def estimate(self, solution):
mesh = solution.function_space().mesh()
# Define cell and facet residuals
R_T = -(self.rhs_f + div(grad(solution)))
n = FacetNormal(mesh)
R_dT = dot(grad(solution), n)
# Will use space of constants to localize indicator form
Constants = FunctionSpace(mesh, "DG", 0)
w = TestFunction(Constants)
h = CellSize(mesh)
# Define form for assembling error indicators
form = (h ** 2 * R_T ** 2 * w * dx + avg(h) * avg(R_dT) ** 2 * 2 * avg(w) * dS)
# + h * R_dT ** 2 * w * ds)
# Assemble error indicators
indicators = assemble(form)
# Calculate error
error_estimate = sqrt(sum(i for i in indicators.array()))
# Take sqrt of indicators
indicators = np.array([sqrt(i) for i in indicators])
# Mark cells for refinement based on maximal marking strategy
largest_error = max(indicators)
cell_markers = MeshFunction("bool", mesh, mesh.topology().dim())
for c in cells(mesh):
cell_markers[c] = indicators[c.index()] > (self.fraction * largest_error)
return error_estimate, cell_markers
def circumradius(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
# Don't lower for non-affine cells, instead leave it to
# form compiler
warning(
"Only know how to compute the circumradius of an affine cell.")
return o
cellname = domain.ufl_cell().cellname()
cellvolume = self.cell_volume(CellVolume(domain))
if cellname == "interval":
r = 0.5 * cellvolume
elif cellname == "triangle":
J = self.jacobian(Jacobian(domain))
trev = CellEdgeVectors(domain)
num_edges = 3
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
r = (elen[0] * elen[1] * elen[2]) / (4.0 * cellvolume)
elif cellname == "tetrahedron":
J = self.jacobian(Jacobian(domain))
trev = CellEdgeVectors(domain)
num_edges = 6
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
# elen[3] = length of edge 3
# la, lb, lc = lengths of the sides of an intermediate triangle
la = elen[3] * elen[2]
lb = elen[4] * elen[1]
lc = elen[5] * elen[0]
# p = perimeter
p = (la + lb + lc)
# s = semiperimeter
s = p / 2
# area of intermediate triangle with Herons formula
triangle_area = sqrt(s * (s - la) * (s - lb) * (s - lc))
r = triangle_area / (6.0 * cellvolume)
else:
error("Unhandled cell type %s." % cellname)
return r
def pseudo_determinant_expr(A):
"""Compute the pseudo-determinant of A."""
m, n = A.ufl_shape
if n == 1:
# Special case 1xm for simpler expression
i = Index()
return sqrt(A[i, 0] * A[i, 0])
elif n == 2 and m == 3:
# Special case 2x3 for simpler expression
c = cross_expr(A[:, 0], A[:, 1])
i = Index()
return sqrt(c[i] * c[i])
else:
# Generic formulation based on A.T*A
return generic_pseudo_determinant_expr(A)
def pseudo_determinant_expr(A):
"""Compute the pseudo-determinant of A."""
m, n = A.ufl_shape
if n == 1:
# Special case 1xm for simpler expression
i = Index()
return sqrt(A[i, 0] * A[i, 0])
elif n == 2 and m == 3:
# Special case 2x3 for simpler expression
c = cross_expr(A[:, 0], A[:, 1])
i = Index()
return sqrt(c[i] * c[i])
else:
# Generic formulation based on A.T*A
return generic_pseudo_determinant_expr(A)
def _reduce_facet_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if domain.ufl_cell().topological_dimension() < 3:
error(
"Facet edge lengths only make sense for topological dimension >= 3."
)
elif not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning(
"Only know how to compute facet edge lengths of P1 or Q1 cell."
)
return o
else:
# P1 tetrahedron or Q1 hexahedron
edges = FacetEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [edges[e, j] * edges[e, j] for e in range(num_edges)]
return sqrt(reduce(reduction_op, elen2))
def max_facet_edge_length(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
# Don't lower for non-affine cells, instead leave it to
# form compiler
warning(
"Only know how to compute the max_facet_edge_length of an affine cell."
