def test_physically_mapped_facet():
mesh = Mesh(VectorElement("P", triangle, 1))
# set up variational problem
U = FiniteElement("Morley", mesh.ufl_cell(), 2)
V = FiniteElement("P", mesh.ufl_cell(), 1)
R = FiniteElement("P", mesh.ufl_cell(), 1)
Vv = VectorElement(BrokenElement(V))
Qhat = VectorElement(BrokenElement(V[facet]))
Vhat = VectorElement(V[facet])
Z = FunctionSpace(mesh, MixedElement(U, Vv, Qhat, Vhat, R))
z = Coefficient(Z)
u, d, qhat, dhat, lam = split(z)
s = FacetNormal(mesh)
trans = as_matrix([[1, 0], [0, 1]])
mat = trans*grad(grad(u))*trans + outer(d, d) * u
J = (u**2*dx +
u**3*dx +
u**4*dx +
inner(mat, mat)*dx +
inner(grad(d), grad(d))*dx +
dot(s, d)**2*ds)
L_match = inner(qhat, dhat - d)
L = J + inner(lam, inner(d, d)-1)*dx + (L_match('+') + L_match('-'))*dS + L_match*ds
compile_form(L)
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 test_facet_normals(cell_type):
"""Test that FacetNormal is outward facing"""
for count in range(5):
mesh = unit_cell(cell_type)
tdim = mesh.topology.dim
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
normal = FacetNormal(mesh)
v = Function(V)
map_f = mesh.topology.index_map(tdim - 1)
num_facets = map_f.size_local + map_f.num_ghosts
indices = np.arange(0, num_facets)
values = np.arange(0, num_facets, dtype=np.intc)
marker = MeshTags(mesh, tdim - 1, indices, values)
# For each facet, check that the inner product of the normal and
# the vector that has a positive normal component on only that
# facet is positive
for i in range(num_facets):
if cell_type == CellType.interval:
co = mesh.geometry.x[i]
v.interpolate(lambda x: x[0] - co[0])
if cell_type == CellType.triangle:
co = mesh.geometry.x[i]
# Vector function that is zero at `co` and points away
# from `co` so that there is no normal component on two
# edges and the integral over the other edge is 1
v.interpolate(lambda x: ((x[0] - co[0]) / 2, (x[1] - co[1]) / 2))
elif cell_type == CellType.tetrahedron:
co = mesh.geometry.x[i]
# Vector function that is zero at `co` and points away
# from `co` so that there is no normal component on
# three faces and the integral over the other edge is 1
v.interpolate(lambda x: ((x[0] - co[0]) / 3, (x[1] - co[1]) / 3, (x[2] - co[2]) / 3))
elif cell_type == CellType.quadrilateral:
# function that is 0 on one edge and points away from
# that edge so that there is no normal component on
# three edges
v.interpolate(lambda x: tuple(x[j] - i % 2 if j == i // 2 else 0 * x[j] for j in range(2)))
elif cell_type == CellType.hexahedron:
# function that is 0 on one face and points away from
# that face so that there is no normal component on five
# faces
v.interpolate(lambda x: tuple(x[j] - i % 2 if j == i // 3 else 0 * x[j] for j in range(3)))
# assert that the integrals these functions dotted with the
# normal over a face is 1 on one face and 0 on the others
ones = 0
for j in range(num_facets):
a = inner(v, normal) * ds(subdomain_data=marker, subdomain_id=j)
result = fem.assemble_scalar(a)
if np.isclose(result, 1):
ones += 1
else:
assert np.isclose(result, 0)
assert ones == 1
def __init__(self, mesh, k: int, omega, c, c0, lumped):
P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), k)
self.V = FunctionSpace(mesh, P)
self.u, self.v = Function(self.V), Function(self.V)
self.g1 = Function(self.V)
self.g2 = Function(self.V)
self.omega = omega
self.c = c
self.c0 = c0
n = FacetNormal(mesh)
