def _build_eval_matrix(V, points):
"""Build the sparse m-by-n matrix that maps a coefficient set for a function in
V to the values of that function at m given points."""
# See <https://www.allanswered.com/post/lkbkm/#zxqgk>
mesh = V.mesh()
bbt = BoundingBoxTree()
bbt.build(mesh)
dofmap = V.dofmap()
el = V.element()
sdim = el.space_dimension()
rows = []
cols = []
data = []
for i, x in enumerate(points):
cell_id = bbt.compute_first_entity_collision(Point(*x))
cell = Cell(mesh, cell_id)
coordinate_dofs = cell.get_vertex_coordinates()
rows.append(np.full(sdim, i))
cols.append(dofmap.cell_dofs(cell_id))
v = el.evaluate_basis_all(x, coordinate_dofs, cell_id)
data.append(v)
rows = np.concatenate(rows)
cols = np.concatenate(cols)
data = np.concatenate(data)
m = len(points)
n = V.dim()
matrix = sparse.csr_matrix((data, (rows, cols)), shape=(m, n))
return matrix
class FEBasisFunction(object):
'''Evaluator of dof of V on functions'''
def __init__(self, V):
self.elm = V.element()
self.mesh = V.mesh()
shape = V.ufl_element().value_shape()
degree = V.ufl_element().degree()
# A fake instanc to talk to with the world
adapter = type('MiroHack',
(Expression, ),
{'value_shape': lambda self_, : shape,
'eval': lambda self_, values, x: self.eval(values, x)})
self.__adapter = adapter(degree=degree)
is_1d_in_3d = self.mesh.topology().dim() == 1 and self.mesh.geometry().dim() == 3
self.orient_cell = cell_orientation(is_1d_in_3d)
# Allocs
self.__cell = Cell(self.mesh, 0)
self.__cell_vertex_x = self.__cell.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(self.__cell)
self.__dof = 0
self.__values = np.zeros(V.ufl_element().value_size())
@property
def dof(self):
return self.__dof
@dof.setter
def dof(self, value):
assert value < self.elm.space_dimension()
self.__dof = value
@property
def cell(self):
return self.__cell
@cell.setter
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def eval(self, values, x):
self.elm.evaluate_basis(self.dof,
values[:],
x,
self.__cell_vertex_x,
self.__cell_orientation)
def __call__(self, x):
self.eval(self.__values, x)
return 1*self.__values
def as_expression(self):
return self.__adapter
class FEBasisFunction(object):
'''Evaluator of dof of V on functions'''
def __init__(self, V):
self.elm = V.element()
self.mesh = V.mesh()
shape = V.ufl_element().value_shape()
degree = V.ufl_element().degree()
# A fake instanc to talk to with the world
adapter = type(
'MiroHack', (Expression, ), {
'value_shape': lambda self_, : shape,
'eval': lambda self_, values, x: self.eval(values, x)
})
self.__adapter = adapter(degree=degree)
is_1d_in_3d = self.mesh.topology().dim() == 1 and self.mesh.geometry(
).dim() == 3
self.orient_cell = cell_orientation(is_1d_in_3d)
# Allocs
self.__cell = Cell(self.mesh, 0)
self.__cell_vertex_x = self.__cell.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(self.__cell)
self.__dof = 0
self.__values = np.zeros(V.ufl_element().value_size())
@property
def dof(self):
return self.__dof
@dof.setter
def dof(self, value):
assert value < self.elm.space_dimension()
self.__dof = value
@property
def cell(self):
return self.__cell
@cell.setter
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def eval(self, values, x):
self.elm.evaluate_basis(self.dof, values[:], x, self.__cell_vertex_x,
self.__cell_orientation)
def __call__(self, x):
self.eval(self.__values, x)
return 1 * self.__values
def as_expression(self):
return self.__adapter
def __init__(self, u, locations, t0=0.0, record=''):
# The idea here is that u(x) means: search for cell containing x,
# evaluate the basis functions of that element at x, restrict
# the coef vector of u to the cell. Of these 3 steps the first
# two don't change. So we cache them
# Check the scalar assumption
assert u.value_rank() == 0 and u.value_size() == 1
# Locate each point
mesh = u.function_space().mesh()
limit = mesh.num_entities_global(mesh.topology().dim())
bbox_tree = mesh.bounding_box_tree()
cells_for_x = [None] * len(locations)
for i, x in enumerate(locations):
cell = bbox_tree.compute_first_entity_collision(Point(*x))
if -1 < cell < limit:
cells_for_x[i] = cell
# Ignore the cells that are not in the mesh. Note that we don't
# care if a node is found in several cells -l think CPU interface
xs_cells = filter(lambda (xi, c): c is not None,
zip(locations, cells_for_x))
V = u.function_space()
element = V.dolfin_element()
coefficients = np.zeros(element.space_dimension())
# I build a series of functions bound to right variables that
# when called compute the value at x
evals = []
locations = []
for x, ci in xs_cells:
basis_matrix = np.zeros(element.space_dimension())
cell = Cell(mesh, ci)
vertex_coords, orientation = cell.get_vertex_coordinates(
), cell.orientation()
# Eval the basis once
element.evaluate_basis_all(basis_matrix, x, vertex_coords,
orientation)
def foo(A=basis_matrix, cell=cell, vc=vertex_coords):
