def compute_functionals(self, velocity, pressure, t):
if self.args.wss:
info('Computing stress tensor')
I = Identity(velocity.geometric_dimension())
T = TensorFunctionSpace(self.mesh, 'Lagrange', 1)
stress = project(-pressure*I + 2*sym(grad(velocity)), T)
info('Generating boundary mesh')
wall_mesh = BoundaryMesh(self.mesh, 'exterior')
# wall_mesh = SubMesh(self.mesh, self.facet_function, 1) # QQ why does not work?
# plot(wall_mesh, interactive=True)
info(' Boundary mesh geometric dim: %d' % wall_mesh.geometry().dim())
info(' Boundary mesh topologic dim: %d' % wall_mesh.topology().dim())
info('Projecting stress to boundary mesh')
Tb = TensorFunctionSpace(wall_mesh, 'Lagrange', 1)
stress_b = interpolate(stress, Tb)
self.fileDict['wss']['file'] << stress_b
if False: # does not work
info('Computing WSS')
n = FacetNormal(wall_mesh)
info(stress_b, True)
# wss = stress_b*n - inner(stress_b*n, n)*n
wss = dot(stress_b, n) - inner(dot(stress_b, n), n)*n # equivalent
Vb = VectorFunctionSpace(wall_mesh, 'Lagrange', 1)
Sb = FunctionSpace(wall_mesh, 'Lagrange', 1)
# wss_func = project(wss, Vb)
wss_norm = project(sqrt(inner(wss, wss)), Sb)
plot(wss_norm, interactive=True)
def nonlinearity(function):
if self.use_ema:
return 2*inner(dot(sym(grad(function)), u_ext), v1) * dx + inner(div(function)*u_ext, v1) * dx
# return 2*inner(dot(sym(grad(function)), u_ext), v) * dx + inner(div(u_ext)*function, v) * dx
# QQ implement this way?
else:
return inner(dot(grad(function), u_ext), v1) * dx
def test_dolfin_expression_compilation_of_dot_with_index_notation(dolfin):
# Define some PyDOLFIN coefficients
mesh = dolfin.UnitSquareMesh(3, 3)
# Using quadratic element deliberately for accuracy
V = dolfin.VectorFunctionSpace(mesh, "CG", 2)
u = dolfin.Function(V)
u.interpolate(dolfin.Expression(("x[0]", "x[1]")))
# Define ufl expression with the simplest possible index notation
uexpr = ufl.dot(u, u) # Needs expand_compounds, not testing here
uexpr = u[0]*u[0] + u[1]*u[1] # Works
i, = ufl.indices(1)
uexpr = u[i]*u[i] # Works
# Define expected output from compilation
# Oneliner version (no reuse in this expression):
xc, yc = ('v_w0[0]', 'v_w0[1]')
expected_lines = ['double s[1];',
'Array<double> v_w0(2);',
'w0->eval(v_w0, x);',
's[0] = pow(%(x)s, 2) + pow(%(y)s, 2);' % {'x':xc, 'y':yc},
'values[0] = s[0];']
# Define expected evaluation values: [(x,value), (x,value), ...]
x, y = (0.6, 0.7)
expected = (x*x+y*y)
expected_values = [((0.0, 0.0), (0.0,)),
((x, y), (expected,)),
]
# Execute all tests
check_dolfin_expression_compilation(uexpr, expected_lines, expected_values, members={'w0':u})
def forms(arguments, coefficients):
v, u = arguments
c, f = coefficients
n = FacetNormal(triangle)
a = u * v * dx
L = f * v * dx
b = u * v * dx(0) + inner(c * grad(u), grad(v)) * \
dx(1) + dot(n, grad(u)) * v * ds + f * v * dx
return (a, L, b)
def compute_functionals(self, velocity, pressure, t, step):
if self.args.wss == 'all' or \
(step >= self.stepsInCycle and self.args.wss == 'peak' and
(self.distance_from_chosen_steps < 0.5 * self.metadata['dt'])):
# TODO check if choosing time steps works properly
# QQ might skip step? change 0.5 to 0.51?
self.tc.start('WSS')
begin('WSS (%dth step)' % step)
if self.args.wss_method == 'expression':
stress = project(self.nu*2*sym(grad(velocity)), self.T)
# pressure is not used as it contributes only to the normal component
stress.set_allow_extrapolation(True) # need because of some inaccuracies in BoundaryMesh coordinates
stress_b = interpolate(stress, self.Tb) # restrict stress to boundary mesh
# self.fileDict['stress']['file'].write(stress_b, self.actual_time)
# info('Saved stress tensor')
info('Computing WSS')
wss = dot(stress_b, self.nb) - inner(dot(stress_b, self.nb), self.nb)*self.nb
wss_func = project(wss, self.Vb)
wss_norm = project(sqrt_ufl(inner(wss, wss)), self.Sb)
info('Saving WSS')
self.fileDict['wss']['file'].write(wss_func, self.actual_time)
self.fileDict['wss_norm']['file'].write(wss_norm, self.actual_time)
if self.args.wss_method == 'integral':
wss_norm = Function(self.SDG)
mS = TestFunction(self.SDG)
scaling = 1/FacetArea(self.mesh)
stress = self.nu*2*sym(grad(velocity))
wss = dot(stress, self.normal) - inner(dot(stress, self.normal), self.normal)*self.normal
wss_norm_form = scaling*mS*sqrt_ufl(inner(wss, wss))*ds # ds is integral over exterior facets only
assemble(wss_norm_form, tensor=wss_norm.vector())
self.fileDict['wss_norm']['file'].write(wss_norm, self.actual_time)
# to get vector WSS values:
# NT this works, but in ParaView for (DG,1)-vector space glyphs are displayed in cell centers
# wss_vector = []
# for i in range(3):
# wss_component = Function(self.SDG)
# wss_vector_form = scaling*wss[i]*mS*ds
# assemble(wss_vector_form, tensor=wss_component.vector())
# wss_vector.append(wss_component)
# wss_func = project(as_vector(wss_vector), self.VDG)
# self.fileDict['wss']['file'].write(wss_func, self.actual_time)
self.tc.end('WSS')
end()
def diffusion(fce):
if self.useLaplace:
return nu * inner(grad(fce), grad(v1)) * dx
else:
form = inner(nu * 2 * sym(grad(fce)), sym(grad(v1))) * dx
if self.bcv == 'CDN':
return form
if self.bcv == 'LAP':
return form - inner(nu * dot(grad(fce).T, n), v1) * problem.get_outflow_measure_form()
if self.bcv == 'DDN':
return form # additional term must be added to non-constant part
def split_vector_laplace(cell, degree):
m = Mesh(VectorElement('CG', cell, 1))
if cell.cellname() in ['interval * interval', 'quadrilateral']:
hcurl_element = FiniteElement('RTCE', cell, degree)
elif cell.cellname() == 'triangle * interval':
U0 = FiniteElement('RT', triangle, degree)
U1 = FiniteElement('CG', triangle, degree)
V0 = FiniteElement('CG', interval, degree)
V1 = FiniteElement('DG', interval, degree - 1)
Wa = HCurlElement(TensorProductElement(U0, V0))
Wb = HCurlElement(TensorProductElement(U1, V1))
hcurl_element = EnrichedElement(Wa, Wb)
elif cell.cellname() == 'quadrilateral * interval':
hcurl_element = FiniteElement('NCE', cell, degree)
RT = FunctionSpace(m, hcurl_element)
CG = FunctionSpace(m, FiniteElement('Q', cell, degree))
sigma = TrialFunction(CG)
u = TrialFunction(RT)
tau = TestFunction(CG)
v = TestFunction(RT)
return [dot(u, grad(tau))*dx, dot(grad(sigma), v)*dx, dot(curl(u), curl(v))*dx]
def to_reference_coordinates(ufl_coordinate_element, parameters):
# Set up UFL form
cell = ufl_coordinate_element.cell()
domain = ufl.Mesh(ufl_coordinate_element)
K = ufl.JacobianInverse(domain)
x = ufl.SpatialCoordinate(domain)
x0_element = ufl.VectorElement("Real", cell, 0)
x0 = ufl.Coefficient(ufl.FunctionSpace(domain, x0_element))
expr = ufl.dot(K, x - x0)
# Translation to GEM
C = ufl.Coefficient(ufl.FunctionSpace(domain, ufl_coordinate_element))
expr = ufl_utils.preprocess_expression(expr)
expr = ufl_utils.simplify_abs(expr)
builder = firedrake_interface.KernelBuilderBase()
builder.domain_coordinate[domain] = C
builder._coefficient(C, "C")
builder._coefficient(x0, "x0")
dim = cell.topological_dimension()
point = gem.Variable('X', (dim,))
context = tsfc.fem.GemPointContext(
interface=builder,
ufl_cell=cell,
precision=parameters["precision"],
point_indices=(),
point_expr=point,
)
translator = tsfc.fem.Translator(context)
ir = map_expr_dag(translator, expr)
# Unroll result
ir = [gem.Indexed(ir, alpha) for alpha in numpy.ndindex(ir.shape)]
# Unroll IndexSums
max_extent = parameters["unroll_indexsum"]
if max_extent:
def predicate(index):
return index.extent <= max_extent
ir = gem.optimise.unroll_indexsum(ir, predicate=predicate)
# Translate to COFFEE
ir = impero_utils.preprocess_gem(ir)
return_variable = gem.Variable('dX', (dim,))
assignments = [(gem.Indexed(return_variable, (i,)), e)
for i, e in enumerate(ir)]
impero_c = impero_utils.compile_gem(assignments, ())
body = tsfc.coffee.generate(impero_c, {}, parameters["precision"])
body.open_scope = False
return body
def diffusion(fce):
if self.useLaplace:
return nu*inner(grad(fce), grad(v1)) * dx
else:
form = inner(nu * 2 * sym(grad(fce)), sym(grad(v1))) * dx
if self.bcv == 'CDN':
# IMP will work only if p=0 on output, or we must add term
# inner(p0*n, v)*problem.get_outflow_measure_form() to avoid boundary layer
return form
if self.bcv == 'LAP':
return form - inner(nu*dot(grad(fce).T, n), v1) * problem.get_outflow_measure_form()
if self.bcv == 'DDN':
# IMP will work only if p=0 on output, or we must add term
# inner(p0*n, v)*problem.get_outflow_measure_form() to avoid boundary layer
return form # additional term must be added to non-constant part
def split_mixed_poisson(cell, degree):
m = Mesh(VectorElement('CG', cell, 1))
if cell.cellname() in ['interval * interval', 'quadrilateral']:
hdiv_element = FiniteElement('RTCF', cell, degree)
elif cell.cellname() == 'triangle * interval':
U0 = FiniteElement('RT', triangle, degree)
U1 = FiniteElement('DG', triangle, degree - 1)
V0 = FiniteElement('CG', interval, degree)
V1 = FiniteElement('DG', interval, degree - 1)
Wa = HDivElement(TensorProductElement(U0, V1))
Wb = HDivElement(TensorProductElement(U1, V0))
hdiv_element = EnrichedElement(Wa, Wb)
elif cell.cellname() == 'quadrilateral * interval':
hdiv_element = FiniteElement('NCF', cell, degree)
RT = FunctionSpace(m, hdiv_element)
DG = FunctionSpace(m, FiniteElement('DQ', cell, degree - 1))
sigma = TrialFunction(RT)
u = TrialFunction(DG)
tau = TestFunction(RT)
v = TestFunction(DG)
return [dot(sigma, tau) * dx, div(tau) * u * dx, div(sigma) * v * dx]
elif dimension == 3:
ellipsoid_mesh(domain_radius, domain_radius, domain_radius, h)
# Read mesh
with XDMFFile(MPI.COMM_WORLD, "mesh.xdmf", "r",
XDMFFile.Encoding.HDF5) as xdmf:
mesh = xdmf.read_mesh(name="Grid")
n = FacetNormal(mesh)
x = ufl.geometry.SpatialCoordinate(mesh)
if dimension == 2:
di = ufl.as_vector([np.cos(angle), np.sin(angle)])
elif dimension == 3:
di = ufl.as_vector([np.cos(angle), np.sin(angle), 0])
ui = ufl.exp(1j * k0 * ufl.dot(di, x))
# Create function space
V = FunctionSpace(mesh, FiniteElement("Lagrange", mesh.ufl_cell(), degree))
def circle_refractive_index(x):
r = np.sqrt(x[0]**2 + x[1]**2)
inside = (r <= radius)
outside = (r > radius)
return inside * ref_index * k0 + outside * k0
def sphere_refractive_index(x):
r = np.sqrt(x[0]**2 + x[1]**2 + x[2]**2)
inside = (r <= radius)
def laplace(cell, degree):
m = Mesh(VectorElement('Q', cell, 1))
V = FunctionSpace(m, FiniteElement('Q', cell, degree, variant='spectral'))
u = TrialFunction(V)
v = TestFunction(V)
return dot(grad(u), grad(v))*dx(rule=gll_quadrature_rule(cell, degree))
def initialize(self, pc):
"""Set up the problem context. Take the original
mixed problem and reformulate the problem as a
hybridized mixed system.