# Pieces for plane wave incident field
x = ufl.geometry.SpatialCoordinate(mesh)
cos_wave = ufl.cos(self.omega / self.c0 * x[0])
sin_wave = ufl.sin(self.omega / self.c0 * x[0])
plane_wave = self.g1 * cos_wave + self.g2 * sin_wave
dv, p = TrialFunction(self.V), TestFunction(self.V)
self.L1 = - inner(grad(self.u), grad(p)) * dx(degree=k) \
- (1 / self.c) * inner(self.v, p) * ds \
- (1 / self.c**2) * (-self.omega**2) * inner(plane_wave, p) * dx \
- inner(grad(plane_wave), grad(p)) * dx \
+ inner(dot(grad(plane_wave), n), p) * ds
# Vector to be re-used for assembly
self.b = None
# TODO: precompile/pre-process Form L
self.lumped = lumped
if self.lumped:
a = (1 / self.c**2) * p * dx(degree=k)
self.M = dolfinx.fem.assemble_vector(a)
self.M.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
else:
a = (1 / self.c**2) * inner(dv, p) * dx(degree=k)
M = dolfinx.fem.assemble_matrix(a)
M.assemble()
self.solver = PETSc.KSP().create(mesh.mpi_comm())
opts = PETSc.Options()
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-8
self.solver.setFromOptions()
self.solver.setOperators(M)
def __init__(self, args, tc, metadata):
self.has_analytic_solution = False
self.problem_code = 'FACB'
super(Problem, self).__init__(args, tc, metadata)
self.name = 'test on real mesh'
self.status_functional_str = 'not selected'
# input parameters
self.factor = args.factor
self.scale_factor.append(self.factor)
self.nu = 0.001 * args.nufactor # kinematic viscosity
# Import gmsh mesh
self.compatible_meshes = ['bench3D_1', 'bench3D_2', 'bench3D_3']
if args.mesh not in self.compatible_meshes:
exit('Bad mesh, should be some from %s' %
str(self.compatible_meshes))
self.mesh = Mesh("meshes/" + args.mesh + ".xml")
self.cell_function = MeshFunction(
"size_t", self.mesh,
"meshes/" + args.mesh + "_physical_region.xml")
self.facet_function = MeshFunction(
"size_t", self.mesh, "meshes/" + args.mesh + "_facet_region.xml")
self.dsIn = Measure("ds",
subdomain_id=2,
subdomain_data=self.facet_function)
self.dsOut = Measure("ds",
subdomain_id=3,
subdomain_data=self.facet_function)
self.dsWall = Measure("ds",
subdomain_id=1,
subdomain_data=self.facet_function)
self.dsCyl = Measure("ds",
subdomain_id=5,
subdomain_data=self.facet_function)
self.normal = FacetNormal(self.mesh)
print("Mesh name: ", args.mesh, " ", self.mesh)
print("Mesh norm max: ", self.mesh.hmax())
print("Mesh norm min: ", self.mesh.hmin())
self.actual_time = None
self.v_in = None
def run_dg_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.vector.set(2.0)
k.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a, [])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Calculate error
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def compute(space, epsilon, weakBnd, skeleton, mol=None):
u = TrialFunction(space)
v = TestFunction(space)
n = FacetNormal(space)
he = avg(CellVolume(space)) / FacetArea(space)
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
exact = uflFunction(space.gridView,
name="exact",
order=3,
ufl=sin(x[0] * x[1]))
uh = space.interpolate(exact, name="solution")
# diffusion factor
eps = Constant(epsilon, "eps")
# transport direction and upwind flux
b = as_vector([1, 0])
hatb = (dot(b, n) + abs(dot(b, n))) / 2.0
# characteristic function for left/right boundary
dD = conditional((1 + x[0]) * (1 - x[0]) < 1e-10, 1, 0)
# penalty parameter
beta = Constant(20 * space.order**2, "beta")
rhs = -(div(eps * grad(exact) - b * exact)) * v * dx