)
return o
cellname = domain.ufl_cell().cellname()
if cellname == "triangle":
return self.facet_area(FacetArea(domain))
elif cellname == "tetrahedron":
J = self.jacobian(Jacobian(domain))
trev = FacetEdgeVectors(domain)
num_edges = 3
i, j, k = indices(3)
elen = [
sqrt((J[i, j] * trev[edge, j]) * (J[i, k] * trev[edge, k]))
for edge in range(num_edges)
]
return max_value(elen[0], max_value(elen[1], elen[2]))
else:
error("Unhandled cell type %s." % cellname)
def cell_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
gdim = domain.geometric_dimension()
tdim = domain.topological_dimension()
if tdim == gdim - 1: # n-manifold embedded in n-1 space
i = Index()
J = self.jacobian(Jacobian(domain))
if tdim == 2:
# Surface in 3D
t0 = as_vector(J[i, 0], i)
t1 = as_vector(J[i, 1], i)
cell_normal = cross_expr(t0, t1)
elif tdim == 1:
# Line in 2D (cell normal is 'up' for a line pointing
# to the 'right')
cell_normal = as_vector((-J[1, 0], J[0, 0]))
else:
error("Cell normal not implemented for tdim %d, gdim %d" %
(tdim, gdim))
# Return normalized vector, sign corrected by cell
# orientation
co = CellOrientation(domain)
return co * cell_normal / sqrt(cell_normal[i] * cell_normal[i])
else:
error("What do you want cell normal in gdim={0}, tdim={1} to be?".
format(gdim, tdim))
def cell_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
gdim = domain.geometric_dimension()
tdim = domain.topological_dimension()
if tdim == gdim - 1: # n-manifold embedded in n-1 space
i = Index()
J = self.jacobian(Jacobian(domain))
if tdim == 2:
# Surface in 3D
t0 = as_vector(J[i, 0], i)
t1 = as_vector(J[i, 1], i)
cell_normal = cross_expr(t0, t1)
elif tdim == 1:
# Line in 2D (cell normal is 'up' for a line pointing
# to the 'right')
cell_normal = as_vector((-J[1, 0], J[0, 0]))
else:
error("Cell normal not implemented for tdim %d, gdim %d" % (tdim, gdim))
# Return normalized vector, sign corrected by cell
# orientation
co = CellOrientation(domain)
return co * cell_normal / sqrt(cell_normal[i]*cell_normal[i])
else:
error("What do you want cell normal in gdim={0}, tdim={1} to be?".format(gdim, tdim))
def facet_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
tdim = domain.topological_dimension()
if tdim == 1:
# Special-case 1D (possibly immersed), for which we say
# that n is just in the direction of J.
J = self.jacobian(Jacobian(domain)) # dx/dX
ndir = J[:, 0]
gdim = domain.geometric_dimension()
if gdim == 1:
nlen = abs(ndir[0])
else:
i = Index()
nlen = sqrt(ndir[i] * ndir[i])
rn = ReferenceNormal(domain) # +/- 1.0 here
n = rn[0] * ndir / nlen
r = n
else:
# Recall that the covariant Piola transform u -> J^(-T)*u
# preserves tangential components. The normal vector is
# characterised by having zero tangential component in
# reference and physical space.
Jinv = self.jacobian_inverse(JacobianInverse(domain))
i, j = indices(2)
rn = ReferenceNormal(domain)
# compute signed, unnormalised normal; note transpose
ndir = as_vector(Jinv[j, i] * rn[j], i)
# normalise
i = Index()
n = ndir / sqrt(ndir[i] * ndir[i])
r = n
if r.ufl_shape != o.ufl_shape:
error("Inconsistent dimensions (in=%d, out=%d)." %
(o.ufl_shape[0], r.ufl_shape[0]))
return r
def facet_normal(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
tdim = domain.topological_dimension()
if tdim == 1:
# Special-case 1D (possibly immersed), for which we say
# that n is just in the direction of J.
J = self.jacobian(Jacobian(domain)) # dx/dX
ndir = J[:, 0]
gdim = domain.geometric_dimension()
if gdim == 1:
nlen = abs(ndir[0])
else:
i = Index()
nlen = sqrt(ndir[i]*ndir[i])
rn = ReferenceNormal(domain) # +/- 1.0 here
n = rn[0] * ndir / nlen
r = n
else:
