
# shape: (dim, nnodes)
unit_nodes_01 = 0.5 + 0.5*unit_nodes
_, nnodes = unit_nodes.shape
from pytools import generate_nonnegative_integer_tuples_below as gnitb
id_tuples = list(gnitb(2, dim))
assert len(id_tuples) == nvertices
vdm = np.empty((nvertices, nvertices))
for i, vertex_tuple in enumerate(id_tuples):
for j, func_tuple in enumerate(id_tuples):
vertex_ref = np.array(vertex_tuple, dtype=np.float64)
vdm[i, j] = np.prod(vertex_ref**func_tuple)
# shape: (dim, nelements, nvertices)
coeffs = np.empty((dim, nelements, nvertices))
for d in range(dim):
coeffs[d] = la.solve(vdm, el_vertices[d].T).T
vdm_nodes = np.zeros((nnodes, nvertices))
for j, func_tuple in enumerate(id_tuples):
vdm_nodes[:, j] = np.prod(
unit_nodes_01 ** np.array(func_tuple).reshape(-1, 1),
axis=0)
nodes = np.einsum("ij,dej->dei", vdm_nodes, coeffs)
else:
raise ValueError("unsupported value for 'group_factory': %s"
% group_factory)
# make contiguous
nodes = nodes.copy()
return group_factory(
order, vertex_indices, nodes,
unit_nodes=unit_nodes)
def unit_nodes(self):
from modepy.nodes import tensor_product_nodes, equidistant_nodes
return tensor_product_nodes(self.dim, equidistant_nodes(1, self.order))
def test_nonhomogeneous_tensor_product_nodes(dim):
nnodes = (3, 7, 5, 4)[:dim]
nodes = nd.tensor_product_nodes([np.arange(n) for n in nnodes])
assert np.allclose(nodes[0],
list(range(nnodes[-1])) * int(np.prod(nnodes[:-1])))
def unit_nodes(self):
from modepy.nodes import tensor_product_nodes
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
return tensor_product_nodes(
self.dim, legendre_gauss_lobatto_nodes(self.order))
def _(shape: Hypercube):
from modepy.nodes import tensor_product_nodes
return tensor_product_nodes(shape.dim, np.array([-1.0, 1.0]))
def make_group_from_vertices(vertices, vertex_indices, order,
group_factory=None):
# shape: (dim, nelements, nvertices)
el_vertices = vertices[:, vertex_indices]
from meshmode.mesh import SimplexElementGroup, TensorProductElementGroup
if group_factory is None:
group_factory = SimplexElementGroup
if issubclass(group_factory, SimplexElementGroup):
el_origins = el_vertices[:, :, 0][:, :, np.newaxis]
# ambient_dim, nelements, nspan_vectors
spanning_vectors = (
el_vertices[:, :, 1:] - el_origins)
nspan_vectors = spanning_vectors.shape[-1]
dim = nspan_vectors
# dim, nunit_nodes
if dim <= 3:
unit_nodes = mp.warp_and_blend_nodes(dim, order)
else:
unit_nodes = mp.equidistant_nodes(dim, order)
unit_nodes_01 = 0.5 + 0.5*unit_nodes
nodes = np.einsum(
"si,des->dei",
unit_nodes_01, spanning_vectors) + el_origins
elif issubclass(group_factory, TensorProductElementGroup):
nelements, nvertices = vertex_indices.shape
dim = 0
while True:
if nvertices == 2**dim:
break
if nvertices < 2**dim:
raise ValueError("invalid number of vertices for tensor-product "
"elements, must be power of two")
dim += 1
from modepy.quadrature.jacobi_gauss import legendre_gauss_lobatto_nodes
from modepy.nodes import tensor_product_nodes
unit_nodes = tensor_product_nodes(dim, legendre_gauss_lobatto_nodes(order))
# shape: (dim, nnodes)
unit_nodes_01 = 0.5 + 0.5*unit_nodes
_, nnodes = unit_nodes.shape
from pytools import generate_nonnegative_integer_tuples_below as gnitb
id_tuples = list(gnitb(2, dim))
assert len(id_tuples) == nvertices
vdm = np.empty((nvertices, nvertices))
for i, vertex_tuple in enumerate(id_tuples):
for j, func_tuple in enumerate(id_tuples):
vertex_ref = np.array(vertex_tuple, dtype=np.float64)
vdm[i, j] = np.prod(vertex_ref**func_tuple)
# shape: (dim, nelements, nvertices)
coeffs = np.empty((dim, nelements, nvertices))
for d in range(dim):
coeffs[d] = la.solve(vdm, el_vertices[d].T).T
vdm_nodes = np.zeros((nnodes, nvertices))
for j, func_tuple in enumerate(id_tuples):
vdm_nodes[:, j] = np.prod(
unit_nodes_01 ** np.array(func_tuple).reshape(-1, 1),
axis=0)
nodes = np.einsum("ij,dej->dei", vdm_nodes, coeffs)
else:
raise ValueError("unsupported value for 'group_factory': %s"
% group_factory)
# make contiguous
nodes = nodes.copy()
return group_factory(
order, vertex_indices, nodes,
unit_nodes=unit_nodes)