aInternal = dot(eps * grad(u) - b * u, grad(v)) * dx
aInternal -= eps * dot(grad(exact), n) * v * (1 - dD) * ds
diffSkeleton = eps*beta/he*jump(u)*jump(v)*dS -\
eps*dot(avg(grad(u)),n('+'))*jump(v)*dS -\
eps*jump(u)*dot(avg(grad(v)),n('+'))*dS
if weakBnd:
diffSkeleton += eps*beta/hbnd*(u-exact)*v*dD*ds -\
eps*dot(grad(exact),n)*v*dD*ds
advSkeleton = jump(hatb * u) * jump(v) * dS
if weakBnd:
advSkeleton += (hatb * u + (dot(b, n) - hatb) * exact) * v * dD * ds
if skeleton:
form = aInternal + diffSkeleton + advSkeleton
else:
form = aInternal
if weakBnd and skeleton:
strongBC = None
else:
strongBC = DirichletBC(space, exact, dD)
if space.storage[0] == "numpy":
solver = {
"solver": ("suitesparse", "umfpack"),
"parameters": {
"newton.verbose": True,
"newton.linear.verbose": False,
"newton.linear.tolerance": 1e-5,
}
}
else:
solver = {
"solver": "bicgstab",
"parameters": {
"newton.linear.preconditioning.method": "ilu",
"newton.linear.tolerance": 1e-13,
"newton.verbose": True,
"newton.linear.verbose": False
}
}
if mol == 'mol':
scheme = molSolutionScheme([form == rhs, strongBC], **solver)
else:
scheme = solutionScheme([form == rhs, strongBC], **solver)
eoc = []
info = scheme.solve(target=uh)
error = dot(uh - exact, uh - exact)
error0 = math.sqrt(integrate(gridView, error, order=5))
print(error0, " # output", flush=True)
for i in range(3):
gridView.hierarchicalGrid.globalRefine(1)
uh.interpolate(exact)
scheme.solve(target=uh)
error = dot(uh - exact, uh - exact)
error1 = math.sqrt(integrate(gridView, error, order=5))
eoc += [math.log(error1 / error0) / math.log(0.5)]
print(i, error0, error1, eoc, " # output", flush=True)
error0 = error1
# print(space.order,epsilon,eoc)
if (eoc[-1] - (space.order + 1)) < -0.1:
print("ERROR:", space.order, epsilon, eoc)
return eoc
has_petsc_complex, solve)
from dolfinx.fem.assemble import assemble_scalar
from dolfinx.io import XDMFFile
from ufl import FacetNormal, TestFunction, TrialFunction, dx, grad, inner
# wavenumber
k0 = 4 * np.pi
# approximation space polynomial degree
deg = 1
# number of elements in each direction of mesh
n_elem = 128
mesh = UnitSquareMesh(MPI.COMM_WORLD, n_elem, n_elem)
n = FacetNormal(mesh)
# Source amplitude
if has_petsc_complex:
A = 1 + 1j
else:
A = 1
# Test and trial function space
V = FunctionSpace(mesh, ("Lagrange", deg))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Function(V)
f.interpolate(lambda x: A * k0**2 * np.cos(k0 * x[0]) * np.cos(k0 * x[1]))
# \begin{align*}
# u(x) = \left(\frac{1}{2}(x^2 + y^2) - \frac{1}{3}(x^3 - y^3)\right) + 1
# \end{align*}
# <codecell>
from ufl import TestFunction, TrialFunction
from dune.ufl import DirichletBC
u = TrialFunction(space)
v = TestFunction(space)
from ufl import dx, grad, div, grad, dot, inner, sqrt, conditional, FacetNormal, ds
a = dot(grad(u), grad(v)) * dx
f = -div( grad(exact) )
g_N = grad(exact)
n = FacetNormal(space)
b = f*v*dx + dot(g_N,n)*conditional(x[0]>=1e-8,1,0)*v*ds
dbc = DirichletBC(space,exact,x[0]<=1e-8)
# <markdowncell>
# With the model described as a ufl form, we can construct a scheme class
# that provides the solve method which we can use to compute the solution:
# <codecell>
from dune.fem import parameter
parameter.append({"fem.verboserank": -1})
from dune.fem.scheme import galerkin as solutionScheme
scheme = solutionScheme([a == b, dbc], solver='cg')
scheme.solve(target = u_h)
# <markdowncell>
def b(tau_S, v):
n = FacetNormal(mesh)
