def test_additivity(mode):
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = dolfinx.FunctionSpace(mesh, ("CG", 1))
f1 = dolfinx.Function(V)
f2 = dolfinx.Function(V)
f3 = dolfinx.Function(V)
with f1.vector.localForm() as f1_local:
f1_local.set(1.0)
with f2.vector.localForm() as f2_local:
f2_local.set(2.0)
with f3.vector.localForm() as f3_local:
f3_local.set(3.0)
j1 = ufl.inner(f1, f1) * ufl.dx(mesh)
j2 = ufl.inner(f2, f2) * ufl.ds(mesh)
j3 = ufl.inner(ufl.avg(f3), ufl.avg(f3)) * ufl.dS(mesh)
# Assemble each scalar form separately
J1 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1), op=MPI.SUM)
J2 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j2), op=MPI.SUM)
J3 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j3), op=MPI.SUM)
# Sum forms and assemble the result
J12 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j2), op=MPI.SUM)
J13 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j3), op=MPI.SUM)
J23 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j2 + j3), op=MPI.SUM)
J123 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j2 + j3), op=MPI.SUM)
# Compare assembled values
assert (J1 + J2) == pytest.approx(J12)
assert (J1 + J3) == pytest.approx(J13)
assert (J2 + J3) == pytest.approx(J23)
assert (J1 + J2 + J3) == pytest.approx(J123)
def test_basic_interior_facet_assembly():
ghost_mode = dolfin.cpp.mesh.GhostMode.none
if (dolfin.MPI.size(dolfin.MPI.comm_world) > 1):
ghost_mode = dolfin.cpp.mesh.GhostMode.shared_facet
mesh = dolfin.RectangleMesh(
dolfin.MPI.comm_world,
[numpy.array([0.0, 0.0, 0.0]),
numpy.array([1.0, 1.0, 0.0])], [5, 5],
cell_type=dolfin.cpp.mesh.CellType.Type.triangle,
ghost_mode=ghost_mode)
V = dolfin.function.FunctionSpace(mesh, ("DG", 1))
u, v = dolfin.TrialFunction(V), dolfin.TestFunction(V)
a = ufl.inner(ufl.avg(u), ufl.avg(v)) * ufl.dS
A = dolfin.fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
L = ufl.conj(ufl.avg(v)) * ufl.dS
b = dolfin.fem.assemble_vector(L)
b.assemble()
assert isinstance(b, PETSc.Vec)
def test_additivity(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
f1 = Function(V)
f2 = Function(V)
f3 = Function(V)
f1.x.array[:] = 1.0
f2.x.array[:] = 2.0
f3.x.array[:] = 3.0
j1 = ufl.inner(f1, f1) * ufl.dx(mesh)
j2 = ufl.inner(f2, f2) * ufl.ds(mesh)
j3 = ufl.inner(ufl.avg(f3), ufl.avg(f3)) * ufl.dS(mesh)
# Assemble each scalar form separately
J1 = mesh.comm.allreduce(assemble_scalar(form(j1)), op=MPI.SUM)
J2 = mesh.comm.allreduce(assemble_scalar(form(j2)), op=MPI.SUM)
J3 = mesh.comm.allreduce(assemble_scalar(form(j3)), op=MPI.SUM)
# Sum forms and assemble the result
J12 = mesh.comm.allreduce(assemble_scalar(form(j1 + j2)), op=MPI.SUM)
J13 = mesh.comm.allreduce(assemble_scalar(form(j1 + j3)), op=MPI.SUM)
J23 = mesh.comm.allreduce(assemble_scalar(form(j2 + j3)), op=MPI.SUM)
J123 = mesh.comm.allreduce(assemble_scalar(form(j1 + j2 + j3)), op=MPI.SUM)
# Compare assembled values
assert (J1 + J2) == pytest.approx(J12)
assert (J1 + J3) == pytest.approx(J13)
assert (J2 + J3) == pytest.approx(J23)
assert (J1 + J2 + J3) == pytest.approx(J123)
def test_basic_interior_facet_assembly():
mesh = create_rectangle(MPI.COMM_WORLD, [np.array([0.0, 0.0]), np.array([1.0, 1.0])],
[5, 5], cell_type=CellType.triangle,
ghost_mode=GhostMode.shared_facet)
V = FunctionSpace(mesh, ("DG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = ufl.inner(ufl.avg(u), ufl.avg(v)) * ufl.dS
a = form(a)
A = assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
L = ufl.conj(ufl.avg(v)) * ufl.dS
L = form(L)
b = assemble_vector(L)
b.assemble()
assert isinstance(b, PETSc.Vec)
def test_ghost_mesh_dS_assembly(mode, dS):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
dS = dS(mesh)
a = form(inner(avg(u), avg(v)) * dS)
# Initial assembly
A = fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
# Check that the norms are the same for all three modes
normA = A.norm()
print(normA)
assert normA == pytest.approx(2.1834054713561906, rel=1.e-6, abs=1.e-12)
def test_coefficents_non_constant():
"Test packing coefficients with non-constant values"
mesh = create_unit_square(MPI.COMM_WORLD, 3, 5)
V = FunctionSpace(
mesh, ("Lagrange", 3)) # degree 3 so that interpolation is exact
u = Function(V)
u.interpolate(lambda x: x[0] * x[1]**2)
x = SpatialCoordinate(mesh)
v = ufl.TestFunction(V)
# -- Volume integral vector
F = form((ufl.inner(u, v) - ufl.inner(x[0] * x[1]**2, v)) * dx)
b0 = assemble_vector(F)
b0.assemble()
assert np.linalg.norm(b0.array) == pytest.approx(0.0)
# -- Exterior facet integral vector
F = form((ufl.inner(u, v) - ufl.inner(x[0] * x[1]**2, v)) * ds)
b0 = assemble_vector(F)
b0.assemble()
assert np.linalg.norm(b0.array) == pytest.approx(0.0)
# -- Interior facet integral vector
V = FunctionSpace(mesh,
("DG", 3)) # degree 3 so that interpolation is exact
u0 = Function(V)
u0.interpolate(lambda x: x[1]**2)
u1 = Function(V)
u1.interpolate(lambda x: x[0])
x = SpatialCoordinate(mesh)
v = ufl.TestFunction(V)
F = (ufl.inner(u1('+') * u0('-'), ufl.avg(v)) -
ufl.inner(x[0] * x[1]**2, ufl.avg(v))) * ufl.dS
F = form(F)
b0 = assemble_vector(F)
b0.assemble()
assert np.linalg.norm(b0.array) == pytest.approx(0.0)
def test_basic_interior_facet_assembly():
mesh = dolfinx.RectangleMesh(
MPI.COMM_WORLD,
[numpy.array([0.0, 0.0, 0.0]),
numpy.array([1.0, 1.0, 0.0])], [5, 5],
cell_type=dolfinx.cpp.mesh.CellType.triangle,
ghost_mode=dolfinx.cpp.mesh.GhostMode.shared_facet)
V = fem.FunctionSpace(mesh, ("DG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = ufl.inner(ufl.avg(u), ufl.avg(v)) * ufl.dS
A = dolfinx.fem.assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
L = ufl.conj(ufl.avg(v)) * ufl.dS
b = dolfinx.fem.assemble_vector(L)
b.assemble()
assert isinstance(b, PETSc.Vec)
def error_indicators(self):
"""
Generate and return linear form defining error indicators
"""
# Extract these to increase readability
R_T = self._R_T
R_dT = self._R_dT
z = self._Ez_h
z_h = self._z_h
# Define linear form for computing error indicators
v = self.module.TestFunction(self._DG0)
eta_T = (v * inner(R_T, z - z_h) * dx(self.domain) +
avg(v)*(inner(R_dT('+'), (z - z_h)('+')) +
inner(R_dT('-'), (z - z_h)('-'))) * dS(self.domain) +
v * inner(R_dT, z - z_h) * ds(self.domain))
return eta_T
def qoi_varf(u,m):
return ufl.avg(ufl.exp(m)*ufl.dot( ufl.grad(u), n) )*dss(1)
# array of displacement values to apply at right boundary
stretchVals = np.hstack((
ldot * timeVals[:len(timeVals) // 2],
ldot * (-timeVals[len(timeVals) // 2:] + 2 * timeVals[len(timeVals) // 2]),
))
svals = np.zeros_like(stretchVals)
plt.plot(timeVals, stretchVals)
plt.savefig("stretchesVisco.png")
plt.close()
# stabilization parameters
h = FacetArea(mesh)
h_avg = avg(h)
# new variable name to take derivatives
FF = Identity(3) + grad(u)
CC = FF.T * FF
Fv = variable(FF)
S = diff(freeEnergy(Fv.T * Fv, CCv), Fv) # first PK stress
dl_interp(CC, C)
dl_interp(CC, Cn)
