def create_cell_circumcenters_and_volumes(self):
'''Computes the center of the circumsphere of each cell.
'''
cell_coords = self.node_coords[self.cells['nodes']]
a = cell_coords[:, 1, :] - cell_coords[:, 0, :]
b = cell_coords[:, 2, :] - cell_coords[:, 0, :]
c = cell_coords[:, 3, :] - cell_coords[:, 0, :]
omega = _row_dot(a, numpy.cross(b, c))
self.cell_circumcenters = cell_coords[:, 0, :] + (
numpy.cross(b, c) * _row_dot(a, a)[:, None] +
numpy.cross(c, a) * _row_dot(b, b)[:, None] +
numpy.cross(a, b) * _row_dot(c, c)[:, None]
) / (2.0 * omega[:, None])
# https://en.wikipedia.org/wiki/Tetrahedron#Volume
self.cell_volumes = abs(omega) / 6.0
return
def compute_control_volumes(self):
'''Compute the control volumes of all nodes in the mesh.
'''
self.control_volumes = numpy.zeros(len(self.node_coords), dtype=float)
# 1/3. * (0.5 * edge_length) * covolume
# = 1/6 * edge_length**2 * ce_ratio_edge_ratio
e = self.node_coords[self.edges['nodes'][:, 1]] - \
self.node_coords[self.edges['nodes'][:, 0]]
vals = _row_dot(e, e) * self.ce_ratios / 6.0
edge_nodes = self.edges['nodes']
numpy.add.at(self.control_volumes, edge_nodes[:, 0], vals)
numpy.add.at(self.control_volumes, edge_nodes[:, 1], vals)
return
def compute_ce_ratios_algebraic(self):
# Precompute edges.
edges = \
self.node_coords[self.edges['nodes'][:, 1]] - \
self.node_coords[self.edges['nodes'][:, 0]]
# create cells -> edges
num_cells = len(self.cells['nodes'])
cells_edges = numpy.empty((num_cells, 6), dtype=int)
for cell_id, face_ids in enumerate(self.cells['faces']):
edges_set = set(self.faces['edges'][face_ids].flatten())
cells_edges[cell_id] = list(edges_set)
self.cells['edges'] = cells_edges
# Build the equation system:
# The equation
#
# |simplex| ||u||^2 = \sum_i \alpha_i <u,e_i> <e_i,u>
#
# has to hold for all vectors u in the plane spanned by the edges,
# particularly by the edges themselves.
cells_edges = edges[self.cells['edges']]
# <http://stackoverflow.com/a/38110345/353337>
A = numpy.einsum('ijk,ilk->ijl', cells_edges, cells_edges)
A = A**2
# Compute the RHS cell_volume * <edge, edge>.
# The dot product <edge, edge> is also on the diagonals of A (before
# squaring), but simply computing it again is cheaper than extracting
# it from A.
edge_dot_edge = _row_dot(edges, edges)
rhs = edge_dot_edge[self.cells['edges']] * self.cell_volumes[..., None]
# Solve all k-by-k systems at once ("broadcast"). (`k` is the number of
# edges per simplex here.)
# If the matrix A is (close to) singular if and only if the cell is
# (close to being) degenerate. Hence, it has volume 0, and so all the
# edge coefficients are 0, too. Hence, do nothing.
sol = numpy.linalg.solve(A, rhs)
return sol
def compute_curl(self, vector_field):
'''Computes the curl of a vector field. While the vector field is
point-based, the curl will be cell-based. The approximation is based on
.. math::
n\cdot curl(F) = \lim_{A\\to 0} |A|^{-1} \int_{dGamma} F dr;
see <https://en.wikipedia.org/wiki/Curl_(mathematics)>. Actually, to
approximate the integral, one would only need the projection of the
vector field onto the edges at the midpoint of the edges.
'''
edge_coords = \
self.node_coords[self.edges['nodes'][:, 1]] - \
self.node_coords[self.edges['nodes'][:, 0]]
barycenters = 1./3. * numpy.sum(
self.node_coords[self.cells['nodes']],
axis=1
)
# Compute the projection of A on the edge at each edge midpoint.
nodes = self.edges['nodes']
x = self.node_coords[nodes]
A = 0.5 * numpy.sum(vector_field[nodes], axis=1)
edge_dot_A = _row_dot(edge_coords, A)
directions = numpy.cross(
x[self.cells['edges'], 0] - barycenters[:, None, :],
x[self.cells['edges'], 1] - barycenters[:, None, :]
)
dir_nrms = numpy.sqrt(numpy.sum(directions**2, axis=2))
directions /= dir_nrms[..., None]
# a: directions scaled with edge_dot_a
a = directions * edge_dot_A[self.cells['edges']][..., None]
# sum over all local edges
curl = numpy.sum(a, axis=1)
# Divide by cell volumes
curl /= self.cell_volumes[..., None]
return curl