A KSP is created for the Lagrange multiplier system.
"""
from firedrake import (FunctionSpace, Function, Constant,
TrialFunction, TrialFunctions, TestFunction,
DirichletBC)
from firedrake.assemble import (allocate_matrix,
create_assembly_callable)
from firedrake.formmanipulation import split_form
from ufl.algorithms.replace import replace
# Extract the problem context
prefix = pc.getOptionsPrefix() + "hybridization_"
_, P = pc.getOperators()
self.ctx = P.getPythonContext()
if not isinstance(self.ctx, ImplicitMatrixContext):
raise ValueError("The python context must be an ImplicitMatrixContext")
test, trial = self.ctx.a.arguments()
V = test.function_space()
mesh = V.mesh()
if len(V) != 2:
raise ValueError("Expecting two function spaces.")
if all(Vi.ufl_element().value_shape() for Vi in V):
raise ValueError("Expecting an H(div) x L2 pair of spaces.")
# Automagically determine which spaces are vector and scalar
for i, Vi in enumerate(V):
if Vi.ufl_element().sobolev_space().name == "HDiv":
self.vidx = i
else:
assert Vi.ufl_element().sobolev_space().name == "L2"
self.pidx = i
# Create the space of approximate traces.
W = V[self.vidx]
if W.ufl_element().family() == "Brezzi-Douglas-Marini":
tdegree = W.ufl_element().degree()
else:
try:
# If we have a tensor product element
h_deg, v_deg = W.ufl_element().degree()
tdegree = (h_deg - 1, v_deg - 1)
except TypeError:
tdegree = W.ufl_element().degree() - 1
TraceSpace = FunctionSpace(mesh, "HDiv Trace", tdegree)
# Break the function spaces and define fully discontinuous spaces
broken_elements = ufl.MixedElement([ufl.BrokenElement(Vi.ufl_element()) for Vi in V])
V_d = FunctionSpace(mesh, broken_elements)
# Set up the functions for the original, hybridized
# and schur complement systems
self.broken_solution = Function(V_d)
self.broken_residual = Function(V_d)
self.trace_solution = Function(TraceSpace)
self.unbroken_solution = Function(V)
self.unbroken_residual = Function(V)
shapes = (V[self.vidx].finat_element.space_dimension(),
np.prod(V[self.vidx].shape))
domain = "{[i,j]: 0 <= i < %d and 0 <= j < %d}" % shapes
instructions = """
for i, j
w[i,j] = w[i,j] + 1
end
"""
self.weight = Function(V[self.vidx])
par_loop((domain, instructions), ufl.dx, {"w": (self.weight, INC)},
is_loopy_kernel=True)
instructions = """
for i, j
vec_out[i,j] = vec_out[i,j] + vec_in[i,j]/w[i,j]
end
"""
self.average_kernel = (domain, instructions)
# Create the symbolic Schur-reduction:
# Original mixed operator replaced with "broken"
# arguments
arg_map = {test: TestFunction(V_d),
trial: TrialFunction(V_d)}
Atilde = Tensor(replace(self.ctx.a, arg_map))
gammar = TestFunction(TraceSpace)
n = ufl.FacetNormal(mesh)
sigma = TrialFunctions(V_d)[self.vidx]
if mesh.cell_set._extruded:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_h
+ gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_v)
else:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS)
# Here we deal with boundaries. If there are Neumann
# conditions (which should be enforced strongly for
# H(div)xL^2) then we need to add jump terms on the exterior
# facets. If there are Dirichlet conditions (which should be
# enforced weakly) then we need to zero out the trace
# variables there as they are not active (otherwise the hybrid
# problem is not well-posed).
# If boundary conditions are contained in the ImplicitMatrixContext:
if self.ctx.row_bcs:
# Find all the subdomains with neumann BCS
# These are Dirichlet BCs on the vidx space
neumann_subdomains = set()
for bc in self.ctx.row_bcs:
if bc.function_space().index == self.pidx:
raise NotImplementedError("Dirichlet conditions for scalar variable not supported. Use a weak bc")
if bc.function_space().index != self.vidx:
raise NotImplementedError("Dirichlet bc set on unsupported space.")
# append the set of sub domains
subdom = bc.sub_domain
if isinstance(subdom, str):
neumann_subdomains |= set([subdom])
else:
neumann_subdomains |= set(as_tuple(subdom, int))
# separate out the top and bottom bcs
extruded_neumann_subdomains = neumann_subdomains & {"top", "bottom"}
neumann_subdomains = neumann_subdomains - extruded_neumann_subdomains
integrand = gammar * ufl.dot(sigma, n)
measures = []
trace_subdomains = []
if mesh.cell_set._extruded:
ds = ufl.ds_v
for subdomain in sorted(extruded_neumann_subdomains):
measures.append({"top": ufl.ds_t, "bottom": ufl.ds_b}[subdomain])
trace_subdomains.extend(sorted({"top", "bottom"} - extruded_neumann_subdomains))
else:
ds = ufl.ds
if "on_boundary" in neumann_subdomains:
measures.append(ds)
else:
measures.extend((ds(sd) for sd in sorted(neumann_subdomains)))
markers = [int(x) for x in mesh.exterior_facets.unique_markers]
dirichlet_subdomains = set(markers) - neumann_subdomains
trace_subdomains.extend(sorted(dirichlet_subdomains))
for measure in measures:
Kform += integrand*measure
trace_bcs = [DirichletBC(TraceSpace, Constant(0.0), subdomain) for subdomain in trace_subdomains]
else:
# No bcs were provided, we assume weak Dirichlet conditions.
# We zero out the contribution of the trace variables on
# the exterior boundary. Extruded cells will have both
# horizontal and vertical facets
trace_subdomains = ["on_boundary"]
if mesh.cell_set._extruded:
trace_subdomains.extend(["bottom", "top"])
trace_bcs = [DirichletBC(TraceSpace, Constant(0.0), subdomain) for subdomain in trace_subdomains]
# Make a SLATE tensor from Kform
K = Tensor(Kform)
# Assemble the Schur complement operator and right-hand side
self.schur_rhs = Function(TraceSpace)
self._assemble_Srhs = create_assembly_callable(
K * Atilde.inv * AssembledVector(self.broken_residual),
tensor=self.schur_rhs,
form_compiler_parameters=self.ctx.fc_params)
mat_type = PETSc.Options().getString(prefix + "mat_type", "aij")
schur_comp = K * Atilde.inv * K.T
self.S = allocate_matrix(schur_comp, bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type,
options_prefix=prefix)
self._assemble_S = create_assembly_callable(schur_comp,
tensor=self.S,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type)
self._assemble_S()
self.S.force_evaluation()
Smat = self.S.petscmat
nullspace = self.ctx.appctx.get("trace_nullspace", None)
if nullspace is not None:
nsp = nullspace(TraceSpace)
Smat.setNullSpace(nsp.nullspace(comm=pc.comm))
# Set up the KSP for the system of Lagrange multipliers
trace_ksp = PETSc.KSP().create(comm=pc.comm)
trace_ksp.setOptionsPrefix(prefix)
trace_ksp.setOperators(Smat)
trace_ksp.setUp()
trace_ksp.setFromOptions()
self.trace_ksp = trace_ksp
split_mixed_op = dict(split_form(Atilde.form))
split_trace_op = dict(split_form(K.form))
# Generate reconstruction calls
self._reconstruction_calls(split_mixed_op, split_trace_op)
def initialize(self, pc):
"""Set up the problem context. Take the original
mixed problem and reformulate the problem as a
hybridized mixed system.
A KSP is created for the Lagrange multiplier system.
"""
from firedrake import (FunctionSpace, Function, Constant,
TrialFunction, TrialFunctions, TestFunction,
DirichletBC)
from firedrake.assemble import (allocate_matrix,
create_assembly_callable)
from firedrake.formmanipulation import split_form
from ufl.algorithms.replace import replace
# Extract the problem context
prefix = pc.getOptionsPrefix() + "hybridization_"
_, P = pc.getOperators()
self.ctx = P.getPythonContext()
if not isinstance(self.ctx, ImplicitMatrixContext):
raise ValueError(
"The python context must be an ImplicitMatrixContext")
test, trial = self.ctx.a.arguments()
V = test.function_space()
mesh = V.mesh()
if len(V) != 2:
raise ValueError("Expecting two function spaces.")
if all(Vi.ufl_element().value_shape() for Vi in V):
raise ValueError("Expecting an H(div) x L2 pair of spaces.")
# Automagically determine which spaces are vector and scalar
for i, Vi in enumerate(V):
if Vi.ufl_element().sobolev_space().name == "HDiv":
self.vidx = i
else:
assert Vi.ufl_element().sobolev_space().name == "L2"
self.pidx = i
# Create the space of approximate traces.
W = V[self.vidx]
if W.ufl_element().family() == "Brezzi-Douglas-Marini":
tdegree = W.ufl_element().degree()
else:
try:
# If we have a tensor product element
h_deg, v_deg = W.ufl_element().degree()
tdegree = (h_deg - 1, v_deg - 1)
except TypeError:
tdegree = W.ufl_element().degree() - 1
TraceSpace = FunctionSpace(mesh, "HDiv Trace", tdegree)
# Break the function spaces and define fully discontinuous spaces
broken_elements = ufl.MixedElement(
[ufl.BrokenElement(Vi.ufl_element()) for Vi in V])
V_d = FunctionSpace(mesh, broken_elements)
# Set up the functions for the original, hybridized
# and schur complement systems
self.broken_solution = Function(V_d)
self.broken_residual = Function(V_d)
self.trace_solution = Function(TraceSpace)
self.unbroken_solution = Function(V)
self.unbroken_residual = Function(V)
shapes = (V[self.vidx].finat_element.space_dimension(),
np.prod(V[self.vidx].shape))
domain = "{[i,j]: 0 <= i < %d and 0 <= j < %d}" % shapes
instructions = """
for i, j
w[i,j] = w[i,j] + 1
end
"""
self.weight = Function(V[self.vidx])
par_loop((domain, instructions),
ufl.dx, {"w": (self.weight, INC)},
is_loopy_kernel=True)
instructions = """
for i, j
vec_out[i,j] = vec_out[i,j] + vec_in[i,j]/w[i,j]
end
"""
self.average_kernel = (domain, instructions)
# Create the symbolic Schur-reduction:
# Original mixed operator replaced with "broken"
# arguments
arg_map = {test: TestFunction(V_d), trial: TrialFunction(V_d)}
Atilde = Tensor(replace(self.ctx.a, arg_map))
gammar = TestFunction(TraceSpace)
n = ufl.FacetNormal(mesh)
sigma = TrialFunctions(V_d)[self.vidx]
if mesh.cell_set._extruded:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_h +
gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_v)
else:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS)
# Here we deal with boundaries. If there are Neumann
# conditions (which should be enforced strongly for
# H(div)xL^2) then we need to add jump terms on the exterior
# facets. If there are Dirichlet conditions (which should be
# enforced weakly) then we need to zero out the trace
# variables there as they are not active (otherwise the hybrid
# problem is not well-posed).
# If boundary conditions are contained in the ImplicitMatrixContext:
if self.ctx.row_bcs:
# Find all the subdomains with neumann BCS
# These are Dirichlet BCs on the vidx space
neumann_subdomains = set()
for bc in self.ctx.row_bcs:
if bc.function_space().index == self.pidx:
raise NotImplementedError(
"Dirichlet conditions for scalar variable not supported. Use a weak bc"
)
if bc.function_space().index != self.vidx:
raise NotImplementedError(
"Dirichlet bc set on unsupported space.")