return inner(tau_S, grad(grad(v))) * dx \
- ufl.dot(ufl.dot(tau_S('+'), n('+')), n('+')) * jump(grad(v), n) * dS \
- ufl.dot(ufl.dot(tau_S, n), n) * ufl.dot(grad(v), n) * ds
# Last changed: 2007-07-15
#
# The bilinear form a(v, u) and linear form L(v) for
# Poisson's equation in a discontinuous Galerkin (DG)
# formulation.
from ufl import (Coefficient, Constant, FacetNormal, FiniteElement,
TestFunction, TrialFunction, avg, dot, dS, ds, dx, grad,
inner, jump, triangle)
element = FiniteElement("Discontinuous Lagrange", triangle, 1)
v = TestFunction(element)
u = TrialFunction(element)
f = Coefficient(element)
n = FacetNormal(triangle)
h = Constant(triangle)
gN = Coefficient(element)
alpha = 4.0
gamma = 8.0
a = inner(grad(v), grad(u)) * dx \
- inner(avg(grad(v)), jump(u, n)) * dS \
- inner(jump(v, n), avg(grad(u))) * dS \
+ alpha / h('+') * dot(jump(v, n), jump(u, n)) * dS \
- inner(grad(v), u * n) * ds \
- inner(v * n, grad(u)) * ds \
+ gamma / h * v * u * ds
def __init__(self, args, tc, metadata):
self.has_analytic_solution = False
self.problem_code = 'REAL'
super(Problem, self).__init__(args, tc, metadata)
self.name = 'test on real mesh'
self.status_functional_str = 'outflow/inflow'
self.last_inflow = 0
# time settings
self.itp_lengths = {
1: 1.0,
2: 0.9375,
}
self.cycle_length = self.itp_lengths[self.args.itp]
# input parameters
self.nu = self.args.nu # kinematic viscosity
self.factor = args.factor
self.metadata['factor'] = self.factor
self.scale_factor.append(self.factor)
self.tc.start('mesh')
# Import mesh
try:
self.mesh, self.facet_function = super(Problem,
self).loadMesh(args.mesh)
info("Mesh name: " + args.mesh + " " + str(self.mesh))
f_ini = open('meshes/' + args.mesh + '.ini', 'r')
reader = csv.reader(f_ini, delimiter=' ', escapechar='\\')
except (EnvironmentError, RuntimeError):
print(
'Unable to open mesh.hdf5 or mesh.ini file. Check if the mesh was prepared to be used '
'with \"real\" problem.')
exit(1)
# load inflows and outflows (interfaces) from mesh.ini file
obj = None
self.interfaces = []
for row in reader:
if not row:
pass
elif row[0] == 'volume':
self.mesh_volume = float(row[1])
elif row[0] == 'in':
if obj is not None:
self.interfaces.append(obj)
obj = {'inflow': True, 'number': row[1]}
elif row[0] == 'out':
if obj is not None:
self.interfaces.append(obj)
obj = {'inflow': False, 'number': row[1]}
else:
if len(row) == 2: # scalar values
obj[row[0]] = row[1]
else: # vector values
obj[row[0]] = [float(f) for f in row[1:]]
self.interfaces.append(obj)
f_ini.close()
self.tc.end('mesh')
# collect inflows and outflows into separate lists
self.outflow_area = 0
self.inflows = []
self.outflows = []
for obj in self.interfaces:
if not obj['inflow']:
self.outflow_area += float(obj['S'])
self.outflows.append(obj)
else:
self.inflows.append(obj)
info('Outflow area: %f' % self.outflow_area)
# self.dsWall = Measure("ds", subdomain_id=1, subdomain_data=self.facet_function)
self.normal = FacetNormal(self.mesh)
# generate measures, collect measure lists
self.inflow_measures = []
for obj in self.interfaces:
obj['measure'] = Measure("ds",
subdomain_id=int(obj['number']),
subdomain_data=self.facet_function)
if obj['inflow']:
self.inflow_measures.append(obj['measure'])
else:
self.outflow_measures.append(obj['measure'])
self.listDict.update({
'outflow': {
'list': [],
'name': 'outflow rate',
'abrev': 'OUT',
'slist': []
},
'inflow': {
'list': [],
'name': 'inflow rate',
'abrev': 'IN',
'slist': []
},
'oiratio': {
'list': [],
'name': 'outflow/inflow ratio (mass conservation)',
'abrev': 'O/I',
'slist': []
},
})
for obj in self.outflows:
n = obj['number']
self.listDict.update({
'outflow' + n: {
'list': [],
'name': 'outflow rate ' + n,
'abrev': 'OUT' + n,
'slist': []
}
})
self.can_force_outflow = True
#
# Author: Marie Rognes
# Modified by: Martin Sandve Alnes
# Date: 2009-02-12
#
from ufl import (FacetNormal, FiniteElement, TestFunctions, TrialFunctions,
div, dot, ds, dx, tetrahedron)
cell = tetrahedron
RT = FiniteElement("Raviart-Thomas", cell, 1)
DG = FiniteElement("DG", cell, 0)
MX = RT * DG
(u, p) = TrialFunctions(MX)
(v, q) = TestFunctions(MX)
n = FacetNormal(cell)
a0 = (dot(u, v) + div(u) * q + div(v) * p) * dx
a1 = (dot(u, v) + div(u) * q + div(v) * p) * dx - p * dot(v, n) * ds
forms = [a0, a1]
def model(space, epsilon, weakBnd, skeleton, useMol):
u = TrialFunction(space)
v = TestFunction(space)
n = FacetNormal(space)
he = avg(CellVolume(space)) / FacetArea(space)
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
#exact = sin(x[0]*x[1]) # atan(1*x[1])
exact = uflFunction(space.gridView,
name="exact",
order=3,
ufl=sin(x[0] * x[1]))
# diffusion factor
eps = 1 # Constant(epsilon,"eps")
# transport direction and upwind flux
b = as_vector([1, 0])