# Restrict for each call using the bound cell, vc ...
u.restrict(coefficients, element, cell, vc, cell)
# A here is bound to the right basis_matri
return np.dot(A, coefficients)
evals.append(foo)
locations.append(x)
self.probes = evals
self.locations = locations
self.rank = MPI.rank(mesh.mpi_comm())
self.data = []
self.record = record
# Make the initial record
self.probe(t=t0)
class DegreeOfFreedom(object):
'''Evaluator of dof of V on functions'''
def __init__(self, V):
self.elm = V.element()
self.mesh = V.mesh()
is_1d_in_3d = self.mesh.topology().dim() == 1 and self.mesh.geometry().dim() == 3
self.orient_cell = cell_orientation(is_1d_in_3d)
# Allocs
self.__cell = Cell(self.mesh, 0)
self.__cell_vertex_x = self.__cell.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(self.__cell)
self.__dof = 0
@property
def dof(self):
return self.__dof
@dof.setter
def dof(self, value):
assert value < self.elm.space_dimension()
self.__dof = value
@property
def cell(self):
return self.__cell
@cell.setter
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def eval(self, f):
return self.elm.evaluate_dofs(f,
self.__cell_vertex_x,
self.__cell_orientation,
self.__cell)[self.dof]
def eval_dofs(self, f):
return self.elm.evaluate_dofs(f,
self.__cell_vertex_x,
self.__cell_orientation,
self.__cell)
class FEBasisFunction(UserExpression):
'''Evaluator of dof of V on functions'''
def __init__(self, V, **kwargs):
super().__init__(self, element=V.ufl_element(), **kwargs)
self.elm = V.element()
self.mesh = V.mesh()
is_1d_in_3d = self.mesh.topology().dim() == 1 and self.mesh.geometry().dim() == 3
self.orient_cell = cell_orientation(is_1d_in_3d)
# Allocs
self.__cell = Cell(self.mesh, 0)
self.__cell_vertex_x = self.__cell.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(self.__cell)
self.__dof = 0
self.__values = np.zeros(V.ufl_element().value_size())
@property
def dof(self):
return self.__dof
@dof.setter
def dof(self, value):
assert value < self.elm.space_dimension()
self.__dof = value
@property
def cell(self):
return self.__cell
@cell.setter
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def eval_cell(self, values, x, cell):
values[:] = self.elm.evaluate_basis(self.dof,
x,
self.__cell_vertex_x,
self.__cell_orientation)
def __call__(self, x):
self.eval(self.__values, x)
return 1*self.__values
def point_trace_matrix(V, TV, x0):
'''
Let u in V; u = ck phi_k then u(x0) \in TV = ck phi_k(x0). So this
is a 1 by dim(V) matrix where the column values are phi_k(x0).