my_identity = grad(SpatialCoordinate(mesh))
dl_interp(my_identity, CCv)
dl_interp(my_identity, Cvn)
dl_interp(my_identity, C_quart)
dl_interp(my_identity, C_thr_quart)
# <markdowncell>
# Error estimator
# <codecell>
fvspace = dune.fem.space.finiteVolume(uh.space.grid)
estimate = fvspace.interpolate([0], name="estimate")
chi = ufl.TestFunction(fvspace)
hT = ufl.MaxCellEdgeLength(fvspace.cell())
he = ufl.MaxFacetEdgeLength(fvspace.cell())('+')
n = ufl.FacetNormal(fvspace.cell())
residual = (u - uh_n) / dt - div(diffusiveFlux) + source(u, u, u, vh)
estimator_ufl = hT**2 * residual**2 * chi * dx +\
he * inner( jump(diffusiveFlux), n('+'))**2 * avg(chi) * dS
estimator = dune.fem.operator.galerkin(estimator_ufl)
# <markdowncell>
# Time loop
# <codecell>
nextSaveTime = saveInterval
count = 0
levelFunction = dune.fem.function.levelFunction(gridView)
gridView.writeVTK("spiral",
pointdata=[uh, vh],
number=count,
celldata=[estimate, levelFunction])
count += 1
# 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
L = v * f * dx + v * gN * ds
def test_manufactured_poisson_dg(degree, filename, datadir):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
with XDMFFile(MPI.COMM_WORLD,
os.path.join(datadir, filename),
"r",
encoding=XDMFFile.Encoding.ASCII) as xdmf:
mesh = xdmf.read_mesh(name="Grid")
V = FunctionSpace(mesh, ("DG", degree))
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)
error = mesh.mpi_comm().allreduce(assemble_scalar((u_exact - uh)**2 * dx),
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
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})
with common.Timer("Compile forms"):
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)
with common.Timer("Assemble vector"):
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
with common.Timer("Assemble matrix"):
A = assemble_matrix(a, [])
A.assemble()
with common.Timer("Solve"):
# 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)
with common.Timer("Error functional compile"):
# Calculate error
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
with common.Timer("Error assembly"):
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
common.list_timings(MPI.COMM_WORLD, [common.TimingType.wall])
assert np.absolute(error) < 1.0e-14
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
# with Dirichlet boundary conditions. Here $\varepsilon$ is a small
# constant and $b$ a given vector.
# <codecell>
gridView = leafGridView([-1, -1], [1, 1], [20, 20])
order = 2
from dune.fem.space import dglegendre as dgSpace
space = dgSpace(gridView, order=order)
from ufl import avg, jump, dS, ds,\
CellVolume, FacetArea, FacetNormal,\
as_vector, atan
u = TrialFunction(space)
v = TestFunction(space)
n = FacetNormal(space)
he = avg( CellVolume(space) ) / FacetArea(space)
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
# diffusion factor
eps = Constant(0.1,"eps")
# transport direction and upwind flux
b = as_vector([1,0])
hatb = (dot(b, n) + abs(dot(b, n)))/2.0
# boundary values (for left/right boundary)
dD = conditional((1+x[0])*(1-x[0])<1e-10,1,0)
g = conditional(x[0]<0,atan(10*x[1]),0)
# penalty parameter
beta = 10*order*order
aInternal = dot(eps*grad(u) - b*u, grad(v)) * dx
# Copyright (C) 2009 Kristian B. Oelgaard
#
# This file is part of UFL.
#
# UFL is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# UFL is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with UFL. If not, see <http://www.gnu.org/licenses/>.