hatb = (dot(b, n) + abs(dot(b, n))) / 2.0
# characteristic function for left/right boundary
dD = conditional((1 + x[0]) * (1 - x[0]) < 1e-10, 1, 0)
# penalty parameter
beta = Constant(10 * space.order**2 if space.order > 0 else 1, "beta")
rhs = (-div(eps * grad(exact) - b * exact) + exact) * v * dx
aInternal = (dot(eps * grad(u) - b * u, grad(v)) + dot(u, v)) * dx
diffSkeleton = eps*beta/he*jump(u)*jump(v)*dS -\
eps*dot(avg(grad(u)),n('+'))*jump(v)*dS -\
eps*jump(u)*dot(avg(grad(v)),n('+'))*dS
diffSkeleton -= eps * dot(grad(exact), n) * v * (1 - dD) * ds
if weakBnd:
diffSkeleton += eps*beta/hbnd*(u-exact)*v*dD*ds -\
eps*dot(grad(exact),n)*v*dD*ds
advSkeleton = jump(hatb * u) * jump(v) * dS
if weakBnd:
advSkeleton += (hatb * u + (dot(b, n) - hatb) * exact) * v * dD * ds
if skeleton:
form = aInternal + diffSkeleton + advSkeleton
else:
form = aInternal
if weakBnd and skeleton:
strongBC = None
else:
strongBC = None # DirichletBC(space,exact,dD)
if space.storage[0] == "fem":
solver = {"solver": ("suitesparse", "umfpack")}
else:
solver = {
"solver": "bicgstab",
"parameters": {
"newton.linear.preconditioning.method": "jacobi",
"newton.linear.tolerance": 1e-13
}
}
if useMol:
scheme = solutionMolScheme([form == rhs, strongBC], **solver)
else:
scheme = solutionScheme([form == rhs, strongBC], **solver)
uh = space.interpolate(exact, name="solution")
A = linear(scheme)
return scheme, uh, A, exact
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
def __init__(self, args, tc, metadata):
self.has_analytic_solution = True
self.problem_code = 'WCYL'
super(Problem, self).__init__(args, tc, metadata)
self.tc.init_watch('assembleSol', 'Assembled analytic solution', True)
self.tc.init_watch('analyticP', 'Analytic pressure', True)
self.tc.init_watch('analyticVnorms',
'Computed analytic velocity norms', True)
self.tc.init_watch('errorP', 'Computed pressure error', True)
self.tc.init_watch('errorForce', 'Computed force error', True)
self.tc.init_watch('computePG', 'Computed pressure gradient', True)
self.name = 'womersley_cylinder'
self.status_functional_str = 'last H1 velocity error'
# input parameters
self.ic = args.ic
self.factor = args.factor
self.metadata['factor'] = self.factor
self.scale_factor.append(self.factor)
# fixed parameters (used in analytic solution and in BC)
self.nu = 3.71 * self.args.nufactor # kinematic viscosity
self.R = 5.0 # cylinder radius
self.mesh_volume = pi * 25. * 20.
# Import gmsh mesh
self.tc.start('mesh')
self.mesh, self.facet_function = super(Problem,
self).loadMesh(args.mesh)
self.dsIn = Measure("ds",
subdomain_id=2,
subdomain_data=self.facet_function)
self.dsOut = Measure("ds",
subdomain_id=3,
subdomain_data=self.facet_function)
self.dsWall = Measure("ds",
subdomain_id=1,
subdomain_data=self.facet_function)
self.normal = FacetNormal(self.mesh)
print("Mesh name: ", args.mesh, " ", self.mesh)
print("Mesh norm max: ", self.mesh.hmax())
print("Mesh norm min: ", self.mesh.hmin())
self.tc.end('mesh')
self.sol_p = None
self.last_analytic_pressure_norm = None
self.v_in = None
self.area = None
choose_note = {1.0: '', 0.1: 'nuL10', 0.01: 'nuL100', 10.0: 'nuH10'}
self.precomputed_filename = args.mesh + choose_note[self.args.nufactor]
print('chosen filename for precomputed solution',
self.precomputed_filename)
# partial Bessel functions and coefficients
self.bessel_parabolic = None
self.bessel_real = []
self.bessel_complex = []
self.coefs_exp = [-8, -6, -4, -2, 2, 4, 6, 8]
self.listDict.update({
'u_H1w': {
'list': [],
'name': 'corrected velocity H1 error on wall',
'abrev': 'CE_H1w',
'scale': self.scale_factor,
'relative': 'av_norm_H1w',
'slist': []
},
'u2H1w': {
'list': [],
'name': 'tentative velocity H1 error on wall',
'abrev': 'TE_H1w',
'scale': self.scale_factor,
'relative': 'av_norm_H1w',
'slist': []
},
'av_norm_H1w': {
'list': [],
'name': 'analytic velocity H1 norm on wall',
'abrev': 'AVN_H1w'
},
'a_force_wall': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AF'
},
'a_force_wall_normal': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AFN'
},
'a_force_wall_shear': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AFS'
},
'force_wall': {
'list': [],
'name': 'force error on wall',
'abrev': 'FE',
'relative': 'a_force_wall',
'slist': []
},
'force_wall_normal': {
'list': [],
'name': 'normal force error on wall',
'abrev': 'FNE',
'relative': 'a_force_wall',
'slist': []
},
'force_wall_shear': {
'list': [],
'name': 'shear force error on wall',
'abrev': 'FSE',
'relative': 'a_force_wall',
'slist': []
},
})
def _compileUFL(integrands, form, *args, tempVars=True):
if isinstance(form, Equation):
form = form.lhs - form.rhs
if not isinstance(form, Form):
raise ValueError("ufl.Form or ufl.Equation expected.")