# append the set of sub domains
subdom = bc.sub_domain
if isinstance(subdom, str):
neumann_subdomains |= set([subdom])
else:
neumann_subdomains |= set(
as_tuple(subdom, numbers.Integral))
# separate out the top and bottom bcs
extruded_neumann_subdomains = neumann_subdomains & {
"top", "bottom"
}
neumann_subdomains = neumann_subdomains - extruded_neumann_subdomains
integrand = gammar * ufl.dot(sigma, n)
measures = []
trace_subdomains = []
if mesh.cell_set._extruded:
ds = ufl.ds_v
for subdomain in sorted(extruded_neumann_subdomains):
measures.append({
"top": ufl.ds_t,
"bottom": ufl.ds_b
}[subdomain])
trace_subdomains.extend(
sorted({"top", "bottom"} - extruded_neumann_subdomains))
else:
ds = ufl.ds
if "on_boundary" in neumann_subdomains:
measures.append(ds)
else:
measures.extend((ds(sd) for sd in sorted(neumann_subdomains)))
markers = [int(x) for x in mesh.exterior_facets.unique_markers]
dirichlet_subdomains = set(markers) - neumann_subdomains
trace_subdomains.extend(sorted(dirichlet_subdomains))
for measure in measures:
Kform += integrand * measure
trace_bcs = [
DirichletBC(TraceSpace, Constant(0.0), subdomain)
for subdomain in trace_subdomains
]
else:
# No bcs were provided, we assume weak Dirichlet conditions.
# We zero out the contribution of the trace variables on
# the exterior boundary. Extruded cells will have both
# horizontal and vertical facets
trace_subdomains = ["on_boundary"]
if mesh.cell_set._extruded:
trace_subdomains.extend(["bottom", "top"])
trace_bcs = [
DirichletBC(TraceSpace, Constant(0.0), subdomain)
for subdomain in trace_subdomains
]
# Make a SLATE tensor from Kform
K = Tensor(Kform)
# Assemble the Schur complement operator and right-hand side
self.schur_rhs = Function(TraceSpace)
self._assemble_Srhs = create_assembly_callable(
K * Atilde.inv * AssembledVector(self.broken_residual),
tensor=self.schur_rhs,
form_compiler_parameters=self.ctx.fc_params)
mat_type = PETSc.Options().getString(prefix + "mat_type", "aij")
schur_comp = K * Atilde.inv * K.T
self.S = allocate_matrix(schur_comp,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type,
options_prefix=prefix,
appctx=self.get_appctx(pc))
self._assemble_S = create_assembly_callable(
schur_comp,
tensor=self.S,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type)
with timed_region("HybridOperatorAssembly"):
self._assemble_S()
Smat = self.S.petscmat
nullspace = self.ctx.appctx.get("trace_nullspace", None)
if nullspace is not None:
nsp = nullspace(TraceSpace)
Smat.setNullSpace(nsp.nullspace(comm=pc.comm))
# Create a SNESContext for the DM associated with the trace problem
self._ctx_ref = self.new_snes_ctx(pc,
schur_comp,
trace_bcs,
mat_type,
self.ctx.fc_params,
options_prefix=prefix)
# dm associated with the trace problem
trace_dm = TraceSpace.dm
# KSP for the system of Lagrange multipliers
trace_ksp = PETSc.KSP().create(comm=pc.comm)
trace_ksp.incrementTabLevel(1, parent=pc)
# Set the dm for the trace solver
trace_ksp.setDM(trace_dm)
trace_ksp.setDMActive(False)
trace_ksp.setOptionsPrefix(prefix)
trace_ksp.setOperators(Smat, Smat)
# Option to add custom monitor
monitor = self.ctx.appctx.get('custom_monitor', None)
if monitor:
monitor.add_reconstructor(self.backward_substitution)
trace_ksp.setMonitor(monitor)
self.trace_ksp = trace_ksp
with dmhooks.add_hooks(trace_dm,
self,
appctx=self._ctx_ref,
save=False):
trace_ksp.setFromOptions()
split_mixed_op = dict(split_form(Atilde.form))
split_trace_op = dict(split_form(K.form))
# Generate reconstruction calls
self._reconstruction_calls(split_mixed_op, split_trace_op)
def test_diff_then_integrate():
# Define 1D geometry
n = 21
mesh = UnitIntervalMesh(MPI.COMM_WORLD, n)
# Shift and scale mesh
x0, x1 = 1.5, 3.14
mesh.coordinates()[:] *= (x1 - x0)
mesh.coordinates()[:] += x0
x = SpatialCoordinate(mesh)[0]
xs = 0.1 + 0.8 * x / x1 # scaled to be within [0.1,0.9]
# Define list of expressions to test, and configure
# accuracies these expressions are known to pass with.
# The reason some functions are less accurately integrated is
# likely that the default choice of quadrature rule is not perfect
F_list = []
def reg(exprs, acc=10):
for expr in exprs:
F_list.append((expr, acc))
# FIXME: 0*dx and 1*dx fails in the ufl-ffcx-jit framework somewhere
# reg([Constant(0.0, cell=cell)])
# reg([Constant(1.0, cell=cell)])
monomial_list = [x**q for q in range(2, 6)]
reg(monomial_list)
reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list])
reg([x**x])
reg([x**(x**2)], 8)
reg([x**(x**3)], 6)
reg([x**(x**4)], 2)
# Special functions:
reg([atan(xs)], 8)
reg([sin(x), cos(x), exp(x)], 5)
reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3)
reg([asin(xs), acos(xs)], 1)
reg([tan(xs)], 7)
try:
import scipy
except ImportError:
scipy = None
if hasattr(math, 'erf') or scipy is not None:
reg([erf(xs)])
else:
print(
"Warning: skipping test of erf, old python version and no scipy.")
# if 0:
# print("Warning: skipping tests of bessel functions, doesn't build on all platforms.")
# elif scipy is None:
# print("Warning: skipping tests of bessel functions, missing scipy.")
# else:
# for nu in (0, 1, 2):
# # Many of these are possibly more accurately integrated,
# # but 4 covers all and is sufficient for this test
# reg([bessel_J(nu, xs), bessel_Y(nu, xs), bessel_I(nu, xs), bessel_K(nu, xs)], 4)
# To handle tensor algebra, make an x dependent input tensor
# xx and square all expressions
def reg2(exprs, acc=10):
for expr in exprs:
F_list.append((inner(expr, expr), acc))
xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]])
x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4])
cc = as_matrix([[2, 3], [4, 5]])
reg2([xx])
reg2([x3v])
reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))])
reg2([xx.T])
reg2([tr(xx)])
reg2([det(xx)])
reg2([dot(xx, 0.1 * xx)])
reg2([outer(xx, xx.T)])
reg2([dev(xx)])
reg2([sym(xx)])
reg2([skew(xx)])
reg2([elem_mult(7 * xx, cc)])
reg2([elem_div(7 * xx, xx + cc)])
reg2([elem_pow(1e-3 * xx, 1e-3 * cc)])
reg2([elem_pow(1e-3 * cc, 1e-3 * xx)])
reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate...
# FIXME: Add tests for all UFL operators:
# These cause discontinuities and may be harder to test in the
# above fashion:
# 'inv', 'cofac',
# 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not',
# 'conditional', 'sign',
# 'jump', 'avg',
# 'LiftingFunction', 'LiftingOperator',
# FIXME: Test other derivatives: (but algorithms for operator
# derivatives are the same!):
# 'variable', 'diff',
# 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative',
# Run through all operators defined above and compare integrals
debug = 0
for F, acc in F_list:
# Apply UFL differentiation
f = diff(F, SpatialCoordinate(mesh))[..., 0]
if debug:
print(F)
print(x)
print(f)
# Apply integration with DOLFINX
# (also passes through form compilation and jit)
M = f * dx
f_integral = assemble_scalar(M) # noqa
f_integral = mesh.mpi_comm().allreduce(f_integral, op=MPI.SUM)
# Compute integral of f manually from anti-derivative F
# (passes through pybind11 interface and uses UFL evaluation)
F_diff = F((x1, )) - F((x0, ))
# Compare results. Using custom relative delta instead
# of decimal digits here because some numbers are >> 1.
delta = min(abs(f_integral), abs(F_diff)) * 10**-acc
assert f_integral - F_diff <= delta
u = ufl.TrialFunction(space)
phi = ufl.TestFunction(space)
dt = dune.ufl.Constant(0.01, "timeStep")
t = dune.ufl.Constant(0.0, "time")
# define storage for discrete solutions
uh = space.interpolate(x, name="uh")
uh_old = uh.copy()
# problem definition
# space form
xForm = inner(grad(u), grad(phi)) * dx
# add time discretization
form = dot(u - uh_old, phi) * dx + dt * xForm
# define dirichlet boundary conditions
bc = dune.ufl.DirichletBC(space, x)
# setup scheme
dune.fem.parameter.append({"fem.verboserank": 0})
solverParameters =\
{"newton.tolerance": 1e-9,
"newton.linear.tolerance": 1e-11,
"newton.linear.preconditioning.method": "ilu",
"newton.verbose": False,
"newton.linear.verbose": False}
scheme = solutionScheme([form == 0, bc],
space,
solver="cg",
def solve(self, problem):
self.problem = problem
doSave = problem.doSave
save_this_step = False
onlyVel = problem.saveOnlyVel
dt = self.metadata['dt']
nu = Constant(self.problem.nu)
self.tc.init_watch('init', 'Initialization', True, count_to_percent=False)
self.tc.init_watch('rhs', 'Assembled right hand side', True, count_to_percent=True)
self.tc.init_watch('applybc1', 'Applied velocity BC 1st step', True, count_to_percent=True)
self.tc.init_watch('applybc3', 'Applied velocity BC 3rd step', True, count_to_percent=True)
self.tc.init_watch('applybcP', 'Applied pressure BC or othogonalized rhs', True, count_to_percent=True)
self.tc.init_watch('assembleMatrices', 'Initial matrix assembly', False, count_to_percent=True)
self.tc.init_watch('solve 1', 'Running solver on 1st step', True, count_to_percent=True)
self.tc.init_watch('solve 2', 'Running solver on 2nd step', True, count_to_percent=True)
self.tc.init_watch('solve 3', 'Running solver on 3rd step', True, count_to_percent=True)
self.tc.init_watch('solve 4', 'Running solver on 4th step', True, count_to_percent=True)
self.tc.init_watch('assembleA1', 'Assembled A1 matrix (without stabiliz.)', True, count_to_percent=True)
self.tc.init_watch('assembleA1stab', 'Assembled A1 stabilization', True, count_to_percent=True)
self.tc.init_watch('next', 'Next step assignments', True, count_to_percent=True)
self.tc.init_watch('saveVel', 'Saved velocity', True)
self.tc.start('init')
# Define function spaces (P2-P1)
mesh = self.problem.mesh
self.V = VectorFunctionSpace(mesh, "Lagrange", 2) # velocity
self.Q = FunctionSpace(mesh, "Lagrange", 1) # pressure
self.PS = FunctionSpace(mesh, "Lagrange", 2) # partial solution (must be same order as V)
self.D = FunctionSpace(mesh, "Lagrange", 1) # velocity divergence space
problem.initialize(self.V, self.Q, self.PS, self.D)
# Define trial and test functions
u = TrialFunction(self.V)
v = TestFunction(self.V)
p = TrialFunction(self.Q)
q = TestFunction(self.Q)
n = FacetNormal(mesh)
I = Identity(find_geometric_dimension(u))
# Initial conditions: u0 velocity at previous time step u1 velocity two time steps back p0 previous pressure
[u1, u0, p0] = self.problem.get_initial_conditions([{'type': 'v', 'time': -dt},
{'type': 'v', 'time': 0.0},
{'type': 'p', 'time': 0.0}])
u_ = Function(self.V) # current tentative velocity
u_cor = Function(self.V) # current corrected velocity
p_ = Function(self.Q) # current pressure or pressure help function from rotation scheme
p_mod = Function(self.Q) # current modified pressure from rotation scheme
# Define coefficients
k = Constant(self.metadata['dt'])
f = Constant((0, 0, 0))
# Define forms
# step 1: Tentative velocity, solve to u_
u_ext = 1.5 * u0 - 0.5 * u1 # extrapolation for convection term
# Stabilisation
h = CellSize(mesh)
if self.args.cbc_tau:
# used in Simula cbcflow project
tau = Constant(self.stabCoef) * h / (sqrt(inner(u_ext, u_ext)) + h)
else:
# proposed in R. Codina: On stabilized finite element methods for linear systems of
# convection-diffusion-reaction equations.
tau = Constant(self.stabCoef) * k * h ** 2 / (
2 * nu * k + k * h * sqrt(DOLFIN_EPS + inner(u_ext, u_ext)) + h ** 2)
# DOLFIN_EPS is added because of FEniCS bug that inner(u_ext, u_ext) can be negative when u_ext = 0
if self.use_full_SUPG:
v1 = v + tau * 0.5 * dot(grad(v), u_ext)
parameters['form_compiler']['quadrature_degree'] = 6
else:
v1 = v
def nonlinearity(function):
if self.args.ema:
return 2 * inner(dot(sym(grad(function)), u_ext), v1) * dx + inner(div(function) * u_ext, v1) * dx
else:
return inner(dot(grad(function), u_ext), v1) * dx
def diffusion(fce):
if self.useLaplace:
return nu * inner(grad(fce), grad(v1)) * dx
else:
form = inner(nu * 2 * sym(grad(fce)), sym(grad(v1))) * dx
if self.bcv == 'CDN':
return form
if self.bcv == 'LAP':
return form - inner(nu * dot(grad(fce).T, n), v1) * problem.get_outflow_measure_form()
if self.bcv == 'DDN':
return form # additional term must be added to non-constant part
def pressure_rhs():
if self.args.bc == 'outflow':
return inner(p0, div(v1)) * dx
else:
return inner(p0, div(v1)) * dx - inner(p0 * n, v1) * problem.get_outflow_measure_form()
a1_const = (1. / k) * inner(u, v1) * dx + diffusion(0.5 * u)
a1_change = nonlinearity(0.5 * u)
if self.bcv == 'DDN':