'''
mesh = V.mesh()
tree = mesh.bounding_box_tree()
cell = tree.compute_first_entity_collision(Point(*x0))
assert cell < mesh.num_cells()
# Cell for restriction
Vcell = Cell(mesh, cell)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
x0 = np.fromiter(x0, dtype=float)
# Columns - get all components at once
all_dofs = V.dofmap().cell_dofs(cell).tolist()
Vel = V.element()
value_size = V.ufl_element().value_size()
basis_values = np.zeros(V.element().space_dimension() * value_size)
Vel.evaluate_basis_all(basis_values, x0, vertex_coordinates,
cell_orientation)
with petsc_serial_matrix(TV, V) as mat:
# Scalar gets all
if value_size == 1:
component_dofs = lambda component: V.dofmap().cell_dofs(cell)
# Slices
else:
component_dofs = lambda component: V.sub(component).dofmap(
).cell_dofs(cell)
for row in map(int, TV.dofmap().cell_dofs(cell)): # R^n components
sub_dofs = component_dofs(row)
sub_dofs_local = [all_dofs.index(dof) for dof in sub_dofs]
print row, sub_dofs, sub_dofs_local, basis_values[sub_dofs_local]
mat.setValues([row], sub_dofs, basis_values[sub_dofs_local],
PETSc.InsertMode.INSERT_VALUES)
return mat
class DegreeOfFreedom(object):
'''Evaluator of dof of V on functions'''
def __init__(self, V):
self.elm = V.element()
self.mesh = V.mesh()
is_1d_in_3d = self.mesh.topology().dim() == 1 and self.mesh.geometry().dim() == 3
self.orient_cell = cell_orientation(is_1d_in_3d)
# Allocs
self.__cell = Cell(self.mesh, 0)
self.__cell_vertex_x = self.__cell.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(self.__cell)
self.__dof = 0
@property
def dof(self):
return self.__dof
@dof.setter
def dof(self, value):
assert value < self.elm.space_dimension()
self.__dof = value
@property
def cell(self):
return self.__cell
@cell.setter
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def eval(self, f):
return self.elm.evaluate_dof(self.dof,
f.as_expression() if isinstance(f, FEBasisFunction) else f,
self.__cell_vertex_x,
self.__cell_orientation,
self.__cell)
def point_trace_matrix(V, TV, x0):
'''
Let u in V; u = ck phi_k then u(x0) \in TV = ck phi_k(x0). So this
is a 1 by dim(V) matrix where the column values are phi_k(x0).
'''
mesh = V.mesh()
tree = mesh.bounding_box_tree()
cell = tree.compute_first_entity_collision(Point(*x0))
assert cell < mesh.num_cells()
# Cell for restriction
Vcell = Cell(mesh, cell)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
x0 = np.fromiter(x0, dtype=float)
# Columns - get all components at once
all_dofs = V.dofmap().cell_dofs(cell).tolist()
Vel = V.element()
value_size = V.ufl_element().value_size()
basis_values = np.zeros(V.element().space_dimension()*value_size)
Vel.evaluate_basis_all(basis_values, x0, vertex_coordinates, cell_orientation)
with petsc_serial_matrix(TV, V) as mat:
# Scalar gets all
if value_size == 1:
component_dofs = lambda component: V.dofmap().cell_dofs(cell)
# Slices
else:
component_dofs = lambda component: V.sub(component).dofmap().cell_dofs(cell)
for row in map(int, TV.dofmap().cell_dofs(cell)): # R^n components
sub_dofs = component_dofs(row)
sub_dofs_local = [all_dofs.index(dof) for dof in sub_dofs]
print row, sub_dofs, sub_dofs_local, basis_values[sub_dofs_local]
mat.setValues([row], sub_dofs, basis_values[sub_dofs_local],
PETSc.InsertMode.INSERT_VALUES)
return mat
def build_grad_matrices(V, points):
"""Build the sparse m-by-n matrices that map a coefficient set for a function in V
to the values of dx and dy at a number m of points.