# Recall that the covariant Piola transform u -> J^(-T)*u
# preserves tangential components. The normal vector is
# characterised by having zero tangential component in
# reference and physical space.
Jinv = self.jacobian_inverse(JacobianInverse(domain))
i, j = indices(2)
rn = ReferenceNormal(domain)
# compute signed, unnormalised normal; note transpose
ndir = as_vector(Jinv[j, i] * rn[j], i)
# normalise
i = Index()
n = ndir / sqrt(ndir[i]*ndir[i])
r = n
if r.ufl_shape != o.ufl_shape:
error("Inconsistent dimensions (in=%d, out=%d)." % (o.ufl_shape[0], r.ufl_shape[0]))
return r
def circumradius(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
error("Circumradius only makes sense for affine simplex cells")
cellname = domain.ufl_cell().cellname()
cellvolume = self.cell_volume(CellVolume(domain))
if cellname == "interval":
# Optimization for square interval; no square root needed
return 0.5 * cellvolume
# Compute lengths of cell edges
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen = [sqrt(edges[e, j] * edges[e, j]) for e in range(num_edges)]
if cellname == "triangle":
return (elen[0] * elen[1] * elen[2]) / (4.0 * cellvolume)
elif cellname == "tetrahedron":
# la, lb, lc = lengths of the sides of an intermediate triangle
# NOTE: Is here some hidden numbering assumption?
la = elen[3] * elen[2]
lb = elen[4] * elen[1]
lc = elen[5] * elen[0]
# p = perimeter
p = (la + lb + lc)
# s = semiperimeter
s = p / 2
# area of intermediate triangle with Herons formula
triangle_area = sqrt(s * (s - la) * (s - lb) * (s - lc))
return triangle_area / (6.0 * cellvolume)
def circumradius(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.is_piecewise_linear_simplex_domain():
error("Circumradius only makes sense for affine simplex cells")
cellname = domain.ufl_cell().cellname()
cellvolume = self.cell_volume(CellVolume(domain))
if cellname == "interval":
# Optimization for square interval; no square root needed
return 0.5 * cellvolume
# Compute lengths of cell edges
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen = [sqrt(edges[e, j]*edges[e, j]) for e in range(num_edges)]
if cellname == "triangle":
return (elen[0] * elen[1] * elen[2]) / (4.0 * cellvolume)
elif cellname == "tetrahedron":
# la, lb, lc = lengths of the sides of an intermediate triangle
# NOTE: Is here some hidden numbering assumption?
la = elen[3] * elen[2]
lb = elen[4] * elen[1]
lc = elen[5] * elen[0]
# p = perimeter
p = (la + lb + lc)
# s = semiperimeter
s = p / 2
# area of intermediate triangle with Herons formula
triangle_area = sqrt(s * (s - la) * (s - lb) * (s - lc))
return triangle_area / (6.0 * cellvolume)
def _reduce_facet_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if domain.ufl_cell().topological_dimension() < 3:
error("Facet edge lengths only make sense for topological dimension >= 3.")
elif not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute facet edge lengths of P1 or Q1 cell.")
return o
else:
# P1 tetrahedron or Q1 hexahedron
edges = FacetEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [edges[e, j]*edges[e, j] for e in range(num_edges)]
return sqrt(reduce(reduction_op, elen2))
def cell_diameter(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() in {1, (1, 1)}:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell diameter of P1 or Q1 cell.")
return o
elif domain.is_piecewise_linear_simplex_domain():
# Simplices
return self.max_cell_edge_length(MaxCellEdgeLength(domain))
else:
# Q1 cells, maximal distance between any two vertices
verts = CellVertices(domain)
verts = [verts[v, ...] for v in range(verts.ufl_shape[0])]
j = Index()
elen2 = (real((v0 - v1)[j] * conj((v0 - v1)[j])) for v0, v1 in combinations(verts, 2))
return real(sqrt(reduce(max_value, elen2)))
def cell_diameter(self, o):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell diameter of P1 or Q1 cell.")