# does not penalize influx for current step, only for the next one
# this can lead to oscilation:
# DDN correct next step, but then u_ext is OK so in next step DDN is not used, leading to new influx...
# u and u_ext cannot be switched, min_value is nonlinear function
a1_change += -0.5 * min_value(Constant(0.), inner(u_ext, n)) * inner(u,
v1) * problem.get_outflow_measure_form()
# NT works only with uflacs compiler
L1 = (1. / k) * inner(u0, v1) * dx - nonlinearity(0.5 * u0) - diffusion(0.5 * u0) + pressure_rhs()
if self.bcv == 'DDN':
L1 += 0.5 * min_value(0., inner(u_ext, n)) * inner(u0, v1) * problem.get_outflow_measure_form()
# Non-consistent SUPG stabilisation
if self.stabilize and not self.use_full_SUPG:
# a1_stab = tau*inner(dot(grad(u), u_ext), dot(grad(v), u_ext))*dx
a1_stab = 0.5 * tau * inner(dot(grad(u), u_ext), dot(grad(v), u_ext)) * dx(None, {'quadrature_degree': 6})
# optional: to use Crank Nicolson in stabilisation term following change of RHS is needed:
# L1 += -0.5*tau*inner(dot(grad(u0), u_ext), dot(grad(v), u_ext))*dx(None, {'quadrature_degree': 6})
outflow_area = Constant(problem.outflow_area)
need_outflow = Constant(0.0)
if self.useRotationScheme:
# Rotation scheme
F2 = inner(grad(p), grad(q)) * dx + (1. / k) * q * div(u_) * dx
else:
# Projection, solve to p_
if self.forceOutflow and problem.can_force_outflow:
info('Forcing outflow.')
F2 = inner(grad(p - p0), grad(q)) * dx + (1. / k) * q * div(u_) * dx
for m in problem.get_outflow_measures():
F2 += (1. / k) * (1. / outflow_area) * need_outflow * q * m
else:
F2 = inner(grad(p - p0), grad(q)) * dx + (1. / k) * q * div(u_) * dx
a2, L2 = system(F2)
# step 3: Finalize, solve to u_
if self.useRotationScheme:
# Rotation scheme
F3 = (1. / k) * inner(u - u_, v) * dx + inner(grad(p_), v) * dx
else:
F3 = (1. / k) * inner(u - u_, v) * dx + inner(grad(p_ - p0), v) * dx
a3, L3 = system(F3)
if self.useRotationScheme:
# Rotation scheme: modify pressure
F4 = (p - p0 - p_ + nu * div(u_)) * q * dx
a4, L4 = system(F4)
# Assemble matrices
self.tc.start('assembleMatrices')
A1_const = assemble(a1_const) # must be here, so A1 stays one Python object during repeated assembly
A1_change = A1_const.copy() # copy to get matrix with same sparse structure (data will be overwritten)
if self.stabilize and not self.use_full_SUPG:
A1_stab = A1_const.copy() # copy to get matrix with same sparse structure (data will be overwritten)
A2 = assemble(a2)
A3 = assemble(a3)
if self.useRotationScheme:
A4 = assemble(a4)
self.tc.end('assembleMatrices')
if self.solvers == 'direct':
self.solver_vel_tent = LUSolver('mumps')
self.solver_vel_cor = LUSolver('mumps')
self.solver_p = LUSolver('mumps')
if self.useRotationScheme:
self.solver_rot = LUSolver('mumps')
else:
# NT 2016-1 KrylovSolver >> PETScKrylovSolver
# not needed, chosen not to use hypre_parasails:
# if self.prec_v == 'hypre_parasails': # in FEniCS 1.6.0 inaccessible using KrylovSolver class
# self.solver_vel_tent = PETScKrylovSolver('gmres') # PETSc4py object
# self.solver_vel_tent.ksp().getPC().setType('hypre')
# PETScOptions.set('pc_hypre_type', 'parasails')
# # this is global setting, but preconditioners for pressure solvers are set by their constructors
# else:
self.solver_vel_tent = PETScKrylovSolver('gmres', self.args.precV) # nonsymetric > gmres
# cannot use 'ilu' in parallel
self.solver_vel_cor = PETScKrylovSolver('cg', self.args.precVC)
self.solver_p = PETScKrylovSolver(self.args.solP, self.args.precP) # almost (up to BC) symmetric > CG
if self.useRotationScheme:
self.solver_rot = PETScKrylovSolver('cg', 'hypre_amg')
# setup Krylov solvers
if self.solvers == 'krylov':
# Get the nullspace if there are no pressure boundary conditions
foo = Function(self.Q) # auxiliary vector for setting pressure nullspace
if self.args.bc == 'nullspace':
null_vec = Vector(foo.vector())
self.Q.dofmap().set(null_vec, 1.0)
null_vec *= 1.0 / null_vec.norm('l2')
self.null_space = VectorSpaceBasis([null_vec])
as_backend_type(A2).set_nullspace(self.null_space)
# apply global options for Krylov solvers
solver_options = {'monitor_convergence': True, 'maximum_iterations': 10000, 'nonzero_initial_guess': True}
# 'nonzero_initial_guess': True with solver.solve(A, u, b) means that
# Solver will use anything stored in u as an initial guess
for solver in [self.solver_vel_tent, self.solver_vel_cor, self.solver_rot, self.solver_p] if \
self.useRotationScheme else [self.solver_vel_tent, self.solver_vel_cor, self.solver_p]:
for key, value in solver_options.items():
try:
solver.parameters[key] = value
except KeyError:
info('Invalid option %s for KrylovSolver' % key)
return 1
if self.args.solP == 'richardson':
self.solver_p.parameters['monitor_convergence'] = False
self.solver_vel_tent.parameters['relative_tolerance'] = 10 ** (-self.args.prv1)
self.solver_vel_tent.parameters['absolute_tolerance'] = 10 ** (-self.args.pav1)
self.solver_vel_cor.parameters['relative_tolerance'] = 10E-12
self.solver_vel_cor.parameters['absolute_tolerance'] = 10E-4
self.solver_p.parameters['relative_tolerance'] = 10 ** (-self.args.prp)
self.solver_p.parameters['absolute_tolerance'] = 10 ** (-self.args.pap)
if self.useRotationScheme:
self.solver_rot.parameters['relative_tolerance'] = 10E-10
self.solver_rot.parameters['absolute_tolerance'] = 10E-10
if self.args.Vrestart > 0:
self.solver_vel_tent.parameters['gmres']['restart'] = self.args.Vrestart
if self.args.solP == 'gmres' and self.args.Prestart > 0:
self.solver_p.parameters['gmres']['restart'] = self.args.Prestart
# boundary conditions
bcu, bcp = problem.get_boundary_conditions(self.args.bc == 'outflow', self.V, self.Q)
self.tc.end('init')
# Time-stepping
info("Running of Incremental pressure correction scheme n. 1")
ttime = self.metadata['time']
t = dt
step = 1
# debug function
if problem.args.debug_rot:
plot_cor_v = Function(self.V)
while t < (ttime + dt / 2.0):
self.problem.update_time(t, step)
if self.MPI_rank == 0:
problem.write_status_file(t)
if doSave:
save_this_step = problem.save_this_step
# assemble matrix (it depends on solution)
self.tc.start('assembleA1')
assemble(a1_change, tensor=A1_change) # assembling into existing matrix is faster than assembling new one
A1 = A1_const.copy() # we dont want to change A1_const
A1.axpy(1, A1_change, True)
self.tc.end('assembleA1')
self.tc.start('assembleA1stab')
if self.stabilize and not self.use_full_SUPG:
assemble(a1_stab, tensor=A1_stab) # assembling into existing matrix is faster than assembling new one
A1.axpy(1, A1_stab, True)
self.tc.end('assembleA1stab')
# Compute tentative velocity step
begin("Computing tentative velocity")
self.tc.start('rhs')
b = assemble(L1)
self.tc.end('rhs')
self.tc.start('applybc1')
[bc.apply(A1, b) for bc in bcu]
self.tc.end('applybc1')
try:
self.tc.start('solve 1')
self.solver_vel_tent.solve(A1, u_.vector(), b)
self.tc.end('solve 1')
if save_this_step:
self.tc.start('saveVel')
problem.save_vel(True, u_)
self.tc.end('saveVel')
if save_this_step and not onlyVel:
problem.save_div(True, u_)
problem.compute_err(True, u_, t)
problem.compute_div(True, u_)
except RuntimeError as inst:
problem.report_fail(t)
return 1
end()
if self.useRotationScheme:
begin("Computing tentative pressure")
else:
begin("Computing pressure")
if self.forceOutflow and problem.can_force_outflow:
out = problem.compute_outflow(u_)
info('Tentative outflow: %f' % out)
n_o = -problem.last_inflow - out
info('Needed outflow: %f' % n_o)
need_outflow.assign(n_o)
self.tc.start('rhs')
b = assemble(L2)
self.tc.end('rhs')
self.tc.start('applybcP')
[bc.apply(A2, b) for bc in bcp]
if self.args.bc == 'nullspace':
self.null_space.orthogonalize(b)
self.tc.end('applybcP')
try:
self.tc.start('solve 2')
self.solver_p.solve(A2, p_.vector(), b)
self.tc.end('solve 2')
except RuntimeError as inst:
problem.report_fail(t)
return 1
if self.useRotationScheme:
foo = Function(self.Q)
foo.assign(p_ + p0)
if save_this_step and not onlyVel:
problem.averaging_pressure(foo)
problem.save_pressure(True, foo)
else:
foo = Function(self.Q)
foo.assign(p_) # we do not want to change p_ by averaging
if save_this_step and not onlyVel:
problem.averaging_pressure(foo)
problem.save_pressure(False, foo)
end()
begin("Computing corrected velocity")
self.tc.start('rhs')
b = assemble(L3)
self.tc.end('rhs')
if not self.args.B:
self.tc.start('applybc3')
[bc.apply(A3, b) for bc in bcu]
self.tc.end('applybc3')
try:
self.tc.start('solve 3')
self.solver_vel_cor.solve(A3, u_cor.vector(), b)
self.tc.end('solve 3')
problem.compute_err(False, u_cor, t)
problem.compute_div(False, u_cor)
except RuntimeError as inst:
problem.report_fail(t)
return 1
if save_this_step:
self.tc.start('saveVel')
problem.save_vel(False, u_cor)
self.tc.end('saveVel')
if save_this_step and not onlyVel:
problem.save_div(False, u_cor)
end()
if self.useRotationScheme:
begin("Rotation scheme pressure correction")
self.tc.start('rhs')
b = assemble(L4)
self.tc.end('rhs')
try:
self.tc.start('solve 4')
self.solver_rot.solve(A4, p_mod.vector(), b)
self.tc.end('solve 4')
except RuntimeError as inst:
problem.report_fail(t)
return 1
if save_this_step and not onlyVel:
problem.averaging_pressure(p_mod)
problem.save_pressure(False, p_mod)
end()
if problem.args.debug_rot:
# save applied pressure correction (expressed as a term added to RHS of next tentative vel. step)
# see comment next to argument definition
plot_cor_v.assign(project(k * grad(nu * div(u_)), self.V))
problem.fileDict['grad_cor']['file'].write(plot_cor_v, t)
# compute functionals (e. g. forces)
problem.compute_functionals(u_cor, p_mod if self.useRotationScheme else p_, t, step)
# Move to next time step
self.tc.start('next')
u1.assign(u0)
u0.assign(u_cor)
u_.assign(u_cor) # use corrected velocity as initial guess in first step
if self.useRotationScheme:
p0.assign(p_mod)
else:
p0.assign(p_)
t = round(t + dt, 6) # round time step to 0.000001
step += 1
self.tc.end('next')
info("Finished: Incremental pressure correction scheme n. 1")
problem.report()
return 0
def damage_dissipation_density(self, alpha):
w_1 = self.w(1)
return self.w(alpha) + w_1 * self.ell**2 * dot(grad(alpha),
grad(alpha))
def nitsche_ufl(mesh: dmesh.Mesh, mesh_data: Tuple[_cpp.mesh.MeshTags_int32, int, int],
physical_parameters: dict = {}, nitsche_parameters: Dict[str, float] = {},
plane_loc: float = 0.0, vertical_displacement: float = -0.1,
nitsche_bc: bool = True, quadrature_degree: int = 5, form_compiler_params: Dict = {},
jit_params: Dict = {}, petsc_options: Dict = {}, newton_options: Dict = {}) -> _fem.Function:
"""
Use UFL to compute the one sided contact problem with a mesh coming into contact
with a rigid surface (not meshed).