#
# Restriction of a finite element.
# The below syntax show how one can restrict a higher order Lagrange element
# to only take into account those DOFs that live on the facets.
from ufl import (FiniteElement, TestFunction, TrialFunction, avg, dS, ds,
triangle)
# Restricted element
CG_R = FiniteElement("Lagrange", triangle, 4)["facet"]
u_r = TrialFunction(CG_R)
v_r = TestFunction(CG_R)
a = avg(v_r) * avg(u_r) * dS + v_r * u_r * ds
top = 1.0
if dim == 3:
top = ufl.conditional(x[2] > 1.25, 1, 0)
initial_u = iu(x[1]) * top + iu(2.5 - x[1]) * (1.0 - top)
initial_v = ufl.conditional(x[0] < 1.25, 0.5, 0)
uh = space.interpolate(initial_u, name="u")
uh_n = uh.copy()
vh = space.interpolate(initial_v, name="v")
vh_n = vh.copy()
u = ufl.TrialFunction(space)
phi = ufl.TestFunction(space)
hT = ufl.MaxCellEdgeLength(space.cell())
hS = ufl.avg(ufl.MaxFacetEdgeLength(space.cell()))
hs = ufl.MaxFacetEdgeLength(space.cell())('+')
n = ufl.FacetNormal(space.cell())
ustar = lambda v: (v + spiral_b) / spiral_a
diffusiveFlux = lambda w, d: spiral_D * d
source = lambda u1, u2, u3, v: -1 / spiral_eps * u1 * (1 - u2) * (u3 - ustar(v)
)
source = lambda u1, u2, u3, v: -1 / spiral_eps * u1 * (1 - u2) * (u3 - ustar(v)
)
xForm = inner(diffusiveFlux(u, grad(u)), grad(phi)) * dx
xForm += ufl.conditional(uh_n < ustar(vh_n), source(u, uh_n, uh_n, vh_n),
source(uh_n, u, uh_n, vh_n)) * phi * dx
# <markdowncell>
uh_pm1 = spcpm.interpolate( initial_u, name="u_p-1" )
uh_n = uh.copy()
vh = space.interpolate( initial_v, name="v" )
vh_n = vh.copy()
# <markdowncell>
# Setting up the model
# <codecell>
u = ufl.TrialFunction(space)
phi = ufl.TestFunction(space)
n = ufl.FacetNormal(space)
penalty = 5 * (maxOrder * ( maxOrder + 1 )) * spiral_D
hT = ufl.MaxCellEdgeLength(space)
hS = ufl.avg( ufl.MaxFacetEdgeLength(space) )
hs = ufl.MaxFacetEdgeLength(space)('+')
ustar = lambda v: (v+spiral_b)/spiral_a
diffusiveFlux = lambda w,d: spiral_D * d
source = lambda u1,u2,u3,v: -1/spiral_eps * u1*(1-u2)*(u3-ustar(v))
xForm = inner(diffusiveFlux(u,grad(u)), grad(phi)) * dx
xForm += ufl.conditional(uh_n<ustar(vh_n), source(u,uh_n,uh_n,vh_n), source(uh_n,u,uh_n,vh_n)) * phi * dx
# dg terms
# xForm -= ( inner( outer(jump(u), n('+')), avg(diffusiveFlux(u,grad(phi)))) +\
# inner( avg(diffusiveFlux(u,grad(u))), outer(jump(phi), n('+'))) ) * dS
# xForm += penalty/hS * inner(jump(u), jump(phi)) * dS
# Test and trial functions
vq = BlockTestFunction(W)
(v, q) = block_split(vq)
up = BlockTrialFunction(W)
(u, p) = block_split(up)
w = BlockFunction(W)
w0 = BlockFunction(W)
(u0, p0) = block_split(w0)
n = FacetNormal(mesh)
vc = CellVolume(mesh)
fc = FacetArea(mesh)
h = vc / fc
h_avg = (vc("+") + vc("-")) / (2 * avg(fc))
penalty1 = 1.0
penalty2 = 10.0
theta = 1.0
# Constitutive parameters
K = 1000.e3
nu = 0.25
E = K_nu_to_E(K, nu) # Pa 14
(mu_l, lmbda_l) = E_nu_to_mu_lmbda(E, nu)
f_stress_y = Constant(-1.e3)
f = Constant((0.0, 0.0)) # sink/source for displacement