"""
# See <https://www.allanswered.com/post/lkbkm/#zxqgk>
mesh = V.mesh()
bbt = BoundingBoxTree()
bbt.build(mesh)
dofmap = V.dofmap()
el = V.element()
rows = []
cols = []
datax = []
datay = []
for i, xy in enumerate(points):
cell_id = bbt.compute_first_entity_collision(Point(*xy))
cell = Cell(mesh, cell_id)
coordinate_dofs = cell.get_vertex_coordinates()
rows.append([i, i, i])
cols.append(dofmap.cell_dofs(cell_id))
v = el.evaluate_basis_derivatives_all(1, xy, coordinate_dofs, cell_id)
v = v.reshape(3, 2)
datax.append(v[:, 0])
datay.append(v[:, 1])
rows = numpy.concatenate(rows)
cols = numpy.concatenate(cols)
datax = numpy.concatenate(datax)
datay = numpy.concatenate(datay)
m = len(points)
n = V.dim()
dx_matrix = sparse.csr_matrix((datax, (rows, cols)), shape=(m, n))
dy_matrix = sparse.csr_matrix((datay, (rows, cols)), shape=(m, n))
return dx_matrix, dy_matrix
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
def cylinder_average_matrix(V, TV, radius, quad_degree):
'''Averaging matrix'''
mesh = V.mesh()
line_mesh = TV.mesh()
# We are going to perform the integration with Gauss quadrature at
# the end (PI u)(x):
# A cell of mesh (an edge) defines a normal vector. Let P be the plane
# that is defined by the normal vector n and some point x on Gamma. Let L
# be the circle that is the intersect of P and S. The value of q (in Q) at x
# is defined as
#
# q(x) = (1/|L|)*\int_{L}g(x)*dL
#
# which simplifies to g(x) = (1/(2*pi*R))*\int_{-pi}^{pi}u(L)*R*d(theta) and
# or = (1/2) * \int_{-1}^{1} u (L(pi*s)) * ds
# This can be integrated no problemo once we figure out L. To this end, let
# t_1 and t_2 be two unit mutually orthogonal vectors that are orthogonal to
# n. Then L(pi*s) = p + R*t_1*cos(pi*s) + R*t_2*sin(pi*s) can be seen to be
# such that i) |x-p| = R and ii) x.n = 0 [i.e. this the suitable
# parametrization]
# Clearly we can scale the weights as well as precompute
# cos and sin terms.
xq, wq = leggauss(quad_degree)
wq *= 0.5
cos_xq = np.cos(np.pi*xq).reshape((-1, 1))
sin_xq = np.sin(np.pi*xq).reshape((-1, 1))
if is_number(radius):
radius = lambda x, radius=radius: radius
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension()*value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent => orthogonal tangent vectors
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
t1 = np.array([n[1]-n[2], n[2]-n[0], n[0]-n[1]])
t2 = np.cross(n, t1)
t1 /= np.linalg.norm(t1)
t2 = t2/np.linalg.norm(t2)
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Get radius and integration points
rad = radius(avg_point)
integration_points = avg_point + rad*t1*sin_xq + rad*t2*cos_xq
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip, vertex_coordinates, cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values*wq[index]
# Add
for col, value in zip(cols_ip, values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value
else:
data[col] = value
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array([data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values, PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def sphere_average_matrix(V, TV, radius, quad_degree):
'''Averaging matrix over the sphere'''
mesh = V.mesh()
line_mesh = TV.mesh()
# Lebedev below need off degrees
if quad_degree % 2 == 0: quad_degree += 1
# NOTE: this is a dependency
from quadpy.sphere import Lebedev
integrator = Lebedev(quad_degree)
xq = integrator.points
wq = integrator.weights
if is_number(radius):
radius = lambda x, radius=radius: radius
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension()*value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Get radius and integration points
rad = radius(avg_point)
# Scale and shift the unit sphere to the point
integration_points = xq*rad + avg_point
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip, vertex_coordinates, cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values*wq[index]
# Add
for col, value in zip(cols_ip, values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value
else:
data[col] = value
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array([data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values, PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def trace_3d1d_matrix(V, TV, reduced_mesh):
'''Trace from 3d to 1d. Makes sense only for CG space'''
assert reduced_mesh.id() == TV.mesh().id()
assert V.ufl_element().family() == 'Lagrange'
mesh = V.mesh()
line_mesh = TV.mesh()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# We use the map to get (1d cell -> [3d edge) -> 3d cell]
if hasattr(reduced_mesh, 'parent_entity_map'):