Parameters
==========
mesh
The input mesh
mesh_data
A triplet with a mesh tag for facets and values v0, v1. v0 should be the value in the mesh tags
for facets to apply a Dirichlet condition on. v1 is the value for facets which should have applied
a contact condition on
physical_parameters
Optional dictionary with information about the linear elasticity problem.
Valid (key, value) tuples are: ('E': float), ('nu', float), ('strain', bool)
nitsche_parameters
Optional dictionary with information about the Nitsche configuration.
Valid (keu, value) tuples are: ('gamma', float), ('theta', float) where theta can be -1, 0 or 1 for
skew-symmetric, penalty like or symmetric enforcement of Nitsche conditions
plane_loc
The location of the plane in y-coordinate (2D) and z-coordinate (3D)
vertical_displacement
The amount of verticial displacment enforced on Dirichlet boundary
nitsche_bc
Use Nitche's method to enforce Dirichlet boundary conditions
quadrature_degree
The quadrature degree to use for the custom contact kernels
form_compiler_params
Parameters used in FFCX compilation of this form. Run `ffcx --help` at
the commandline to see all available options. Takes priority over all
other parameter values, except for `scalar_type` which is determined by
DOLFINX.
jit_params
Parameters used in CFFI JIT compilation of C code generated by FFCX.
See https://github.com/FEniCS/dolfinx/blob/main/python/dolfinx/jit.py
for all available parameters. Takes priority over all other parameter values.
petsc_options
Parameters that is passed to the linear algebra backend
PETSc. For available choices for the 'petsc_options' kwarg,
see the `PETSc-documentation
<https://petsc4py.readthedocs.io/en/stable/manual/ksp/>`
newton_options
Dictionary with Newton-solver options. Valid (key, item) tuples are:
("atol", float), ("rtol", float), ("convergence_criterion", "str"),
("max_it", int), ("error_on_nonconvergence", bool), ("relaxation_parameter", float)
"""
# Compute lame parameters
plane_strain = physical_parameters.get("strain", False)
E = physical_parameters.get("E", 1e3)
nu = physical_parameters.get("nu", 0.1)
mu_func, lambda_func = lame_parameters(plane_strain)
mu = mu_func(E, nu)
lmbda = lambda_func(E, nu)
sigma = sigma_func(mu, lmbda)
# Nitche parameters and variables
theta = nitsche_parameters.get("theta", 1)
gamma = nitsche_parameters.get("gamma", 1)
(facet_marker, top_value, bottom_value) = mesh_data
assert(facet_marker.dim == mesh.topology.dim - 1)
# Normal vector pointing into plane (but outward of the body coming into contact)
# Similar to computing the normal by finding the gap vector between two meshes
n_vec = np.zeros(mesh.geometry.dim)
n_vec[mesh.geometry.dim - 1] = -1
n_2 = ufl.as_vector(n_vec) # Normal of plane (projection onto other body)
# Scaled Nitsche parameter
h = ufl.CellDiameter(mesh)
gamma_scaled = gamma * E / h
# Mimicking the plane y=-plane_loc
x = ufl.SpatialCoordinate(mesh)
gap = x[mesh.geometry.dim - 1] + plane_loc
g_vec = [i for i in range(mesh.geometry.dim)]
g_vec[mesh.geometry.dim - 1] = gap
V = _fem.VectorFunctionSpace(mesh, ("CG", 1))
u = _fem.Function(V)
v = ufl.TestFunction(V)
metadata = {"quadrature_degree": quadrature_degree}
dx = ufl.Measure("dx", domain=mesh)
ds = ufl.Measure("ds", domain=mesh, metadata=metadata,
subdomain_data=facet_marker)
a = ufl.inner(sigma(u), epsilon(v)) * dx
zero = np.asarray([0, ] * mesh.geometry.dim, dtype=_PETSc.ScalarType)
L = ufl.inner(_fem.Constant(mesh, zero), v) * dx
# Derivation of one sided Nitsche with gap function
n = ufl.FacetNormal(mesh)
def sigma_n(v):
# NOTE: Different normals, see summary paper
return ufl.dot(sigma(v) * n, n_2)
F = a - theta / gamma_scaled * sigma_n(u) * sigma_n(v) * ds(bottom_value) - L
F += 1 / gamma_scaled * R_minus(sigma_n(u) + gamma_scaled * (gap - ufl.dot(u, n_2))) * \
(theta * sigma_n(v) - gamma_scaled * ufl.dot(v, n_2)) * ds(bottom_value)
# Compute corresponding Jacobian
du = ufl.TrialFunction(V)
q = sigma_n(u) + gamma_scaled * (gap - ufl.dot(u, n_2))
J = ufl.inner(sigma(du), epsilon(v)) * ufl.dx - theta / gamma_scaled * sigma_n(du) * sigma_n(v) * ds(bottom_value)
J += 1 / gamma_scaled * 0.5 * (1 - ufl.sign(q)) * (sigma_n(du) - gamma_scaled * ufl.dot(du, n_2)) * \
(theta * sigma_n(v) - gamma_scaled * ufl.dot(v, n_2)) * ds(bottom_value)
# Nitsche for Dirichlet, another theta-scheme.
# https://doi.org/10.1016/j.cma.2018.05.024
if nitsche_bc:
disp_vec = np.zeros(mesh.geometry.dim)
disp_vec[mesh.geometry.dim - 1] = vertical_displacement
u_D = ufl.as_vector(disp_vec)
F += - ufl.inner(sigma(u) * n, v) * ds(top_value)\
- theta * ufl.inner(sigma(v) * n, u - u_D) * \
ds(top_value) + gamma_scaled / h * ufl.inner(u - u_D, v) * ds(top_value)
bcs = []
J += - ufl.inner(sigma(du) * n, v) * ds(top_value)\
- theta * ufl.inner(sigma(v) * n, du) * \
ds(top_value) + gamma_scaled / h * ufl.inner(du, v) * ds(top_value)
else:
# strong Dirichlet boundary conditions
def _u_D(x):
values = np.zeros((mesh.geometry.dim, x.shape[1]))
values[mesh.geometry.dim - 1] = vertical_displacement
return values
u_D = _fem.Function(V)
u_D.interpolate(_u_D)
u_D.name = "u_D"
u_D.x.scatter_forward()
tdim = mesh.topology.dim
dirichlet_dofs = _fem.locate_dofs_topological(V, tdim - 1, facet_marker.find(top_value))
bc = _fem.dirichletbc(u_D, dirichlet_dofs)
bcs = [bc]
# DEBUG: Write each step of Newton iterations
# Create nonlinear problem and Newton solver
# def form(self, x: _PETSc.Vec):
# x.ghostUpdate(addv=_PETSc.InsertMode.INSERT, mode=_PETSc.ScatterMode.FORWARD)
# self.i += 1
# xdmf.write_function(u, self.i)
# setattr(_fem.petsc.NonlinearProblem, "form", form)
problem = _fem.petsc.NonlinearProblem(F, u, bcs, J=J, jit_params=jit_params,
form_compiler_params=form_compiler_params)
# DEBUG: Write each step of Newton iterations
# problem.i = 0
# xdmf = _io.XDMFFile(mesh.comm, "results/tmp_sol.xdmf", "w")
# xdmf.write_mesh(mesh)
solver = _nls.petsc.NewtonSolver(mesh.comm, problem)
null_space = rigid_motions_nullspace(V)
solver.A.setNearNullSpace(null_space)
# Set Newton solver options
solver.atol = newton_options.get("atol", 1e-9)
solver.rtol = newton_options.get("rtol", 1e-9)
solver.convergence_criterion = newton_options.get("convergence_criterion", "incremental")
solver.max_it = newton_options.get("max_it", 50)
solver.error_on_nonconvergence = newton_options.get("error_on_nonconvergence", True)
solver.relaxation_parameter = newton_options.get("relaxation_parameter", 0.8)
def _u_initial(x):
values = np.zeros((mesh.geometry.dim, x.shape[1]))
values[-1] = -0.01 - plane_loc
return values
# Set initial_condition:
u.interpolate(_u_initial)
# Define solver and options
ksp = solver.krylov_solver
opts = _PETSc.Options()
option_prefix = ksp.getOptionsPrefix()
# Set PETSc options
opts = _PETSc.Options()
opts.prefixPush(option_prefix)
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
ksp.setFromOptions()
# Solve non-linear problem
_log.set_log_level(_log.LogLevel.INFO)
num_dofs_global = V.dofmap.index_map_bs * V.dofmap.index_map.size_global
with _common.Timer(f"{num_dofs_global} Solve Nitsche"):
n, converged = solver.solve(u)
u.x.scatter_forward()
if solver.error_on_nonconvergence:
assert(converged)
print(f"{num_dofs_global}, Number of interations: {n:d}")
return u
def sigma_n(v):
# NOTE: Different normals, see summary paper
return ufl.dot(sigma(v) * n, n_2)
import math
from ufl import SpatialCoordinate, sin, pi, dot, as_vector
from dune.grid import structuredGrid, gridFunction
from dune.fem.space import lagrange
grid = structuredGrid([0, 0], [1, 1], [5, 5])
space = lagrange(grid, order=3, dimRange=2)
x = SpatialCoordinate(space)
df = space.interpolate([
sin(2**i * pi * x[0] * x[1] / (0.25**(2 * i) + dot(x, x)))
for i in range(1, 3)
],
name="femgf")
@gridFunction(grid)
def f(x):
return [
math.sin(2**i * math.pi * x[0] * x[1] /
(0.25**(2 * i) + x[0] * x[0] + x[1] * x[1]))
for i in range(1, 3)
]
from dune.vtk import vtkWriter
writer = vtkWriter(grid,
"testFem",
pointScalar={
"f":
f,
ds = Measure("ds", subdomain_data=boundary_parts)
# Define forms (dont redefine functions used here)
# step 1
u0 = Function(V)
u1 = Function(V)
p0 = Function(Q)