# added for dirichlet treatment same as conservationlaw model
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
uflExpr = [form] + [bc.ufl_value for bc in dirichletBCs]
if len(form.arguments()) < 2:
raise ValueError(
"Integrands model requires form with at least two arguments.")
x = SpatialCoordinate(form.ufl_cell())
n = FacetNormal(form.ufl_cell())
cellVolume = CellVolume(form.ufl_cell())
maxCellEdgeLength = MaxCellEdgeLength(form.ufl_cell())
minCellEdgeLength = MinCellEdgeLength(form.ufl_cell())
facetArea = FacetArea(form.ufl_cell())
maxFacetEdgeLength = MaxFacetEdgeLength(form.ufl_cell())
minFacetEdgeLength = MinFacetEdgeLength(form.ufl_cell())
phi, u = form.arguments()
ubar = Coefficient(u.ufl_function_space())
derivatives = gatherDerivatives(form, [phi, u])
derivatives_phi = derivatives[0]
derivatives_u = derivatives[1]
derivatives_ubar = map_expr_dags(Replacer({u: ubar}), derivatives_u)
try:
integrands.field = u.ufl_function_space().field
except AttributeError:
pass
integrals = splitForm(form, [phi])
dform = apply_derivatives(derivative(action(form, ubar), ubar, u))
linearizedIntegrals = splitForm(dform, [phi, u])
if not set(
integrals.keys()) <= {'cell', 'exterior_facet', 'interior_facet'}:
raise Exception('unknown integral encountered in ' +
str(set(integrals.keys())) + '.')
if 'cell' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.interior = generateUnaryCode(predefined,
derivatives_phi,
integrals['cell'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedInterior = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('cell'),
tempVars=tempVars)
if 'exterior_facet' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.boundary = generateUnaryCode(predefined,
derivatives_phi,
integrals['exterior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedBoundary = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('exterior_facet'),
tempVars=tempVars)
if 'interior_facet' in integrals.keys():
argIn = Variable(integrands.domainValueTuple, 'uIn')
argOut = Variable(integrands.domainValueTuple, 'uOut')
predefined = {
derivatives_u[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.skeleton = generateBinaryCode(predefined,
derivatives_phi,
integrals['interior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.linearizedSkeleton = generateBinaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('interior_facet'),
tempVars=tempVars)
if dirichletBCs:
integrands.hasDirichletBoundary = True
predefined = {}
# predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + integrands.arg_x.name + ' ) )')
predefined[x] = UnformattedExpression(
'auto', 'intersection.geometry().center( )')
integrands.predefineCoefficients(predefined, False)
maxId = 0
codeDomains = []
bySubDomain = dict()
neuman = []
wholeDomain = None
for bc in dirichletBCs:
if bc.subDomain in bySubDomain:
raise Exception(
'Multiply defined Dirichlet boundary for subdomain ' +
str(bc.subDomain))
if not isinstance(bc.functionSpace,
(FunctionSpace, FiniteElementBase)):
raise Exception(
'Function space must either be a ufl.FunctionSpace or a ufl.FiniteElement'
)
if isinstance(bc.functionSpace, FunctionSpace) and (
bc.functionSpace != u.ufl_function_space()):
raise Exception(
'Space of trial function and dirichlet boundary function must be the same - note that boundary conditions on subspaces are not available, yet'
)
if isinstance(bc.functionSpace, FiniteElementBase) and (
bc.functionSpace != u.ufl_element()):
raise Exception(
'Cannot handle boundary conditions on subspaces, yet')
if isinstance(bc.value, list):
neuman = [i for i, x in enumerate(bc.value) if x == None]
else:
neuman = []
value = ExprTensor(u.ufl_shape)