return o
elif domain.is_piecewise_linear_simplex_domain():
# Simplices
return self.max_cell_edge_length(MaxCellEdgeLength(domain))
else:
# Q1 cells, maximal distance between any two vertices
verts = CellVertices(domain)
verts = [verts[v, ...] for v in range(verts.ufl_shape[0])]
j = Index()
elen2 = ((v0-v1)[j]*(v0-v1)[j] for v0, v1 in combinations(verts, 2))
return sqrt(reduce(max_value, elen2))
def _reduce_cell_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell edge lengths of P1 or Q1 cell.")
return o
elif domain.ufl_cell().cellname() == "interval":
# Interval optimization, square root not needed
return self.cell_volume(CellVolume(domain))
else:
# Other P1 or Q1 cells
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [edges[e, j]*edges[e, j] for e in range(num_edges)]
return sqrt(reduce(reduction_op, elen2))
def _reduce_cell_edge_length(self, o, reduction_op):
if self._preserve_types[o._ufl_typecode_]:
return o
domain = o.ufl_domain()
if not domain.ufl_coordinate_element().degree() == 1:
# Don't lower bendy cells, instead leave it to form compiler
warning("Only know how to compute cell edge lengths of P1 or Q1 cell.")
return o
elif domain.ufl_cell().cellname() == "interval":
# Interval optimization, square root not needed
return self.cell_volume(CellVolume(domain))
else:
# Other P1 or Q1 cells
edges = CellEdgeVectors(domain)
num_edges = edges.ufl_shape[0]
j = Index()
elen2 = [real(edges[e, j] * conj(edges[e, j])) for e in range(num_edges)]
return real(sqrt(reduce(reduction_op, elen2)))
def acos(self, f):
r".. math:: \\frac{d}{dx} f(x) = \frac{d}{dx} \arccos(x) = \frac{-1}{\sqrt{1 - x^2}}"
x, = f.operands()
return (-1.0/sqrt(1.0 - x**2),)
def erf(self, o, a):
f, fp = a
o = self.reuse_if_possible(o, f)
op = fp*(2.0/sqrt(pi)*exp(-f**2))
return (o, op)
def asin(self, o, a):
f, fp = a
o = self.reuse_if_possible(o, f)
op = fp/sqrt(1.0 - f**2)
return (o, op)
def erf(self, o, fp):
f, = o.ufl_operands
return fp * (2.0 / sqrt(pi) * exp(-f**2))
def acos(self, o, fp):
f, = o.ufl_operands
return -fp / sqrt(1.0 - f**2)
def asin(self, o, a):
f, fp = a
o = self.reuse_if_possible(o, f)
op = fp / sqrt(1.0 - f**2)
return (o, op)
def asin(self, o, fp):
f, = o.ufl_operands
return fp / sqrt(1.0 - f**2)
def erf(self, f):
"d/dx erf x = 2/sqrt(pi)*exp(-x^2)"
x, = f.operands()
return (2.0/sqrt(pi)*exp(-x**2),)
def erf(self, o, a):
f, fp = a
o = self.reuse_if_possible(o, f)
op = fp * (2.0 / sqrt(pi) * exp(-f**2))
return (o, op)
def generic_pseudo_determinant_expr(A):
"""Compute the pseudo-determinant of A: sqrt(det(A.T*A))."""
i, j, k = indices(3)
ATA = as_tensor(A[k, i] * A[k, j], (i, j))
return sqrt(determinant_expr(ATA))
def asin(self, o, fp):
f, = o.ufl_operands
return fp / sqrt(1.0 - f**2)
def asin(self, f):
"d/dx asin x = 1/sqrt(1 - x^2)"
x, = f.operands()
return (1.0/sqrt(1.0 - x**2),)
def acos(self, o, fp):
f, = o.ufl_operands
return -fp / sqrt(1.0 - f**2)
def generic_pseudo_determinant_expr(A):
"""Compute the pseudo-determinant of A: sqrt(det(A.T*A))."""
i, j, k = indices(3)
ATA = as_tensor(A[k, i] * A[k, j], (i, j))
return sqrt(determinant_expr(ATA))
def erf(self, o, fp):
f, = o.ufl_operands
return fp * (2.0 / sqrt(pi) * exp(-f**2))
def compute_stress(mesh: Mesh, displacement: Function) -> Function:
V = FunctionSpace(mesh, 'P', 1)
s = sigma(displacement) - tr(sigma(displacement)) * Identity(3) / 3
return project(sqrt(1.5 * inner(s, s)), V)