u_tent = TrialFunction(V)
v = TestFunction(V)
# U_ = 1.5*u0 - 0.5*u1
# nonlinearity = inner(dot(0.5 * (u_tent.dx(0) + u0.dx(0)), U_), v) * dx
# F_tent = (1./dt)*inner(u_tent - u0, v) * dx + nonlinearity\
# + nu*inner(0.5 * (u_tent.dx(0) + u0.dx(0)), v.dx(0)) * dx + inner(p0.dx(0), v) * dx\
# - inner(f, v)*dx # solve to u_
# using explicite scheme: so LHS has interpretation as heat equation, RHS are sources
F_tent = (1./dt)*inner(u_tent - u0, v)*dx + inner(dot(u0.dx(0), u0), v)*dx + nu*inner((u_tent.dx(0) + u0.dx(0)), v.dx(0)) * dx + inner(p0.dx(0), v) * dx\
- inner(f, v)*dx # solve to u_
a_tent, L_tent = system(F_tent)
# step 2
u_tent_computed = Function(V)
p = TrialFunction(Q)
q = TestFunction(Q)
F_p = inner(grad(p - p0), grad(q)) * dx + (1. / dt) * u_tent_computed.dx(
0) * q * dx # + 2*p.dx(0)*q*ds(1) # tried to force dp/dn=0 on inflow
# TEST: prescribe Neumann outflow BC
# F_p = inner(grad(p-p0), grad(q))*dx + (1./dt)*u_tent_computed.dx(0)*q*dx + (1./dt)*(v_in_expr-u_tent_computed)*q*ds(2)
a_p, L_p = system(F_p)
A_p = assemble(a_p)
as_backend_type(A_p).set_nullspace(null_space)
print(A_p.array())
# step 2 rotation
#
# Author: Martin Sandve Alnes
# Date: 2008-10-03
#
from ufl import Coefficient, FiniteElement, dot, dx, grad, triangle
element = FiniteElement("Lagrange", triangle, 1)
f = Coefficient(element)
a = (f * f + dot(grad(f), grad(f))) * dx
def nonlinearity(function):
if self.args.ema:
return 2 * inner(dot(sym(grad(function)), u_ext), v1) * dx + inner(div(function) * u_ext, v1) * dx
else:
return inner(dot(grad(function), u_ext), v1) * dx
def solve(self, problem):
self.problem = problem
doSave = problem.doSave
save_this_step = False
onlyVel = problem.saveOnlyVel
dt = self.metadata['dt']
nu = Constant(self.problem.nu)
# TODO check proper use of watches
self.tc.init_watch('init', 'Initialization', True, count_to_percent=False)
self.tc.init_watch('rhs', 'Assembled right hand side', True, count_to_percent=True)
self.tc.init_watch('updateBC', 'Updated velocity BC', True, count_to_percent=True)
self.tc.init_watch('applybc1', 'Applied velocity BC 1st step', True, count_to_percent=True)
self.tc.init_watch('applybc3', 'Applied velocity BC 3rd step', True, count_to_percent=True)
self.tc.init_watch('applybcP', 'Applied pressure BC or othogonalized rhs', True, count_to_percent=True)
self.tc.init_watch('assembleMatrices', 'Initial matrix assembly', False, count_to_percent=True)
self.tc.init_watch('solve 1', 'Running solver on 1st step', True, count_to_percent=True)
self.tc.init_watch('solve 2', 'Running solver on 2nd step', True, count_to_percent=True)
self.tc.init_watch('solve 3', 'Running solver on 3rd step', True, count_to_percent=True)
self.tc.init_watch('solve 4', 'Running solver on 4th step', True, count_to_percent=True)
self.tc.init_watch('assembleA1', 'Assembled A1 matrix (without stabiliz.)', True, count_to_percent=True)
self.tc.init_watch('assembleA1stab', 'Assembled A1 stabilization', True, count_to_percent=True)
self.tc.init_watch('next', 'Next step assignments', True, count_to_percent=True)
self.tc.init_watch('saveVel', 'Saved velocity', True)
self.tc.start('init')
# Define function spaces (P2-P1)
mesh = self.problem.mesh
self.V = VectorFunctionSpace(mesh, "Lagrange", 2) # velocity
self.Q = FunctionSpace(mesh, "Lagrange", 1) # pressure
self.PS = FunctionSpace(mesh, "Lagrange", 2) # partial solution (must be same order as V)
self.D = FunctionSpace(mesh, "Lagrange", 1) # velocity divergence space
if self.bc == 'lagrange':
L = FunctionSpace(mesh, "R", 0)
QL = self.Q*L
problem.initialize(self.V, self.Q, self.PS, self.D)
# Define trial and test functions
u = TrialFunction(self.V)
v = TestFunction(self.V)
if self.bc == 'lagrange':
(pQL, rQL) = TrialFunction(QL)
(qQL, lQL) = TestFunction(QL)
else:
p = TrialFunction(self.Q)
q = TestFunction(self.Q)
n = FacetNormal(mesh)
I = Identity(u.geometric_dimension())
# Initial conditions: u0 velocity at previous time step u1 velocity two time steps back p0 previous pressure
[u1, u0, p0] = self.problem.get_initial_conditions([{'type': 'v', 'time': -dt},
{'type': 'v', 'time': 0.0},
{'type': 'p', 'time': 0.0}])
if doSave:
problem.save_vel(False, u0, 0.0)
problem.save_vel(True, u0, 0.0)
u_ = Function(self.V) # current tentative velocity
u_cor = Function(self.V) # current corrected velocity
if self.bc == 'lagrange':
p_QL = Function(QL) # current pressure or pressure help function from rotation scheme
pQ = Function(self.Q) # auxiliary function for conversion between QL.sub(0) and Q
else:
p_ = Function(self.Q) # current pressure or pressure help function from rotation scheme
p_mod = Function(self.Q) # current modified pressure from rotation scheme
# Define coefficients
k = Constant(self.metadata['dt'])
f = Constant((0, 0, 0))
# Define forms
# step 1: Tentative velocity, solve to u_
u_ext = 1.5*u0 - 0.5*u1 # extrapolation for convection term
# Stabilisation
h = CellSize(mesh)
# CBC delta:
if self.cbcDelta:
delta = Constant(self.stabCoef)*h/(sqrt(inner(u_ext, u_ext))+h)
else:
delta = Constant(self.stabCoef)*h**2/(2*nu*k + k*h*inner(u_ext, u_ext)+h**2)
if self.use_full_SUPG:
v1 = v + delta*0.5*k*dot(grad(v), u_ext)
parameters['form_compiler']['quadrature_degree'] = 6
else:
v1 = v
def nonlinearity(function):
if self.use_ema:
return 2*inner(dot(sym(grad(function)), u_ext), v1) * dx + inner(div(function)*u_ext, v1) * dx
# return 2*inner(dot(sym(grad(function)), u_ext), v) * dx + inner(div(u_ext)*function, v) * dx
# QQ implement this way?
else:
return inner(dot(grad(function), u_ext), v1) * dx
def diffusion(fce):
if self.useLaplace:
return nu*inner(grad(fce), grad(v1)) * dx
else:
form = inner(nu * 2 * sym(grad(fce)), sym(grad(v1))) * dx
if self.bcv == 'CDN':
# IMP will work only if p=0 on output, or we must add term
# inner(p0*n, v)*problem.get_outflow_measure_form() to avoid boundary layer
return form
if self.bcv == 'LAP':
return form - inner(nu*dot(grad(fce).T, n), v1) * problem.get_outflow_measure_form()
if self.bcv == 'DDN':
# IMP will work only if p=0 on output, or we must add term
# inner(p0*n, v)*problem.get_outflow_measure_form() to avoid boundary layer
return form # additional term must be added to non-constant part
def pressure_rhs():
if self.useLaplace or self.bcv == 'LAP':
return inner(p0, div(v1)) * dx - inner(p0*n, v1) * problem.get_outflow_measure_form()
# NT term inner(inner(p, n), v) is 0 when p=0 on outflow
else:
return inner(p0, div(v1)) * dx
a1_const = (1./k)*inner(u, v1)*dx + diffusion(0.5*u)
a1_change = nonlinearity(0.5*u)
if self.bcv == 'DDN':
# IMP Problem: Does not penalize influx for current step, only for the next one
# IMP this can lead to oscilation: DDN correct next step, but then u_ext is OK so in next step DDN is not used, leading to new influx...
# u and u_ext cannot be switched, min_value is nonlinear function
a1_change += -0.5*min_value(Constant(0.), inner(u_ext, n))*inner(u, v1)*problem.get_outflow_measure_form()
# IMP works only with uflacs compiler
L1 = (1./k)*inner(u0, v1)*dx - nonlinearity(0.5*u0) - diffusion(0.5*u0) + pressure_rhs()
if self.bcv == 'DDN':
L1 += 0.5*min_value(0., inner(u_ext, n))*inner(u0, v1)*problem.get_outflow_measure_form()
# Non-consistent SUPG stabilisation
if self.stabilize and not self.use_full_SUPG:
# a1_stab = delta*inner(dot(grad(u), u_ext), dot(grad(v), u_ext))*dx
a1_stab = 0.5*delta*inner(dot(grad(u), u_ext), dot(grad(v), u_ext))*dx(None, {'quadrature_degree': 6})
# NT optional: use Crank Nicolson in stabilisation term: change RHS
# L1 += -0.5*delta*inner(dot(grad(u0), u_ext), dot(grad(v), u_ext))*dx(None, {'quadrature_degree': 6})
outflow_area = Constant(problem.outflow_area)
need_outflow = Constant(0.0)
if self.useRotationScheme:
# Rotation scheme
if self.bc == 'lagrange':
F2 = inner(grad(pQL), grad(qQL))*dx + (1./k)*qQL*div(u_)*dx + pQL*lQL*dx + qQL*rQL*dx
else:
F2 = inner(grad(p), grad(q))*dx + (1./k)*q*div(u_)*dx
else:
# Projection, solve to p_
if self.bc == 'lagrange':
F2 = inner(grad(pQL - p0), grad(qQL))*dx + (1./k)*qQL*div(u_)*dx + pQL*lQL*dx + qQL*rQL*dx
else:
if self.forceOutflow and problem.can_force_outflow:
info('Forcing outflow.')