# ( )
mapping = reduced_mesh.parent_entity_map[mesh.id()][1]
# [ ]
mesh.init(1)
mesh.init(1, 3)
e2c = mesh.topology()(1, 3)
# From 1d cell (by index)
get_cell3d = lambda c, d1d3=mapping, d3d3=e2c: d3d3(d1d3[c.index()])[0]
# Tree collision by midpoint
else:
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
get_cell3d = lambda c, tree=tree, bound=limit: (
lambda index: index if index<bound else None
)(tree.compute_first_entity_collision(c.midpoint()))
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension()*value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent => orthogonal tangent vectors
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
# Let's get a 3d cell to use for getting the V values
# CG assumption allows taking any
tet_cell = get_cell3d(line_cell)
if tet_cell is None: continue
Vcell = Cell(mesh, tet_cell)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = 0
# Columns are determined by V cell! I guess the sparsity
# could be improved if for x_dofs of TV only x_dofs of V
# were considered
column_indices = np.array(V_dm.cell_dofs(tet_cell), dtype='int32')
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# 3d at point
Vel.evaluate_basis_all(basis_values, avg_point, vertex_coordinates, cell_orientation)
# The thing now is that with data we can assign to several
# rows of the matrix. Shift determines the (x, y, ... ) or
# (xx, xy, yx, ...) component of Q
data = basis_values.reshape((-1, value_size)).T
for shift, column_values in enumerate(data):
row = scalar_row + shift
mat.setValues([row], column_indices, column_values, PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def cylinder_average_matrix(V, TV, radius, quad_degree):
'''Averaging matrix'''
mesh = V.mesh()
line_mesh = TV.mesh()
# We are going to perform the integration with Gauss quadrature at
# the end (PI u)(x):
# A cell of mesh (an edge) defines a normal vector. Let P be the plane
# that is defined by the normal vector n and some point x on Gamma. Let L
# be the circle that is the intersect of P and S. The value of q (in Q) at x
# is defined as
#
# q(x) = (1/|L|)*\int_{L}g(x)*dL
#
# which simplifies to g(x) = (1/(2*pi*R))*\int_{-pi}^{pi}u(L)*R*d(theta) and
# or = (1/2) * \int_{-1}^{1} u (L(pi*s)) * ds
# This can be integrated no problemo once we figure out L. To this end, let
# t_1 and t_2 be two unit mutually orthogonal vectors that are orthogonal to
# n. Then L(pi*s) = p + R*t_1*cos(pi*s) + R*t_2*sin(pi*s) can be seen to be
# such that i) |x-p| = R and ii) x.n = 0 [i.e. this the suitable
# parametrization]
# Clearly we can scale the weights as well as precompute
# cos and sin terms.
xq, wq = leggauss(quad_degree)
wq *= 0.5
cos_xq = np.cos(np.pi * xq).reshape((-1, 1))
sin_xq = np.sin(np.pi * xq).reshape((-1, 1))
if is_number(radius):
radius = lambda x, radius=radius: radius
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent => orthogonal tangent vectors
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
t1 = np.array([n[1] - n[2], n[2] - n[0], n[0] - n[1]])
t2 = np.cross(n, t1)
t1 /= np.linalg.norm(t1)
t2 = t2 / np.linalg.norm(t2)
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Get radius and integration points
rad = radius(avg_point)
integration_points = avg_point + rad * t1 * sin_xq + rad * t2 * cos_xq
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip,
vertex_coordinates,
cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(cols_ip,
values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value
else:
data[col] = value
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def sphere_average_matrix(V, TV, radius, quad_degree):
'''Averaging matrix over the sphere'''
mesh = V.mesh()
line_mesh = TV.mesh()
# Lebedev below need off degrees
if quad_degree % 2 == 0: quad_degree += 1
# NOTE: this is a dependency
from quadpy.sphere import Lebedev
integrator = Lebedev(quad_degree)
xq = integrator.points
wq = integrator.weights
if is_number(radius):
radius = lambda x, radius=radius: radius
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Get radius and integration points
rad = radius(avg_point)
# Scale and shift the unit sphere to the point
integration_points = xq * rad + avg_point
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip,
vertex_coordinates,
cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(cols_ip,
values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value
else:
data[col] = value
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def trace_3d1d_matrix(V, TV, reduced_mesh):
'''Trace from 3d to 1d. Makes sense only for CG space'''
assert reduced_mesh.id() == TV.mesh().id()