for key in value.keys():
value[key] = Indexed(
bc.ufl_value,
MultiIndex(tuple(FixedIndex(k) for k in key)))
if bc.subDomain is None:
wholeDomain = value, neuman
elif isinstance(bc.subDomain, int):
bySubDomain[bc.subDomain] = value, neuman
maxId = max(maxId, bc.subDomain)
else:
domain = ExprTensor(())
for key in domain.keys():
domain[key] = Indexed(
bc.subDomain,
MultiIndex(tuple(FixedIndex(k) for k in key)))
codeDomains.append((value, neuman, domain))
defaultCode = []
if len(codeDomains) > 0:
defaultCode.append(Declaration(Variable('int', 'domainId')))
# defaultCode.append(Declaration(Variable('auto', 'x'),
# initializer=UnformattedExpression('auto','intersection.geometry().center()')))
for i, v in enumerate(codeDomains):
block = Block()
block.append(
generateDirichletDomainCode(predefined,
v[2],
tempVars=tempVars))
block.append('if (domainId)')
ifBlock = UnformattedBlock()
ifBlock.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + i + 2) + ' );')
if len(v[1]) > 0:
[
ifBlock.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
ifBlock.append('return true;')
block.append(ifBlock)
defaultCode.append(block)
if wholeDomain is not None:
block = UnformattedBlock()
block.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + 1) + ' );')
if len(wholeDomain[1]) > 0:
[
block.append('dirichletComponent[' + str(c) + '] = 0;')
for c in wholeDomain[1]
]
block.append('return true;')
defaultCode.append(block)
defaultCode.append(return_(False))
bndId = Variable('const int', 'bndId')
getBndId = UnformattedExpression(
'int', 'BoundaryIdProviderType::boundaryId( ' +
integrands.arg_i.name + ' )')
# getBndId = UnformattedExpression('int', 'boundaryIdGetter_.boundaryId( ' + integrands.arg_i.name + ' )')
switch = SwitchStatement(bndId, default=defaultCode)
for i, v in bySubDomain.items():
code = []
if len(v[1]) > 0:
[
code.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
code.append(return_(True))
switch.append(i, code)
integrands.isDirichletIntersection = [
Declaration(bndId, initializer=getBndId),
UnformattedBlock(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ bndId.name + ' );'), switch
]
predefined[x] = UnformattedExpression(
'auto', 'entity().geometry().global( Dune::Fem::coordinate( ' +
integrands.arg_x.name + ' ) )')
if wholeDomain is None:
defaultCode = assign(integrands.arg_r, construct("RRangeType", 0))
else:
defaultCode = generateDirichletCode(predefined,
wholeDomain[0],
tempVars=tempVars)
switch = SwitchStatement(integrands.arg_bndId, default=defaultCode)
for i, v in bySubDomain.items():
switch.append(
i, generateDirichletCode(predefined, v[0], tempVars=tempVars))
for i, v in enumerate(codeDomains):
switch.append(
i + maxId + 2,
generateDirichletCode(predefined, v[0], tempVars=tempVars))
integrands.dirichlet = [switch]
return integrands
def __init__(self, args, tc, metadata):
self.has_analytic_solution = True
self.problem_code = 'SCYL'
super(Problem, self).__init__(args, tc, metadata)
self.tc.init_watch('errorP', 'Computed pressure error', True)
self.tc.init_watch('errorV', 'Computed velocity error', True)
self.tc.init_watch('errorForce', 'Computed force error', True)
self.tc.init_watch('errorVtest', 'Computed velocity error test', True)
self.tc.init_watch('computePG', 'Computed pressure gradient', True)
self.name = 'steady_cylinder'
self.status_functional_str = 'last H1 velocity error'
# input parameters
self.ic = args.ic
self.factor = args.factor
self.metadata['factor'] = self.factor
self.scale_factor.append(self.factor)
# fixed parameters (used in analytic solution and in BC)
self.nu = 3.71 # kinematic viscosity
self.R = 5.0 # cylinder radius
self.mesh_volume = pi * 25. * 20.