F2 = inner(grad(p - p0), grad(q))*dx + (1./k)*q*div(u_)*dx
for m in problem.get_outflow_measures():
F2 += (1./k)*(1./outflow_area)*need_outflow*q*m
else:
F2 = inner(grad(p - p0), grad(q))*dx + (1./k)*q*div(u_)*dx
a2, L2 = system(F2)
# step 3: Finalize, solve to u_
if self.useRotationScheme:
# Rotation scheme
if self.bc == 'lagrange':
F3 = (1./k)*inner(u - u_, v)*dx + inner(grad(p_QL.sub(0)), v)*dx
else:
F3 = (1./k)*inner(u - u_, v)*dx + inner(grad(p_), v)*dx
else:
if self.bc == 'lagrange':
F3 = (1./k)*inner(u - u_, v)*dx + inner(grad(p_QL.sub(0) - p0), v)*dx
else:
F3 = (1./k)*inner(u - u_, v)*dx + inner(grad(p_ - p0), v)*dx
a3, L3 = system(F3)
if self.useRotationScheme:
# Rotation scheme: modify pressure
if self.bc == 'lagrange':
pr = TrialFunction(self.Q)
qr = TestFunction(self.Q)
F4 = (pr - p0 - p_QL.sub(0) + nu*div(u_))*qr*dx
else:
F4 = (p - p0 - p_ + nu*div(u_))*q*dx
# TODO zkusit, jestli to nebude rychlejsi? nepocitat soustavu, ale p.assign(...), nutno project(div(u),Q) coz je pocitani podobne soustavy
# TODO zkusit v project zadat solver_type='lu' >> primy resic by mel byt efektivnejsi
a4, L4 = system(F4)
# Assemble matrices
self.tc.start('assembleMatrices')
A1_const = assemble(a1_const) # need to be here, so A1 stays one Python object during repeated assembly
A1_change = A1_const.copy() # copy to get matrix with same sparse structure (data will be overwriten)
if self.stabilize and not self.use_full_SUPG:
A1_stab = A1_const.copy() # copy to get matrix with same sparse structure (data will be overwriten)
A2 = assemble(a2)
A3 = assemble(a3)
if self.useRotationScheme:
A4 = assemble(a4)
self.tc.end('assembleMatrices')
if self.solvers == 'direct':
self.solver_vel_tent = LUSolver('mumps')
self.solver_vel_cor = LUSolver('mumps')
self.solver_p = LUSolver('umfpack')
if self.useRotationScheme:
self.solver_rot = LUSolver('umfpack')
else:
# NT not needed, chosen not to use hypre_parasails
# if self.prec_v == 'hypre_parasails': # in FEniCS 1.6.0 inaccessible using KrylovSolver class
# self.solver_vel_tent = PETScKrylovSolver('gmres') # PETSc4py object
# self.solver_vel_tent.ksp().getPC().setType('hypre')
# PETScOptions.set('pc_hypre_type', 'parasails')
# # this is global setting, but preconditioners for pressure solvers are set by their constructors
# else:
self.solver_vel_tent = KrylovSolver('gmres', self.prec_v) # nonsymetric > gmres
# IMP cannot use 'ilu' in parallel (choose different default option)
self.solver_vel_cor = KrylovSolver('cg', 'hypre_amg') # nonsymetric > gmres
self.solver_p = KrylovSolver('cg', self.prec_p) # symmetric > CG
if self.useRotationScheme:
self.solver_rot = KrylovSolver('cg', self.prec_p)
solver_options = {'monitor_convergence': True, 'maximum_iterations': 1000, 'nonzero_initial_guess': True}
# 'nonzero_initial_guess': True with solver.solbe(A, u, b) means that
# Solver will use anything stored in u as an initial guess
# Get the nullspace if there are no pressure boundary conditions
foo = Function(self.Q) # auxiliary vector for setting pressure nullspace
if self.bc in ['nullspace', 'nullspace_s']:
null_vec = Vector(foo.vector())
self.Q.dofmap().set(null_vec, 1.0)
null_vec *= 1.0/null_vec.norm('l2')
self.null_space = VectorSpaceBasis([null_vec])
if self.bc == 'nullspace':
as_backend_type(A2).set_nullspace(self.null_space)
# apply global options for Krylov solvers
self.solver_vel_tent.parameters['relative_tolerance'] = 10 ** (-self.precision_rel_v_tent)
self.solver_vel_tent.parameters['absolute_tolerance'] = 10 ** (-self.precision_abs_v_tent)
self.solver_vel_cor.parameters['relative_tolerance'] = 10E-12
self.solver_vel_cor.parameters['absolute_tolerance'] = 10E-4
self.solver_p.parameters['relative_tolerance'] = 10**(-self.precision_p)
self.solver_p.parameters['absolute_tolerance'] = 10E-10
if self.useRotationScheme:
self.solver_rot.parameters['relative_tolerance'] = 10**(-self.precision_p)
self.solver_rot.parameters['absolute_tolerance'] = 10E-10
if self.solvers == 'krylov':
for solver in [self.solver_vel_tent, self.solver_vel_cor, self.solver_p, self.solver_rot] if \
self.useRotationScheme else [self.solver_vel_tent, self.solver_vel_cor, self.solver_p]:
for key, value in solver_options.items():
try:
solver.parameters[key] = value
except KeyError:
info('Invalid option %s for KrylovSolver' % key)
return 1
solver.parameters['preconditioner']['structure'] = 'same'
# matrices A2-A4 do not change, so we can reuse preconditioners
self.solver_vel_tent.parameters['preconditioner']['structure'] = 'same_nonzero_pattern'
# matrix A1 changes every time step, so change of preconditioner must be allowed
if self.bc == 'lagrange':
fa = FunctionAssigner(self.Q, QL.sub(0))
# boundary conditions
bcu, bcp = problem.get_boundary_conditions(self.bc == 'outflow', self.V, self.Q)
self.tc.end('init')
# Time-stepping
info("Running of Incremental pressure correction scheme n. 1")
ttime = self.metadata['time']
t = dt
step = 1
while t < (ttime + dt/2.0):
info("t = %f" % t)
self.problem.update_time(t, step)
if self.MPI_rank == 0:
problem.write_status_file(t)
if doSave:
save_this_step = problem.save_this_step
# DDN debug
# u_ext_in = assemble(inner(u_ext, n)*problem.get_outflow_measure_form())
# DDN_triggered = assemble(min_value(Constant(0.), inner(u_ext, n))*problem.get_outflow_measure_form())
# print('DDN: u_ext*n dSout = ', u_ext_in)
# print('DDN: negative part of u_ext*n dSout = ', DDN_triggered)
# assemble matrix (it depends on solution)
self.tc.start('assembleA1')
assemble(a1_change, tensor=A1_change) # assembling into existing matrix is faster than assembling new one
A1 = A1_const.copy() # we dont want to change A1_const
A1.axpy(1, A1_change, True)
self.tc.end('assembleA1')
self.tc.start('assembleA1stab')
if self.stabilize and not self.use_full_SUPG:
assemble(a1_stab, tensor=A1_stab) # assembling into existing matrix is faster than assembling new one
A1.axpy(1, A1_stab, True)
self.tc.end('assembleA1stab')
# Compute tentative velocity step
begin("Computing tentative velocity")
self.tc.start('rhs')
b = assemble(L1)
self.tc.end('rhs')
self.tc.start('applybc1')
[bc.apply(A1, b) for bc in bcu]
self.tc.end('applybc1')
try:
self.tc.start('solve 1')
self.solver_vel_tent.solve(A1, u_.vector(), b)
self.tc.end('solve 1')
if save_this_step:
self.tc.start('saveVel')
problem.save_vel(True, u_, t)
self.tc.end('saveVel')
if save_this_step and not onlyVel:
problem.save_div(True, u_)
problem.compute_err(True, u_, t)
problem.compute_div(True, u_)
except RuntimeError as inst:
problem.report_fail(t)
return 1
end()
# DDN debug
# u_ext_in = assemble(inner(u_, n)*problem.get_outflow_measure_form())
# DDN_triggered = assemble(min_value(Constant(0.), inner(u_, n))*problem.get_outflow_measure_form())
# print('DDN: u_tent*n dSout = ', u_ext_in)
# print('DDN: negative part of u_tent*n dSout = ', DDN_triggered)
if self.useRotationScheme:
begin("Computing tentative pressure")
else:
begin("Computing pressure")
if self.forceOutflow and problem.can_force_outflow:
out = problem.compute_outflow(u_)
info('Tentative outflow: %f' % out)
n_o = -problem.last_inflow-out
info('Needed outflow: %f' % n_o)
need_outflow.assign(n_o)
self.tc.start('rhs')
b = assemble(L2)
self.tc.end('rhs')
self.tc.start('applybcP')
[bc.apply(A2, b) for bc in bcp]
if self.bc in ['nullspace', 'nullspace_s']:
self.null_space.orthogonalize(b)
self.tc.end('applybcP')
try:
self.tc.start('solve 2')
if self.bc == 'lagrange':
self.solver_p.solve(A2, p_QL.vector(), b)
else:
self.solver_p.solve(A2, p_.vector(), b)
self.tc.end('solve 2')
except RuntimeError as inst:
problem.report_fail(t)
return 1
if self.useRotationScheme:
foo = Function(self.Q)
if self.bc == 'lagrange':
fa.assign(pQ, p_QL.sub(0))
foo.assign(pQ + p0)
else:
foo.assign(p_+p0)
problem.averaging_pressure(foo)
if save_this_step and not onlyVel:
problem.save_pressure(True, foo)
else:
if self.bc == 'lagrange':
fa.assign(pQ, p_QL.sub(0))
problem.averaging_pressure(pQ)
if save_this_step and not onlyVel:
problem.save_pressure(False, pQ)
else:
# we do not want to change p=0 on outflow, it conflicts with do-nothing conditions
foo = Function(self.Q)
foo.assign(p_)
problem.averaging_pressure(foo)
if save_this_step and not onlyVel:
problem.save_pressure(False, foo)
end()
begin("Computing corrected velocity")
self.tc.start('rhs')
b = assemble(L3)
self.tc.end('rhs')
if not self.B:
self.tc.start('applybc3')
[bc.apply(A3, b) for bc in bcu]
self.tc.end('applybc3')
try:
self.tc.start('solve 3')
self.solver_vel_cor.solve(A3, u_cor.vector(), b)
self.tc.end('solve 3')
problem.compute_err(False, u_cor, t)
problem.compute_div(False, u_cor)
except RuntimeError as inst:
problem.report_fail(t)
return 1
if save_this_step:
self.tc.start('saveVel')
problem.save_vel(False, u_cor, t)
self.tc.end('saveVel')
if save_this_step and not onlyVel:
problem.save_div(False, u_cor)
end()
# DDN debug
# u_ext_in = assemble(inner(u_cor, n)*problem.get_outflow_measure_form())
# DDN_triggered = assemble(min_value(Constant(0.), inner(u_cor, n))*problem.get_outflow_measure_form())
# print('DDN: u_cor*n dSout = ', u_ext_in)
# print('DDN: negative part of u_cor*n dSout = ', DDN_triggered)
if self.useRotationScheme:
begin("Rotation scheme pressure correction")
self.tc.start('rhs')
b = assemble(L4)
self.tc.end('rhs')
try:
self.tc.start('solve 4')
self.solver_rot.solve(A4, p_mod.vector(), b)
self.tc.end('solve 4')
except RuntimeError as inst:
problem.report_fail(t)
return 1
problem.averaging_pressure(p_mod)
if save_this_step and not onlyVel:
problem.save_pressure(False, p_mod)
end()
# compute functionals (e. g. forces)
problem.compute_functionals(u_cor,
p_mod if self.useRotationScheme else (pQ if self.bc == 'lagrange' else p_), t)
# Move to next time step
self.tc.start('next')
u1.assign(u0)
u0.assign(u_cor)
u_.assign(u_cor) # use corretced velocity as initial guess in first step
if self.useRotationScheme:
p0.assign(p_mod)
else:
if self.bc == 'lagrange':
p0.assign(pQ)
else:
p0.assign(p_)
t = round(t + dt, 6) # round time step to 0.000001
step += 1
self.tc.end('next')
info("Finished: Incremental pressure correction scheme n. 1")
problem.report()
return 0
def helmholtz(cell, degree):
m = Mesh(VectorElement('CG', cell, 1))
V = FunctionSpace(m, FiniteElement('CG', cell, degree))
u = TrialFunction(V)
v = TestFunction(V)
return (u*v + dot(grad(u), grad(v)))*dx
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho*self.wrho*self.dx
self.dU_integral = sum(
[tdUi*wUi*self.dx for tdUi,wUi in zip(self.tdUs, self.wUs)]
)
self.A = fe.assemble(self.drho_integral + self.dU_integral)
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drho, self.dUs = dsolsplit[0], dsolsplit[1:]
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iUs, self.irho)) - (
self.s2*ufl.grad(self.irho)/ufl.max_value(self.irho,
self.rho_min)
)
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (
self.vn('+')*ufl.max_value(self.irho('+'), 0.0) -
self.vn('-')*ufl.max_value(self.irho('-'), 0.0)
)
self.rho_flux_jump = -self.facet_flux*ufl.jump(self.wrho)*self.dS
self.rho_grad_move = ufl.dot(self.flux,
ufl.grad(self.wrho))*self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.irho, self.n),
ufl.jump(self.wrho, self.n)) * self.dS
)
self.grho_penalty = -(
self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) * self.dS
)
self.rho_terms = (
self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty
)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = sum(
[-lig.gamma * iUi * wUi * self.dx for
lig,iUi,wUi in
zip(self.ligands.ligands(), self.iUs, self.wUs)]
)
self.U_secretion = sum(
[lig.s * self.irho * wUi * self.dx for
lig,wUi in zip(self.ligands.ligands(), self.wUs)]
)
self.jump_gUw = sum(
[lig.D * ufl.jump(wUi * ufl.grad(iUi), self.n) * self.dS
for lig,wUi,iUi in
zip(self.ligands.ligands(), self.wUs, self.iUs)]
)
self.U_diffusion = sum(
[-lig.D * ufl.dot(ufl.grad(iUi), ufl.grad(wUi))*self.dx for
lig,iUi,wUi in
zip(self.ligands.ligands(), self.iUs, self.wUs)]
)
self.U_penalty = sum(
[-(self.Upen*self.degree**2/self.havg) *
ufl.dot(ufl.jump(iUi, self.n),
ufl.jump(wUi, self.n))*self.dS for
iUi,wUi in zip(self.iUs, self.wUs)]
)
self.gU_penalty = -self.gUpen * self.degree**2 * sum(
[ufl.jump(ufl.grad(iUi), self.n) *
ufl.jump(ufl.grad(wUi), self.n) * self.dS for
iUi,wUi in zip(self.iUs, self.wUs)]
)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty
)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
# <codecell>
from ufl import pi, atan, atan_2, tan, grad, as_vector, inner, dot
psi = pi / 8.0 + atan_2(grad(u_h_n[0])[1], (grad(u_h_n[0])[0]))
Phi = tan(N / 2.0 * psi)
beta = (1.0 - Phi * Phi) / (1.0 + Phi * Phi)
dbeta_dPhi = -2.0 * N * Phi / (1.0 + Phi * Phi)
fac = 1.0 + c * beta
diag = fac * fac
offdiag = -fac * c * dbeta_dPhi
d0 = as_vector([diag, offdiag])
d1 = as_vector([-offdiag, diag])
m = u[0] - 0.5 - kappa1 / pi * atan(kappa2 * u[1])
s = as_vector([dt / tau * u[0] * (1.0 - u[0]) * m, u[0]])