assert V.ufl_element().family() == 'Lagrange'
mesh = V.mesh()
line_mesh = TV.mesh()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# We use the map to get (1d cell -> [3d edge) -> 3d cell]
if hasattr(reduced_mesh, 'parent_entity_map'):
# ( )
mapping = reduced_mesh.parent_entity_map[mesh.id()][1]
# [ ]
mesh.init(1)
mesh.init(1, 3)
e2c = mesh.topology()(1, 3)
# From 1d cell (by index)
get_cell3d = lambda c, d1d3=mapping, d3d3=e2c: d3d3(d1d3[c.index()])[0]
# Tree collision by midpoint
else:
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
get_cell3d = lambda c, tree=tree, bound=limit: (
lambda index: index if index < bound else None)(
tree.compute_first_entity_collision(c.midpoint()))
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent => orthogonal tangent vectors
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
# Let's get a 3d cell to use for getting the V values
# CG assumption allows taking any
tet_cell = get_cell3d(line_cell)
if tet_cell is None: continue
Vcell = Cell(mesh, tet_cell)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = 0
# Columns are determined by V cell! I guess the sparsity
# could be improved if for x_dofs of TV only x_dofs of V
# were considered
column_indices = np.array(V_dm.cell_dofs(tet_cell), dtype='int32')
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# 3d at point
Vel.evaluate_basis_all(basis_values, avg_point,
vertex_coordinates, cell_orientation)
# The thing now is that with data we can assign to several
# rows of the matrix. Shift determines the (x, y, ... ) or
# (xx, xy, yx, ...) component of Q
data = basis_values.reshape((-1, value_size)).T
for shift, column_values in enumerate(data):
row = scalar_row + shift
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def average_matrix(V, TV, shape):
'''
Averaging matrix for reduction of g in V to TV by integration over shape.
'''
# We build a matrix representation of u in V -> Pi(u) in TV where
#
# Pi(u)(s) = |L(s)|^-1*\int_{L(s)}u(t) dx(s)
#
# Here L is the shape over which u is integrated for reduction.
# Its measure is |L(s)|.
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
mesh = V.mesh()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
line_mesh = TV.mesh()
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent (normal of the plane which cuts the virtual
# surface to yield the bdry curve
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Avg point here has the role of 'height' coordinate
quadrature = shape.quadrature(avg_point, n)
integration_points = quadrature.points
wq = quadrature.weights
curve_measure = sum(wq)
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
cs = tree.compute_entity_collisions(Point(*ip))
# assert False
for c in cs[:1]:
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
basis_values[:] = Vel.evaluate_basis_all(
ip, vertex_coordinates, cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(
cols_ip, values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value / curve_measure
else:
data[col] = value / curve_measure
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(list(data.keys()), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return mat
def surface_average_matrix(V, TV, bdry_curve):
'''Averaging matrix'''
mesh = V.mesh()
line_mesh = TV.mesh()
# We build a matrix representation of u in V -> Pi(u) in TV where
#
# Pi(u)(s) = |L(s)|^-1*\int_{L(s)}u(t) dL(s)
#
# Here L represents a curve bounding the surface at 'height' s.
#
# We do this numerically as |L(s)|^-1*\sum_q u(x_q)*w_q
# Weights remaing fixed
wq = bdry_curve.weights
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent (normal of the plane which cuts the virtual
# surface to yield the bdry curve
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
# We can specialize quadrature points; we can have several
# height points with same normal
pts_at_n = bdry_curve.points(n)
len_at_n = bdry_curve.length(n)
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Avg point here has the role of 'height' coordinate
integration_points = pts_at_n(avg_point)
len_bdry_curve = len_at_n(avg_point)
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip,
vertex_coordinates,
cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(cols_ip,
values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value / len_bdry_curve
else:
data[col] = value / len_bdry_curve
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def cell(self, value):
cell_ = Cell(self.mesh, value)
self.__cell_vertex_x[:] = cell_.get_vertex_coordinates()
self.__cell_orientation = self.orient_cell(cell_)
self.__cell = cell_