# Import gmsh mesh
self.tc.start('mesh')
self.mesh, self.facet_function = super(Problem,
self).loadMesh(args.mesh)
self.dsIn = Measure("ds",
subdomain_id=2,
subdomain_data=self.facet_function)
self.dsOut = Measure("ds",
subdomain_id=3,
subdomain_data=self.facet_function)
self.dsWall = Measure("ds",
subdomain_id=1,
subdomain_data=self.facet_function)
self.normal = FacetNormal(self.mesh)
print("Mesh name: ", args.mesh, " ", self.mesh)
print("Mesh norm max: ", self.mesh.hmax())
print("Mesh norm min: ", self.mesh.hmin())
self.tc.end('mesh')
self.sol_p = None
self.analytic_gradient = None
self.analytic_pressure_norm = None
self.v_in = None
self.area = None
self.analytic_v_norm_L2 = None
self.analytic_v_norm_H1 = None
self.analytic_v_norm_H1w = None
self.listDict.update({
'u_H1w': {
'list': [],
'name': 'corrected velocity H1 error on wall',
'abrev': 'CE_H1w',
'scale': self.scale_factor,
'relative': 'av_norm_H1w',
'slist': []
},
'u2H1w': {
'list': [],
'name': 'tentative velocity H1 error on wall',
'abrev': 'TE_H1w',
'scale': self.scale_factor,
'relative': 'av_norm_H1w',
'slist': []
},
'av_norm_H1w': {
'list': [],
'name': 'analytic velocity H1 norm on wall',
'abrev': 'AVN_H1w'
},
'a_force_wall': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AF'
},
'a_force_wall_normal': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AFN'
},
'a_force_wall_shear': {
'list': [],
'name': 'analytic force on wall',
'abrev': 'AFS'
},
'force_wall': {
'list': [],
'name': 'force error on wall',
'abrev': 'FE',
'relative': 'a_force_wall',
'slist': []
},
'force_wall_normal': {
'list': [],
'name': 'normal force error on wall',
'abrev': 'FNE',
'relative': 'a_force_wall',
'slist': []
},
'force_wall_shear': {
'list': [],
'name': 'shear force error on wall',
'abrev': 'FSE',
'relative': 'a_force_wall',
'slist': []
},
})
def readin_mesh(self):
if self.meshfile_type=='ASCII':
encoding = XDMFFile.Encoding.ASCII
elif self.meshfile_type=='HDF5':
encoding = XDMFFile.Encoding.HDF5
else:
raise NameError('Choose either ASCII or HDF5 as meshfile_type, or add a different encoding!')
# read in xdmf mesh - domain
with XDMFFile(self.comm, self.mesh_domain, 'r', encoding=encoding) as infile:
self.mesh = infile.read_mesh(name="Grid")
self.mt_d = infile.read_meshtags(self.mesh, name="Grid")
# read in xdmf mesh - boundary
# here, we define b1 BCs as BCs associated to a topology one dimension less than the problem (most common),
# b2 BCs two dimensions less, and b3 BCs three dimensions less
# for a 3D problem - b1: surface BCs, b2: edge BCs, b3: point BCs
# for a 2D problem - b1: edge BCs, b2: point BCs
# 1D problems not supported (currently...)
if self.mesh.topology.dim == 3:
try:
self.mesh.topology.create_connectivity(2, self.mesh.topology.dim)
with XDMFFile(self.comm, self.mesh_boundary, 'r', encoding=encoding) as infile:
self.mt_b1 = infile.read_meshtags(self.mesh, name="Grid")
except:
pass
try:
self.mesh.topology.create_connectivity(1, self.mesh.topology.dim)
with XDMFFile(self.comm, self.mesh_boundary, 'r', encoding=encoding) as infile:
self.mt_b2 = infile.read_meshtags(self.mesh, name="Grid_b2")
except:
pass
try:
self.mesh.topology.create_connectivity(0, self.mesh.topology.dim)
with XDMFFile(self.comm, self.mesh_boundary, 'r', encoding=encoding) as infile:
self.mt_b3 = infile.read_meshtags(self.mesh, name="Grid_b3")
except:
pass
elif self.mesh.topology.dim == 2:
try:
self.mesh.topology.create_connectivity(1, self.mesh.topology.dim)
with XDMFFile(self.comm, self.mesh_boundary, 'r', encoding=encoding) as infile:
self.mt_b1 = infile.read_meshtags(self.mesh, name="Grid")
except:
pass
try:
self.mesh.topology.create_connectivity(0, self.mesh.topology.dim)
with XDMFFile(self.comm, self.mesh_boundary, 'r', encoding=encoding) as infile:
self.mt_b2 = infile.read_meshtags(self.mesh, name="Grid_b2")
except:
pass
else:
raise AttributeError("Your mesh seems to be 1D! Not supported!")
# useful fields:
# facet normal
self.n0 = FacetNormal(self.mesh)
# cell diameter
self.h0 = CellDiameter(self.mesh)