a_im = (alpha * alpha * dt / tau *
(inner(dot(d0, grad(u[0])),
grad(v[0])[0]) + inner(dot(d1, grad(u[0])),
grad(v[0])[1])) + 2.25 * dt *
inner(grad(u[1]), grad(v[1])) + inner(u, v) - inner(s, v)) * dx
# <markdowncell>
# We set up the scheme with some parameters.
# <codecell>
from dune.fem.scheme import galerkin as solutionScheme
solverParameters = {
"newton.tolerance": 1e-8,
"newton.linear.tolerance": 1e-10,
"newton.linear.preconditioning.method": "jacobi",
"newton.verbose": False,
"newton.linear.verbose": False
ds = Measure("ds", subdomain_data=boundary_parts)
# Define forms (dont redefine functions used here)
# step 1
u0 = Function(V)
u1 = Function(V)
p0 = Function(Q)
u_tent = TrialFunction(V)
v = TestFunction(V)
# U_ = 1.5*u0 - 0.5*u1
# nonlinearity = inner(dot(0.5 * (u_tent.dx(0) + u0.dx(0)), U_), v) * dx
# F_tent = (1./dt)*inner(u_tent - u0, v) * dx + nonlinearity\
# + nu*inner(0.5 * (u_tent.dx(0) + u0.dx(0)), v.dx(0)) * dx + inner(p0.dx(0), v) * dx\
# - inner(f, v)*dx # solve to u_
# using explicite scheme: so LHS has interpretation as heat equation, RHS are sources
F_tent = (1./dt)*inner(u_tent - u0, v)*dx + inner(dot(u0.dx(0), u0), v)*dx + nu*inner((u_tent.dx(0) + u0.dx(0)), v.dx(0)) * dx + inner(p0.dx(0), v) * dx\
- inner(f, v)*dx # solve to u_
a_tent, L_tent = system(F_tent)
# step 2
u_tent_computed = Function(V)
p = TrialFunction(Q)
q = TestFunction(Q)
F_p = inner(grad(p-p0), grad(q))*dx + (1./dt)*u_tent_computed.dx(0)*q*dx # + 2*p.dx(0)*q*ds(1) # tried to force dp/dn=0 on inflow
# TEST: prescribe Neumann outflow BC
# F_p = inner(grad(p-p0), grad(q))*dx + (1./dt)*u_tent_computed.dx(0)*q*dx + (1./dt)*(v_in_expr-u_tent_computed)*q*ds(2)
a_p, L_p = system(F_p)
A_p = assemble(a_p)
as_backend_type(A_p).set_nullspace(null_space)
print(A_p.array())
# step 2 rotation
# F_p_rot = inner(grad(p), grad(q))*dx + (1./dt)*u_tent_computed.dx(0)*q*dx + (1./dt)*(v_in_expr-u_tent_computed)*q*ds(2)
uD_i = dolfinx.Function(V_i)
uD_i.interpolate(lambda x: 1 + x[0]**2 + 2 * x[1]**2)
uD_i.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
fdim_i = mesh_i.topology.dim - 1
mesh_i.topology.create_connectivity(fdim_i, mesh_i.topology.dim)
boundary_facets_i = np.where(
np.array(dolfinx.cpp.mesh.compute_boundary_facets(mesh_i.topology)) ==
1)[0]
boundary_dofs_i = dolfinx.fem.locate_dofs_topological(
V_i, fdim_i, boundary_facets_i)
bc_i = dolfinx.DirichletBC(uD_i, boundary_dofs_i)
u_i = ufl.TrialFunction(V_i)
v_i = ufl.TestFunction(V_i)
f_i = dolfinx.Constant(mesh_i, -6)
a_i = ufl.dot(ufl.grad(u_i), ufl.grad(v_i)) * ufl.dx
A_i = dolfinx.fem.assemble_matrix(a_i, bcs=[bc_i])
A_i.assemble()
assert isinstance(A_i, PETSc.Mat)
ai, aj, av = A_i.getValuesCSR()
A_sp_i = scp.sparse.csr_matrix((av, aj, ai))
del A_i, av, ai, aj
# Stores the Sparse version of the Stiffness Matrices
A_sp_dict[i] = (A_sp_i, i)
L_i = f_i * v_i * ufl.dx
b_i = dolfinx.fem.create_vector(L_i)
with b_i.localForm() as loc_b:
loc_b.set(0)
dolfinx.fem.assemble_vector(b_i, L_i)
dolfinx.fem.apply_lifting(b_i, [a_i], [[bc_i]])
b_i.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
N = dolfinx.fem.VectorFunctionSpace(mesh, ("DG", q), mesh.geometry.dim)
B = dolfinx.fem.TensorFunctionSpace(mesh, ("DG", q),
(mesh.topology.dim, mesh.topology.dim))
# Normal vector (gdim x 1) and curvature tensor (tdim x tdim)
n0i = dolfinx.fem.Function(N)
B0i = dolfinx.fem.Function(B)
# Jacobi matrix of map reference -> undeformed
J0 = ufl.geometry.Jacobian(mesh)
# Tangent basis
gs = J0[:, 0]
gη = ufl.as_vector([0, 1, 0]) # unit vector e_y (assume curve in x-z plane)
gξ = ufl.cross(gs, gη)
# Unit tangent basis
gs /= ufl.sqrt(ufl.dot(gs, gs))
gη /= ufl.sqrt(ufl.dot(gη, gη))
gξ /= ufl.sqrt(ufl.dot(gξ, gξ))
# Interpolate normal vector
dolfiny.interpolation.interpolate(gξ, n0i)
# Contravariant basis
K0 = ufl.geometry.JacobianInverse(mesh).T
# Curvature tensor
B0 = -ufl.dot(ufl.dot(ufl.grad(n0i), J0).T,
K0) # = ufl.dot(n0i, ufl.dot(ufl.grad(K0), J0))
# Interpolate curvature tensor
dolfiny.interpolation.interpolate(B0, B0i)
# ----------------------------------------------------------------------------
# DERIVATIVE with respect to arc-length coordinate s of straight reference configuration: du/ds = du/dx * dx/dr * dr/ds
I3_C = J**2
# Green strain tensor
E = (C - I) / 2
# Mapping of strain in fiber directions
Ef = A * E * A.T
# Strain energy function W(Q(Ef))
Q = c00 * Ef[0, 0]**2 + c11 * Ef[1, 1]**2 + c22 * Ef[2, 2]**2 # FIXME: insert some simple law here
W = (K / 2) * (exp(Q) - 1) # + p stuff
# First Piola-Kirchoff stress tensor
P = diff(W, F)
# Acceleration term discretized with finite differences
k = dt / rho
acc = (u - 2 * up + upp)
# Residual equation # FIXME: Can contain errors, not tested!
a_F = inner(acc, v) * dx \
+ k * inner(P, grad(v)) * dx \
- k * dot(J * Finv * T, v) * ds(0) \
- k * dot(J * Finv * p0 * N, v) * ds(1)
# Jacobi matrix of residual equation
a_J = derivative(a_F, u, w)
# Export forms
forms = [a_F, a_J]
colorbar()
fig.set_label(name)
u = TrialFunction(Q1D)
u_ext = interpolate(Expression("1."), Q1D)
v = TestFunction(Q1D)
a_mass = inner(u, v)*dx
A_mass = assemble(a_mass)
# plot_matrix(A_mass)
a_diffusion_1D = inner(u.dx(0), v.dx(0))*dx
A_diffusion_1D = assemble(a_diffusion_1D)
# plot_matrix(A_diffusion_1D)
# plot_matrix(A_diffusion_1D)
a_convection_1D = inner(dot(u.dx(0), u_ext), v)*dx
A_convection_1D = assemble(a_convection_1D)
# plot_matrix(A_convection_1D)
u_exts = [interpolate(Expression(('1.', '1.')), V2D), interpolate(Expression(('1.', '1.', '1.')), V3D)]
Qspaces = [Q2D, Q3D]
Vspaces = [V2D, V3D]
names = ['2D', '3D']
for i in [0, 1]:
Qspace = Qspaces[i]
Vspace = Vspaces[i]
u = TrialFunction(Vspace)
v = TestFunction(Vspace)
a_mass = inner(u, v)*dx
A_mass = assemble(a_mass)
def GRAD(u):
return ufl.dot(ufl.grad(u),
J0[:, 0]) * 1 / ufl.geometry.JacobianDeterminant(mesh)
def test_div_grad_then_integrate_over_cells_and_boundary():
# Define 2D geometry
n = 10
mesh = RectangleMesh(
[numpy.array([0.0, 0.0, 0.0]),
numpy.array([2.0, 3.0, 0.0])], 2 * n, 3 * n)
x, y = SpatialCoordinate(mesh)
xs = 0.1 + 0.8 * x / 2 # scaled to be within [0.1,0.9]
# ys = 0.1 + 0.8 * y / 3 # scaled to be within [0.1,0.9]
n = FacetNormal(mesh)
# Define list of expressions to test, and configure accuracies
# these expressions are known to pass with. The reason some
# functions are less accurately integrated is likely that the
# default choice of quadrature rule is not perfect
F_list = []
def reg(exprs, acc=10):
for expr in exprs:
F_list.append((expr, acc))
# FIXME: 0*dx and 1*dx fails in the ufl-ffcx-jit framework somewhere
# reg([Constant(0.0, cell=cell)])
# reg([Constant(1.0, cell=cell)])
monomial_list = [x**q for q in range(2, 6)]
reg(monomial_list)
reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list])
reg([xs**xs])
reg(
[xs**(xs**2)], 8
) # Note: Accuracies here are from 1D case, not checked against 2D results.
reg([xs**(xs**3)], 6)
reg([xs**(xs**4)], 2)
# Special functions:
reg([atan(xs)], 8)
reg([sin(x), cos(x), exp(x)], 5)
reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3)
reg([asin(xs), acos(xs)], 1)
reg([tan(xs)], 7)
# To handle tensor algebra, make an x dependent input tensor
# xx and square all expressions
def reg2(exprs, acc=10):
for expr in exprs:
F_list.append((inner(expr, expr), acc))
xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]])
xxs = as_matrix([[2 * xs**2, 3 * xs**3], [11 * xs**5, 7 * xs**4]])
x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4])
cc = as_matrix([[2, 3], [4, 5]])
reg2(
[xx]
) # TODO: Make unit test for UFL from this, results in listtensor with free indices
reg2([x3v])
reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))])
reg2([xx.T])
reg2([tr(xx)])
reg2([det(xx)])
reg2([dot(xx, 0.1 * xx)])
reg2([outer(xx, xx.T)])
reg2([dev(xx)])
reg2([sym(xx)])
reg2([skew(xx)])
reg2([elem_mult(7 * xx, cc)])
reg2([elem_div(7 * xx, xx + cc)])
reg2([elem_pow(1e-3 * xxs, 1e-3 * cc)])
reg2([elem_pow(1e-3 * cc, 1e-3 * xx)])
reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate...
# FIXME: Add tests for all UFL operators:
# These cause discontinuities and may be harder to test in the
# above fashion:
# 'inv', 'cofac',
# 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not',
# 'conditional', 'sign',
# 'jump', 'avg',
# 'LiftingFunction', 'LiftingOperator',
# FIXME: Test other derivatives: (but algorithms for operator
# derivatives are the same!):
# 'variable', 'diff',
# 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative',
# Run through all operators defined above and compare integrals
debug = 0
if debug:
F_list = F_list[1:]
for F, acc in F_list:
if debug:
print('\n', "F:", str(F))
# Integrate over domain and its boundary
int_dx = assemble(div(grad(F)) * dx(mesh)) # noqa
int_ds = assemble(dot(grad(F), n) * ds(mesh)) # noqa
if debug:
print(int_dx, int_ds)
# Compare results. Using custom relative delta instead of
# decimal digits here because some numbers are >> 1.
delta = min(abs(int_dx), abs(int_ds)) * 10**-acc
assert int_dx - int_ds <= delta
destC = space.interpolate(as_vector([
0,
] * dimR), name='destC')
destD = space.interpolate(as_vector([
0,
] * dimR), name='destD')
destE = space.interpolate(as_vector([
0,
] * dimR), name='destE')
u = TrialFunction(space)
v = TestFunction(space)
x = SpatialCoordinate(space.cell())
ubar = space.interpolate(as_vector([
dot(x, x),
] * dimR), name="ubar")
a = (inner(0.5 * dot(u, u), v[0]) + inner(u[0] * grad(u), grad(v))) * dx
op = create.operator("galerkin", a, space)
A = linearOperator(op)
op.jacobian(ubar, A)
A(arg, destA)
da = apply_derivatives(derivative(action(a, ubar), ubar, u))
dop = create.operator("galerkin", da, space)
dop(arg, destB)
err = integrate(grid, (destA - destB)**2, 5)
# print("error=",err)
assert (err < 1e-15)
A = linearOperator(dop)
